A Concise

Logic Introduction to

Fourteenth Edition

Patrick J. Hurley

University of San Diego

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Printed in the United States of America

Print Number: 01 Print Year: 2023

To Linda

It is wrong always, everywhere, and for anyone,

to believe anything upon insufficient evidence.

�W. K. Clifford

Nothing can be more important than the art of

formal reasoning according to true logic.

�Gottfried Wilhelm Leibniz

Brief Contents

Part I Informal Logic

1 Basic Concepts 1

2 Language: Meaning and Definition 70

3 Informal Fallacies 107

Part II Formal Logic

4 Categorical Propositions 175

5 Categorical Syllogisms 231

6 Propositional Logic 275

7 Natural Deduction in�Propositional Logic 336

8 Predicate Logic 390

Part III Inductive Logic

9 Analogy and Legal and Moral Reasoning 447

10 Causality and Mill�s Methods 467

11 Probability 490

12 Statistical Reasoning 506

13 Hypothetical/Scientific Reasoning 526

14 Science and Superstition 543

Answers to Selected Exercises 571

Glossary/Index 615

Contents

Preface xi

1 Basic Concepts 1

1.1 Arguments, Premises, and Conclusions 1

Exercise 1.1 6

1.2 Recognizing Arguments 12

Exercise 1.2 21

1.3 Deduction and Induction 29

Exercise 1.3 35

1.4 Validity, Truth, Soundness, Strength, and Cogency 39

Exercise 1.4 48

1.5 Argument Forms: Proving Invalidity 52

Exercise 1.5 56

1.6 Extended Arguments 58

Exercise 1.6 62

2 Language: Meaning and Definition 70

2.1 Varieties of Meaning 70

Exercise 2.1 74

2.2 The Intension and Extension of Terms 80

Exercise 2.2 83

2.3 Definitions and Their Purposes 84

Exercise 2.3 89

2.4 Definitional Techniques 92

Exercise 2.4 97

2.5 Criteria for Lexical Definitions 101

Exercise 2.5 103

3 Informal Fallacies 107

3.1 Fallacies in General 107

Exercise 3.1 109

3.2 Fallacies of Relevance 110

Exercise 3.2 120

3.3 Fallacies of Weak Induction 125

Exercise 3.3 135

3.4 Fallacies of Presumption, Ambiguity,

and Illicit Transference 141

Exercise 3.4 152

3.5 Fallacies in Ordinary Language 159

Exercise 3.5 164

4 Categorical Propositions 175

4.1 The Components of�Categorical Propositions 175

Exercise 4.1 178

4.2 Quality, Quantity, and Distribution 179

Exercise 4.2 182

4.3 Venn Diagrams and the Modern Square of

Opposition 183

Exercise 4.3 193

4.4 Conversion, Obversion, and Contraposition 194

Exercise 4.4 200

4.5 The Traditional Square of Opposition 203

Exercise 4.5 208

4.6 Venn Diagrams and�the� Traditional

Standpoint 213

Exercise 4.6 217

vi Contents

4.7 Translating Ordinary Language Statements into

Categorical Form 219

Exercise 4.7 224

5 Categorical Syllogisms 231

5.1 Standard Form, Mood, and Figure 231

Exercise 5.1 235

5.2 Venn Diagrams 238

Exercise 5.2 245

5.3 Rules and Fallacies 248

Exercise 5.3 253

5.4 Reducing the Number of Terms 255

Exercise 5.4 257

5.5 Ordinary Language Arguments 258

Exercise 5.5 260

5.6 Enthymemes 261

Exercise 5.6 263

5.7 Sorites 266

Exercise 5.7 269

6 Propositional Logic 275

6.1 Symbols and Translation 275

Exercise 6.1 283

6.2 Truth Functions 287

Exercise 6.2 294

6.3 Truth Tables for Propositions 296

Exercise 6.3 301

6.4 Truth Tables for Arguments 304

Exercise 6.4 306

6.5 Indirect Truth Tables 310

Exercise 6.5 316

6.6 Argument Forms and Fallacies 318

Exercise 6.6 327

Contents vii

7 Natural Deduction in�Propositional Logic 336

7.1 Rules of Implication I 336

Exercise 7.1 341

7.2 Rules of Implication II 347

Exercise 7.2 350

7.3 Rules of Replacement I 355

Exercise 7.3 359

7.4 Rules of Replacement II 366

Exercise 7.4 369

7.5 Conditional Proof 377

Exercise 7.5 380

7.6 Indirect Proof 382

Exercise 7.6 384

7.7 Proving Logical Truths 387

Exercise 7.7 388

8 Predicate Logic 390

8.1 Symbols and Translation 390

Exercise 8.1 396

8.2 Using the Rules of Inference 398

Exercise 8.2 404

8.3 Quantifier Negation Rule 408

Exercise 8.3 411

8.4 Conditional and Indirect Proof 413

Exercise 8.4 415

8.5 Proving Invalidity 418

Exercise 8.5 421

8.6 Relational Predicates and�Overlapping Quantifiers 423

Exercise 8.6 429

8.7 Identity 432

Exercise 8.7 439

viii Contents

9 Analogy and Legal and Moral Reasoning 447

9.1 Analogical Reasoning 447

9.2 Legal Reasoning 450

9.3 Moral Reasoning 453

Exercise 9 457

10 Causality and Mill�s Methods 467

10.1 �Cause� and Necessary and�Sufficient Conditions 467

10.2 Mill�s Five Methods 469

10.3 Mill�s Methods and Science 477

Exercise 10 482

11 Probability 490

11.1 Theories of Probability 490

11.2 The Probability Calculus 494

Exercise 11 502

12 Statistical Reasoning 506

12.1 Evaluating Statistics 506

12.2 Samples 507

12.3 The Meaning of �Average� 511

12.4 Dispersion 512

12.5 Graphs and Pictograms 516

12.6 Percentages 518

Exercise 12 520

13 Hypothetical/Scientific Reasoning 526

13.1 The Hypothetical Method 526

13.2 Hypothetical Reasoning: Four Examples from

Science 529

Contents ix

13.3 The Proof of Hypotheses 534

13.4 The Tentative Acceptance of Hypotheses 536

Exercise 13 538

14 Science and Superstition 543

14.1 Distinguishing Between Science and Superstition 543

14.2 Evidentiary Support 544

14.3 Objectivity 548

14.4 Integrity 552

14.5 Abusing Science 556

Exercise 14560

Answers to Selected Exercises 571

Glossary/Index 615

x Contents

Preface

The most immediate benefit derived from the study of logic is the skill needed to construct sound arguments of one�s own and to evaluate the arguments of others. In accomplishing this goal, logic instills a sensitivity for the formal component in lan-guage, a thorough command of which is indispensable to clear, effective, and meaning-ful communication. On a broader scale, by focusing attention on the requirement for reasons or evidence to support our views, logic provides a fundamental defense against the prejudiced and uncivilized attitudes that threaten the foundations of our demo-cratic society. Finally, through its attention to inconsistency as a fatal flaw in any theory or point of view, logic proves a useful device in disclosing ill-conceived policies in the political sphere and, ultimately, in distinguishing what makes sense from what makes no sense. This book is written with the aim of securing these benefits.

Note to the Instructor

The image on the front cover is intended to convey the message that logic is the key to all learning. Keys open doors. Logic is the key that opens the door to reasoned dis-course and dialogue, unlocking an important opportunity for learning. Through logic, students learn to support their views with reasons and to open their minds to the rea-sons of others. Logic creates a common foundation upon which individuals who hold opposing points of view can learn from each other. What might otherwise devolve into a shouting match of conflicting opinions becomes a venue for the rational exchange of ideas.

To promote the achievement of this goal, this new edition adopts the theme that learning logic is empowering. In this context, saying that logic is empowering does not mean that logic is properly used to overpower one�s opponents, to smash their arguments. Rather, it means that logic empowers both the arguer and the listener to enter into a rational exchange where each is free to explore the strengths and weaknesses of the arguments presented on both sides. Logic lays the foundation for a meeting of minds and is therefore one of the great civilizing elements in human society. To implement the empowerment theme, each chapter begins with a brief selection explaining how the content of the chapter is empowering. While the inclu-sion of these selections may be the most visible change in the new edition, as you proceed through the book, you will encounter additional improvements that are less visible.

New to This Edition

Each chapter of the book now begins with a signature paragraph explaining how and why the content of the chapter is empowering. These �Empowerment� selections replace the �How Logical Are You?� selections in the prior edition. Examples and exercises throughout the book have been updated, featuring current situations and young personalities drawn from an ethnically diverse population that students will readily recognize. In the section relating to induction, more treatment is given to probabilities and odds, and how to com-pute the one from the other. Further, in the final chapter, new treatment is given to the cor-rupting influence of corporate money on what we take to be scientific truth. From the start, a chief motivator of this new edition has been Diversity and Inclusion. The book has been especially tuned to respect and value all individuals, regardless of marital status, gender, ethnicity, religious orientation, or socioeconomic status. We are committed to the view that such a perspective constitutes the best environment for learning anything, and it will help students feel at home when studying logic.

Hallmark Features

● Chapters are organized so that earlier sections provide the foundation for later ones.

Later sections can be skipped by instructors opting to do so.

● The main points are always presented up front so students cannot possibly miss

them.

● Relevant and up-to-date examples are used extensively.

● Key terms are introduced in boldface type and defined in the Glossary/Index.

● Central concepts are illustrated in graphic boxes.

● Numerous exercises, many drawn from real-life sources such as newspapers, text-

books, and magazines, are included to perfect student skills�the current edition includes over 2,700 exercises.

● Biographical vignettes of prominent logicians are included to give logic a human face.

● Dialogue exercises illustrate the application of logical principles to real-life situations.

● Venn diagrams for syllogisms are presented in a novel and more effective way, using

color to identify the relevant areas.

● End-of-chapter summaries facilitate student review.

● The solution to every third exercise is provided in the Answers to Selected Exercises

section, so students can easily check their work.

● A robust digital platform offers thousands of autograded practice questions that

boost students� confidence in mastering logic. These are accompanied by video and tutorial help resources.

● Multimedia resources, such as Learning Logic, offer students stepped-out tutorials

on challenging Logic concepts and applications.

xii Preface

Ways of Approaching This Textbook

In general, the material in each chapter is arranged so that certain later sections can be skipped without affecting subsequent chapters. For example, those wishing a brief treatment of natural deduction in both propositional and predicate logic may want to skip the last three sections of Chapter 7 and the last four (or even five) sections of Chapter 8. Chapter 2 can be skipped altogether, although some may want to cover the first section of that chapter as an introduction to Chapter 3. Finally, Chapters 9 through 14 depend only slightly on earlier chapters, so these can be treated in any order one chooses. However, Chapter 14 does depend in part on Chapter 13.

Type of Course

Informal logic

Traditional course, critical-Course emphasizing logic coursereasoning coursemodern formal logic

Recommended Chapter 3 Chapter 1 Chapter 1 material Chapter 4 Chapter 2 Sections 4.1�4.3

Chapter 5 Chapter 3 Section 4.7 Chapter 6 Chapter 4 Sections 6.1�6.5 Sections Sections 5.1�5.3 Chapter 7 7.1�7.4 Sections 5.5�5.6 Chapter 8

Sections 6.1�6.4 Truth Trees supplement Section 6.6

Chapter 9

Chapter 12

Chapter 13

Chapter 14

Critical Think-

ing and Writing

supplement

Optional Chapter 2 Section 5.4 Chapter 3 material Sections Section 5.7 Sections 4.4�4.6

7.5�7.7 Section 6.5 Sections 5.1�5.2 Chapters 9�14 Chapter 10 Section 5.7

Chapter 11 Section 6.6

To the Student

Imagine that you are interviewing for a job. The person across the desk asks about your strengths, and you reply that you are energetic, enthusiastic, and willing to work long hours. Also, you are creative and innovative, and you have good leadership skills. Then the interviewer asks about your reasoning abilities. You reply that you have always con-sidered your reasoning abilities to be excellent.

�Can you point to any evidence of this ability?� the interviewer asks.

�Well,� you reply, �in college I took a course in logic, and I easily earned an �A.��

�I�m impressed,� says the interviewer. �You sound like exactly the kind of job can-didate we are looking for. Your reasoning abilities will combine with your other talents for a powerful effect.�

Preface xiii

�I�m happy to hear you say that,� you reply. �But could you elaborate on that last point?�

�I�ll be happy to,� the interviewer says. �Reasoning skills are essential to good judgment. Your reasoning ability and good judgment will combine with your cre-ativity to yield projects that truly benefit our company, they will convince our other employees that you are leading them in the right direction, and they will blend with your enthusiasm to inspire others to do their best. Further, your willingness to work long hours will guarantee that your efforts will pay off in the end.�

�Then, does that mean that you have a place for me?� you ask.

�Absolutely,� says the interviewer. �I will be delighted to recommend you for a highly responsible position in our company.�

The point of this brief dialogue is that good reasoning skills are essential to doing any-thing right. The businessperson uses reasoning skills in writing a report or preparing a presentation, the scientist uses them in designing an experiment or clinical trial, the department manager uses them in maximizing worker efficiency, and the lawyer uses them in composing an argument to a judge or jury. And that�s where logic comes in. The chief purpose of logic is to develop good reasoning skills. In fact, logic is so impor-tant that when the liberal arts program of studies was formulated fifteen hundred years ago, logic was selected as one of the original seven liberal arts. Logic remains to this day a central component of a college or university education.

From a more pragmatic angle, logic is important to earning a good score on any of the several tests required for admission to graduate professional schools�the LSAT, GMAT, MCAT, GRE, and so on. Obviously, the designers of these tests recognize that the ability to reason logically is a prerequisite to success in these fields. Also, logic is a useful tool in relieving what has come to be called math anxiety. For whatever reason, countless students today are terrified of any form of reasoning that involves abstract symbols. If you happen to be one of these students, you should find it relatively easy to master the use of logical sym-bols, which are closely tied to ordinary language, and your newly found comfort with these symbols will carry over into the other, more difficult fields.

In addition to the existing and new features described in the �New to This Edition� section of this preface, WebAssign also includes the following supplements. Critical Thinking and Writing offers practice in writing arguments about real-life topics; Truth Trees present a standard introduction to the method of truth trees, which can be used as a supplement or alternative to the truth-table method; and Logic and Graduate-Level Admission Tests shows how the principles learned in studying logic can be used to answer questions on the LSAT, GMAT, MCAT, and GRE. Finally, Existential Import traces the history of existential import through the logic of Aristotle and George Boole.

Why Learning Logic Is Empowering

In 2019, Stanford Graduate School of Education published the results of a study of 11,000 letters written by teenaged students to the 2016 U.S. presidential candidates.

xiv Preface

One might have expected that the letters would have been filled with bombastic lan-guage or emotionally charged appeals to the candidates. But no. What the researchers found is that the majority of letters appealed to logic in expressing their point. The stu-dents used reasons and evidence to present their positions. There had been no required format to follow in writing these letters, so why the appeal to logic?

As we will see throughout this textbook, logic empowers both arguer and listener. Non-logical strategies are often coercive, manipulative, and dishonest. Sometimes evidence is withheld, or the arguer attempts to make the listener feel threatened or emotionally overwhelmed. The goal of such strategies is to trick listeners into believing something they might not believe if all the relevant evidence were made available. Trickery fails to respect the listener�s ability to evaluate evidence rationally. And it is counterproductive for the arguer, since the listener typically ends up feeling cheated rather than convinced.

Empowerment through logic is very different from attempting to gain power over other people by manipulating or coercing them. For example, suppose someone tries to get you to believe something by making you feel afraid�afraid of losing something like social status or money. In the end, if you are taken in by the manipulator�s scheme, the manipulator �wins� and you �lose.� You believe something, not on the basis of freely understanding that a conclusion follows on the basis of evidence, but rather as an out-come of coercion. This type of manipulation is a zero-sum game. There is only one winner. If the manipulation succeeds, the manipulator wins and you lose.

Logical reasoning, on the other hand, is a win-win strategy. The arguer presents evi-dence or reasons relevant to the conclusion, and, if the reasoning is solid, the listener is persuaded and freely consents to the conclusion. In effect, the arguer presents evi-dence in a way that invites listeners to open their minds to new ways of evaluating the evidence presented. The effectiveness of logical persuasion is manifest by the listener�s free embrace of the conclusion advanced. Once listeners understand how the conclu-sion follows from the evidence, they freely accept the conclusion and make it their own. The listeners are then motivated to present the same line of reasoning to others. In this manner, logical reasoning provides a solid basis for consensus building.

Logic provides a set of basic principles that are the same for everyone. It resembles math-ematics in this respect. Mathematical reasoning is so convincing because, if you follow the rules, you can�t go astray. It provides a sure and solid foundation for arriving at conclusions. Logic does the same thing. Because the rules of logic are universal in the same way that the rules of math are universal, there is no opportunity for manipulation or fabrication. The rules work equally to everyone�s advantage. Everyone wins. In this way, logic is the founda-tion of all productive negotiations. Good arguments facilitate a meeting of minds. Good arguments facilitate mutual understanding and encourage cooperation. In effect, logic may be the most powerful engine for preserving peace yet created by the human mind.

Online Learning Platform: WebAssign

Built by educators, WebAssign provides flexible settings at every step to customize your course with online activities and secure testing to meet learners� unique needs. Students

Preface xv

get everything in one place, including rich content and study resources designed to fuel deeper understanding, plus access to a dynamic, interactive eTextbook. Proven to help hone problem-solving skills, WebAssign helps you help learners in any course format. For more information, visit https://www.cengage.com/webassign.

Study Smarter

Ever wonder if you studied enough? WebAssign from Cengage can help.

WebAssign is an online learning platform for your math, statistics, physical sciences, and engineering courses. It helps you practice, focus your study time, and absorb what you learn. When class comes�you�re way more confident.

With WebAssign, you will:

● Get instant feedback and grading on your assignments ● Know how well you understand concepts ● Watch videos and tutorials when you�re stuck ● Perform better on in-class activities

Ancillaries for the Instructor

Additional instructor resources for this product are available online at the Cengage Instructor Center�an all-in-one resource for class preparation, presentation, and test-ing. Resources available for download from the Cengage Instructor Center include:

Instructor�s Manual

Includes solutions to all the exercises in the textbook.

Solutions and Answer Guide

This manual contains solutions to all exercises from the text, including Chapter Review Exercises and Chapter Tests, located in the instructor companion site and the WebAssign Resources tab.

Cengage Testing Powered by Cognero

Cognero is a flexible online system that allows you to author, edit, and manage test bank content online. You can create multiple tests in an instant and deliver them from your LMS, your classroom, or export to printable PDF or Word format for in-class assessment. This is available online via the Cengage Instructor Center.

PowerPoint Slides

The PowerPoint� slides are ready-to-use, visual outlines of each section that can be easily customized for your lectures.

Presentations include activities, examples, and opportunities for student engagement and interaction.

xvi Preface

Educator�s Guide

Describes the content and activities available in the accompanying WebAssign course� including videos, prebuilt assignments, and other exercise types�that you can integrate into your course to support your learning outcomes and enhance student engagement and success.

Acknowledgments

For their reviews and suggestions leading to this fourteenth edition I want to thank the following:

Benjamin Arah, Bowie State University; Philip Blosser, Sacred Heart Major Seminary; Michael Clabaugh, College of St. Scholastica; Steven Gerrard, Williams College; Yan Mikhaylov, College of Southern Nevada; Dr. Teresa I. Reed, Quincy University; Robert Robinson, City College of New York-CUNY; Adam R. Thompson, University of Nebraska-Lincoln; Elizabeth J.M. Wakeman, The College of Idaho.

Of course any errors or omissions that may remain are the result of my own oversight.

Those who have contributed reviews and suggestions leading to the thirteen previous editions, and to whom I express my continued thanks, are the following: Karl Aho, Baylor University; James T. Anderson, University of San Diego; Carol Anthony, Villanova University; Benjamin Arah, Bowie State University; Joseph Asike, Howard University; Harriet E. Baber, University of San Diego; Kent Baldner, Western Michigan University; James Baley, Mary Washington College; Jerome Balmuth, Colgate University; Victor Balowitz, State University of New York, College at Buffalo; Ida Baltikauskas, Century College; Gary Baran, Los Angeles City College; Robert Barnard, University of Mississippi; Peter Barry, Saginaw Valley State University; Gregory Bassham, Kings College; Thora Bayer, Xavier University of Louisiana; David Behan, Agnes Scott College; John Bender, Ohio University, Athens; James O. Bennett, University of Tennessee, Knoxville; Victoria Berdon, IUPU Columbus; Robert Berman, Xavier University of Louisana; Kevin Berry, Ohio University; Joseph Bessie, Normandale Com-munity College; John R. Bosworth, Oklahoma State University; Andrew Botterell, University of Toronto; Jeff Broome, Arapahoe Community College; Tom Browder, University of Nevada, Las Vegas; Harold Brown, Northern Illinois University; Kevin Browne, Indiana University Southeast; Ken Buckman, University of Texas, Pan American; Robert Burch, Texas A&M University; Keith Burgess-Jackson, University of Texas, Arlington; Michael Byron, Kent State University; Scott Calef, Ohio Wesleyan University; Gabriel Camacho, El Paso Community College; James Campbell, University of Toledo; Joseph Keim Campbell, Washington State University; Loren Cannon, Humboldt State University; Charles Carr, Arkansas State University; William Carroll, Coppin State University; Jennifer Caseldine-Bracht, IUPU Fort Wayne; John Casey, Northern Illinois University; Robert Greg Cavin, Cypress College; Ping-Tung Chang, University of Alaska; Prakash Chenjeri, Southern Oregon University; Drew Christie, University of New Hampshire; Timothy Christion, University of North Texas; Ralph W. Clarke, West Virginia University; David Clowney, Rowan University; Darryl Cohen, Mesa Community College; Michael Cole, College of William and Mary; Michael Coledu, Reedley College; Louis Colombo, Bethune-Cookman University; Michael J. Colson, Merced College; William F. Cooper, Baylor University; William Cornwell, Salem State College; Victor Cosculluela, Polk Community College;

Preface xvii

Mike�Coste, Front Range Community College; Ronald R. Cox, San Antonio College; Houston A. Craighead, Winthrop University; Donald Cress, Northern Illinois University, DeKalb; Richard La Croix, State University College at Buffalo; Jack Crumley, University of San Diego; Rosibel Cruz, Harold Washington College; Drew Dalton, Florida Southern College; Linda Damico, Kennesaw State University; Ray Darr, Southern Illinois University; William J. DeAngelis, North-eastern University; Joseph DeMarco, Cleveland State University; Paul DeVries, Wheaton Col-lege; Jill Dieterle, Eastern Michigan University; Mary Domski, University of New Mexico; Beverly R. Doss and Richard W. Doss, Orange Coast College; Paul Draper, Purdue University; William A. Drumin, King�s College, Pennsylvania; Clinton Dunagan, Saint Philips College; Jeffrey Easlick, Saginaw Valley State University; Benjamin Easton, Azusa Pacific University; Paul Eckstein, Bergen Community College; Anne M. Edwards, Austin Peay State University; Samantha Emswiler, John Tyler Community College; Michael Epperson, California State Uni-versity, Sacramento; Lenore Erickson, Cuesta College; Cassandra Evans, San Diego City Col-lege; Frank Fair, Sam Houston State University; Evan Fales, University of Iowa; Thompson Faller, University of Portland; William Ferraiolo, San Joaquin Delta College; Albert Filice, Mesa Community College; Lewis S. Ford, Old Dominion University; Gary Foulk, Indiana State University, Terre Haute; Timothy C. Fout, University of Louisville; LeAnn Fowler, Slippery Rock University; Craig Fox, California University of Pennsylvania; Thomas H. Franks, Eastern Michigan University; Bernard D. Freydberg, Slippery Rock University; Thomas J. Frost, Biola University/Long Beach City College; Dick Gaffney, Siena College; George Gale, University of Missouri�Kansas City; Pieranna Garavaso, University of Minnesota at Morris; Paul Gass, Coppin State University; Dimitria Gatzia, The University of Akron Wayne College; Dimitria Gatzia, University of Akron Wayne College; Joseph Georges, El Camino College; Kevin Gibson, Uni-versity of Colorado; James Granitto, Santiago Canyon College; Victor Grassian, Los Angeles Harbor College; Catherine Green, Rockhurst University; James Greene, Northern Michigan University; Harold Greenstein, SUNY Brockport; J. Randall Groves, Ferris State University; Shahrokh Haghighi, California State University; Alexander W. Hall, Clayton State University; Dean Hamden, Montclair State University; Courtney Hammond, Cuyamaca College; Ken Hanly, Brandon University; Anthony Hanson, West Valley College; Merle Harton, Edward Waters College; Larry Hauser, Alma College; Deborah Heikes, University of Alabama in Hunts-ville; Ryan Hickerson, Western Oregon University; Douglas Hill, Saddleback College and Golden West College; Ronald Hill, University of San Diego; Lawrence Hinman, University of San Diego; Lynn Holt, Mississippi State University; Jeremy Hovda, Minneapolis Community & Technical College; John B. Howell, III, Southwestern Baptist Theological Seminary; R. I. G. Hughes, University of South Carolina, Columbia; Peter Hutcheson, Texas State University; Debby Hutchins, Gonzaga University; William H. Hyde, Golden West College; Ron Jackson, Clayton State University; William Jamison, University of Alaska Anchorage; Sandra Johanson, Green River Community College; Richard A. Jones, Howard University; Russel Jones, Univer-sity of Oklahoma; Gary Jones, University of San Diego; Glenn C. Joy, Texas State University, San Marcos; Olin Joynton, North Harris County College; Grant Julin, St. Francis University; Patrick Kenny, Onondaga Community College; Glen Kessler, University of Virginia; Charles F. Kielkopf, Ohio State University; Moya Kinchla, Bakersfield College; Kristin Klamm-Doneen, Anoka Ramsey Community College; Bernard W. Kobes, Arizona State University; Keith W. Krasemann, College of DuPage; Sandra LaFave, West Valley College, Saratoga, California; William Lawhead, University of Mississippi; Stephen Leach, University of Texas�Pan American; Stephen Leach, UTPA; Richard Lee, University of Arkansas; Lory Lemke, University of Minnesota, Morris; Robert Levis, Pasadena City College; Chenyang Li, Monmouth College,

xviii Preface

Monmouth, Illinois; Chris M. Lorkowski, Kent State University; Chris Lorkowski, Kent State University�Trumbull Campus; Keane Lundt, Massachusetts College of Liberal Arts; Ardon Lyon, City University of London; Scott MacDonald, University of Iowa; Ian MacKinnon, The University of Akron; Krishna Mallick, Salem State College; Thomas Manig, University of Missouri�Columbia; James Manns, University of Kentucky; Dalman Mayer, Bellevue Commu-nity College; Larry D. Mayhew, Western Kentucky University; Leemon McHenry, California State University Northridge; Robert McKay, Norwich University; Rick McKita, Colorado State University; Phillip McReynolds, Pennsylvania State University; Erik Meade, Southern Illinois University�Edwardsville; Noel Merino, Humboldt State University; Kenneth R. Merrill, University of Oklahoma; Daniel Metz, Metropolitan State College of Denver (Program chair at CCCOnline); Daniel Metz, Metropolitan State College of Denver; Thomas Michaud, Wheeling Jesuit College; George D. Miller, DePaul University; Richard Miller, East Carolina University; Alexander Miller, Piedmont Technical College; Dolores Miller, University of Missouri�Kansas City; Frederick Mills, Bowie State University; Jeff Mitchell, Arkansas Tech University; John Mize, Long Beach City College; Mike Monge, Orange Coast College; Allyson Mount, Keene State College; Seyed Mousavian, University of Alberta; Dwayne Mulder, California State University, Fresno; John D. Mullen, Dowling College; Madeline Muntersbjorn, University of Toledo; Henry Nardone, Kings College; Nathaniel Nicol, Washington State University; Theresa Norman, South Texas Community College; David O�Connor, Seton Hall University; Tim O�Neill, Roches-ter Community and Technical College; Elane O�Rourke, Moorpark College; Brendan O�Sullivan, Rhodes College; Len Olsen, Georgia Southern University; Stephen Pacheco, University of San Diego; Joseph Pak, Los Angeles City College; Christopher Pallotti, Los Angeles Valley College/CSUN; Christopher Pearson, Southern Illinois University Edwardsville; Rodney Peffer, University of San Diego; Linda Peterson, University of San Diego; Robert G. Pielke, El Camino College; Cassandra Pinnick, Western Kentucky University; Nelson Pole, Cleveland State Uni-versity; Norman Prigge, Bakersfield State University; Gray Prince, West Los Angeles College; R. Puligandla, University of Toledo; T. R. Quigley, Oakland University; Nani Rankin, Indiana University at Kokomo; Robert Redmon, Virginia Commonwealth University; Bruce Reichenbach, Augsburg College; Herminia Reyes, San Diego State University; Fernando Rincon-Tellez, Piedmont Technical College; David Ring, Southern Methodist University; Tony Roark, Boise State University; Matthew Roberts, Patrick Henry College; Phyllis Rooney, Oakland University; Michael Rooney, Pasadena City College; Beth Rosdatter, University of Kentucky; Michelle M. Rotert, Rock Valley College; Paul A. Roth, University of Missouri�Saint Louis; Daniel Rothbart, George Mason University; Robert Rupert, University of Colorado, Boulder; Sam Russo, El Camino College; Frank Ryan, Kent State University; Eric Saidel, George Washington University; Kelly Salsbery, Stephen F. Austin State University; Paul Santelli, Siena College; Stephen Satris, Clemson University; Philip Schneider, George Mason University; James D. Schumaker, University of North Carolina at Charlotte; Stephanie Semler, Radford University; Pat Sewell, University of North Texas; Elizabeth Shadish, El Camino College; Joseph G. Shay, Boston College; Candice Shelby, University of Colorado Denver; Janet Simpson, Suffolk County Community College; Robert Skipper, St. Mary�s University; Aeon Skoble, Bridgewater State College; Matthew Slater, Bucknell University; Kent Slinker, Pima Community College; Dennis L. Slivinski, California State University, Channel Islands; Joshua Smith, Central Michigan University; Arnold Smith, Youngstown State University; John-Christian Smith, Youngstown State University; Paula Smithka, University of Southern Mississippi; Eric W. Snider, University of Toledo; Joseph Snyder, Anne Arundel Community College; Bob Snyder, Humboldt University; Lynne Spellman, University of Arkansas;

Preface xix

David�Stern, University of Iowa; Sean Stidd, Wayne State University; James Stuart, Bowling Green State University; Paul Studtmann, Davidson College; Krys Sulewski, Edmonds Commu-nity College; John Sullins, Sonoma State University; Weimin Sun, California State University, Northridge; Corine Sutherland, Cerritos College; Robert Sutton, Cape Fear Community Col-lege; Catherine S. Sutton, Virginia Commonwealth University; John Sweigart, James Madison University; Clarendon Swift, Moorpark College; Wayne Swindall, California Baptist College; Mojgan Taheri, California State University, Northridge; Brian Tapia, Foothill College; Bangs Tapscott, University of Utah; J. Ramon Tello, Shasta College; Mark Thames, El Centro College; Jan Thomas, University of Arkansas at Little Rock; Phil Thompson, Eastern Illinois University; Michael Thune, Joliet Junior College; Richard Tieszen, San Jose State University; Larry Udell, West Chester University; Ted Ulrich, Purdue University; Robert Urekew, University of Louisville; William Uzgalis, Oregon State University; William Vanderburgh, Wichita State University; Michael Ventimiglia, Sacred Heart University; Susan Vineberg, Wayne State University; Mark Vopat, Youngstown State University; Thomas H. Warren, Solano College; Andrew J. Waskey, Dalton State University; Roy Weatherford, University of South Florida; Chris Weigand, Our Lady of the Lake University; David Weinburger, Stockton State College; Paul Weirich, Univer-sity of Missouri�Columbia; David Weise, Gonzaga University; Dennis Weiss, York College of Pennsylvania; Robert Wengert, University of Illinois at Urbana�Champaign; Shannon Grace Werre, Edmonds Community College; Gerald Joseph Williams, Seton Hall University; Derrick Willis, Temple College; Frank Wilson, Bucknell University; W. Kent Wilson, University of Illinois, Chicago; Katherine D. Witzig, Southwestern Illinois College; Sandra Woien, Mesa Community College; Stephen Wykstra, Calvin College; Julie Yoo, California State University, Northridge; Elaine Yoshikawa, Arizona State University; Marie Zaccaria, Georgia Perimeter College; Jeffrey Zents, University of Texas; Xiaoyu Zhu, Antelope Valley College.

Finally, it has been a pleasure working with the Senior Portfolio Product Manager for Philosophy, Vanessa Coloura, Senior Content Manager Michael Lepera, and Senior Learning Designer Powell Vacha. Most of all, I want to thank my wife, Linda Peterson, for her countless suggestions and support.

xx Preface

1Basic Concepts

1.1 Arguments, Premises, and Conclusions

1.2 Recognizing Arguments

1.3 Deduction and Induction

1.4 Validity, Truth, Soundness, Strength, and Cogency

1.5 Argument Forms: Proving Invalidity

1.6 Extended Arguments

1.1 Arguments, Premises, and Conclusions

Learning how to construct good arguments is empowering. Suppose there is an upcoming election for Sheriff, and one of the candidates is Timberly Brown. One of your friends urges that you vote against Brown. �I just don�t trust her,� she says. �I get a bad feeling when I see her. I don�t like her attitude. And I also don�t like her smile or the way she dresses.� Are these good reasons to vote against Brown? Has your friend given you a good argument?

Logic may be defined as the organized body of knowledge, or science, that evaluates arguments. All of us encounter arguments in our day-to-day experi-ence. We read them in books and newspapers, hear them on television, and for-mulate them when communicating with friends and associates. The aim of logic is to develop a system of methods and principles that we may use as criteria for evaluating the arguments of others and as guides in constructing arguments of our own. Among the benefits to be expected from the study of logic is an increase in confidence that we are making sense when we criticize the arguments of others and when we advance arguments of our own.

An argument, in its simplest form, is a group of Where is Khartoum? (question)

statements, one or more of which (the premises) are Let�s go to a movie tonight. (proposal) claimed to provide support for, or reasons to believe, I suggest you get contact lenses. (suggestion) one of the others (the conclusion). Every argument Turn off the TV right now. (command) may be placed in either of two basic groups: those Fantastic! (exclamation)

in which the premises really do support the conclu- The statements that make up an argument are sion and those in which they do not, even though divided into one or more premises and exactly they are claimed to. The former are said to be good one conclusion. The premises are the state-arguments (at least to that extent), the latter bad ments that set forth the reasons or evidence, and arguments. The purpose of logic, as the science that the conclusion is the statement that the evidence evaluates arguments, is thus to develop methods and is claimed to support or imply. In other words, techniques that allow us to distinguish good argu-the conclusion is the statement that is claimed to ments from bad. follow from the premises. Here is an example of As is apparent from the given definition, the term an argument: argument has a very specific meaning in logic. It does not mean, for example, a mere verbal fight, as one All film stars are celebrities. might have with one�s parent, spouse, or friend. Let Zendaya is a film star. us examine the features of this definition in greater Therefore, Zendaya is a celebrity.

A The first two statements are the premises; the third is detail. First of all, an argument is a group of statements.

a sentence component that could stand as a declarative or imply the conclusion is indicated by the word �therefore.�) In this argument the premises really sentence. The following sentences are statements: do support the conclusion, and so the argument is a good one. But consider this argument: in other words, typically a declarative sentence or the conclusion. (The claim that the premises support statement is a sentence that is either true or false�

Chocolate truffles are loaded with calories.

Melatonin helps relieve jet lag. Some film stars are men. Political candidates always tell the complete Zoe Saldana is a film star. truth. Therefore, Zoe Saldana is a man.

No wives ever cheat on their husbands. In this argument the premises do not support the Naomi Osaka plays tennis and Kevin Durant conclusion, even though they are claimed to, and so

The first two statements are true, and the second One of the most important tasks in the analysis of arguments is being able to distinguish premises plays basketball. the argument is not a good one.

the two possible sion is really a premise, and vice versa, the subse-truth values of a statement. Thus, quent analysis cannot possibly be correct. Many the truth value of the first two statements is true, the arguments contain indicator words that provide truth value of the second two is false, and the truth both of which are true. Truth and falsity are called from conclusions. If what is thought to be a conclu-two false. The last one expresses two statements,

value of the last statement, as well as that of its com- clues in identifying premises and conclusion. Some typical conclusion indicators are ponents, is true.

Unlike statements, many sentences cannot be therefore accordingly entails that

said to be either true or false. Questions, proposals, wherefore we may conclude hence suggestions, commands, and exclamations usually thus it must be that it follows that cannot, and so are not usually classified as state- consequently for this reason implies that ments. The following sentences are not statements: we may infer so as a result

2Chapter1 Basic Concepts

Whenever a statement follows one of these indica- and the conclusion is �Pregnant women should never tors, it can usually be identified as the conclusion. By use recreational drugs.� process of elimination, the other statements in the In reviewing the list of indicators, note that �for argument are the premises. Example: this reason� is a conclusion indicator, whereas �for the

reason that� is a premise indicator. �For this reason�

Tortured prisoners will say anything just to (except when followed by a colon) means for the relieve the pain. Consequently, torture is not reason (premise) that was just given, so what follows a reliable method of interrogation. is the conclusion. On the other hand, �for the reason

The conclusion of this argument is �Torture is not a that� announces that a premise is about to be stated. Sometimes a single indicator can be used to iden-reliable method of interrogation,� and the premise is tify more than one premise. Consider the following �Tortured prisoners will say anything just to relieve argument: the pain.�

It is vitally important that wilderness areas be

preserved, for wilderness provides essential

habitat for wildlife, including endangered

Claimed species, and it is a natural retreat from the

Premises evidence stress of daily life.

The premise indicator �for� goes with both �Wil-

derness provides essential habitat for wildlife,

including endangered species,� and �It is a natural

retreat from the stress of daily life.� These are the

Conclusion premises. By method of elimination, �It is vitally What is claimed

to follow from important that wilderness areas be preserved� is the evidence the conclusion.

Some arguments contain no indicators. With

If an argument does not contain a conclusion these, the reader/listener must ask such questions

indicator, it may contain a premise indicator. Some as: What single statement is claimed (implicitly) to typical premise indicators are follow from the others? What is the arguer trying to

prove? What is the main point in the passage? The

since as answers to these questions should point to the con-as indicated by given that clusion. Example: because seeing that Modernizing our nation�s crumbling infra-for for the reason that structure is long overdue. Many of our in that inasmuch as bridges are practically falling down, and our may be inferred from owing to transit system is in dire ned of repair. Further-

Any statement following one of these indicators can more, making these improvements would create jobs for millions of workers. usually be identified as a premise. Example:

Pregnant women should never use recre- The conclusion of this argument is the first state-ational drugs, since the use of these drugs ment, and all of the other statements are premises. can jeopardize the development of the The argument illustrates the pattern found in most fetus. arguments that lack indicator words: The intended

conclusion is stated first, and the remaining state-

The premise of this argument is �The use of these ments are then offered in support of this first state-drugs can jeopardize the development of the fetus,� ment. When the argument is restructured according

Section 1.1 Arguments, Premises, and Conclusions 3

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). to logical principles, however, the conclusion is In the first argument, the opening statement serves

always listed after the premises: merely to introduce the topic, so it is not part of the P argument. The premise is the second statement, and : Many of our bridges are practically fall-1 the conclusion is the last statement. In the second

P ing comment, so it is not part of the argument. The : Making these improvements would 3 premises are the first three statements, and the state-P : Our transit system is in dire need of repair. 2 ing down. argument, the final statement merely makes a pass-

create jobs for millions of workers.

C: Modernizing our nation�s crumbling infra- ment following �for these reasons� is the conclusion.

structure is long overdue. Closely related to the concepts of argument and

When restructuring arguments such as this, one An statement are those of inference and proposition.

version, while at the same time attending to the In the broad sense of the term, �inference� is used requirement that premises and conclusion be com-interchangeably with �argument.� Analogously, a plete sentences that are meaningful in the order in proposition , in the narrow sense, is the meaning or which they are listed. should remain as close as possible to the original the reasoning process expressed by an argument. inference, in the narrow sense of the term, is

Note that the first two premises are included of this book, however, �proposition� and �statement� information content of a statement. For the purposes

within the scope of a single sentence in the original are used interchangeably. argument. For the purposes of this chapter, com-

pound arrangements of statements in which the

various components are all claimed to be true will be Note on the History of Logic

considered as separate statements. The person who is generally credited as the father

Passages that contain arguments sometimes con- of logic is the ancient Greek philosopher Aristotle

tain statements that are neither premises nor con- (384�322 b.c.e.). Aristotle�s predecessors had been clusions. Only statements that are actually intended interested in the art of constructing persuasive argu-to support the conclusion should be included in the ments and in techniques for refuting the arguments of list of premises. If, for example, a statement serves others, but it was Aristotle who first devised system-merely to introduce the general topic, or merely atic criteria for analyzing and evaluating arguments. makes a passing comment, it should not be taken as Aristotle�s chief accomplishment is called syllo-part of the argument. Examples: gistic logic, a kind of logic in which the fundamen-

The claim is often made that malpractice tal elements are terms, and arguments are evaluated lawsuits drive up the cost of health care. as good or bad depending on how the terms are But if such suits were outlawed or severely arranged in the argument. Chapters 4 and 5 of this restricted, then patients would have no textbook are devoted mainly to syllogistic logic. But means of recovery for injuries caused by Aristotle also deserves credit for originating modal negligent doctors. Hence, the availability of logic , a kind of logic that involves such concepts malpractice litigation should remain intact. as possibility, necessity, belief, and doubt. In addi-Massive federal deficits push up interest rates tion, Aristotle catalogued several informal fallacies, for everyone. Servicing the debt gobbles up a topic treated in Chapter 3 of this book. a huge portion of the federal budget, which After Aristotle�s death, another Greek philosopher, lowers our standard of living. And big deficits Chrysippus (280�206 b.c.e.), one of the founders of also weaken the value of the dollar. For these the Stoic school, developed a logic in which the funda-reasons, Congress must make a determined mental elements were whole propositions. Chrysippus effort to cut overall spending and raise taxes. treated every proposition as either true or false and Politicians who ignore this reality imperil the developed rules for determining the truth or falsity future of the nation. of compound propositions from the truth or falsity of

4Chapter1 Basic Concepts

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). their components. In the course of doing so, he laid the had already begun to lose its unique identity in the foundation for the truth-functional interpretation of Middle Ages, was ignored altogether, and the logic the logical connectives presented in Chapter 6 of this of Aristotle was studied only in highly simplistic book and introduced the notion of natural deduction, presentations. A reawakening did not occur until treated in Chapter 7. two hundred years later through the work of Gott-

For thirteen hundred years after the death of fried Wilhelm Leibniz (1646�1716).

Chrysippus, relatively little creative work was done in Leibniz, a genius in numerous fields, attempted logic. The physician Galen (c.e. 129�ca. 199) devel- to develop a symbolic language or �calculus� that oped the theory of the compound categorical syllo- could be used to settle all forms of disputes, whether gism, but for the most part philosophers confined in theology, philosophy, or international relations. themselves to writing commentaries on the works of As a result of this work, Leibniz is sometimes cred-Aristotle and Chrysippus. Boethius (ca. 480�524) is ited with being the father of symbolic logic. Leibniz�s a noteworthy example. efforts to symbolize logic were carried into the nine-

The first major logician of the Middle Ages was teenth century by Bernard Bolzano (1781�1848).

Peter Abelard (1079�1142). Abelard reconstructed In the middle of the nineteenth century, logic and refined the logic of Aristotle and Chrysippus commenced an extremely rapid period of develop-as communicated by Boethius, and he originated a ment that has continued to this day. Work in symbolic theory of universals that traced the universal charac- logic was done by many philosophers and mathema-ter of general terms to concepts in the mind rather ticians, including Augustus De Morgan (1806�1871), than to �natures� existing outside the mind, as Aristo- George Boole (1815�1864), William Stanley Jevons tle had held. In addition, Abelard distinguished argu- (1835�1882), and John Venn (1834�1923). The rule ments that are�valid because of their form from those bearing De Morgan�s name is used in Chapter 7 of that are valid because of their content, but he held that this book. Boole�s interpretation of categorical prop-only formal validity is the �perfect� or conclusive vari- ositions and Venn�s method for diagramming them ety. This textbook follows Abelard on this point. are covered in Chapters 4 and 5. At the same time a

After Abelard, the study of logic during the revival in inductive logic was initiated by the British

Middle Ages flourished through the work of numer- philosopher John Stuart Mill (1806�1873), whose ous philosophers. A logical treatise by William of methods of induction are presented in Chapter 10. Sherwood (ca. 1200�1271) contains the first expres- Across the Atlantic, the American philosopher sion of the �Barbara, Celarent . . .� poem quoted in Charles Sanders Peirce (1839�1914) developed a Section 5.1 of this book, and the Summulae Logi- logic of relations, invented symbolic quantifiers, cales of Peter of Spain (ca. 1205�1277) became the and suggested the truth-table method for formu-standard textbook in logic for three hundred years. las in propositional logic. These topics are covered However, the most original contributions from this in Chapters 6 and 8 of this book. The truth-table period were made by William of Ockham (ca. 1285� method was completed independently by Emil Post 1347). Ockham extended the theory of modal logic, (1897�1954) and Ludwig Wittgenstein (1889�1951). conducted an exhaustive study of the forms of valid Toward the end of the nineteenth century, the and invalid syllogisms, and further developed the foundations of modern mathematical logic were laid idea of a metalanguage, a higher-level language used by Gottlob Frege (1848�1925). His Begriffsschrift to discuss linguistic entities such as words, terms, sets forth the theory of quantification presented in and propositions. Chapter 8 of this text. Frege�s work was continued

Toward the middle of the fifteenth century, a into the twentieth century by Alfred North White-

reaction set in against the logic of the Middle Ages. head (1861�1947) and Bertrand Russell (1872�1970), Rhetoric largely displaced logic as the primary whose monumental Principia Mathematica attempted focus of attention; the� logic of Chrysippus, which to reduce the whole of pure mathematics to logic.

Section 1.1 Arguments, Premises, and Conclusions 5

The Principia is the source of much of the symbolism theory there exists an undecidable formula�that that appears in Chapters 6, 7, and 8 of this text. is, a formula such that neither it nor its negation

During the twentieth century, much of the work is derivable from the axioms of the system. Other

in logic focused on the formal ization of logical sys- developments included multivalued logics and the tems and on questions dealing with the complete- formalization of modal logic. Most recently, logic ness and consistency of such systems. A now-famous has made a major contribution to technology by theorem proved by Kurt G�del (1906�1978) states providing the conceptual foundation for the elec-that in any formal system adequate for number tronic circuitry of digital computers.

Exercise 1.1

I. Each of the following passages contains a single argument. Using the letters �P� and

�C,� identify the premises and conclusion of each argument, writing premises first and conclusion last. List the premises in the order in which they make the most sense (usu-ally the order in which they occur), and write both premises and conclusion in the form of separate declarative sentences. Indicator words may be eliminated once premises and conclusion have been appropriately labeled. The exercises marked with a star are answered in the back of the book.

★1. Carbon monoxide molecules happen to be just the right size and shape, and

happen to have just the right chemical properties, to fit neatly into cavities within hemoglobin molecules in blood that are normally reserved for oxygen molecules. Consequently, carbon monoxide diminishes the oxygen-carrying capacity of blood.

(Nivaldo J. Tro, Chemistry: A Molecular Approach, 2nd ed.)

2. Since the good, according to Plato, is that which furthers a person�s real

interests, it follows that in any given case when the good is known, men will seek it.

(Avrum Stroll and Richard Popkin, Philosophy

and the Human Spirit)

3. As the denial or perversion of justice by the sentences of courts, as well as in

any other manner, is with reason classed among the just causes of war, it will follow that the federal judiciary ought to have cognizance of all causes in which the citizens of other countries are concerned.

(Alexander Hamilton, Federalist Papers, No. 80)

★4. When individuals voluntarily abandon property, they forfeit any expectation of

privacy in it that they might have had. Therefore, a warrantless search or seizure of abandoned property is not unreasonable under the Fourth Amendment.

( Judge Stephanie Kulp Seymour, United States v. Jones)

5. Artists and poets look at the world and seek relationships and order. But they

translate their ideas to canvas, or to marble, or into poetic images. Scientists try to find relationships between different objects and events. To express the

6Chapter1 Basic Concepts

order they find, they create hypotheses and theories. Thus the great scientific theories are easily compared to great art and great literature.

(Douglas C. Giancoli, The Ideas of Physics, 3rd ed.)

6. The fact that there was never a land bridge between Australia and mainland

Asia is evidenced by the fact that the animal species in the two areas are very different. Asian placental mammals and Australian marsupial mammals have not been in contact in the last several million years.

( T. Douglas Price and Gary M. Feinman, Images of the Past)

★ 7. It really does matter if you get enough sleep. We need sleep to think clearly,

react quickly, and create memories. Studies show that people who are taught mentally challenging tasks do better after a good night�s sleep. Other research suggests that sleep is needed for creative problem solving.

(U.S. National Institutes of Health,

�Your Guide to Healthy Sleep�)

8. The classroom teacher is crucial to the development and academic success

of the average student, and administrators simply are ancillary to this effort. For this reason, classroom teachers ought to be paid at least the equivalent of administrators at all levels, including the superintendent.

(Peter F. Falstrup, letter to the editor)

9. An agreement cannot bind unless both parties to the agreement know what

they are doing and freely choose to do it. This implies that the seller who intends to enter a contract with a customer has a duty to disclose exactly what the customer is buying and what the terms of the sale are.

(Manuel G. Velasquez, �The Ethics of Consumer Production�)

★10. Punishment, when speedy and specific, may suppress undesirable behavior,

but it cannot teach or encourage desirable alternatives. Therefore, it is cru-cial to use positive techniques to model and reinforce appropriate behavior that the person can use in place of the unacceptable response that has to be �suppressed.

( Walter Mischel and Harriet Mischel, Essentials of Psychology)

11. Profit serves a very crucial function in a free-enterprise economy, such as our

own. High profits are the signal that consumers want more of the output of the industry. High profits provide the incentive for firms to expand output and for more firms to enter the industry in the long run. For a firm of above-average efficiency, profits represent the reward for greater efficiency.

(Dominic Salvatore, Managerial Economics, 3rd ed.)

12. Cats can think circles around dogs! My cat regularly used to close and lock the

door to my neighbor�s doghouse, trapping their sleeping Doberman inside. Try telling a cat what to do, or putting a leash on him�he�ll glare at you and say, �I don�t think so. You should have gotten a dog.�

(Kevin Purkiser, letter to the editor)

★13. Since private property helps people define themselves, since it frees people

from mundane cares of daily subsistence, and since it is finite, no individual

Section 1.1 Arguments, Premises, and Conclusions 7

should accumulate so much property that others are prevented from accumu-lating the necessities of life.

(Leon P. Baradat, Political Ideologies,

Their Origins and Impact)

14. To every existing thing God wills some good. Hence, since to love any thing is

nothing else than to will good to that thing, it is manifest that God loves every-thing that exists.

( Thomas Aquinas, Summa Theologica)

15. Women of the working class, especially wage workers, should not have more

than two children at most. The average working man can support no more and the average working woman can take care of no more in decent fashion.

(Margaret Sanger, Family Limitations)

★16. Radioactive fallout isn�t the only concern in the aftermath of nuclear explo-

sions. The nations of planet Earth have acquired nuclear weapons with an explosive power equal to more than a million Hiroshima bombs. Studies sug-gest that explosion of only half these weapons would produce enough soot, smoke, and dust to blanket the earth, block out the sun, and bring on a nuclear winter that would threaten the survival of the human race.

( John W. Hill and Doris K. Kolb, Chemistry

for Changing Times, 7th ed.)

17. An ant releases a chemical when it dies, and its fellows then carry it away to the com-

post heap. Apparently the communication is highly effective; a healthy ant painted with the death chemical will be dragged to the funeral heap again and again.

(Carol R. Ember and Melvin Ember,

Cultural Anthropology, 7th ed.)

18. Every art and every inquiry, and similarly every action and pursuit, is thought

to aim at some good; and for this reason the good has rightly been declared to be that at which all things aim.

(Aristotle, Nicomachean Ethics)

★19. Poverty offers numerous benefits to the nonpoor. Antipoverty programs pro-

vide jobs for middle-class professionals in social work, penology, and public health. Such workers� future advancement is tied to the continued growth of bureaucracies dependent on the existence of poverty.

( J. John Palen, Social Problems)

20. Corn is an annual crop. Butcher�s meat, a crop which requires four or five years

to grow. As an acre of land, therefore, will produce a much smaller quantity of the one species of food than the other, the inferiority of the quantity must be compensated by the superiority of the price.

(Adam Smith, The Wealth of Nations)

21. Neither a borrower nor lender be

For loan oft loses both itself and friend,

And borrowing dulls the edge of husbandry.

( William Shakespeare, Hamlet I, 3)

8Chapter1 Basic Concepts

★22. The stakes in whistleblowing are high. Take the nurse who alleges that phy-

sicians enrich themselves in her hospital through unnecessary surgery; the engineer who discloses safety defects in the braking systems of a fleet of new rapid-transit vehicles; the Defense Department official who alerts Congress to military graft and overspending: all know that they pose a threat to those whom they denounce and that their own careers may be at risk.

(Sissela Bok, �Whistleblowing and Professional Responsibility�)

23. If a piece of information is not �job relevant,� then the employer is not entitled qua

employer to know it. Consequently, since sexual practices, political beliefs, asso-ciational activities, etc., are not part of the description of most jobs, that is, since they do not directly affect one�s job performance, they are not legitimate informa-tion for an employer to know in the determination of the hiring of a job applicant.

(George G. Brenkert, �Privacy, Polygraphs, and Work�)

24. Many people believe that a dark tan is attractive and a sign of good health, but

mounting evidence indicates that too much sun can lead to health problems. One of the most noticeable effects is premature aging of the skin. The sun also contributes to certain types of cataracts, and, what is most worrisome, it plays a role in skin cancer.

( Joseph M. Moran and Michael D. Morgan,

Meteorology, 4th ed.)

★25. Contrary to the tales of some scuba divers, the toothy, gaping grin on the

mouth of an approaching shark is not necessarily anticipatory. It is generally accepted that by constantly swimming with its mouth open, the shark is simply avoiding suffocation. This assures a continuous flow of oxygen-laden water into their mouths, over their gills, and out through the gill slits.

(Robert A. Wallace et al., Biology: The Science of Life)

26. Not only is the sky blue [as a result of scattering], but light coming from it

is also partially polarized. You can readily observe this by placing a piece of Polaroid (for example, one lens of a pair of Polaroid sunglasses) in front of your eye and rotating it as you look at the sky on a clear day. You will notice a change in light intensity with the orientation of the Polaroid.

(Frank J. Blatt, Principles of Physics, 2nd ed.)

27. Since the secondary light [from the moon] does not inherently belong to the

moon and is not received from any star or from the sun, and since in the whole universe there is no other body left but the earth, what must we conclude? What is to be proposed? Surely we must assert that the lunar body (or any other dark and sunless orb) is illuminated by the earth.

(Galileo Galilei, The Starry Messenger)

★28. Anyone familiar with our prison system knows that there are some inmates

who behave little better than brute beasts. But the very fact that these prisoners exist is a telling argument against the efficacy of capital punishment as a deter-rent. If the death penalty had been truly effective as a deterrent, such prisoners would long ago have vanished.

(�The Injustice of the Death Penalty,� America)

Section 1.1 Arguments, Premises, and Conclusions 9

29. Though it is possible that REM sleep and dreaming are not necessary in the

adult, REM deprivation studies seem to suggest otherwise. Why would REM pressure increase with deprivation if the system is unimportant in the adult?

(Herbert L. Petri, Motivation: Theory and Research, 2nd ed.)

30. We say that an end pursued in its own right is more complete than an end pursued

because of something else, and that an end that is never choiceworthy because of something else is more complete than ends that are choiceworthy both in their own right and because of this end. Hence, an end that is always choiceworthy in its own right, and never because of something else, is complete without qualification.

(Aristotle, Nicomachean Ethics)

II. The following arguments were taken from magazine and newspaper editorials and let-

ters to the editor. In most instances the main conclusion must be rephrased to capture the full intent of the author. Write out what you interpret the main conclusion to be.

★1. University administrators know well the benefits that follow notable success in

college sports: increased applications for admissions, increased income from licensed logo merchandise, more lucrative television deals, postseason game revenue, and more successful alumni fund drives. The idea that there is some-thing ideal and pure about the amateur athlete is self-serving bunk.

(Michael McDonnell, letter to the editor)

2. In a nation of immigrants, people of diverse ethnic backgrounds must have a common

bond through which to exchange ideas. How can this bond be accomplished if there is no common language? It is those who shelter the immigrant from learning English by encouraging the development of a multilingual society who are creating a xeno-phobic atmosphere. They allow the immigrant to surround himself with a cocoon of language from which he cannot escape and which others cannot penetrate.

(Rita Toften, letter to the editor)

3. The health and fitness of our children has become a problem partly because

of our attitude toward athletics. The purpose of sports, especially for children, should be to make healthy people healthier. The concept of team sports has failed to do this. Rather than learning to interact and cooperate with others, youngsters are taught to compete. Team sports have only reinforced the notion that the team on top is the winner, and all others are losers. This approach does not make sports appealing to many children, and some, especially among the less fit, burn out by the time they are twelve.

(Mark I. Pitman, �Young Jocks�)

★4. College is the time in which a young mind is supposed to mature and acquire

wisdom, and one can only do this by experiencing as much diverse intellectual stimuli as possible. A business student may be a whiz at accounting, but has he or she ever experienced the beauty of a Shakespearean sonnet or the bound-less events composing Hebrew history? Most likely not. While many of these neoconservatives will probably go on to be financially successful, they are rob-bing themselves of the true purpose of collegiate academics, a sacrifice that outweighs the future salary checks.

( Robert S. Griffith, �Conservative College Press�)

10Chapter1 Basic Concepts

5. History has shown repeatedly that you cannot legislate morality, nor does

anyone have a right to. The real problem is the people who have a vested inter-est in sustaining the multibillion-dollar drug industry created by the laws against drugs. The legalization of drugs would remove the thrill of breaking the law; it would end the suffering caused by unmetered doses, impurities, and substandard paraphernalia. A huge segment of the underground and extra-legal�economy would move into a legitimate economy, taking money away from criminals, eliminating crime and violence, and restoring many talented people to useful endeavor.

( Thomas L. Wayburn, letter to the editor)

6. Infectious disease is no longer the leading cause of death in this country,

thanks to antibiotics, but there are new strains of bacteria that are resistant to�and others that grow only in the presence of�antibiotics. Yet Congress wants to cut the National Institutes of Health budget. Further cuts would leave us woefully unprepared to cope with the new microbes Mother Nature has cooking in her kitchen.

( Valina L. Dawson, letter to the editor)

★ 7. A person cannot reject free will and still insist on criminality and codes of

moral behavior. If people are compelled by forces beyond their control (genes or environment), then their actions, no matter how vile, are excusable. That means the Nuremberg trials of Nazi murderers were invalid, and all prison gates should be flung open. The essence of our humanity is the ability to choose between right and wrong, good and evil, and act accordingly. Strip that from us and we are mere animals.

(Rabbi Gilbert S. Rosenthal, letter to the editor)

8. Ideally, decisions about health care should be based on the doctor�s clinical

judgment, patient preference, and scientific evidence. Patients should always be presented with options in their care. Elective cesarean section, however, is not used to treat a problem but to avoid a natural process. An elective surgery like this puts the patient at unnecessary risk, increases the risk for complica-tions in future deliveries, and increases health-care costs.

(Anne Foster-Rosales, M.D., letter to the editor)

9. Parents who feel guilty for the little time they can (or choose to) spend with

their children �pick up� after them�so the children don�t learn to face the con-sequences of their own choices and actions. Parents who allow their children to fail are showing them greater love and respect.

(Susan J. Peters, letter to the editor)

★10. Most of the environmental problems facing us stem, at least in part, from the

sheer number of Americans. The average American produces three quarters of a ton of garbage every year, consumes hundreds of gallons of gasoline, and uses large amounts of electricity (often from a nuclear power plant, coal burn-ing, or a dam). The least painful way to protect the environment is to limit population growth.

(Craig M. Bradley, letter to the editor)

Section 1.1 Arguments, Premises, and Conclusions 11

III. Define the following terms:

logic premise inference argument conclusion proposition statement conclusion indicator syllogistic logic truth value premise indicator modal logic

IV. Answer �true� or �false� to the following statements:

1. The purpose of the premise or premises is to set forth the reasons or evidence

given in support of the conclusion.

2. Some arguments have more than one conclusion.

3. All arguments must have more than one premise.

4. The words �therefore,� �hence,� �so,� �since,� and �thus� are all conclusion indicators.

5. The words �for,� �because,� �as,� and �for the reason that� are all premise indicators.

6. In the strict sense of the terms, inference and argument have exactly the same

meaning.

7. In most (but not all) arguments that lack indicator words, the conclusion is the

first statement.

8. Any sentence that is either true or false is a statement.

9. Every statement has a truth value.

10. Aristotle is the person usually credited with being the father of logic.

V. Create the following arguments:

1. An argument having one or more premise indicators and the conclusion:

�Uniforms should/should not be required in high schools.�

2. An argument having a conclusion indicator and the conclusion: �The legal

drinking age should/should not be lowered to 18.�

3. An argument having no indicator words and the conclusion: �Renewable forms

of energy should/should not be subsidized by the government.�

1.2 Recognizing Arguments

Not all passages contain arguments. Because logic In the previous section of this book we learned deals with arguments, it is important to be able to that every argument has at least one premise and distinguish passages that contain arguments from exactly one conclusion. The premise or premises set those that do not. In general, a passage contains an forth the alleged evidence or reasons, and the con-argument if it purports to prove something; if it does clusion asserts what is claimed to follow from the not do so, it does not contain an argument. alleged evidence or reasons. This definition of an

12Chapter1 Basic Concepts

Prominent Aristotle 384�322 Logiciansb.c.e.

A ristotle was born in Stagira, a small Greek town situated on the northern coast of the Aegean Sea. His father was a physician in the court of King Amyntas II of Macedonia, and the young Aristotle was a friend of the king�s son, Philip, who was later

to become king himself and the father of Alexander

the Great. When he was about seventeen, Aristotle

was sent to Athens to further his education in Plato�s

Academy, the finest institution of higher learning in

the Greek world. After Plato�s death Aristotle left for

Assos, a small town on the coast of Asia Minor, where

he married the niece of the local ruler.

Six years later, Aristotle accepted an invitation to

return to Macedonia to serve as tutor of the young

Alexander. When Alexander ascended the throne fol-

lowing his father�s assassination, Aristotle�s tutorial

job was finished, and he departed for Athens where Science History Images/Alamy Stock Photo he set up a school near the temple of Apollo Lyceus.

The school came to be known as the Lyceum, and three categorical propositions. He showed how cat-Alexander supported it with contributions of money egorical syllogisms can be catalogued in terms of and specimens of flora and fauna derived from his mood and figure, and he developed a set of rules far-flung conquests. After Alexander�s death, an anti- for determining the validity of categorical syllo-Macedonian rebellion forced Aristotle to leave Athens gisms. Also, he showed how the modal concepts for Chalcis, about thirty miles to the north, where he of possibility and necessity apply to categorical died one year later at the age of sixty-two. propositions. In addition to the theory of the syl-

Aristotle is universally recognized as the originator logism, Aristotle advanced the theory of definition

of logic. He defined logic as the study of the process by genus and difference, and he showed how argu-by which a statement follows by necessity from one or ments could be defective in terms of thirteen forms more other statements. The most fundamental kind of of informal fallacy. statement, he thought, is the categorical proposition, Aristotle made profound contributions to many and he classified the four kinds of categorical proposi- areas of human learning including biology, physics, tions in terms of their being universal, particular, affir- metaphysics, epistemology, psychology, aesthetics, mative, and negative. He also developed the square ethics, and politics. However, his accomplishments in of opposition, which shows how one such proposition logic were so extensive and enduring that two thou-implies the truth or falsity of another, and he identified sand years after his death, the great philosopher the relations of conversion, obversion, and contrapo- Immanuel Kant said that Aristotle had discovered sition, which provide the basis for various immediate everything that could be known about logic. His logic inferences. was not superseded until the end of the nineteenth

His crowning achievement is the theory of the cat- century when Frege, Whitehead, and Russell devel-

egorical syllogism, a kind of argument consisting of oped modern mathematical logic.

Section 1.2 Recognizing Arguments 13

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). argument expresses what is needed for a passage to justified in calling the passage an argument. The first

contain an argument: statement is the conclusion, and the other two are 1. At least one of the statements must claim to the premises. In deciding whether there is a claim that evidence 2. There must be a claim that the alleged evidence present evidence or reasons. supports or implies something, keep an eye out for (1) premise and conclusion indicator words and supports or implies something�that is, a claim (2) the presence of an inferential relationship that something follows from the alleged evidence between the statements. In connection with these or reasons.

points, however, a word of caution is in order. First,

It is not necessary that the premises present actual the mere occurrence of an indicator word by no

evidence or true reasons nor that the premises actu- means guarantees the presence of an argument. For ally support the conclusion. But at least the premises example, consider the following passages:

must claim to present evidence or reasons, and there Since Edison invented the phonograph, there must be a claim that the evidence or reasons support have been many technological innovations. or imply something. Also, you should recognize that Since Edison invented the phonograph, he the second claim is not equatable with the intentions deserves credit for a major technological of the arguer. Intentions are subjective and, as such, innovation. are usually not accessible to the evaluator. Rather, this claim is an objective feature of an argument In the first passage the word �since� is used in a tem-grounded in its language or structure. poral sense. It means �from the time that.� Thus, the

In deciding whether a passage contains an argu- first passage is not an argument. In the second pas-

ment, the claim that the alleged reasons or evidence sage �since� is used in a logical sense, and so the pas-supports or implies something is usually the more sage is an argument. important of the two. Such a claim can be either explicit The second cautionary point is that it is not always or implicit. An explicit claim is usually asserted by easy to detect the occurrence of an inferential rela-premise or conclusion indicator words (�thus,� �since,� tionship between the statements in a passage, and �because,� �hence,� �therefore,� and so on). Example: one may have to review a passage several times before

The Ebola virus has yet to be eradicated, making a decision. In reaching such a decision, one and it kills on average 50 percent of those may find it helpful to mentally insert the word �there-it infects. Thus, Ebola remains a threat to fore� before the various statements to see whether it human health. makes sense to interpret one of them as following

The word �thus� expresses the claim that something from the others. Even with this mental aid, however, the decision whether a passage contains an inferential is being inferred, so the passage is an argument. relationship (as well as the decision about indicator An implicit claim exists if there is an inferential words) often involves a heavy dose of interpretation. relationship between the statements in a passage, but As a result, not everyone will agree about every pas-the passage contains no indicator words. Example: sage. Sometimes the only answer possible is a condi-The genetic modification of food is risky tional one: � If this passage contains an argument, then business. Genetic engineering can intro-these are the premises and that is the conclusion.� duce unintended changes into the DNA of To assist in distinguishing passages that contain the food-producing organism, and these arguments from those that do not, let us now inves-changes can be toxic to the consumer. tigate some typical kinds of nonarguments. These The inferential relationship between the first state-include simple noninferential passages, expository ment and the other two constitutes an implicit passages, illustrations, explanations, and conditional claim that evidence supports something, so we are statements.

14Chapter1 Basic Concepts

Simple Noninferential Passages at an adequate profit and that the services and

Simple noninferential passages are unproblematic products we offer must be better than those passages that lack a claim that anything is being offered by competitors. proved. Such passages contain statements that could (Robert D. Hay and Edmund R. be premises or conclusions (or both), but what is Gray, �Introduction to Social

Responsibility�)

ports a conclusion or that any potential conclusion is After 400 years of racism toward Blacks, the treatment of Asians and the antipathy toward missing is a claim that any potential premise sup-

opinion, loosely associated statements, and reports. more equitable, not scrambling to preserve A warning is a form of expression that is their ability to lord over everyone. Putting up intended to put someone on guard against a danger-with a little discomfort is a small price to pay for ous or detrimental situation. Examples: warnings, pieces of advice, statements of belief or any religion besides Christianity, the privileged class should be working to make the country supported by premises. Passages of this sort include

a country where you are judged on your merits,

Watch out that you don�t slip on the ice. not your skin.

Whatever you do, never confide personal Say Is Grow Up�) (Susan Polifronio, �All I Can secrets to Blabbermouth Bob.

If no evidence is given to prove that such statements Because neither of these statements asserts any claim that a belief or opinion is supported by evidence, or are true, then there is no argument. that it supports some conclusion, there is no argument. A piece of advice is a form of expression that Loosely associated statements may be about makes a recommendation about some future deci-the same general subject, but they lack a claim that sion or course of conduct. Examples: one of them is proved by the others. Example:

You should keep a few things in mind before Not to honor men of worth will keep the people buying a used car. Test drive the car at vary- from contention; not to value goods that are ing speeds and conditions, examine the oil hard to come by will keep them from theft; not in the crankcase, ask to see service records, to display what is desirable will keep them from and, if possible, have the engine and power being unsettled of mind. train checked by a mechanic. (Lao-Tzu, Thoughts from the Before accepting a job after class hours, I Tao Te Ching)

would suggest that you give careful consid- Because there is no claim that any of these state-eration to your course load. Will you have ments provides evidence or reasons for believing sufficient time to prepare for classes and another, there is no argument. tests, and will the job produce an excessive A report consists of a group of statements that drain on your energies? convey information about some topic or event.

As with warnings, if there is no evidence that is Example: intended to prove anything, then there is no argument.

A The period of 1648�1789 was one of competi-statement of belief or opinion is an expres-

sion about what someone happens to believe or tion among the primary monarchs of Europe.

think about something. Examples: Wars among the great powers were frequent but limited. France made major efforts to

We believe that our company must develop and become paramount, but the balance of power produce outstanding products that will perform operated to block French expansion. a great service or fulfill a need for our custom- (Steven L. Spiegel, World ers. We believe that our business must be run Politics in a New Era)

Section 1.2 Recognizing Arguments 15

These statements could serve as the premises There are three familiar states of matter: solid,

of an argument, but because the author makes no liquid, and gas. Solid objects ordinarily maintain claim that they support or imply anything, there is their shape and volume regardless of their loca-no argument. Another type of report is the news tion. A liquid occupies a definite volume, but report: assumes the shape of the occupied portion of

its container. A gas maintains neither shape nor

Witnesses said they heard a loud crack before volume. It expands to fill completely whatever a balcony gave way at a popular nightspot, container it is in. dropping dozens of screaming people fourteen (John W. Hill and Doris K. feet. At least eighty people were injured at the Kolb, Chemistry for Changing Diamond Horseshoe casino when they fell onto Times, 7th ed.) broken glass and splintered wood. Investiga-

deck�s occupancy load. There is a stylized relation of artist to mass audi-tors are waiting for an engineer�s report on the

(Newspaper clipping) ence in the sports, especially in baseball. Each player develops a style of his own�the swagger

Again, because the reporter makes no claim that as he steps to the plate, the unique windup a these statements imply anything, there is no pitcher has, the clean-swinging and hard-driving argument. hits, the precision quickness and grace of infield

One must be careful, though, with reports about and outfield, the sense of surplus power behind

arguments: whatever is done.

(Max Lerner,

�The Air Force faces a serious shortage of America as a Civilization) experienced pilots in the years ahead, because

repeated overseas tours and the allure of high- In each passage the topic sentence is stated first, and paying jobs with commercial airlines are winning the remaining sentences merely develop and flesh out over lucrative bonuses to stay in the ser- out this topic sentence. These passages are not argu-vice,� says a prominent Air Force official. ments, because they lack an inferential claim. How-

(Newspaper clipping) ever, expository passages differ from simple nonin-

Properly speaking, this passage is not an argument, ferential passages (such as warnings and pieces of advice) in that many of them can also be taken as because the author of the passage does not claim arguments. If the purpose of the subsequent sen-that anything is supported by evidence. Rather, the tences in the passage is not only to flesh out the topic author reports the claim by the Air Force official sentence but also to prove it, then the passage is an that something is supported by evidence. If such argument. Example: passages are interpreted as �containing� arguments, it must be made clear that the argument is not the Skin and the mucous membrane lining the author�s but one made by someone about whom the respiratory and digestive tracts serve as author is reporting. mechanical barriers to entry by microbes. Oil-

gland secretions contain chemicals that weaken

Expository Passages or kill bacteria on skin. The respiratory tract is

lined by cells that sweep mucus and trapped

An expository passage is a kind of discourse that particles up into the throat, where they can begins with a topic sentence followed by one or more be swallowed. The stomach has an acidic pH, sentences that develop the topic sentence. If the which inhibits the growth of many types of objective is not to prove the topic sentence but only bacteria. to expand it or elaborate it, then there is no argument. (Sylvia S. Mader, Examples: Human Biology, 4th ed.)

16Chapter1 Basic Concepts

In this passage the topic sentence is stated first, first selection, the word �thus� indicates how some-

and the purpose of the remaining sentences is not thing is done�namely, how chemical elements and only to show how the skin and mucous membranes compounds can be represented by formulas. In the serve as barriers to microbes but also to prove that second, the examples cited are intended to illustrate they do this. Thus, the passage can be taken as both the meaning of the word �deciduous.� It pins down an expository passage and an argument. the meaning by providing concrete instances.

In deciding whether an expository passage should However, as with expository passages, many illus-

be interpreted as an argument, try to determine trations can be taken as arguments. Such arguments whether the purpose of the subsequent sentences in are often called arguments from example. Here the passage is merely to develop the topic sentence is an instance of one: or also to prove that it is true. In borderline cases,

situations authors rarely try to prove something is noma, the most common of all skin cancers, can produce disfigurement, but it almost true when everyone already accepts it. However, if never results in death. the topic sentence makes a claim that many people do not accept or have never thought about, then the In this passage the example given is intended to purpose of the remaining sentences may be both to prove the truth of �Not all cancers are life threat-prove the topic sentence is true as well as to develop claim that everyone accepts or agrees with. If it does, can cause death, not all cancers are life threatening. For example, basal cell carci-the passage is probably not an argument. In real-life ask yourself whether the topic sentence makes a Although most forms of cancer, if untreated,

ening.� Thus, the passage is best interpreted as an

it. If this is so, the passage is an argument. argument.

Finally, if even this procedure yields no definite In deciding whether an illustration should be

answer, the only alternative may be to say that if interpreted as an argument, determine whether the passage is taken as an argument, then the first the passage merely shows how something is done statement is the conclusion and the others are the or what something means, or whether it also pur-premises. ports to prove something. In borderline cases it

helps to note whether the claim being illustrated

Illustrations with. If it is, the passage is probably not an argu-is one that practically everyone accepts or agrees An illustration ment. As already noted, in real-life situations is an expression involving one or more examples that is intended to show what some- authors rarely attempt to prove what everyone thing means or how it is done. Illustrations are often already accepts. But if the claim being illustrated confused with arguments because many illustrations is one that many people do not accept or have contain indicator words such as �thus.� Examples: never thought about, then the passage may be

Chemical elements, as well as compounds, interpreted as an argument.

by �H O,� and sodium chloride by �NaCl.� 2 cally everyone knows that water is H O�and most 2 A deciduous tree is any tree that loses its people have at least a vague idea of what a deciduous leaves during the winter. For example, tree is. But they may not have ever considered maples are deciduous. And so are elms, pop-whether some forms of cancer are not life threaten-lars, hawthorns, and alders. ing. This is one of the reasons for evaluating the first Thus, oxygen is represented by �O considered, most people are aware that elements and ,� water 2 compounds can be expressed by formulas�practi-can be represented by molecular formulas. Thus, in reference to the first two examples we

These selections are not arguments, because they two examples as mere illustrations and the last one make no claim that anything is being proved. In the as an argument.

Section 1.2 Recognizing Arguments 17

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). the explanation. An is an expression ment, the purpose of the premises is to prove explanationwhy something is the case, whereas in an argu-that that purports to shed light on some event or phe-something is the case. nomenon. The event or phenomenon in question is In the first example given, the fact that the sky One of the most important kinds of nonargument is In other words, the purpose of the explanans is to show Explanations explanandum event�not to prove that it occurred. usually accepted as a matter of fact. Examples: is blue is readily apparent to everyone. The state-

The sky appears blue from the earth�s surface ment that light rays from the sun are scattered because light rays from the sun are scattered by particles in the atmosphere is not intended to by particles in the atmosphere. prove that the sky is blue, but rather to show why Golf balls have a dimpled surface because it is blue. In the second example, practically every-the dimples reduce air drag, causing the ball one knows that golf balls have a dimpled surface. to travel farther. The�purpose of the passage is to explain why they Navel oranges are called by that name have a dimpled surface�not to prove that they do. because they have a growth that resembles Similarly, in the third example, it is obvious that a human navel on the end opposite the naval oranges are called naval oranges. The pur-stem. pose of the passage is to shed light on why they

components: the explanandum and explanans. ments, identify the statement that is either the The explanandum is the statement that describes explanandum or the conclusion (usually this is the the event or phenomenon to be explained, and statement that precedes the word �because�). If this Every explanation is composed of two distinct Thus, to distinguish explanations from argu-have this name.

the explanans is the statement or group of state- statement describes an accepted matter of fact, and ments that purports to do the explaining. In the first if the remaining statements purport to shed light on example, the explanandum is the statement �The this statement, then the passage is an explanation. sky appears blue from the earth�s surface� and the This method usually works to distinguish argu-explanans is �Light rays from the sun are scattered ments from explanations. However, some passages by particles in the atmosphere.� can be interpreted as both explanations and argu-

ments. Examples:

ArgumentExplanation Women become intoxicated by drinking a

smaller amount of alcohol than men because

Premises Accepted men metabolize part of the alcohol before facts Explanans

it reaches the bloodstream, whereas women

Claimed to to shed prove Claimed do not. Household bleach should never be mixed

Conclusion Accepted Explanandum tion releases chlorine gas, which is highly fact light on with ammonia because the combina-

poisonous.

The purpose of these passages could be to prove

Explanations are sometimes mistaken for argu- the first statement to those who do not accept it as

ments because they often contain the indicator fact, and to shed light on that fact to those who do word �because.� Yet explanations are not argu- accept it. Alternately, the passage could be intended ments, because in an explanation the purpose of the to prove the first statement to a person who accepts explanans is to shed light on, or to make sense of, the its truth on blind faith or incomplete experience,

18Chapter1 Basic Concepts

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). and simultaneously to shed light on this truth. a conditional statement as those about chocolate Thus, these passages can be correctly interpreted as and school boards. both an explanation and an argument.

Perhaps the greatest problem confronting the

effort to distinguish explanations from argu- Conditional statements

ments lies in determining whether something Antecedent Consequent is an accepted matter of fact. Obviously, what is

accepted by one person may not be accepted by If then . another. Thus, the effort often involves determin-

ing which person or group of people the passage is Consequent Antecedent directed to�the intended audience. Sometimes the

source of the passage (textbook, newspaper, tech- if . nical journal, etc.) will decide the issue. But when

the passage is taken totally out of context, ascer-

taining the source may prove impossible. In those Conditional statements are not arguments, circumstances the only possible answer may be to because they fail to meet the criteria given earlier. In say that if the passage is an argument, then such- an argument, at least one statement must claim to and-such is the conclusion and such-and-such are present evidence, and there must be a claim that this the premises. evidence implies something. In a conditional state-

ment, there is no claim that either the antecedent or

the consequent presents evidence. In other words,

Conditional Statements there is no assertion that either the antecedent or the

A consequent is true. Rather, there is only the assertion conditional statement is an �if . . . then . . .� statement; for example: that if the antecedent is true, then so is the conse-

quent. Of course, a conditional statement as a whole

If chocolate contains caffeine, then chocolate may present evidence because it asserts a relation-impairs sleep. ship between statements. Yet when conditional state-If school boards ban books, then they jeopar- ments are taken in this sense, there is still no argu-dize the educational process. ment, because there is then no separate claim that

Every conditional statement is made up of two this evidence implies anything. component statements. The component statement Some conditional statements are similar to argu-immediately following the �if � is called the ments, however, in that they express the outcome ante-cedent of a reasoning process. As such, they may be said , and the one following the �then� is called the to have a certain inferential content. Consider the consequent . (Occasionally, the word �then� is left out, and occasionally the order of anteced- following:

ent and consequent is reversed.) In the first exam- If sugary drinks cause heart disease and ple, the antecedent is �Professional football games diabetes, then sugary drinks should be incite violence in the home,� and the consequent regulated. is �The widespread approval given to this sport should be reconsidered.� In both of these exam- The link between the antecedent and consequent ples, there is a meaningful relationship between resembles the inferential link between the prem-antecedent and consequent. However, such a rela- ises and conclusion of an argument. Yet there is a tionship need not exist for a statement to count difference because the premises of an argument as conditional. The statement �If Miley Cyrus is a are claimed to be true, whereas no such claim is singer, then Denver is in Colorado� is just as much made for the antecedent of a conditional statement.

Section 1.2 Recognizing Arguments 19

Accordingly, conditional statements are not argu- argument, depending on such factors as the pres-ments.* Yet their inferential content may be reex- ence of indicator words and an inferential relation-pressed to form arguments: ship between the statements.

Sugary drinks cause heart disease and diabetes. Conditional statements are especially important Therefore, sugary drinks should be regulated. in logic (and many other fields) because they express

the relationship between necessary and sufficient

Finally, while no single conditional statement is conditions. A is said to be a sufficient condition

an argument, a conditional statement may serve as for B whenever the occurrence of A is all that is either the premise or the conclusion (or both) of an needed for the occurrence of B. For example, being argument, as the following examples illustrate: a dog is a sufficient condition for being an animal.

If North Korea is developing ballistic missiles, On the other hand, B is said to be a necessary then North Korea is a threat to world peace. condition for A whenever A cannot occur without North Korea is developing ballistic missiles. the occurrence of B. Thus, being an animal is a nec-Therefore, North Korea is a threat to world essary condition for being a dog. peace. The difference between sufficient and necessary

conditions is a bit tricky. So, to clarify the idea fur-

If our borders are porous, then terrorists can ther, suppose you are given a large, closed cardboard enter the country at will. box. Also, suppose you are told there is a dog in the If terrorists can enter the country at will, then all box. Then you know for sure there is an animal in

of us are less secure. the box. No additional information is needed to draw Therefore, if our borders are porous, then all of this conclusion. This means that being a dog is suffi-us are less secure. cient for being an animal. However, being a dog is not The relation between conditional statements and necessary for being an animal, because if you are told

arguments may now be summarized as follows: that the box contains a cat, you can conclude with 1. A single conditional statement is not an argument. equal certainty that it contains an animal. In other words, it is not necessary for the box to contain a dog

2. A conditional statement may serve as either for it to contain an animal. It might equally well con-

the premise or the conclusion (or both) of an tain a cat, a mouse, a squirrel, or any other animal. argument. On the other hand, suppose you are told that

3. The inferential content of a conditional state- whatever might be in the box is not an animal. Then

ment may be reexpressed to form an argument. you know for certain there is no dog in the box. The

recognition of arguments. According to the first The first two rules are especially pertinent to the reason you can draw this conclusion is that being an animal is necessary for being a dog. If there is no animal, there is no dog. However, being an animal is rule, if a passage consists of a single conditional not sufficient for being a dog, because if you are told statement, it is not an argument. But if it consists of that the box contains an animal, you cannot, from this a conditional statement together with some other information alone, conclude that it contains a dog. It statement, then, by the second rule, it may be an might contain a cat, a mouse, a squirrel, and so�on.

These ideas are expressed in the following condi-

*In saying this we are temporarily ignoring the possibility of these state- tional statements: ments being enthymemes. As we shall see in Chapter 5, an enthymeme

is an argument in which a premise or conclusion (or both) is implied If X is a dog, then X is an animal. but not stated. If, to this example, we add the premise �Sugary drinks If X is not an animal, then X is not a dog. cause heart disease and diabetes� and the conclusion �Therefore,

sugary drinks should be regulated,� we have a complete argument. To The first statement says that being a dog is a familiar with the context in which it occurs. sufficient condition for being an animal, and the decide whether a conditional statement is an enthymeme, we must be

20Chapter1 Basic Concepts

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). second that being an animal is a necessary condition condition and in another way a sufficient condition. for being a dog. However, a little reflection reveals The terminology of sufficient and necessary condi-that these two statements say exactly the same tions will be used in later chapters to express defini-thing. Thus, each expresses in one way a necessary tions and causal connections.

Summary

In deciding whether a passage contains an argument, you should look for three things: (1) indicator words such as �therefore,� �since,� �because,� and so on; (2) an inferential relation-ship between the statements; and (3) typical kinds of nonarguments. But remember that the mere occurrence of an indicator word does not guarantee the presence of an argument. You must check to see that the statement identified as the conclusion is claimed to be supported by one or more of the other statements. Also keep in mind that in many arguments that lack indicator words, the con-clusion is the first statement. Furthermore, it helps to mentally insert the word �therefore� before the various statements before deciding that a statement should be interpreted as a conclusion. The typical kinds of nonarguments that we have surveyed are as follows:

warnings reports

pieces of advice expository passages statements of belief illustrations statements of opinion explanations loosely associated statements conditional statements

Keep in mind that these kinds of nonargument are not mutually exclusive, and that, for example,

one and the same passage can sometimes be interpreted as both a report and a statement of opin-ion, or as both an expository passage and an illustration. The precise kind of nonargument a pas-sage might be is nowhere near as important as correctly deciding whether or not it is an argument.

After working the exercises in this section, you may, if you wish, proceed directly to Section 1.6

[�Extended Arguments�].

Exercise 1.2

I. Determine which of the following passages are arguments. For those that are, identify

the conclusion. For those that are not, determine the kind of nonargument.

★ 1. The turkey vulture is called by that name because its red featherless head

resembles the head of a wild turkey.

2. If public education fails to improve the quality of instruction in both primary

and secondary schools, then it is likely that it will lose additional students to the private sector in the years ahead.

3. Freedom of the press is the most important of our constitutionally guaranteed

freedoms. Without it, our other freedoms would be immediately threatened. Furthermore, it provides the fulcrum for the advancement of new freedoms.

Section 1.2 Recognizing Arguments 21

★4. A mammal is a vertebrate animal that nurses its offspring. Thus, cats and dogs

are mammals, as are sheep, monkeys, rabbits, and bears.

5. It is strongly recommended that you have your house inspected for termite

damage at the earliest possible opportunity.

6. Mosquito bites are not always the harmless little irritations most of us take

them to be. For example, some mosquitoes carry West Nile virus, and people who are infected can become very sick or even die.

★7. If stem-cell research is restricted, then future cures will not materialize.

If future cures do not materialize, then people will die prematurely. Therefore, if�stem-cell research is restricted, then people will die prematurely.

8. Fictional characters behave according to the same psychological probabilities

as real people. But the characters of fiction are found in exotic dilemmas that real people hardly encounter. Consequently, fiction provides us with the opportunity to ponder how people react in uncommon situations, and to deduce moral lessons, psychological principles, and philosophical insights from their behavior.

( J. R. McCuen and A. C. Winkler, Readings for Writers, 4th ed.)

9. I believe that it must be the policy of the United States to support free peoples

who are resisting attempted subjugation by armed minorities or by outside pressures. I believe that we must assist free peoples to work out their own desti-nies in their own way. I believe that our help should be primarily through eco-nomic and financial aid, which is essential to economic stability and orderly political processes.

(President Truman, Address to Congress, 1947)

★10. Five college students who were accused of sneaking into the Cincinnati Zoo

and trying to ride the camels pleaded no contest to criminal trespass yesterday. The students scaled a fence to get into the zoo and then climbed another fence to get into the camel pit before security officials caught them, zoo officials said.

(Newspaper clipping)

11. Mortality rates for women undergoing early abortions, where the procedure

is legal, appear to be as low as or lower than the rates for normal childbirth. Consequently, any interest of the state in protecting the woman from an inher-ently hazardous procedure, except when it would be equally dangerous for her to forgo it, has largely disappeared.

( Justice Blackmun, Roe v. Wade)

12. The pace of reading, clearly, depends entirely upon the reader. He may read

as slowly or as rapidly as he can or wishes to read. If he does not understand something, he may stop and reread it, or go in search of elucidation before continuing. The reader can accelerate his pace when the material is easy or less than interesting, and can slow down when it is difficult or enthralling. If�what he reads is moving he can put down the book for a few moments and cope with his emotions without fear of losing anything.

(Marie Winn, The Plug-In Drug)

22Chapter1 Basic Concepts

★13. Any unit of length, when cubed, becomes a unit of volume. Thus, the cubic

meter, cubic centimeter, and cubic millimeter are all units of volume.

(Nivaldo J. Tro, Chemistry: A Molecular Approach, 2nd ed.)

14. Lions at Kruger National Park in South Africa are dying of tuberculosis. �All

of�the lions in the park may be dead within ten years because the disease is incurable, and the lions have no natural resistance,� said the deputy director of the Department of Agriculture.

(Newspaper clipping)

15. Economics is of practical value in business. An understanding of the overall

operation of the economic system puts the business executive in a better posi-tion to formulate policies. The executive who understands the causes and con-sequences of inflation is better equipped during inflationary periods to make more-intelligent decisions than otherwise.

(Campbell R. McConnell, Economics, 8th ed.)

★16. Bear one thing in mind before you begin to write your paper: Famous

literary works, especially works regarded as classics, have been thoroughly studied to the point where prevailing opinion on them has assumed the character of orthodoxy.

( J. R. McCuen and A. C. Winkler, Readings for Writers,

4th ed.)

17. Young people at universities study to achieve knowledge and not to learn a

trade. We must all learn how to support ourselves, but we must also learn how to live. We need a lot of engineers in the modern world, but we do not want a world of modern engineers.

( Winston Churchill, A Churchill Reader, ed. Colin R. Coote)

18. No business concern wants to sell on credit to a customer who will prove

unable or unwilling to pay his or her account. Consequently, most business organizations include a credit department which must reach a decision on the credit worthiness of each prospective customer.

( Walter B. Meigs and Robert F. Meigs, Accounting)

★19. For organisms at the sea surface, sinking into deep water usually means death.

Plant cells cannot photosynthesize in the dark depths. Fishes and other ani-mals that descend lose contact with the main surface food supply and them-selves become food for strange deep-living predators.

(David H. Milne, Marine Life and the Sea)

20. Since the 1950s a malady called whirling disease has invaded U.S. fishing

streams, frequently attacking rainbow trout. A parasite deforms young fish, which often chase their tails before dying, hence the name.

(�Trout Disease�A Turn for the Worse,� National Geographic)

21. Dachshunds are ideal dogs for small children, as they are already stretched and

pulled to such a length that the child cannot do much harm one way or the other.

(Robert Benchley, quoted in Cold Noses and Warm Hearts)

Section 1.2 Recognizing Arguments 23

★22. Atoms are the basic building blocks of all matter. They can combine to form

molecules, whose properties are generally very different from those of the con-stituent atoms. Table salt, for example, a simple chemical compound formed from chlorine and sodium, resembles neither the poisonous gas nor the highly reactive metal.

(Frank J. Blatt, Principles of Physics, 2nd ed.)

23. The coarsest type of humor is the practical joke: pulling away the chair from

the dignitary�s lowered bottom. The victim is perceived first as a person of consequence, then suddenly as an inert body subject to the laws of physics: authority is debunked by gravity, mind by matter; man is degraded to a mechanism.

(Arthur Koestler, Janus: A Summing Up)

24. If a man holding a belief which he was taught in childhood or persuaded of

afterwards keeps down and pushes away any doubts which arise about it in his mind, purposely avoids the reading of books and the company of men that call in question or discuss it, and regards as impious those questions which cannot easily be asked without disturbing it�the life of that man is one long sin against mankind.

( W. K. Clifford, The Ethics of Belief)

★25. It is usually easy to decide whether or not something is alive. This is because

living things share many common attributes, such as the capacity to extract energy from nutrients to drive their various functions, the power to actively respond to changes in their environment, and the ability to grow, to differenti-ate, and to reproduce.

(Donald Voet and Judith G. Voet, Biochemistry, 2nd ed.)

26. Words are slippery customers. The full meaning of a word does not appear

until it is placed in its context. . . . And even then the meaning will depend upon the listener, upon the speaker, upon their entire experience of the language, upon their knowledge of one another, and upon the whole situation.

(C. Cherry, On Human Communication)

27. Haydn developed the string quartet from the eighteenth-century diverti-

mento, giving more substance to the light, popular form and scoring it for two violins, a viola, and a cello. His eighty-three quartets, written over the course of his creative lifetime, evolved slowly into a sophisticated form. Together they constitute one of the most important bodies of chamber-music literature.

(Robert Hickok, Exploring Music)

★28. A person never becomes truly self-reliant. Even though he deals effectively

with things, he is necessarily dependent upon those who have taught him to do so. They have selected the things he is dependent upon and determined the kinds and degrees of dependencies.

(B. F. Skinner, Beyond Freedom and Dignity)

29. There is no doubt that some businessmen conspire to shorten the useful life

of their products in order to guarantee replacement sales. There is, similarly,

24Chapter1 Basic Concepts

no doubt that many of the annual model changes with which American (and other) consumers are increasingly familiar are not technologically substantive.

(Alvin Toffler, Future Shock)

30. The brain and the nervous system are composed of two types of cells� neurons

and glial cells. Neurons are responsible for information transmission through-out the nervous system. Glial cells constitute the support system for the neu-rons. For example, glial cells take away the waste products of neurons, keep the neurons� chemical environment stable, and insulate them, allowing neurons to do their work more efficiently.

(Richard Griggs, Psychology: A Concise Introduction)

★31. In areas where rats are a problem, it is very difficult to exterminate them with bait

poison. That�s because some rats eat enough poison to die but others eat only enough to become sick and then learn to avoid that particular poison taste in the future.

(Rod Plotnik, Introduction to Psychology, 4th ed.)

32. Although it is customary to think of human population as increasing continu-

ously without declines or fluctuations, population growth has not been a steady march. For example, great declines occurred during the time of the Black Death, during the fourteenth century. Entire towns were abandoned, production of food declined, and in England, one-third of the population died within a single decade.

(Daniel B. Botkin and Edward A. Keller, Environmental Science)

33. If someone avoids and is afraid of everything, standing firm against nothing,

he becomes cowardly; if he is afraid of nothing at all and goes to face every-thing, he becomes rash. Similarly, if he gratifies himself with every pleasure and abstains from none, he becomes intemperate; if he avoids them all, he becomes some sort of insensible person. Temperance and bravery, then, are ruined by excess and deficiency, but preserved by the mean.

(Aristotle, Nicomachean Ethics)

★34. Nations are made in two ways, by the slow working of history or the galvanic

force of ideas. Most nations are made the former way, emerging slowly from the mist of the past, gradually coalescing within concentric circles of shared sympathies, with an accretion of consensual institutions. But a few nations are formed and defined by the citizens� assent to a shared philosophy.

(George Will, �Lithuania and South Carolina�)

35. One form of energy can be converted to another. For example, when an elec-

tric motor is connected to a battery, chemical energy is converted to electrical energy, which in turn is converted to mechanical energy.

(Raymond A . Serway, Physics for Scientists and Engineers, 4th ed.)

II. The following selections were originally submitted as letters to the editor of newspapers

and magazines. Determine which of them can, with good reason, be considered argu-ments. In those that can, identify the conclusion.

★ 1. What this country needs is a return to the concept of swift and certain justice. If

we need more courts, judges and prisons, then so be it. And as for capital pun-ishment, I say let the punishment fit the crime. When criminals behave more

Section 1.2 Recognizing Arguments 25

like humans, then we can start to treat them more humanely. In the meantime, I would like to see the Night Stalkers of our society swiftly executed rather than coddled by our courts and prisons.

( John Pearson)

2. Social security is not merely a retirement program. Six and a half million chil-

dren in the United States are kept out of poverty each year because of assistance from Social Security�s survivors benefits program�which protects virtually all American children in the tragic event of the death of a parent. Beneficiaries include spouses and children of workers who have died or become disabled; grandparents raising grandchildren; severely disabled children; and families of fallen service members.

(Donna Butts)

3. Is there any country in the world that worries more about its kids having fun

in school, making lessons exciting and relevant, and then is more disappointed with the result than the United States? We think learning is like buying a car or smoking a cigarette. Just get into the thing or draw a breath and you will be effortlessly transported to lands of pleasure and excitement.

(Charles M. Breinin)

★4. High school students should be shown what a ballot looks like, how to com-

plete it correctly, and exactly what the process is for counting it. They need to understand that they must complete the task accurately to have a voice in whom they are choosing to make decisions about their lives. This is the point of education: to participate effectively in the society in which you live.

( Judy Novey, letter to the editor)

5. Once the basic necessities have been achieved, future income is only lightly

connected to well-being. Democrats generally seek to tax future income to finance programs that meet basic needs, including food, clothing shelter, retirement security, and health care. Republicans, in contrast, seek to protect future income from taxation, often at the expense of meeting the basic needs of the less fortunate. So which of our two main political parties is more con-cerned with achieving broad happiness, and which party is more concerned with fulfilling selfishness?

( Jonathan Carey)

6. Animal abusers are cowards who take their issues out on �easy victims��and

their targets often include their fellow humans. I cannot begin to say how many incidents I�ve seen involving animal abusers who commit violent acts against humans, and animal neglecters who have neglected their children or other human dependents. Treating cruelty to animals with the seriousness it deserves doesn�t only protect animals, it also makes the entire community safer.

(Martin Mersereau)

★7. The creation of a third political party�the independent party�would allow

Congressional aspirants who desire to think for themselves to claim a high ground that is currently vacant. The new party would provide a more effective forum to discuss the right course for this country and might compel the other

26Chapter1 Basic Concepts

two parties to do likewise. The pressure such a movement would put on those now stagnating in cozy sinecures would, at the least, prove entertaining for a weary, frustrated public.

(Bill Cannon)

8. I agree that when religious institutions exclude woman from their hierarchies

and rituals, the inevitable implication is that females are inferior. But it is important to note that when women�s voices are silenced, it is not only the mes-sage that such discrimination sends that is damaging. The institutions them-selves suffer. By disempowering women, religious institutions, and the broader societies in which they operate, lose the invaluable input of 51 percent of their constituents.

( Jessie Cronan)

9. It looks like India and China are going to compete for a manned landing on the

moon by 2030 while America is muddling along with no real future space plan. Let�s do something significant in space�say, go to Mars by 2030. We could have done it 30 years ago. Planning for a Mars mission was well along. But the nation turned away from space after we landed on the moon, even canceling the three remaining flights to the moon. These Saturn 5 rockets now sit in museums.

(Bill Ketchum)

★10. Teenage bullying is all about power. One person has it, one person does not.

Reluctant to seek help, victims feel ashamed and powerless, and they fear retali-ation should they �rat out� the bully. Strong anti-bullying programs are needed to provide a means to report bullying anonymously, to train all school person-nel to take reports of bullying seriously, and to offer workshops for children on how to respond to being bullied.

(Karen Schulte O�Neill)

III. The following statements represent conclusions for arguments. Each is expressed in the

form of two alternatives. Select one of the alternatives for each conclusion, and then jot down several reasons that support it. Finally, incorporate your reasons into a written argument of at least 100 words that supports the conclusion. Include premise and conclu-sion indicators in some of your arguments, but not in all of them.

1. A constitutional amendment that outlaws flag burning should/should not be

adopted.

2. Street drugs should/should not be legalized.

3. The death penalty should/should not be abolished.

4. Sanctions should/should not be imposed on students for using speech that is

offensive to minorities.

5. Free health care should/should not be guaranteed to all citizens.

6. Same-sex marriages should/should not be recognized by the state.

7. The possession, ownership, and sale of handguns should/should not be outlawed.

8. E-cigarettes should/should not be regulated as an addictive drug.

Section 1.2 Recognizing Arguments 27

9. Affirmative action programs should/should not be abolished.

10. Doctors should/should not be allowed to assist terminally ill patients in com-

mitting suicide.

IV. Define the following terms:

expository passage conditional statement illustration antecedent

argument from example consequent explanation sufficient condition explanandum necessary condition explanans

V. Answer �true� or �false� to the following statements:

1. Any passage that contains an argument must contain a claim that something is

supported by evidence or reasons.

2. In an argument, the claim that something is supported by evidence or reasons

is always explicit.

3. Passages that contain indicator words such as �thus,� �since,� and �because� are

always arguments.

4. In deciding whether a passage contains an argument, we should always keep

an eye out for indicator words and the presence of an inferential relationship between the statements.

5. Some expository passages can be correctly interpreted as arguments.

6. Some passages containing �for example� can be correctly interpreted as

arguments.

7. In deciding whether an expository passage or an illustration should be inter-

preted as an argument, it helps to note whether the claim being developed or illustrated is one that is accepted by everyone.

8. Some conditional statements can be reexpressed to form arguments.

9. In an explanation, the explanandum usually describes an accepted matter of fact.

10. In an explanation, the explanans is the statement or group of statements that

does the explaining.

VI. Fill in the blanks with �necessary� or �sufficient� to make the following statements

true. After the blanks have been filled in, express the result in terms of conditional statements.

★1. Being a tiger is a condition for being an animal.

2. Being an animal is a condition for being a tiger.

3. Drinking a glass of water is a condition for quenching one�s

thirst.

28Chapter1 Basic Concepts

★ 4. Having a racket is a condition for playing tennis.

5. Heating water is a condition for brewing coffee.

6. Stepping on a cat�s tail is a condition for making the cat yowl.

★7. Burning leaves is a condition for producing smoke.

8. Paying attention is a condition for understanding a lecture.

9. Being exactly divisible by 4 is a condition for a number being even.

★10. Intending to deceive someone is a condition for telling a lie.

VII. Look through a book, magazine, newspaper, or electronic publication and find two

arguments, one with indicator words, the other without. Copy the arguments as written, giving the appropriate reference. Then identify the premises and conclusion of each.

VIII. Create the following nonarguments:

1. A conditional statement about a topic in astronomy. Identify the antecedent

and the consequent.

2. An explanation about a topic in history. Explain why your explanation is not

an argument.

3. An expository passage about a topic in chemistry. Explain why your expository

passage is not an argument.

4. An illustration about a topic in mathematics. Explain why your illustration is

not an argument from example.

1.3 Deduction and Induction

The idea that arguments come in two forms, while inductive arguments are those that rest on deductive and inductive, was first asserted by Aris- probabilistic reasoning. totle. In the intervening centuries, deduction and Stated more precisely, a deductive argument is induction have become a settled fixture not only an argument incorporating the claim that it is impos-in logic but in our intellectual culture. Countless sible for the conclusion to be false given that the books, both fiction and nonfiction, have referred premises are true. On the other hand, an inductive to it. Einstein wrote a paper on it. And a huge argument is an argument incorporating the claim number of textbooks ranging from philosophy to that it is improbable that the conclusion be false given education, business to psychology, and chemistry that the premises are true. Two examples:

to anthropology explore the subject. So what is the The meerkat is closely related to the suricat. difference between a deductive and an inductive The suricat thrives on beetle larvae. argument? Briefly we can say that deductive argu-Therefore, probably the meerkat thrives on ments are those that rest on necessary reasoning, beetle larvae.

Section 1.3 Deduction and Induction 29

Prominent 1921�2012Ruth Barcan Marcus Logicians

R uth Barcan was born in New York City in 1921. Her mother stayed at home, and her father was a typesetter at, and contributor to, the Jewish Daily Forward. After completing her primary and sec-

ondary education at public schools, she enrolled in

New York University, where, in addition to her academic

pursuits, she won praise as an outstanding fencer. In

1941 she earned a bachelor�s degree in mathematics

and philosophy, and five years later she received a PhD

in philosophy from Yale University. In 1942 she married

Jules Alexander Marcus, a physicist, and the couple

had four children, two boys and two girls.

After graduating from Yale, Barcan Marcus�s early

career was spent holding several postdoctoral fellow-

ships (including a Guggenheim) and visiting professor-

ships. In 1959 she accepted a position at Roosevelt

University, followed by positions at the University of Courtesy Michael Marsland Illinois, Chicago (where she was founding depart-

ment chair) and Northwestern University. In 1973 she exist in the actual world. This could be taken to imply returned to Yale as professor of philosophy. that nothing new can be created.

Commencing early in her career, Barcan Marcus Personally, Ruth Barcan Marcus was fearless,

made pioneering contributions to the area of quan- down to earth, unpretentious, and a constant sup-tified modal logic. She proposed, as an axiom, porter of others. She had a great sense of humor� the widely discussed Barcan formula, which asserts, in and she was also endearingly absentminded. On one symbols, (x)hFx . h(x) Fx. In English, this means that occasion, while in the midst of a frantic search, she if everything is nec essarily F, then it is necessary that received a call from the local supermarket informing everything is F. The formula is controversial because it her that her final exams had been found amid the implies that all objects that exist in every possible world frozen meats.

The meerkat is a member of the mongoose family. special indicator words, (2) the actual strength of the All members of the mongoose family are inferential link between premises and conclusion, and carnivores. (3) the form or style of argumentation. However, we Therefore, it necessarily follows that the must acknowledge at the outset that many arguments in meerkat is a carnivore. ordinary language are incomplete, and because of this,

deductive. preted as deductive or inductive may be impossible. The occurrence of special indicator words is illus-The first of these arguments is inductive, the second deciding whether the argument should best be inter-

In deciding whether an argument is inductive or

deductive, we look to certain objective features of the trated in the examples we just considered. The word argument. These features include (1) the occurrence of �probably� in the conclusion of the first argument

30Chapter1 Basic Concepts

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). suggests that the argument should be taken as induc- example, the conclusion does not follow from the tive, and the word �necessarily� in the conclusion of premises with strict necessity, but it does follow with the second suggests that the second argument be some degree of probability. If we assume that the taken as deductive. Additional inductive indicators premises are true, then based on that assumption it are �improbable,� �plausible,� �implausible,� �likely,� is probable that the conclusion is true. Thus, it is best �unlikely,� and �reasonable to conclude.� Additional to interpret the second argument as inductive. deductive indicators are �certainly,� �absolutely,� and Occasionally, an argument contains no special �definitely.� (Note that the phrase �it must be the indicator words, and the conclusion does not follow case that� is simply a conclusion indicator that can either necessarily or probably from the premises; in occur in either deductive or inductive argments.) other words, it does not follow at all. This situation

Inductive and deductive indicator words often points to the need for the third factor to be taken

suggest the correct interpretation. However, if they into account, which is the character or form of argu-conflict with one of the other criteria (discussed mentation the arguer uses. shortly), we should probably ignore them. Arguers

often use phrases such as �it certainly follows that�

for rhetorical purposes to add impact to their conclu- Deductive Argument Forms

sion and not to suggest that the argument be taken Many arguments have a distinctive character or as deductive. Similarly, some arguers, not knowing form that indicates that the premises are supposed the distinction between inductive and deductive, to provide absolute support for the conclusion. Five will claim to �deduce� a conclusion when their argu- examples of such forms or kinds of argumentation ment is more correctly interpreted as inductive. are arguments based on mathematics, arguments

The second factor that bears on our interpreta- from definition, and categorical, hypothetical, and

tion of an argument as inductive or deductive is the disjunctive syllogisms. actual strength of the inferential link between prem- An argument based on mathematics is an ises and conclusion. If the conclusion actually does argument in which the conclusion depends on some follow with strict necessity from the premises, the purely arithmetic or geometric computation or mea-argument is clearly deductive. In such an argument surement. For example, a shopper might place two it is impossible for the premises to be true and the apples and three oranges into a paper bag and then conclusion false. On the other hand, if the conclu- conclude that the bag contains five pieces of fruit. sion does not follow with strict necessity but does Or a surveyor might measure a square piece of land follow probably, it is often best to consider the argu- and, after determining that it is 100 feet on each ment inductive. Examples: side, conclude that it contains 10,000 square feet.

All entertainers are extroverts. Since all arguments in pure mathematics are deduc-Stephen Colbert is an entertainer. tive, we can usually consider arguments that depend Therefore, Stephen Colbert is an extrovert. on mathematics to be deductive as well. However, The vast majority of entertainers are extroverts. arguments that depend on statistics are a notewor-Stephen Colbert is an entertainer. thy exception. As we will see shortly, such arguments Therefore, Stephen Colbert is an extrovert. are usually best interpreted as inductive.

An argument from definition is an argument

In the first example, the conclusion follows with in which the conclusion is claimed to depend merely strict necessity from the premises. If we assume that on the definition of some word or phrase used in the all entertainers are extroverts and that Stephen Col- premise or conclusion. For example, someone might bert is an entertainer, then it is impossible that Ste- argue that because Claudia is mendacious, it follows phen Colbert not be an extrovert. Thus, we should that she tells lies, or that because a certain para-interpret this argument as deductive. In the second graph is prolix, it follows that it is excessively wordy.

Section 1.3 Deduction and Induction 31

These arguments are deductive because their con- As with hypothetical syllogisms, such arguments clusions follow with necessity from the definitions are usually best taken as deductive. Hypothetical of �mendacious� and �prolix.� and disjunctive syllogisms will be treated in greater

A syllogism, in general, is an argument consisting of depth in Chapter 6.

exactly two premises and one conclusion. Categorical

syllogisms will be treated in greater depth in Chapter 5,

but for now we will say that a categorical syllogism Inductive Argument Forms

is a syllogism in which each statement begins with one In general, inductive arguments are such that the of the words �all,� �no,� or �some.� Example: content of the conclusion is in some way intended to

All ancient forests are sources of wonder. �go beyond� the content of the premises. The prem-Some ancient forests are targets of the timber ises of such an argument typically deal with some industry. subject that is relatively familiar, and the conclusion Therefore, some sources of wonder are targets then moves beyond this to a subject that is less famil-of the timber industry. iar or that little is known about. Such an argument

treated as deductive. izations, arguments from authority, arguments based A hypothetical syllogism is a syllogism having on signs, and causal inferences, to name just a few. a conditional (�if . . . then�) statement for one or both A Arguments such as these are nearly always best future, arguments from analogy, inductive general-may take any of several forms: predictions about the of its premises. Examples: our knowledge of the past to a claim about the future. prediction is an argument that proceeds from

If estate taxes are abolished, then wealth will For example, someone might argue that because cer-accumulate disproportionately. tain meteorological phenomena have been observed If wealth accumulates disproportionately, then to develop over a certain region of central Missouri, democracy will be threatened. a storm will occur there in six hours. Or again, Therefore, if estate taxes are abolished, then one might argue that because certain fluctuations democracy will be threatened. occurred in the prime interest rate on Friday, the If Fox News is a propaganda machine, then it value of the dollar will decrease against foreign cur-misleads its viewers. rencies on Monday. Nearly everyone realizes that Fox News is a propaganda machine. the future cannot be known with certainty; thus, Therefore, Fox News misleads its viewers. whenever an argument makes a prediction about

the future, one is usually justified in considering the

Later in this book, the first of these arguments will argument inductive. be given the more specific name of pure hypothetical An argument from analogy is an argument syllogism because it is composed exclusively of con- that depends on the existence of an analo gy,�or simi-ditional (hypothetical) statements. The second argu- larity, between two things or states of affairs. Because ment is called a mixed hypothetical syllogism because of the existence of this analogy, a certain condition only one of its component statements is a conditional. that affects the better-known thing or situation is Later in this book, the second argument will be given concluded to affect the similar, lesser-known thing the more specific Latin name modus ponens. or situation. For example, someone might argue�that

A disjunctive syllogism is a syllogism having because Christina�s Porsche is a great-handling car,

a disjunctive (�either . . . or�. . .�) statement. Example: it follows that Angela�s Porsche must also�be a great-

Either climate change will be arrested, or hur- handling car. The argument depends on the exis-ricanes will become more intense. tence of a similarity, or analo gy,� between the two Climate change will not be arrested. cars. The certitude attending such an inference is Therefore, hurricanes will become more intense. probabilistic at best.

32Chapter1 Basic Concepts

A generalization is an argument that proceeds knowledge that a bottle of wine had been acciden-

from the knowledge of a selected sample to some tally left in the freezer overnight, someone might claim about the whole group. Because the mem- conclude that it had frozen (cause to effect). Con-bers of the sample have a certain characteristic, it versely, after tasting a piece of chicken and finding is argued that all the members of the group have it dry and tough, one might conclude that it had that same characteristic. For example, one might been overcooked (effect to cause). Because specific argue that because three oranges selected from instances of cause and effect can never be known a certain crate were especially tasty and juicy, all with absolute certainty, one may usually interpret the oranges from that crate are especially tasty such arguments as inductive. and juicy. Or again, one might argue that because

six out of a total of nine members sampled from Further Considerations

a certain labor union intend to vote for Johnson It should be noted that the various subspecies of for union president, two-thirds of the entire inductive arguments listed here are not intended to membership intend to vote for Johnson. These be mutually exclusive. Overlaps can and do occur. examples illustrate the use of statistics in inductive For example, many causal inferences that proceed argumentation. from cause to effect also qualify as predictions. The

An argument from authority is an argu- purpose of this survey is not to demarcate in pre-

ment that concludes something is true because a cise terms the various forms of induction but rather presumed expert or witness has said that it is. For to provide guidelines for distinguishing induction example, a person might argue that earnings for from deduction. Spotify will be up in the coming quarter because Keeping this in mind, we should take care not to of a statement to that effect by an investment coun- confuse arguments in geometry, which are always selor. Or a lawyer might argue that Mack the Knife deductive, with arguments from analogy or induc-committed the murder because an eyewitness testi- tive generalizations. For example, an argument fied to that effect under oath. Because the invest- concluding that a triangle has a certain attribute ment counselor and the eyewitness could be either (such as a right angle) because another triangle, mistaken or lying, such arguments are essentially with which it is congruent, also has that attribute probabilistic. might be mistaken for an argument from analogy.

An argument based on signs is an argument Similarly, an argument that concludes that all trian-

that proceeds from the knowledge of a sign to a gles have a certain attribute (such as angles totaling claim about the thing or situation that the sign sym- two right angles) because any particular triangle bolizes. The word �sign,� as it is used here, means has that attribute might be mistaken for an induc-any kind of message (usually visual) produced by tive generalization. Arguments such as these, how-an intelligent being. For example, when driving on ever, are always deductive, because the conclusion an unfamiliar highway one might see a sign indi- follows necessarily and with complete certainty cating that the road makes several sharp turns one from the premises. mile ahead. Based on this information, one might One broad classification of arguments not listed argue that the road does indeed make several sharp in this survey is scientific arguments. Arguments turns one mile ahead. Because the sign might be that occur in science can be either inductive or misplaced or in error about the turns, the conclu- deductive, depending on the circumstances. In gen-sion is only probable. eral, arguments aimed at the discovery of a law of

A causal inference is an argument that pro- nature are usually considered inductive. Suppose,

ceeds from knowledge of a cause to a claim about for example, that we want to discover a law that gov-an effect, or, conversely, from knowledge of an effect erns the time required for a falling body to strike to a claim about a cause. For example, from the the earth. We drop bodies of various weights from

Section 1.3 Deduction and Induction 33

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). various heights and measure the time it takes them all the members of a class.) It is true, of course, that to fall. Comparing our measurements, we notice that many inductive and deductive arguments do work the time is approximately proportional to the square in this way; but this fact should not be used as a root of the distance. From this we conclude that the criterion for distinguishing induction from deduc-time required for any body to fall is proportional to tion. As a matter of fact, there are deductive argu-the square root of the distance through which it falls. ments that proceed from the general to the gen-Such an argument is best interpreted as an inductive eral, from the particular to the particular, and from generalization. the particular to the general, as well as from the

Another type of argument that occurs in science general to the particular; and there are inductive

has to do with the application of known laws to spe- arguments that do the same. For example, here is cific circumstances. Scientific laws are widely con- a deductive argument that proceeds from the par-sidered to be generalizations that hold for all times ticular to the general:

and all places. As so understood, their application to Three is a prime number. a specific situation is always deductive, even though Five is a prime number. it might relate to the future. Suppose, for example, Seven is a prime number. that we want to apply Boyle�s law for ideal gases to a Therefore, all odd numbers between two and container of gas in our laboratory. Boyle�s law states eight are prime numbers.

container is inversely proportional to the volume. And here is one that proceeds from the particular to that the pressure exerted by a gas on the walls of its Applying this law, we conclude that when we reduce the particular: the volume of our laboratory sample by half, the Gabriel is a wolf. pressure will double. This application of Boyle�s law Gabriel has a tail. is deductive, even though it pertains to the future. Therefore, Gabriel�s tail is the tail of a wolf.

A final point needs to be made about the dis- Here is an inductive argument that proceeds from

tinction between inductive and deductive argu- the general to the particular:

time of Aristotle that holds that inductive argu- All emeralds previously found have been green. Therefore, the next emerald found will be ments. There is a tradition extending back to the

ments are those that proceed from the particular to green. the general, while deductive arguments are those that proceed from the general to the particular. (A The other varieties are easy to construct. Thus, the particular statement is one that makes a claim progression from particular to general, and vice about one or more particular members of a class, versa, cannot be used as a criterion for distinguishing while a general statement makes a claim about induction from deduction.

Summary

To distinguish deductive arguments from inductive arguments, we attempt to evaluate the strength of the argument�s inferential claim�how strongly the conclusion is claimed to follow from the premises. This claim is an objective feature of an argument, and it may or may not be related to the subjective intentions of the arguer.

To interpret an argument�s inferential claim we look at three factors: special indicator words,

the actual strength of the inferential link between premises and conclusion, and the form or style of argumentation. Given that we have more than one factor to look at, it is possible in a single

34Chapter1 Basic Concepts

argument for the occurrence of two of these factors to conflict with each other, leading to opposite interpretations. For example, in drawing a conclusion to a categorical syllogism (which is clearly deductive), an arguer might say, �It probably follows that . . .� (which suggests induction). To help alleviate this conflict, we can list the factors in order of importance:

1. Arguments in which the premises provide absolute support for the conclusion. Such argu-

ments are always deductive.

2. Arguments having a specific deductive character or form (e.g., categorical syllogism). This

factor is often of equal importance to the first, and, when present, it provides a clear-cut indi-cation that the argument is deductive.

3. Arguments having a specific inductive character or form (e.g., a prediction). Arguments of this

sort are nearly always best interpreted as inductive.

4. Arguments containing inductive indicator language (e.g., �It probably follows

that . . .�). Since arguers rarely try to make their argument appear weaker than it really is, such language can usually be trusted. But if this language conflicts with one of the first two factors, it should be ignored.

5. Arguments containing deductive indicator language (e.g., �It necessarily follows that . . .�).

Arguers occasionally use such language for r hetorical purposes, to make their argument appear stronger than it really is, so such language should be evaluated carefully.

6. Arguments in which the premises provide only probable support for the conclusion. This is the

least important factor, and if it conflicts with any of the earlier ones, it should probably be ignored.

Unfortunately, many arguments in ordinary language are incomplete, so it often happens that

none of these factors are clearly present. Determining the inductive or deductive character of such arguments may be impossible.

Exercise 1.3

I. Determine whether the following arguments are best interpreted as being inductive or

deductive. Also state the criteria you use in reaching your decision (i.e., the presence of indicator words, the nature of the inferential link between premises and conclusion, or the character or form of argumentation).

★ 1. Because triangle A is congruent with triangle B, and triangle A is isosceles, it

follows that triangle B is isosceles.

2. The plaque on the Leaning Tower of Pisa says that Galileo performed experi-

ments there with falling objects. It must be the case that Galileo did indeed perform those experiments there.

3. The rainfall in Seattle has been more than 15 inches every year for the past

thirty years. Therefore, the rainfall next year will probably be more than 15�inches.

★4. No e-mail messages are eloquent creations. Some love letters are eloquent cre-

ations. Therefore, some love letters are not e-mail messages.

Section 1.3 Deduction and Induction 35

5. Chevron, ExxonMobil, and Valero are all listed on the New York Stock Exchange.

It must be the case that all American oil companies are listed on the New York Stock Exchange.

6. The longer a pendulum is, the longer it takes to swing. Therefore, when the

pendulum of a clock is lengthened, the clock slows down.

★7. Paying off terrorists in exchange for hostages is not a wise policy, since such

action will only lead them to take more hostages in the future.

8. The Matterhorn is higher than Mount Whitney, and Mount Whitney is higher

than Mount Rainier. The obvious conclusion is that the Matterhorn is higher than Mount Rainier.

9. Although both front and rear doors were found open after the burglary, there

were pry marks around the lock on the rear door and deposits of mud near the threshold. It must be the case that the thief entered through the rear door and left through the front.

★10. The Wall Street Journal has an article on the new banking regulations. The

Financial Times, like the Wall Street Journal, is a highly respected business pub-lication. Therefore, the Financial Times probably also has an article on the new banking regulations.

11. Cholesterol is endogenous with humans. Therefore, it is manufactured inside

the human body.

12. Either classical culture originated in Greece, or it originated in Egypt. Classi-

cal culture did not originate in Egypt. Therefore, classical culture originated in Greece.

★13. World-renowned physicist Stephen Hawking said that the condition of the

universe at the instant of the big bang was more highly ordered than it is today. In view of Hawking�s stature in the scientific community, we should conclude that this description of the universe is correct.

14. If Elon Musk is right about artificial intelligence, then artificial intelligence

poses a risk to human survival. Elon Musk is right about artificial intelligence. Therefore, artificial intelligence poses a risk to human survival.

15. Crater Lake, the deepest lake in the United States, was caused by a huge volca-

nic eruption 7,700 years ago. Since human beings have lived around the moun-tain for more than 10,000 years, it is likely that people witnessed that eruption.

(National Park Service, �Crater Lake�Its History�)

★16. Each element, such as hydrogen and iron, has a set of gaps�wavelengths that

it absorbs rather than radiates. So if those wavelengths are missing from the spectrum, you know that that element is present in the star you are observing.

(Rick Gore, �Eyes of Science�)

17. Because the apparent daily movement which is common to both the planets and

the fixed stars is seen to travel from the east to the west, but the far-slower single movements of the single planets travel in the opposite direction from west to

36Chapter1 Basic Concepts

east, it is therefore certain that these movements cannot depend on the common movement of the world but should be assigned to the planets themselves.

( Johannes Kepler, Epitomy of Copernican Astronomy)

18. Contrary to the common notion that women tend to be chatty compared to

men, little difference exists between the sexes in terms of talkativeness. Over a five-year period researchers placed unobtrusive microphones on 396 college students in various fields, at campuses in Mexico as well as in the United States. They found that both men and women spoke about 16,000 words per day.

(Richard T. Schaefer, Sociology: A Brief Introduction, 8th ed.)

★19. When the Romans occupied England, coal was burned. Since coal produces

quite a bit of soot and sulfur dioxide, there must have been days almost 2,000 years ago when the air in the larger towns was badly polluted.

(Stanley Gedzelman, The Science and Wonders of the

Atmosphere)

20. The graphical method for solving a system of equations is an approximation,

since reading the point of intersection depends on the accuracy with which the lines are drawn and on the ability to interpret the coordinates of the point.

(Karl J. Smith and Patrick J. Boyle, Intermediate Algebra

for College Students)

21. That [the moons of Jupiter] revolve in unequal circles is manifestly deduced

from the fact that at the longest elongation from Jupiter it is never possible to see two of these moons in conjunction, whereas in the vicinity of Jupiter they are found united two, three, and sometimes all four together.

(Galileo Galilei, The Starry Messenger)

★22. Lenses function by refracting light at their surfaces. Consequently, their action

depends not only on the shape of the lens surfaces, but also on the indices of refraction of the lens material and the surrounding medium.

(Frank J. Blatt, Principles of Physics, 2nd ed.)

23. Given present growth rates in underdeveloped countries, the limited practice

of birth control, and the difficulty of slowing the current growth momentum, it can be said with virtual certainty that none of the people now reading this book will ever live in a world where the population is not growing.

( J. John Palen, Social Problems)

24. The interpretation of the laws is the proper and peculiar province of the courts.

A constitution is, in fact, and must be regarded by the judges, as a fundamental law. It therefore belongs to them to ascertain its meaning, as well as the mean-ing of any particular act proceeding from the legislative body.

(Alexander Hamilton, Federalist Papers, No. 78)

★25. The Simpson incident had shown me that a dog was kept in the stables, and yet,

though someone had been in and had fetched out a horse, he had not barked enough to arouse the two lads in the loft. Obviously the midnight visitor was someone whom the dog knew well.

(A. Conan Doyle, Memoirs of Sherlock Holmes)

Section 1.3 Deduction and Induction 37

26. Eternity is simultaneously whole. But time has a before and an after. Therefore

time and eternity are not the same thing.

( Thomas Aquinas, Summa Theologica)

27. Ordinary things that we encounter every day are electrically neutral. Therefore,

since negatively charged electrons are a part of everything, positively charged particles must also exist in all matter.

( James E. Brady and Gerard E. Humiston, General Chemistry)

★ 28. Animals that live on plant foods must eat large quantities of vegetation, and this

consumes much of their time. Meat eaters, by contrast, have no need to eat so much or so often. Consequently, meat-eating hominines [early humans] may have had more leisure time available to explore and manipulate their environment; like lions and leopards, they would have time to spend lying around and playing.

( William A. Haviland, Cultural Anthropology, 8th ed.)

29. We tell people not to speed, but equip cars with air bags in case they do. So what�s

wrong with telling kids not to have sex, but making Plan B available in case they do?

(Susan Beck, letter to the editor)

30. Because the moon moves relative to the earth so that it returns to the same

position overhead after about 25 hours, there are two high and two low tides at any point every 25 hours.

(Douglas C. Giancoli, The Ideas of Physics, 3rd ed.)

II. Define the following terms:

deductive argument prediction inductive argument argument from analogy argument based on generalization mathematics argument from authority argument from definition argument based on signs categorical syllogism causal inference hypothetical syllogism particular statement disjunctive syllogism general statement

III. Answer �true� or �false� to the following statements:

1. In an inductive argument, it is intended that the conclusion contain more

information than the premises.

2. In a deductive argument, the conclusion is not supposed to contain more

information than the premises.

3. The form of argumentation the arguer uses may allow one to determine

whether an argument is inductive or deductive.

4. The actual strength of the link between premises and conclusion may allow

one to determine whether an argument is inductive or deductive.

5. A geometrical proof is an example of an inductive argument.

6. Most arguments based on statistical reasoning are deductive.

7. If the conclusion of an argument follows merely from the definition of a word

used in a premise, the argument is deductive.

38Chapter1 Basic Concepts

8. An argument that draws a conclusion about a thing based on that thing�s simi-

larity to something else is a deductive argument.

9. An argument that draws a conclusion that something is true because someone

has said that it is, is a deductive argument.

10. An argument that presents two alternatives and eliminates one, leaving the

other as the conclusion, is an inductive argument.

11. An argument that proceeds from knowledge of a cause to knowledge of an

effect is an inductive argument.

12. If an argument contains the phrase �it definitely follows that,� then we know

for certain that the argument is deductive.

13. An argument that predicts what will happen in the future, based on what has

happened in the past, is an inductive argument.

14. Inductive arguments always proceed from the particular to the general.

15. Deductive arguments always proceed from the general to the particular.

IV. Look through a book, magazine, newspaper, or electronic publication and find two

arguments, one inductive and the other deductive. Copy the arguments as written, giving the appropriate reference. Then identify the premises and conclusion of each.

V. Create the following deductive and inductive arguments:

1. A categorical syllogism involving groups of people

2. A disjunctive syllogism dealing with corporations

3. A hypothetical syllogism relating to the environment

4. A prediction relating to the economy

5. A generalization relating to students in a math class

6. An argument from analogy relating to two individual people

7. A causal inference about a car accident

1.4 Validity, Truth, Soundness,

Strength, and Cogency

This section introduces the central ideas and termi- the conclusion? (2) Are all the premises true? The nology needed to evaluate arguments�to distinguish answer to the first question is the more important good arguments from bad arguments. Regardless of one, because if the premises fail to support the con-the type of argument, whether deductive or inductive, clusion (that is, if the reasoning is bad), the argument the evaluation of any argument involves answering is worthless. The material that follows first considers two distinct questions: (1) Do the premises support deductive arguments and then inductive.

Section 1.4 Validity, Truth, Soundness, Strength, and Cogency 39

Deductive Arguments Therefore, United Airlines is a pharmaceutical

The previous section defined a deductive argument company. as one incorporating the claim that it is impossible In this argument, both premises are actually false, but for the conclusion to be false given that the premises it is easy to assume that they are true. Every automaker are true. If this claim is true, the argument is said to could have a corporate division that manufactures be valid. Thus, a valid deductive argument is an pharmaceuticals. Also, in addition to flying airplanes, argument in which it is impossible for the conclu- United Airlines could make cars. Next, in light of these sion to be false given that the premises are true. In assumptions, we determine if it is possible for the con-these arguments the conclusion follows with strict clusion to be false. Again, we see that this is not possible, necessity from the premises. Conversely, an invalid by the same reasoning as the previous example. Assum-deductive argument is a deductive argument in ing the premises to be true and the conclusion false which it is possible for the conclusion to be false entails a contradiction. Thus, the argument is valid. given that the premises are true. In these arguments Another example:

the conclusion does not follow with strict necessity All banks are financial institutions. from the premises, even though it is claimed to. Wells Fargo is a financial institution.

An immediate consequence of these definitions Therefore, Wells Fargo is a bank.

is that there is no middle ground between valid and

invalid. There are no arguments that are �almost� As in the first example, both premises of this argument valid and �almost� invalid. If the conclusion follows are true, so it is easy to assume they are true. Next we with strict necessity from the premises, the argu- determine, in light of this assumption, if it is possible ment is valid; if not, it is invalid. for the conclusion to be false. In this case it is possi-

To test an argument for validity we begin by ble. If banks were included in one part of the group of

assuming that all the premises are true, and then we financial institutions and Wells Fargo were included in determine if it is possible, in light of that assumption, another part, then Wells Fargo would not be a bank. In

for the conclusion to be false. Here is an example: other words, assuming the premises to be true and the

conclusion false does not involve any contradiction,

All television networks are media companies. and so the argument is invalid. NBC is a television network. In addition to illustrating the basic idea of validity, Therefore, NBC is a media company. these examples suggest an important point about valid-

In this argument both premises are actually true, ity and truth. In general, validity is not something that so it is easy to is uniformly determined by the actual truth or falsity of assume that they are true. Next we determine, in light of this assumption, if it is possible the premises and conclusion. Both the NBC example for the conclusion to be false. Clearly this is not pos- and the Wells Fargo example have actually true prem-sible. If NBC is included in the group of television ises and an actually true conclusion, yet one is valid networks (second premise) and if the group of tele- and the other invalid. The United Airlines example has vision networks is included in the group of media actually false premises and an actually false conclusion, companies (first premise), it necessarily follows that yet the argument is valid. Rather, validity is something NBC is included in the group of media companies that is determined by the relationship between prem-(conclusion). In other words, assuming the premises ises and conclusion. The question is not whether the to be true and the conclusion false entails a strict premises and conclusion are true or false, but whether contradiction. the premises support the conclusion. In the examples of Thus, the argument is valid.

Here is another example: valid arguments the premises do support the conclu-

sion, and in the invalid case they do not.

All automakers are pharmaceutical companies. Nevertheless, there is one arrangement of truth United Airlines is an automaker. and falsity in the premises and conclusion that does

40Chapter1 Basic Concepts

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). determine the issue of validity. Any deductive argu- Table 1.1 presents examples of categorical syllo-ment having actually true premises and an actually gisms (deductive arguments) that illustrate the vari-false conclusion is invalid. The reasoning behind ous combinations of truth and falsity in the premises this fact is fairly obvious. If the premises are actually and conclusion. In the examples having false prem-true and the conclusion is actually false, then it cer- ises, both premises are false, but it is easy to con-tainly is possible for the premises to be true and the struct other examples having only one false prem-conclusion false. Thus, by the definition of invalid- ise. When examining this table, note that the only ity, the argument is invalid. combination of truth and falsity that does not allow

The idea that any deductive argument having actu- for both valid and invalid arguments is true premises

ally true premises and a false conclusion is invalid and false conclusion. As we have just seen, any argu-may be the most important point in all of deductive ment having this combination is necessarily invalid. logic. The entire system of deductive logic would be The relationship between the validity of a deduc-quite useless if it accepted as valid any inferential pro- tive argument and the truth or falsity of its premises cess by which a person could start with truth in the and conclusion, as illustrated in Table 1.1, is summa-premises and arrive at falsity in the conclusion. rized as follows:

Table 1.1 Deductive Arguments

ValidInvalid

True premises All flowers are plants. All flowers are plants.

All daisies are flowers. All daisies are plants.

True conclusion Therefore, all daisies are plants. Therefore, all daisies are flowers.

[sound][unsound]

True premises None exist All roses are flowers.

All daisies are flowers.

False conclusion Therefore, all daisies are roses.

[unsound]

False premises All flowers are dogs. All dogs are flowers.

All poodles are flowers. All poodles are flowers.

True conclusion Therefore, all poodles are dogs. Therefore, all poodles are dogs.

[unsound][unsound]

False premises All flowers are dogs. All roses are cats.

All tigers are flowers. All daisies are cats.

False conclusion Therefore, all tigers are dogs. Therefore, all daisies are roses.

[unsound][unsound]

Premises This short summary table reinforces the point that ConclusionValidity T merely knowing the truth or falsity of the premises T ? and conclusion tells us nothing about validity except

T F Invalid in the one case of true premises and false conclusion. F T ? Any deductive argument having true premises and a F F ? false conclusion is necessarily invalid.

Section 1.4 Validity, Truth, Soundness, Strength, and Cogency 41

A sound argument is a deductive argument criterion of evaluation that is distinct from valid-

that is valid and has all true premises. Both condi- ity and that the evaluator be ever vigilant never to tions must be met for an argument to be sound; if confuse soundness with validity. either is missing the argument is unsound. Thus,

an unsound argument is a deductive argument that is invalid, has one or more false premises, Inductive Arguments

or both. Because a valid argument is one such Section 1.3 defined an inductive argument as one that it is impossible for the premises to be true incorporating the claim that it is improbable that and the conclusion false, and because a sound the conclusion be false given that the premises are argument does in fact have true premises, it fol- true. If this claim is true, the argument is said to lows that every sound argument, by definition, be strong. Thus, a strong inductive argument will have a true conclusion as well. A sound argu- is an inductive argument in which it is improbable ment, therefore, is what is meant by a good, or that the conclusion be false given that the premises successful, deductive argument in the fullest sense are true. In such arguments, the conclusion does in of the term. fact follow probably from the premises. Conversely,

a weak inductive argument is an argument in which the conclusion does not follow probably

Sound Valid All true = + from the premises, even though it is claimed to. argument argument premises All inductive arguments depend on what philos-ophers call the uniformity of nature. According to

this principle, the future tends to replicate the past,

In connection with this definition of sound- and regularities that prevail in one spatial region

ness, a single proviso is required: For an argument tend to prevail in other regions. For example, in the to be unsound, the false premise or premises must past, sugar has always tasted sweet. According to actually be needed to support the conclusion. the uniformity of nature, sugar will continue to taste An argument having a conclusion that is validly sweet in the future. Also, just as sugar tastes sweet supported by true premises but having a super- in Los Angeles, so does it in New York, London, fluous false premise would still be sound. By and everywhere else. The uniformity of nature is similar reasoning, no addition of a false premise the ultimate basis for our judgments about what to an originally sound argument can make the we naturally expect to occur. Good inductive argu-argument unsound. Such a premise would be ments are those that accord with the uniformity of superfluous and should not be considered part of nature. They have conclusions that we naturally the argument. Analogous remarks, incidentally, expect to turn out true. If the conclusion of such an extend to induction. argument should turn out to be false, in violation of

Since (at least from the standpoint of logic) our expectations, this occurrence would cause us to

every premise is either true or false, and every react with surprise. deductive argument is either valid or invalid, it The procedure for testing the strength of induc-follows that every deductive argument is either tive arguments runs parallel to the procedure for sound or unsound. However, given that many, deduction. First we assume the premises are true, if not most, premises have truth values that are and then we determine whether, based on that unknown or impossible to determine, it is not assumption, the conclusion is probably true. This always possible to determine the soundness of a determination is accomplished by linking up the deductive argument. But that does not mean that premises with regularities that exist in our experi-soundness is unimportant in logic. It is crucially ential background. For example, if the argument is important that soundness be recognized as a a causal inference, we link the information in the

42Chapter1 Basic Concepts

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). premises with known causal patterns. If the argu- argument is weak. The conclusion, incidentally, hap-ment is an argument from signs, we connect the pens to be false. information in the premises with what we know Another example:

others are not. If the argument is a generalization, During the past fifty years, inflation has con-sistently reduced the value of the American about signs: some kinds of signs are trustworthy,

we connect the information in the premises with dollar. Therefore, industrial productivity will what we know about a sample being representa-probably increase in the years ahead. tive of a population. All of these regularities are

instances of the uniformity of nature. Here is an In this argument, the premise is actually true and the example of a prediction: conclusion is probably true in the actual world, but

All dinosaur bones discovered to this day the probability of the conclusion is in no way based have been at least 50 million years old. on the assumption that the premise is true. Because Therefore, probably the next dinosaur there is no direct connection between inflation and bone to be found will be at least 50 million increased industrial productivity, the premise is irrel-years old. evant to the conclusion and it provides no probabi-

In this argument the premise is actually true. Given listic support for it. The conclusion is probably true independently of the premise. As a result, the argu-that all dinosaur bones discovered to date have ment is weak. been over 50 million years old (and that thousands This last example illustrates an important distinc-of such bones have been discovered), the unifor-tion between strong inductive arguments and valid mity of nature dictates that the next one to be dis-deductive arguments. As we will see in later chap-covered will also be over 50 million years old. This ters, if the conclusion of a deductive argument is is what we would naturally expect, and anything to necessarily true independently of the premises, the the contrary would be highly surprising. Thus, the argument is still considered valid. But if the conclu-conclusion is probably true, and so the argument sion of an inductive argument is probably true inde-is strong. pendently of the premises, the argument is weak. Here is another example: These four examples show that in general the

All meteorites found to this day have con- strength or weakness of an inductive argument results tained salt. Therefore, probably the next not from the actual truth or falsity of the premises meteor ite to be found will contain salt. and conclusion, but from the probabilistic support the

The premise of this argument is clearly false; but premises give to the conclusion. The dinosaur-bone if we assume it to be true, then we would naturally argument has a true premise and a probably true con-expect that the next meteorite to be found would clusion, and the meteorite argument has a false prem-contain salt. Thus, the argument is strong. ise and a probably false conclusion; yet both are strong

The next example is an argument from analogy: because the premise of each provides probabilistic

support for the conclusion. The industrial productivity

Dom P�rignon champagne, which is made argument has a true premise and a probably true con-

in France, sells for over $100 per bottle. clusion, but the argument is weak because the premise Marquis de la Tour is also a French cham-provides no probabilistic support for the conclusion. pagne. Therefore probably it, too, sells for As in the evaluation of deductive arguments, the only over $100 per bottle. arrangement of truth and falsity that establishes any-

In this argument the premises are actually true, but thing is true premises and probably false conclusion our background experience tells us that the mere fact (as in the Dom P�rignon argument). Any inductive that two wines come from the same country does argument having true premises and a probably false not imply that they sell for the same price. Thus, the conclusion is weak.

Section 1.4 Validity, Truth, Soundness, Strength, and Cogency 43

Before proceeding further, however, we must false in the actual world in light of all the known

qualify and explain this last statement. When we evidence. speak of the premises being true, we mean �true� in Table 1.2 presents several predictions (inductive a complete sense. The premises must not exclude arguments) that illustrate the various combinations or overlook some crucial piece of evidence that of truth and falsity in the premises and conclusion. undermines the stated premises and requires a dif- Note that the only arrangement of truth and falsity ferent conclusion. This proviso is otherwise called that is missing for strong arguments is true premises the total evidence requirement. If the total evidence and probably false conclusion. requirement is not met, an argument might have The relationship between the strength of an induc-literally true premises and a probably false conclu- tive argument and the truth or falsity of its premises sion and still be strong. Also, when we speak of the and conclusion, as illustrated in Table 1.2, is summa-conclusion being probably false, we mean probably rized as follows:

Table 1.2 Inductive Arguments

StrongWeak

True premise Every previous U.S. president A few U.S. presidents were

was older than 40. lawyers.

Probably true conclusion Therefore, probably the next Therefore, probably the next

U.S. president will be older than U.S. president will be older 40. [cogent] than 40. [uncogent]

True premise None exist A few U.S. presidents were unmarried.

Therefore, probably the next U.S.

Probably false conclusion president will be unmarried.

[uncogent]

False premise Every previous U.S. president A few U.S. presidents were dentists.

was a TV debater. Therefore, probably the next U.S.

Probably true conclusion Therefore, probably the next U.S. president will be a TV debater.

president will be a TV debater. [uncogent][uncogent]

False premise Every previous U.S. president A few U.S. presidents were

died in office. dentists.

Probably false conclusion Therefore, probably the next Therefore, probably the next

U.S. president will die in office. U.S. president will be a dentist. [uncogent][uncogent]

Premises Like the summary table for deduction, this brief ConclusionStrength T table reinforces the point that merely knowing the probably T ? truth values of the premises and conclusion tells us

T probably F Weak nothing about the strength of an argument except F probably T ? in the one case of true premises and probably false F probably F ? conclusion. Any inductive argument having true

44Chapter1 Basic Concepts

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). premises (in the sense just explained) and a prob- or any combination of these. A cogent argument ably false conclusion is weak. is the inductive analogue of a sound deductive

Unlike the validity and invalidity of deductive argument and is what is meant by a good, or suc-

arguments, the strength and weakness of induc- cessful, inductive argument without qualification. tive arguments allow for degrees. To be considered Because the conclusion of a cogent argument is strong, an inductive argument must have a con- genuinely supported by true premises, it follows clusion that is more probable than improbable. In that the conclusion of every cogent argument is other words, given that the premises are true, the probably true in the actual world in light of all the likelihood that the conclusion is true must be more known�evidence. than 50 percent, and as the probability increases,

the argument becomes stronger. For this purpose,

consider the following pair of arguments: Cogent Strong All true = + argument argument premises This barrel contains 100 apples.

Three apples selected at random

were found to be ripe. As an illustration of the need for the total evidence Therefore, probably all 100 apples requirement, consider the following argument: are ripe.

This barrel contains 100 apples. Swimming in the Caribbean is usually lots Eighty apples selected at random were found of fun. Today the water is warm, the surf is to be ripe. gentle, and on this beach there are no dan-Therefore, probably all 100 apples are ripe. gerous currents. Therefore, it would be fun

to go swimming here now.

The first argument is weak and the second is

strong. However, the first is not absolutely weak If the premises reflect all the important factors, nor the second absolutely strong. Both argu- then the argument is cogent. But if they ignore the ments would be strengthened or weakened by the fact that several large dorsal fins are cutting through random selection of a larger or smaller sample. the water (suggesting sharks), then obviously the For example, if the size of the sample in the second argument is not cogent. Thus, for cogency the argument were reduced to seventy apples, the premises must not only be true but also not over-argument would be weakened. The incorporation look some important fact that requires a different of additional premises into an inductive argument conclusion. will also generally tend to strengthen or weaken Finally, just as it is not always possible to deter-it. For example, if the premise �One unripe apple mine the soundness of a deductive argument, it is that had been found earlier was removed� were not always possible to determine the cogency of added to either argument, the argument would be an inductive argument. And this follows for two weakened. reasons. Many inductive arguments, especially

A cogent argument is an inductive argument those about complex real-life subjects, are not

that is strong and has all true premises. Also, the susceptible to being evaluated as clearly strong premises must be true in the sense of meeting the or clearly weak. And many premises have truth total evidence requirement. If any one of these con- values that are unknown or impossible to deter-ditions is missing, the argument is uncogent. Thus, mine. Yet, it remains important that cogency be an uncogent argument is an inductive argu- recognized as a criterion for evaluating induc-ment that is weak, has one or more false prem- tive arguments and that it not be confused with ises, fails to meet the total evidence requirement, strength and weakness.

Section 1.4 Validity, Truth, Soundness, Strength, and Cogency 45

Summary

For both deductive and inductive arguments, two separate questions need to be answered: (1) Do the premises support the conclusion? (2) Are all the premises true? To answer the first question, we begin by assuming the premises to be true. Then, for deductive arguments we determine whether, in light of this assumption, it necessarily follows that the conclusion is true. If it does, the argument is valid; if not, it is invalid. For inductive arguments, we determine whether it probably follows that the conclusion is true. If it does, the argument is strong; if not, it is weak . For inductive arguments, we keep in mind the requirements that the premises actu-ally support the conclusion and that they not ignore important evidence. Finally, if the argument is either valid or strong, we turn to the second question and determine whether the premises are actually true. If all the premises are true, the argument is sound (in the case of deduction) or cogent (in the case of induction). All invalid deductive arguments are unsound, and all weak inductive arguments are uncogent.

The various alternatives open to statements and arguments may be diagrammed as follows.

Note that in logic one never speaks of an argument as being �true� or �false,� and one never speaks of a statement as being �valid,� �invalid,� �strong,� or �weak .�

True

Statements

False

Deductive

Arguments

Groups of statements

Inductive

Nonarguments

Sound

Valid

Deductive arguments

Unsound

Invalid

(all are unsound)

Cogent

Strong

Inductive arguments Uncogent

Weak

(all are uncogent)

46Chapter1 Basic Concepts

Prominent 280�206 Chrysippus b.c.Logicianse.

C hrysippus was born in Soli, a city located on the southeast coast of Asia Minor. Early in life he moved to Athens, where he studied under the Stoic philosopher Cleanthes, who in turn was a

student of Zeno of Citium, the founder of Stoicism.

Upon Cleanthes� death in 232 b.c.e., Chrysippus took over as leader of the school, and he produced over

700�trea tises that systematized Stoic teaching. All of

these works have been lost, but fragments survive in

the writings of Cicero, Seneca, and others. Because

of his extraordinary contribution, Chrysippus is consid-

ered to be the second founder of Stoicism.

Stoicism derives its name from the Greek word

stoa, which means porch; Stoic philosophers used to

gather on a porch in the Agora (public square) in

Athens to discuss their views. The Stoics prized the ork Public Library/Science Source virtue of self-sufficiency, and they emphasized the New Y importance of not allowing oneself to be carried away

by emotions or passions such as fear or love. The prop ositions inc luded negation, conjunc tion, exclusive Stoics considered emotions to be false judgments disjunction, and implication, and Chrysippus showed about the goodness or badness of something. The how the truth value of a compound statement is a proper therapy for those victimized by emotions is to function of the truth values of its simple components. persuade them that these judgments are indeed false Chrysippus also introduced a set of rules of inference because they constitute obstacles to true happiness. including what is today called modus ponens, modus

Chrysippus is often considered to be the originator tollens, disjunctive syllogism, and a rule similar to

of propositional logic. Unlike Aristotelian logic, where De Morgan�s rule. Finally, he introduced the theory the fundamental components are terms, in proposi- of natural deduction by which the conclusion of an tional logic the fundamental components are whole argument can be derived from its premises through a propositions or statements. Aristotle had overlooked series of discrete steps. this kind of logic, but his close friend and successor The broader philosophy of Chrysippus is character-Theophrastus worked out some of the logic of the ized by monism and determinism. While most of us pure hypothetical syllogism (If A then B; If B, then C; think that the universe is made up of millions of dis-therefore, If A, then C ). Also, Philo of Megara intro- crete entities, Chrysippus argued that in fact only one duced the truth-functional interpretation of the mate- substance exists, and what appear to be individual rial conditional (If A, then B). Beginning at this point, substances are really parts of this one primary sub-Chry sippus advanced propositional logic to a high stance. Furthermore, everything that occurs is strictly level of development. governed by fate. Yet, in the face of this rigid causal

Chrysippus div ided propositions into simple and determinism, Chrysippus held that humans are respon-

compound, and he introduced a set of connectives sible for their actions, and he tried in many ways to that were used to produce� compound propositions prove that the two viewpoints are in fact compatible from one or more simple propositions. The compound with each other.

Section 1.4 Validity, Truth, Soundness, Strength, and Cogency 47

Exercise 1.4

I. The following arguments are deductive. Determine whether each is valid or invalid, and

note the relationship between your answer and the truth or falsity of�the premises and conclusion. Finally, determine whether the argument is sound or unsound.

★1. Since Moby Dick was written by Shakespeare, and Moby Dick is a science-

fiction novel, it follows that Shakespeare wrote a science-fiction novel.

2. Since London is north of Paris and south of Edinburgh, it follows that Paris is

south of Edinburgh.

3. If George Washington was beheaded, then George Washington died. George

Washington died. Therefore, George Washington was beheaded.

★ 4. The longest river in South America is the Amazon, and the Amazon flows

through Brazil. Therefore, the longest river in South America flows through Brazil.

5. Since the Spanish-American War occurred before the U.S. Civil War, and the

U.S. Civil War occurred after the Korean War, it follows that the Spanish-American War occurred before the Korean War.

6. The Empire State Building is taller than the Statue of Liberty, and the Statue of

Liberty is taller than the Eiffel Tower. Therefore, the Empire State Building is taller than the Eiffel Tower.

★7. All leopards with lungs are carnivores. Therefore, all leopards are carnivores.

8. Chicago is a city in Michigan and Michigan is part of the United States. There-

fore, Chicago is a city in the United States.

9. If Prince Harry was born in Los Angeles, then he is a California native. Prince

Harry is not a California native. Therefore, Prince Harry was not born in Los Angeles.

★10. Every province in Canada has exactly one city as its capital. Therefore, since

there are thirty provinces in Canada, there are thirty provincial capitals.

11. Since the Department of Defense Building outside Washington, D.C., has the

shape of a hexagon, it follows that it has seven sides.

12. Since Winston Churchill was English, and Winston Churchill was a famous states-

man, we may conclude that at least one Englishman was a famous statesman.

★13. Since some fruits are green, and some fruits are apples, it follows that some

fruits are green apples.

14. All physicians are individuals who have earned degrees in political science, and

some lawyers are physicians. Therefore, some lawyers are persons who have earned degrees in political science.

15. The U.S. Congress has more members than there are days in the year. There-

fore, at least two members of Congress have the same birthday.

48Chapter1 Basic Concepts

II. The following arguments are inductive. Determine whether each is strong or weak, and

note the relationship between your answer and the truth or falsity of the premise(s) and conclusion. Then determine whether each argument is cogent or uncogent.

★ 1. The grave marker at Arlington National Cemetery says that John F. Kennedy

is buried there. It must be the case that Kennedy really is buried in that cemetery.

2. The ebb and flow of the tides has been occurring every day for millions of

years. But nothing lasts forever. Therefore, probably the motion of the tides will die out within a few years.

3. The vast majority of Rose Bowl games (in Pasadena, California) have been

played in freezing-cold weather. Therefore, probably the next Rose Bowl game will be played in freezing-cold weather.

★4. Franklin Delano Roosevelt said that we have nothing to fear but fear itself.

Therefore, people have no reason to fear serial killers.

5. Most popular film stars are millionaires. Shailene Woodley is a popular film

star. Therefore, probably Shailene Woodley is a millionaire.

6. Constructing the great pyramid at Giza required lifting massive stone blocks

to great heights. Probably the ancient Egyptians had some antigravity device to accomplish this feat.

★7. People have been listening to rock and roll music for over a hundred years.

Probably people will still be listening to it a year from now.

8. Paleontologists have unearthed the fossilized bones of huge reptiles, which

we have named dinosaurs. Tests indicate that these bones are more than 50 million years old. Therefore, probably dinosaurs really did roam the earth more than 50 million years ago.

9. The Declaration of Independence says that all men are endowed by their

creator with certain unalienable rights. Therefore it probably follows that a cre-ator exists.

★10. Coca-Cola is an extremely popular soft drink consumed all over the world.

Therefore, probably someone, somewhere, is drinking a Coke right this minute.

11. Every map of the United States shows that Alabama is situated on the Pacific

coast. Therefore, Alabama must be a western state.

12. When Neil Armstrong landed on the moon, he left behind a gold-plated

Schwinn bicycle, which he used to ride around on the moon�s surface. Probably that bicycle is still up there on the moon.

★ 13. Children love green M&M�s. But brussels sprouts are green, too. Therefore, it

must be the case that children love brussels sprouts.

14. Unlike monkeys, today�s humans have feet that are not suited for grasping

objects. Therefore, a thousand years from now, probably humans will still have feet that are not suited for grasping objects.

Section 1.4 Validity, Truth, Soundness, Strength, and Cogency 49

15. In a random sample of twenty billionaires, including Jeff Bezos and Mark

Zuckerberg, all of those sampled turned out to be registered Democrats. There-fore, it must be the case that most billionaires are registered Democrats.

III. Determine whether the following arguments are inductive or deductive. If an argu-

ment is inductive, determine whether it is strong or weak. If it is deductive, determine whether it is valid or invalid.

★1. Since Tom is the brother of Agatha, and Agatha is the mother of Raquel, it

follows that Tom is the uncle of Raquel.

2. When a cook cannot recall the ingredients in a recipe, it is appropriate that she

refresh her memory by consulting the recipe book. Similarly, when a student cannot recall the answers during a final exam, it is appropriate that she refresh her memory by consulting the textbook.

3. The Broadway Theater marquee says that Hamilton is playing nightly. There-

fore, it must be that case that Hamilton is playing there tonight.

★4. Since Christmas is always on a Thursday, it follows that the day after Christmas

is always a Friday.

5. Suppose figure A is a triangle having two equal angles. It follows that figure A

has two equal sides.

6. By accident Karen baked her brownies two hours longer than she should have.

Therefore, they have probably been ruined.

★7. After taking LSD, Alice said she saw a flying saucer land in the shopping center

parking lot. Since Alice has a reputation for always telling the truth, we must conclude that a flying saucer really did land there.

8. Since Phyllis is the cousin of Denise, and Denise is the cousin of Harriet, it fol-

lows necessarily that Harriet is the cousin of Phyllis.

9. The picnic scheduled in the park for tomorrow will most likely be cancelled.

It�s been snowing for six days straight.

★10. Circle A has exactly twice the diameter of circle B. From this we may conclude

that circle A has exactly twice the area of circle B.

11. Robert has lost consistently at blackjack every day for the past several days.

Therefore, it is very likely that he will win today.

12. Since John loves Nancy and Nancy loves Peter, it follows necessarily that John

loves Peter.

★13. This cash register drawer contains over 100 coins. Three coins selected at

random were found to have dates earlier than 1960. Therefore, probably all of the coins in the drawer have dates earlier than 1960.

14. The Japanese attack on Pearl Harbor happened in either 1941 or 1951. But it

didn�t happen in 1941. Therefore, it happened in 1951.

15. Harry will never be able to solve that difficult problem in advanced calculus in

the limited time allowed. He has never studied anything beyond algebra, and in that he earned only a C.

50Chapter1 Basic Concepts

★16. Since x  y  10, and x  7, it follows that y  4.

17. If acupuncture is hocus pocus, then acupuncture cannot relieve chronic pain.

But acupuncture can relieve chronic pain. Therefore, acupuncture is not hocus pocus.

18. If inflation heats up, then interest rates will rise. If interest rates rise, then

bond prices will decline. Therefore, if inflation heats up, then bond prices will decline.

★19. Statistics reveal that 86 percent of those who receive flu shots do not get the flu.

Jack received a flu shot one month ago. Therefore, he should be immune, even though the flu is going around now.

20. Since Michael is a Pisces, it necessarily follows that he was born in March.

IV. Define the following terms:

valid deductive argument strong inductive argument invalid deductive argument weak inductive argument sound argument cogent argument unsound argument uncogent argument

V. Answer �true� or �false� to the following statements:

1. Some arguments, while not completely valid, are almost valid.

2. Inductive arguments allow for varying degrees of strength and weakness.

3. Invalid deductive arguments are basically the same as inductive arguments.

4. If a deductive argument has true premises and a false conclusion, it is necessar-

ily invalid.

5. A valid argument may have a false premise and a false conclusion.

6. A valid argument may have a false premise and a true conclusion.

7. A sound argument may be invalid.

8. A sound argument may have a false conclusion.

9. A strong argument may have false premises and a probably false conclusion.

10. A strong argument may have true premises and a probably false conclusion.

11. A cogent argument may have a probably false conclusion.

12. A cogent argument must be inductively strong.

13. If an argument has true premises and a true conclusion, we know that it is a

perfectly good argument.

14. A statement may legitimately be spoken of as �valid� or �invalid.�

15. An argument may legitimately be spoken of as �true� or �false.�

Section 1.4 Validity, Truth, Soundness, Strength, and Cogency 51

VI. Create the following sound and cogent arguments:

1. A sound argument about musicians

2. A valid unsound argument about painters

3. An invalid argument about writers

4. A cogent argument relating to politics

5. A weak argument relating to cooking

1.5 Argument Forms: Proving Invalidity

This section explores the idea that the validity of a words or phrases being substituted must refer to deductive argument is determined by its form. This groups of things. Thus, if we substitute �sport-idea was suggested by the arguments in Table 1.1 in ing events,� �engaging pastimes,� and �baseball the previous section. All the arguments in the Valid games� in the place of A, B, and C, respectively, in column have the same valid form, and all the argu- the argument form (left), we obtain the following ments in the Invalid column have the same invalid argument (right):

form. The form of an argument illustrates the argu- All A are B. All sporting events are ment�s internal structure or pattern of reasoning. If engaging pastimes. the pattern of reasoning is good, the argument will All C are A. All baseball games are be valid; if not, it will be invalid. sporting events. In reference to Table 1.1, all the valid arguments All C are B. All baseball games are have this form: engaging pastimes.

All A are B. This argument is called a substitution instance

of things, it is easy to see that this form is valid. All C are A. of the argument form. Any substitution instance of a All C are B. valid argument form is a valid argument. Before proceeding to invalid arguments, we If A, B, and C are thought of as referring to groups must briefly consider valid arguments in which the form is not apparent. Many of the arguments in the Assume, by the second premise, that the C s (what-previous set of exercises were like this. How can ever they might be) are included in the A s, and, by we reconcile the existence of such arguments with the first premise, that the A s (whatever they might be) are included in the B s. Then it necessarily fol-the claim that validity is determined by form? The answer is that these arguments are incomplete, lows that the C s are included in the B s, which is what so the form is not explicit. But once such argu-the conclusion asserts. ments are completed and correctly phrased (which We can use this example to define what we mean by an argument form. An argument form , we address later in this book), the form becomes apparent. For example, consider the following for the present purpose, is an arrangement of let-valid argument: ters (in this case A, B, and C ) and words (in this

case �all� and �are�) such that the uniform sub- Geese are migratory waterfowl, so they fly stitution of words or phrases in the place of the south for the winter. letters results in an argument. For this form, the

52Chapter1 Basic Concepts

This argument is missing a premise: All A are B. All cats are animals. True

Migratory waterfowl fly south for the winter. All A are C. Therefore, all cats are dogs. False All C are B. All dogs are animals. True

form apparent: This substitution instance proves the form The argument can now be rephrased to make its

invalid, because it provides a concrete example of a

All migratory waterfowl are birds that fly south case where the As are in the Bs, the Cs are in the Bs, for the winter. but the As are not in the Cs. All geese are migratory waterfowl. Now, since the form is invalid, can we say that Therefore, all geese are birds that fly south for any argument that has this form is invalid? Unfor-the winter. tunately, the situation with invalid forms is not quite

The form of the argument is as simple as it is�with valid forms. Every substitution

instance of a valid form is a valid argument, but it

All A are B. is not the case that every substitution instance of an All C are A. invalid form is an invalid argument. The reason is All C are B. that some substitution instances of invalid forms are This form is identical to the form we just consid- also substitution in stances�of valid forms.* However,

ered and is valid. we can say that any substitution instance of an invalid

Let us now consider an invalid argument form: form is an invalid argument provided that it is not a All A are B. substitution instance of any valid form. Thus, we will All C are B. say that an argument actually has an invalid form if it All A are C. is a substitution instance of that form and it is not a

substitution instance of any valid form.

In this argument form, if we assume that the As The fact that some substitution instances of

are in the Bs and that the Cs are in�the Bs, it does not invalid forms are also substitution instances of valid necessarily follow that the As are in the Cs. It would forms means simply that we must exercise caution in not follow if the� As were in one part of the Bs and the identifying the form of an argument. However, cases Cs were in another part, as the following diagram of ordinary language arguments that can be inter-illustrates: preted as substitution instances of both valid and

invalid forms are so rare that this book chooses to

ignore them. With this in mind, consider the follow-

ing argument:

As Cs All romantic novels are literary pieces.

All works of fiction are literary pieces.

Therefore, all romantic novels are works of fiction.

* For example, the following valid argument is a substitution instance of

This diagram suggests that we can prove the form the invalid form we have been discussing:

invalid if we can find a substitution instance having All bachelors are persons. actually true premises and an actually false conclu- All unmarried men are persons.

CHowever, because �bachelors� is equivalent in meaning to �unmarried s would be separated from each other, but they men,� the argument is also a substitution instance of this valid form: would both be included in the B s. If we substitute All A are B. sion. In such a substitution instance the Therefore, all bachelors are unmarried men. A s and the �cats� for A, �animals� for B, and �dogs� for C, we All A are B. have such a substitution instance: All A are A.

Section 1.5 Argument Forms: Proving Invalidity 53

This argument clearly has the invalid form just dis- The resulting substitution instance is this:

cussed. This invalid form captures the reasoning Some animals are not mammals. process of the argument, which is obviously defec-All dogs are animals. tive. Therefore, the argument is invalid, and it is Therefore, some dogs are not mammals. invalid precisely because it has an invalid form. The substitution instance has true premises and a false

a false conclusion (like the cats-and-dogs exam- In applying the counterexample method to categori-ple just constructed) is called a counterexample, cal syllogisms, it is useful to keep in mind the follow-and the method we have just used to prove the ing set of terms: �cats,� �dogs,� �mammals,� �fish,� and romantic-novels argument invalid is called the �animals.� Most invalid syllogisms can be proven invalid counterexample method A substitution instance having true premises and Because the substitution instance is invalid, the form is invalid, and therefore the original argument is invalid. Counterexample Method conclusion and is therefore, by definition, invalid.

ing the form of an argument and then constructing . It consists of isolat- by strategically selecting three of these terms and using them to construct a counterexample. Because everyone a substitution instance having true premises and a agrees about these terms, everyone will agree about the false conclusion. This proves the form invalid, which truth or falsity of the premises and conclusion of the in turn proves the argument invalid. The�counter -counterexample. Also, in constructing the counterex-example�method can be used to prove the invalid-ample, it often helps to begin with the conclusion. First, ity of any invalid argument, but it�cannot prove the select two terms that yield a false conclusion, and then validity of any valid argument. Thus, before the select a third term that yields true premises. Another method is applied to an argument, the argument point to keep in mind is that the word �some� in logic must be known or suspected to be invalid in the always means �at least one.� For example, the statement first place. Let us apply the counter example method �Some dogs are animals� means �At least one dog is an to the following invalid categorical syllogism: animal��which is true. Also note that this statement

Since some employees are not social climb- does not imply that some dogs are not animals. ers and all vice presidents are employees, we Not all deductive arguments, of course, are cat-may conclude that some vice presidents are egorical syllogisms. Consider, for example, the fol-not social climbers. lowing hypothetical syllogism:

This argument is invalid because the employ- If the government imposes import restric-

ees who are not social climbers might not be vice tions, the price of automobiles will rise. presidents. Accordingly, we can prove the argu- Therefore, since the government will not ment invalid by constructing a substitution instance impose import restrictions, it follows that having true premises and a false conclusion. We the price of automobiles will not rise.

begin by isolating the form of the argument: This argument is invalid because the price of auto-

Some E are not S. mobiles might rise even though import restrictions All V are E. are not imposed. It has the following form: Some V are not S.

If G, then P.

Next, we select three terms to substitute in place Not G.

of the letters that will make the premises true and the Not P.

conclusion false. The following selection will work: This form differs from the previous one in that its

E  animals letters stand for complete statements. G, for exam-S  mammals ple, stands for �The government imposes import V  dogs restrictions.� If we make the substitution

54Chapter1 Basic Concepts

G  Abraham Lincoln committed suicide. Being able to identify the form of an argument

we obtain the following substitution instance: deductive argument forms. The first task consists in distinguishing the premises from the conclusion. P  Abraham Lincoln is dead. with ease requires a familiarity with the basic

If Abraham Lincoln committed suicide, then Always write the premises first and the conclusion Abraham Lincoln is dead. last. The second task involves distinguishing what Abraham Lincoln did not commit suicide. we may call �form words� from �content words.� Therefore, Abraham Lincoln is not dead. To reduce an argument to its form, leave the form

the substitution instance is clearly invalid. There- �all,� �no,� �some,� �are,� and �not� are form words, fore, the form is invalid, and this proves the original and for hypothetical syllogisms the words �if,� argument invalid. �then,� and �not� are form words. Additional form Since the premises are true and the conclusion false, with letters. For categorical syllogisms, the words words as they are, and replace the content words

argument having a conditional statement as a premise When applying the counterexample method to an words for other types of arguments are �either,� �or,� �both,� and �and.� For various kinds of hybrid argu-(such as the one just discussed), it is recommended ments, a more intuitive approach may be needed. that the statement substituted in place of the condi-Here is an example: tional statement express some kind of necessary con-nection. In the Lincoln example, the first premise All movie stars are actors who are famous, asserts the necessary connection between suicide and because all movie stars who are famous are death. There can be no doubt about the truth of such a actors.

statement. Furthermore, if it should turn out that the If we replace �movie stars,� �actors,� and �famous� conclusion is a conditional statement, note that one with the letters M, A, and F, this argument has the sure way of producing a false conditional statement is following form: by joining a true antecedent with a false consequent. For example, the conditional statement �If Lassie is a All M who are F are A. dog, then Lassie is a cat� is clearly false. All M are A who are F.

Here is one possible substitution instance for this

Counterexample method form:

Given: an All humans who are fathers are men. invalid extract Form of Therefore, all humans are men who are fathers. argument argument

Because the premise is true and the conclusion false,

construct the form is invalid and so is the original argument.

Using the counterexample method to prove

Substitution instance arguments invalid requires a little ingenuity having true premises, because there is no rule that will automatically false conclusion produce the required term or statement to be sub-

proves stituted into the form. Any term or statement will

work, of course, provided that it yields a substitu-

tion instance that has premises that are indisput-

ably true and a conclusion that is indisputably false.

Form is proves Given argument Ideally, the truth value of these statements should invalid. is invalid. be known to the average individual; otherwise, the substitution instance cannot be depended on

Section 1.5 Argument Forms: Proving Invalidity 55

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). to prove anything. If, for example, P in the earlier is produced having true premises and a true con-hypothetical syllogism had been replaced by the clusion, it does not prove that the argument is valid. statement �George Wilson is dead,� the substitution Furthermore, the method is useful only for deduc-instance would be useless, because nobody knows tive arguments because the strength and weakness whether this statement is true or false. of inductive arguments is only partially dependent

The counterexample method is useful only for on the form of the argument. Accordingly, no

proving invalidity, because the only arrangement of method that relates exclusively to the form of an truth and falsity that proves anything is true prem- inductive argument can be used to prove the argu-ises and false conclusion. If a substitution instance ment weak.

Exercise 1.5

I. Use the counterexample method to prove the following categorical syllogisms invalid.

In doing so, follow the suggestions given in the text.

★1. All galaxies are structures that contain black holes in the center, so all galaxies

are quasars, since all quasars are structures that contain black holes in the center.

2. Some evolutionists are not people who believe in the Bible, for no creationists

are evolutionists, and some people who believe in the Bible are not creationists.

3. No patents are measures that discourage research and development, and all

patents are regulations that protect intellectual property. Thus, no measures that discourage research and development are regulations that protect intel-lectual property.

★4. Some farmworkers are not people who are paid decent wages, because no

undocumented individuals are people who are paid decent wages, and some undocumented individuals are not farmworkers.

5. Some politicians are people who will stop at nothing to win an election, and no

people who will stop at nothing to win an election are true statesmen. Hence, no politicians are true statesmen.

6. All meticulously constructed timepieces are true works of art, for all Swiss

watches are true works of art and all Swiss watches are meticulously constructed timepieces.

★7. No patrons of fast-food restaurants are health-food addicts. Consequently, no

patrons of fast-food restaurants are connoisseurs of fine desserts, since no con-noisseurs of fine desserts are health-food addicts.

8. Some toxic dumps are sites that emit hazardous wastes, and some sites that

emit hazardous wastes are undesirable places to live near. Thus, some toxic dumps are undesirable places to live near.

9. All persons who assist others in suicide are people guilty of murder. Accord-

ingly, some individuals motivated by compassion are not persons guilty of

5656ChapterChapter11 Basic Concepts Basic Concepts

murder, inasmuch as some people who assist others in suicide are individuals motivated by compassion.

★10. Some school boards are not groups that oppose critical race theory, because

some school boards are not organizations with a sense for social justice, and some groups that oppose critical race theory are not organizations with a sense for social justice.

11. All super PACs are unlimited spenders. For this reason, some big-time

power brokers are not super PACs, inasmuch as some unlimited spenders are big-time power brokers.

12. No movie producers are uncompetitive business executives, and some

Hollywood moguls are movie producers. It follows that no Hollywood moguls are uncompetitive business executives.

★13. Some improvers of humankind are not exploiters of personal information. As a

result, some corporate social networks are not improvers of humankind, seeing that all corporate social networks are exploiters of personal information.

14. Some drone attacks are assaults on human life, given that some stealth opera-

tions are assaults on human life and all drone attacks are stealth operations.

15. Some near-death experiences are supernatural phenomena, and no near-death

experiences are easily forgotten happenings. Consequently some easily forgot-ten happenings are not supernatural phenomena.

II. Use the counterexample method to prove each of the following arguments invalid.

★1. If animal species are fixed and immutable, then evolution is a myth. Therefore,

evolution is not a myth, since animal species are not fixed and immutable.

2. If carbon dioxide is present in the atmosphere, then plants have a source of

carbon. Hence, since plants have a source of carbon, carbon dioxide is present in the atmosphere.

3. If human rights are recognized, then civilization flourishes. If equality prevails,

then civilization flourishes. Thus, if human rights are recognized, then equality prevails.

★4. If energy taxes are increased, then either the deficit will be reduced or conser-

vation will be taken seriously. If the deficit is reduced, then inflation will be checked. Therefore, if energy taxes are increased, then inflation will be checked.

5. All unhoused people who are panhandlers are impoverished individuals.

Therefore, all unhoused people are impoverished individuals.

6. Some wrestlers are colorful hulks, since some wrestlers are colorful and some

wrestlers are hulks.

★7. All community colleges with low tuition are either schools with large enroll-

ments or institutions supported by taxes. Therefore, all community colleges are institutions supported by taxes.

Section 1.5 Argument Forms: Proving Invalidity 57

8. All merchandisers that are retailers are businesses that are inventory rotators.

Therefore, all merchandisers are inventory rotators.

9. All diabetics are either insulin takers or glucose eliminators. Accordingly, some

diabetics are glucose eliminators, since some diabetics are insulin takers.

★10. All FHA loans are living-standard enhancers for the following reasons. All

reverse mortgages that are FHA loans are either living-standard enhancers or home-equity depleters, and all reverse mortgages are home-equity depleters.

III. Create the following arguments:

1. Two valid arguments, having the same form, about jobs.

2. Two invalid arguments, having the same form, about fast food.

1.6 Extended Arguments

The logical analysis of extended arguments, such as drinking water, which comes from these aqui-those found in editorials, essays, and lengthy letters fers, is being poisoned by chemical wastes to newspaper editors, involves numerous difficulties. dumped into the soil for generations.

Such arguments are often mixed together with frag- This argument is diagrammed as follows: ments of reports, pieces of expository writing, illus-

trations, explanations, and statements of opinion. 2 Proper analysis involves weeding out the extraneous

material and isolating premises and conclusions. 1 Another problem stems from the fact that lengthy arguments often involve complex arrangements of The diagram says that statement 2 , the premise, subarguments that feed into the main argument in supports statement 1 , the conclusion. various ways. Distinguishing one subargument from In extended argument, we can identify two distinct another is often a complicated task. And then there patterns of argumentation, which we will name the are some argumentative passages that involve com-vertical pattern and the horizontal pattern. The vertical pletely separate strands of argumentation leading pattern consists of a series of arguments in which a to separate conclusions. Again, distinguishing the conclusion of a logically prior argument becomes a strands and assigning premises to the right conclu-premise of a subsequent argument. Example: sion is not only problematic but often involves an The selling of human organs, such as element of creativity on the part of the analyst. 1 hearts, kidneys, and corneas, should be To facilitate the analysis of extended arguments, outlawed. 2 Allowing human organs to we will assign numerals to the various statements in be sold will inevitably lead to a situation the passage and use arrows to represent the inferen-in which only the rich will be able to afford tial links. Example: transplants. This is so because 3 whenever

1 The contamination of underground something scarce is bought and sold as a

aquifers represents a pollution problem of commodity, the price always goes up. 4 The catastrophic proportions. 2 Half the nation�s law of supply and demand requires it.

58Chapter1 Basic Concepts

This argument is diagrammed as follows: destroyed. The following argument illustrates the

Vertical occurrence of conjoint premises: 4

pattern 1 Getting people who are poor off the

3 welfare rolls requires that we modify their

behavior patterns. 2 The vast majority of people on welfare are high school dropouts,

2 single parents, or people who abuse alcohol

and drugs. 3 These behavior patterns frus-

1 trate any desire that people who are poor

may have to get a job and improve their

The diagram says that statement 1 condition in life. , which is the main conclusion, is supported by Statement 1 2 is the conclusion. Taken separately, , which in turn is supported by statements 2 3 and , which in turn is supported by 3 4 provide little�or no support for .

The horizontal pattern consists of a single argu- 1 , but taken together they do provide support. That

ment in which two or more premises provide inde- is, 2 and 3 support 1 conjointly. This relationship pendent support for a single conclusion. If one of the between the premises is illustrated by the use of the premises were omitted, the other(s) would continue brace�in the following diagram:

to support the conclusion in the same way. Example: Conjoint 2 3

1 premises The selling of human organs, such as

hearts, kidneys, and corneas, should be out-

lawed. 2 If this practice is allowed to get a 1 foothold, people in desperate financial straits

will start selling their own organs to pay their The next example illustrates the occurrence of a

bills. Alternately, 3 those with a criminal multiple conclusion: bent will take to killing healthy young people 1 White police officers killing Black men is a and selling their organs on the black market. major cause of civil unrest. Therefore,

of human organs comes just too close to the preserve civil order, we must revisit the way 3 police officers are trained. Also. we must buying and selling of life itself. In the final analysis, the buying and selling 2 to

loosen restrictions on prosecuting errant

The diagram for this argument is as follows: police officers.

Horizontal 2 In this passage, statement 1 supports both 2 and 3 4 pattern 3 . Since no single argument can have� more than

one conclusion, the passage is correctly evaluated

1 as consisting of two arguments. For our purposes,

This diagram says that statements , 3 , and 4 sup- ment by joining the two conclusions with a bracket: 2 however, we will treat it as if it were a single argu-port 1 , independently.

Two variations on the horizontal and vertical pat- Multiple 1

terns occur when two or more premises support a conclusion conclusion conjointly, and when one or more prem-ises support multiple conclusions. The first variation 2 3 occurs when the premises depend on one another

in such a way that if one were omitted, the support Our symbolism is now sufficiently developed that the others provide would be diminished or to analyze most arguments found in editorials and

Section 1.6 Extended Arguments 59

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). letters to the editor of newspapers and magazines. supports 1 and 3 supports 4 . Finally, we see that Consider the following argument, taken from a 1 , 4 , and 6 support 5 . Yet these supporting state-newspaper editorial: ments depend on one another for their effect. Thus,

economy $20 billion annually. 1 they support the final conclusion conjointly. The Antibiotic resistant bacteria cost the argument pattern is as follows: 2 We need

new antibiotics to combat these bacteria, but 2 3

3 developing new antibiotics is not profit-

able. 4 Antibiotics tend to be given for only 1 4 6 a short time, and 5 antibiotics are used spar-

ingly under controlled circumstances. 5

6 Thus, to encourage the development of

new antibiotics, a government prize of $2 The next argument is taken from a newspaper editorial:

billion should be given to any company that 1 Chickens raised on factory farms are sub-develops an effective one. ject to terrible cruelty, but 2 there is a way

(Ezekiel J. Emanuel, �How to of avoiding it. 3 Egg-laying hens are kept

Develop New Antibiotics�) in cages so tiny that they cannot even turn

The main conclusion is 6 around, and 4 chickens raised for meat live , which is supported inde-lives filled with hectic misery. 5 They must pendently by 1 and conjointly by 2 and 3 . Then, have their beaks cut off to keep them from 3 is supported independently by 4 and 5 . The argu-pecking one another to death. 6 The United ment pattern is as follows: States could follow the European Union in

4 outlawing these methods for chicken farming. 5 Also, 7 chicken substitutes are available. 8

1 New techniques can produce something that 2 3 resembles chicken meat from plants, and 9

breast meat can be cultivated directly from

6 chicken cells. 10 Today�s treatment of animals

The next argument is taken from a letter to the on factory farms is the moral failing of our age. editor: (Ezra Klein, �We Will Look

Back on This Age of Cruelty to

1 Rhinos in Kenya are threatened with Animals in Horror�)

extinction because 2 poachers are killing

them for their horn. Since 3 the rhino has After reading the argument, we see that 1 and 2 no natural predators, 4 it does not need comprise a multiple conclusion. Statements 3 and its horn to survive. Thus 5 there should be 4 support the first part of the conclusion indepen-an organized program to capture rhinos in dently, and 5 supports 4 . Statements 6 and 7 sup-the wild and remove their horn. 6 Such a port the second part of the conclusion independently, program would eliminate the incentive of the and 8 and 9 support 7 independently. Statement poachers. 10 makes a passing comment on the subject matter

(Pamela C. Wagner, of the argument, so it is not included in the diagram:

�Rhino Poaching�) 8 9 5

First we search for the final conclusion. We select

5 3 , because it is the ultimate point that the passage 4 6 7

attempts to establish. Next, we survey the premise

and conclusion indicators. From this, we see that 2 1 2

60Chapter1 Basic Concepts

The next argument is taken from the science column instructors, but 2 the extensive use of these of a newspaper: instructors is a disadvantage to the students.

3 Most part-time instructors are paid only

in the length of our calendar year for an and as a result, 4 they are forced to teach in definite time to come. 2 This is true for five or six courses just to survive. 1 We can expect small changes to occur 60 percent of what a full-time teacher earns,

tides causes a continual slowing down of the earth�s mass distribution is continually the lower pay demoralizes the part-timer, subject to change. For example, 7 earth-and 8 the lack of tenure makes for constant quakes alter the location of the tectonic financial insecurity. 9 Obviously these condi-plates. Also, 8 the liquid core of the earth tions render the instructor less receptive sloshes as the earth turns, and 9 rainfall to student needs. Lastly, because 10 these redistributes water from the oceans. The part-timers are burning the candle from second reason is that 10 the motion of the both ends, 11 they have no spare energy to improve their courses, and 12 many lack the Earth�s rotation. 11 Tidal motion produces enthusiasm to motivate their students. why is this so? earth exhibits certain irregularities. 4 And students outside the classroom. To make 5 The rotation of any body is matters worse, 6 many part-timers are not affected by its distribution of mass, and 6 even given office space. Furthermore, two reasons. 3 First , the rotation of the detracts from the opportunity to consult with 5 This

heat, and 12 the loss of this heat removes As a result, 13 the educational process is energy from the system. impaired.

(Isaac Asimov, (Gordon Dossett et al., �Part-

�As the World Turns�) Time College Instructors�)

not add any support, we can leave it out of the diagram. Preliminary analysis reveals that the final conclusion Preliminary analysis reveals that the main conclu-is 1 . Also, 2 tells us that the supporting statements sion is not 1 but 2 . Also, we see three main reasons are divided into two basic groups, but since 2 does why part-timers are a disadvantage to students: They have little opportunity to consult with students, they In the first group, 5 and 6 support 3 conjointly, are less receptive to student needs, and the educa-while 7 , 8 , and 9 support 6 independently. 4 will tional process is impaired by 11 and 13 . In the first not appear in the diagram, because it serves merely main branch, the indicator �as a result� shows that as a premise indicator. In the second group, 11 and 3 supports 4 , and 4 and 6 independently sup-12 support 10 conjointly. Thus, the argument pattern port 5 . In the second branch, 7 and 8 indepen-is as follows: dently support 9 . In the third, 10 supports both 11

7 8 and 12 , which in turn support 13 9 independently.

Here is the argument pattern:

5 6 11 12

3 10

4 6 7 8 11 12

5 9 13

Our last example is taken from a letter to the editor

of a newspaper: 2

1 Community college districts save a great

deal of money by hiring untenured part-time

Section 1.6 Extended Arguments 61

Exercise 1.6

I. The following arguments were abstracted from newspaper articles, editorials, and let-

ters to the editor. Use the method presented in this section to construct argument pat-terns. If a statement is redundant or plays no role in the argument, do not include it in the pattern.

★1. 1 The conditions under which many food animals are raised are unhealthy

for humans. 2 To keep these animals alive, large quantities of drugs must be administered. 3 These drugs remain in the animals� flesh and are passed on to the humans who eat it.

(Philip D. Oliver, �We Can Eat Ribs and Still Be Humane�)

2. 1 The development of carbon-embedded plastics, otherwise called �com-

posits,� is an important new technology because 2 it holds the key for new aircraft and spacecraft designs. This is so because 3 these composits are not only stronger than steel but lighter than aluminum.

( Thomas H. Maugh II, �Composits�The Lightweight Champs

of Aircraft Industry�)

3. 1 Homework stifles the thrill of learning in the mind of the student.

2 3 It instills an oppressive learn-or-else discipline. It quenches the desire for

knowledge and the love of truth. For these reasons 4 homework should never be assigned.

(Colman McCarthy, �Homework�s Tyranny Hobbles

Promising Minds�)

★4. 1 When parents become old and destitute, the obligation of caring for them

should be imposed on their children. 2 Clearly, children owe a debt to their parents. 3 Their parents brought them into the world and cared for them when they were unable to care for themselves. 4 This debt could be appropri-ately discharged by having grown children care for their parents.

(Gary Jones, �The Responsibility of Parents�)

5. 1 Defending the war on drugs may not be fashionable, but the fact remains

that 2 hard-core drugs should remain illegal. 3 As long as hard-core drugs are illegal, they are harder to get, and 4 the social stigma of being arrested deters many users.

(Charles Van DeVenter, �I�m Proof: The War on

Drugs Is Working�)

6. 1 The rain forest of Brazil produces oxygen for the whole world, yet 2 it

yields no monetary return to that country. Given that 3 the industrialized nations consume the most oxygen, 4 those nations ought to pay Brazil an annual fee for the use of its rain forest.

(Diane B. Robinson, letter to the editor)

★7. 1 It appears that animals may be able to predict earthquakes. 2 Prior to a

major quake in China, hundreds of snakes suddenly appeared from hiberna-tion and froze to death in the snow, 3 fish were seen leaping from rivers and

62Chapter1 Basic Concepts

lakes, and 4 cows and horses refused to enter barns. Also, 5 prior to a quake in Fremont, California, a flood of callers reported strange behavior from their pets and domestic animals.

(Michael Bowker, �Can Animals Really Predict Earthquakes?�)

8. 1 Contributions to relief organizations are often wasted. 2 Food sent to

war-torn countries rarely reaches its destination, because 3 food distribution is controlled by the warring groups, and 4 these groups sell the food to buy weapons and ammunition.

(Michael Maren, �The Faces of Famine�)

9. 1 Research leading to the development of a scramjet engine is worthwhile.

2 Commercial aircraft incorporating such an engine could cross the

Pacific in as little as two hours. 3 This would relieve the fatigue of flights from New York to Tokyo. Also, 4 such an engine could power future orbiting spacecraft.

( T. A. Heppenheimer, �A Plane for Space�)

★10. 1 There is a lot of pressure on untenured college teachers to dumb down their

courses. 2 Administrators tend to rehire teachers who bring in more money, and 3 teachers who dumb down their classes do precisely this. Why? Because

4 5 easier classes attract more students, and more students means more

money for the school.

(Lynne Drury Lerych, �Meeting the Bottom Line in the

College Biz�)

11. 1 The use of ethanol as an additive in gasoline is really a bad idea. 2 Ethanol

is produced from corn, and 3 when 40 percent of the annual corn crop is burned up in car engines, that obviously raises the price of corn for food. 4 5 Also, adding ethanol to gasoline is highly inefficient. Ethanol yields about 30 percent less energy than gasoline, and 6 it�s more expensive to transport and handle than gasoline.

(Colin A. Carter and Henry I. Miller, �Corn for Food,

Not Fuel�)

12. 1 Unscrupulous for-profit colleges are not only cheating their students but

2 they are also ripping off the American taxpayer to the tune of billions of

dollars. 3 Ninety-six percent of the students who attend these colleges are forced to take out loans. 4 Most of these students leave without degrees.

5 And the students then default on their loans, leaving a $32 billion annual

debt to the taxpayer.

(Newspaper editorial, �False Promises�)

★13. 1 Numerous cancer patients on chemotherapy suffer from constant nausea,

loss of appetite, and inability to sleep. 2 For many of these patients, inhaled marijuana is the only medicine that provides relief. 3 Also, it causes no harm-ful side effects. 4 For these reasons, medical marijuana should be legalized in every state and made available to these patients. 5 This is not a law and order issue; it is a medical and human-rights issue.

(Gustin L. Reichbach, �A Judge�s Plea for Pot�)

Section 1.6 Extended Arguments 63

14. 1 The United States is currently experiencing the greatest economic

inequality since the 1920s, and 2 this inequality is depressing the economy.

3 The true sustainers of the economy are the bottom 99 percent of house-

holds, but 4 the top 1 percent takes in 93 percent of the growth in income. Also, 5 economic inequality is undercutting tax receipts, which are vital for infrastructure, education, and research. This is because 6 income earned by the rich is taxed at a far lower rate than income earned by the middle class.

( Joseph E. Stiglitz, �Inequality Is Holding Back the Recovery�)

15. 1 Programs to develop clean energy have seen some early notable successes.

2 The price of solar and wind power have both fallen sharply in recent

years. 3 The nation�s largest wind farm, sprawling across eastern Oregon, is now open and running, and 4 there is even good news from the natural gas front. 5 The use of natural gas, which is far cleaner than coal, to gener-ate electricity has jumped substantially in the last few years, and 6 thanks to government-sponsored research, energy companies are now able to extract much more gas from a tract of land than once seemed possible.

(David Leonhardt, �There�s Still Hope for the Planet�)

II. The following arguments gradually increase in difficulty. Use the method presented in

this section to construct argument patterns. If a statement is redundant or plays no role in the argument, do not include it in the pattern.

★1. 1 Many people believe that the crime of bribery cannot extend to campaign

contributions. 2 From a legal standpoint, however, countless campaign con-tributions are in fact bribes. 3 A bribe is anything of value or advantage given with the intent to unlawfully influence the person to whom it is given in his official capacity. 4 A campaign contribution is certainly something of value or advantage. Furthermore, 5 every contribution from a lobbyist or special inter-est group is given with the intent to influence voting, and 6 thousands of such contributions are made in every important election.

(Daniel Hays Lowenstein, �Can Candidates Run for Political

Office Without Taking Bribes?�)

2. 1 America�s farm policy desperately needs revamping. 2 Seventy-three cents

of every farm-program dollar ends up in the pockets of the nation�s super-farmers. As a result, 3 the mid-sized family farms are being squeezed out of existence. Also, 4 our farm policy courts environmental disaster. 5 Fed-eral subsidies encourage farmers to use enormous amounts of fertilizer and pesticides. 6 These chemicals percolate down through the soil and pollute limited groundwater.

(Osha Gray Davidson, �Rise of America�s Rural Ghetto�)

3. 1 Social media disseminates news that fuels polarization. 2 It favors sensa-

tional news over constructive news. 3 It is working to destroy responsible journalism. 4 Clearly, social media is harming our democratic society. 5 Fur-ther, the public sphere is shaped by social media, but 6 the public has no say in

64Chapter1 Basic Concepts

how data gets collected, who sees it, or how advertising is regulated. 7 Social media should be regulated now.

(Anne Applebaum, �Regulate Social Media Now. The Future

of our Democracy is at Stake�)

★4. 1 Powerful new particle accelerators are important in high-energy physics,

and 2 they are worth their cost because 3 they will allow scientists to pro-duce and capture significant quantities of Z particles. 4 Z particles result from the collision of positrons and electrons, and 5 particle accelerators are needed to achieve significant numbers of these collisions. 6 Z particles are thought to be the bearers of the weak nuclear force, and 7 learning the nature of this force may lead to the development of entirely new sources of energy.

(Lee Dye, �Linear Collider: Bold Gamble in Atomic Physics�)

5. 1 Although Thomas Jefferson�s contributions to the foundation of the coun-

try were profound, 2 as the master of Monticello, Jefferson was a monster.

3 Simultaneously with his writing the Declaration of Independence, which

announced the self-evident truth that all men are created equal, he owned 175 slaves. 4 Clearly the man was a creepy hypocrite. 5 But Jefferson was also cruel. 6 He sometimes punished slaves by selling them away from their family and friends, and 7 for slaves and free blacks who ran afoul of the law, he advocated harsh, almost barbaric forms of punishment.

(Paul Finkelman, �The Monster of Monticello�)

6. 1 All men crave material success because 2 it serves as an insurance policy

against sexual rejection. This is true because 3 women love men who are suc-cessful. 4 Both men and women want power, and 5 success is the form of power women feel most deprived of. Thus, 6 women try to achieve it vicari-ously through men. 7 As the 5-foot 6-inch Dustin Hoffman once put it, �When I was in high school, women wouldn�t touch me with a 10-foot pole. Now I can�t keep them away with a 10-foot pole.�

( Warren Farrell, �Success Story: From Frog to Prince�)

★7. 1 Using e-cigarettes is a dangerous activity, and 2 if you have any sense you

will avoid it. 3 Vaping causes popcorn lung, and 4 this disease is deadly. Also,

5 6 e-cigarettes contain significant amounts of nicotine, so a single hit can lead to

addiction. Furthermore, 7 to deceive you the manufacturers of e-cigarettes have made their product seductively attractive. 8 The product comes in colorful pack-aging and 9 it is laced with appealing flavors.

(The Daily Targum, �For Your Sake, Say No to Vape�)

8. 1 Flesh food is not a necessity in the human diet, as 2 nutritionally adequate

alternatives are readily available. 3 Many people in the world thrive on a non-meat diet. 4 For example, vegetarian Seventh-Day Adventists in this country live an average of six years longer than their meat-eating counterparts. 5 The National Academy of Science warns that our fat-laden diet is directly responsi-ble for much of today᾽s heart disease and cancer. Furthermore, 6 eating meat is a terribly inefficient way of feeding people. 7 A steer must consume sixteen pounds of grain and soy to produce one pound of meat. 8 The grain and soy-beans we feed our meat-producing animals would feed every hungry mouth in

Section 1.6 Extended Arguments 65

the world many times over. 9 Clearly, a reassessment of the whole concept of killing and eating animals is in order.

(Suzanne Sutton, �Killing Animals for Food�Time for a

Second Look�)

9. 1 The argument has been made that to cut down on teenage drunk driving

we should increase the federal excise tax on beer. 2 Such a measure, however, would almost certainly fail to achieve its intended result. 3 Teenagers are noto-riously insensitive to cost. 4 They gladly accept premium prices for the latest style in clothes or the most popular albums. And then, 5 those who drink and drive already risk arrest and loss of driving privileges. 6 They would not think twice about paying a little more for a six-pack. Finally, 7 the situation is not as bleak as it has been made to appear. 8 The fatality rate for teenage drivers is lower today than it has been in years.

( James C. Sanders, �Increased U.S. Tax on Beer�)

★10. 1 It has been widely acknowledged that the quality of undergraduate educa-

tion in this country is diminishing. 2 An often unrecognized cause of this malady is the exploitative way that universities as employers treat their part-time and temporary faculty members. 3 In many universities there are no formal guidelines for evaluating the work of these instructors. As a result, 4 poor instructors who solicit the favor of the department chairman are often retained over better ones who do not. 5 Another factor is the low pay given to these instructors. 6 In order to survive, many of them must accept heavy teaching loads spread out over three or four institutions. 7 The quality of instruction can only suffer when faculty members stretch themselves so thin. Lastly, because 8 part-time and temporary faculty are rarely members of the faculty senate, 9 they have no voice in university governance. But 10 without a voice, the shoddy conditions under which they work are never brought to�light.

(Michael Schwalbe, �Part-Time Faculty Members Deserve

a Break�)

11. 1 Doctors who attend elderly people in nursing homes often prescribe tran-

quilizers to keep these people immobile. 2 This practice is often unwarranted, and 3 it often impairs the health of the patients. 4 These tranquilizers often have damaging side effects in that 5 they accentuate the symptoms of senility, and 6 they increase the likelihood of a dangerous fall because 7 they produce unsteadiness in walking. Furthermore, since 8 these medications produce immobility, 9 they increase the risk of bedsores. 10 Doctors at the Center for Aging and Health say that physicians who care for the elderly are simply pre-scribing too much medication.

(Hal Willard, �At 90, the Zombie Shuffle�)

12. 1 All of us have encountered motorists who will go to any length to get a park-

ing spot within 20 feet of the door they expect to enter. 2 This obsession with good parking spots transcends all logic. 3 It might take 5 minutes to secure the ideal spot in a store parking lot, 4 while a more distant spot that is immediately available is only a 40-second walk from the door. 5 Waiting for that ideal spot also results in frenzied nerves and skyrocketing blood pressure. 6 Inevitably

66Chapter1 Basic Concepts

the occupant of the desired space will preen her hair before departing, and 7 all the while the cars backed up behind the waiting driver are blaring their horns.

8 9 Parking a little farther away is usually easier and safer because you can pull

out more quickly, and 10 it avoids damage to car doors by adjacent parkers.

(Gwinn Owens, �A Ridiculous Addiction�)

★13. 1 The state has a right to intervene on behalf of unborn children, and 2 this

right should be implemented immediately. 3 While it may be true that a mere fetus has no rights, 4 surely a born child does have rights, and 5 these rights project backward to the time it was in the womb. This is true because 6 what happens to the child in the womb can have an impact throughout the child�s life. 7 It is well known that alcohol and drug abuse by expectant mothers cause birth defects, and 8 these defects are not correctable after birth. 9 Granted, an expectant mother has the right to treat her own body as she chooses, but 10 11 this right does not extend to her unborn child. Once a pregnant woman decides to give birth, she effectively transfers part of her rights over to her unborn child. 12 Unfortunately, however, the unborn child is incapable of secur-ing these rights for itself. Thus, 13 the intervention of a higher power is justified.

(Alan Dershowitz, �Drawing the Line on Prenatal Rights�)

14. 1 A manned trip to Mars is a justified scientific goal because 2 it affords a

unique opportunity to explore the origins of the solar system and the emer-gence of life. However, 3 from a scientific standpoint, an initial landing on the tiny Martian moons, Phobos and Deimos, would be more rewarding than a landing on the planet itself. Because 4 the Martian terrain is rugged,

5 6 humans would not be able to venture far, nor could they operate a robot

vehicle for great distances, since 7 Mars�s mountains would block their view.

8 Explorers on Phobos and Deimos could easily send robot vehicles to the

planet�s surface. 9 Using Mars�s moons as a base would also be better than unmanned exploration directed from the Houston space center. Because 10 the distance is so great, 11 radio signals to and from Mars can take as long as an hour. Thus, 12 driving an unmanned rover from Earth, step by step, would be a time-consuming operation. 13 Sample returns to Earth would take months instead of hours, and 14 follow-on missions would be years apart instead of days, further slowing the process of exploration.

(S. Fred Singer, �The Case for Going to Mars�)

15. 1 A small group of people still believe in nuclear weapons, but 2 the facts

dictate that nuclear weapons not only should be eliminated but 3 they can be eliminated. 4 The opposing arguments are that nuclear weapons hastened the end of World War II, they deter nations from going to war, they have kept the peace, and nuclear technology is irreversible. But 5 nuclear weapons did not persuade the Japanese to surrender. 6 New research shows that the real reason Japan surrendered is because the Soviets renounced neutrality and entered the war. And 7 before losing two cities to nuclear weapons, Japan had already lost sixty-six to conventional weapons. 8 Two more made little differ-ence. Also, 9 history shows that nuclear weapons are ineffective as a deterrent. 10 They did not deter the United States during the Cuban missile crisis, and 11 they did not deter Argentina in the conflict with Britain over the Falklands.

Section 1.6 Extended Arguments 67

Furthermore, 12 there is no good reason to think that nuclear weapons cause peace. 13 14 Peace means the absence of war, but the absence of something can be explained in countless ways. Lastly, 15 the argument that you can�t stuff the nuclear genie back in the bottle is specious. 16 The question is not whether nuclear weapons can be disinvented but whether they are useful. 17 This is doubtful given that 18 not one has been used since 1945 and 19 the trend in modern warfare is toward smaller, more effective precision-guided weapons. 20 21 Banning nuclear weapons will not be easy, but the existing ban on chemi-cal and biological weapons shows that it can be done.

( Ward Wilson, �The Myth of Nuclear Necessity�)

III. Turn to the editorial pages of a newspaper or look at an online publication and select

an editorial that contains an argument. Keep in mind that some editorials are really reports and contain no arguments at all. Also, few editorials are as neat and straight-forward as the selections presented in Parts I and II of this exercise. Guest editorials on the opinion�editorial page (usually opposite the editorial page) are often better written than those on the editorial page. Analyze the argument (or arguments) according to the method presented in this section. Begin by placing a numeral at the beginning of each statement. Compound statements having components that are claimed to be true may be broken up into parts and the parts enumerated accordingly. Numerals should usu-ally be placed after genuine premise and conclusion indicators even when they occur in the middle of a statement. Do not, however, break up conditional statements into ante-cedent and consequent. Proceed to identify the main conclusion (or conclusions) and determine how the other statements provide support. Any statement that does not play a direct role in the argument should be left out of the final argument pattern.

IV. Create an extended argument, containing at least six statements, about climate change.

Summary

Logic: The science that evaluates arguments

Argument: A group of statements comprising one or more premises and one conclusion

Premises: The statements that set forth the reasons or evidence

Conclusion: The statement that is claimed to follow from the premises

To distinguish premises from conclusion, look for:

■ Indicator words (�hence,� �therefore,� �since,� �because,� etc.)

■ An inferential relationship among the statements

Not all groups of statements are arguments. To distinguish arguments from nonarguments, look for:

■ Indicator words (�hence,� �since,� etc.)

■ An inferential relationship among the statements

■ Typical kinds of nonarguments (warnings, reports, expository passages, etc.)

68Chapter1 Basic Concepts

The most problematic kinds of nonarguments:

■ Expository passages (Is the topic sentence proved by the other statements?)

■ Illustrations (Could the passage be an argument from example?) ■ Explanations (Could the explanandum also be a conclusion?)

Conditional statements express the relation between sufficient conditions and necessary conditions:

■ A is a sufficient condition for B: The occurrence of A is all that is needed for the occurrence of B.

■ A is a necessary condition for B: B cannot occur without the occurrence of A.

Arguments are traditionally divided into deductive and inductive:

■ Deductive argument: The conclusion is claimed to follow necessarily from the premises.

■ Inductive argument: The conclusion is claimed to follow probably from the premises.

To distinguish deductive arguments from inductive arguments, look for:

■ Special indicator phrases (�it necessarily follows that,� �it probably follows that,� etc.)

■ The actual strength of the inferential relationship between premises and conclusion ■ Typical forms or styles of argumentation:

● Deductive forms: Arguments based on mathematics; arguments from definition; and cat-

egorical, hypothetical, and disjunctive syllogisms

● Inductive forms: Predictions, arguments from analogy, generalizations, arguments from

authority, arguments based on signs, and causal inferences

Evaluating an argument (either deductive or inductive) involves two steps:

■ Evaluating the link between premises and conclusion

■ Evaluating the truth of the premises

Deductive arguments are valid, invalid, sound, or unsound.

■ Valid: The conclusion actually follows from the premises.

■ Sound: The argument is valid and has all true premises.

Inductive arguments are strong, weak, cogent, or uncogent.

■ Strong: The conclusion actually follows from the premises.

■ Cogent: The argument is strong and has all true premises.

The validity of a deductive argument is determined by the argument�s form. An invalid form allows for a substitution instance having true premises and a false conclusion.

■ Counterexample method:

● Is used to prove invalidity.

● Consists in identifying the form of a given invalid argument and producing a substitution

instance having true premises and a false conclusion.

● This proves the form invalid, which proves the given argument invalid.

The structure of longer arguments can be disclosed by a diagramming method. Four basic argument patterns are:

■ Vertical pattern

■ Horizontal pattern

■ Conjoint premises

■ Multiple conclusion

Summary69

2Language: Meaning and Definition

2.1 Varieties of Meaning

2.2 The Intension and Extension of Terms

2.3 Definitions and Their Purposes

2.4 Definitional Techniques

2.5 Criteria for Lexical Definitions

2.1 Varieties of Meaning

Learning about definitions is empowering. Definitions pin down the meaning of words, and this can be crucial to resolving disputes. For example, the current dispute about abortion can be seen as a disagreement about the mean-ing of the words �human life.� If this phrase designates something that begins at conception and has a right to life, then ending its life is problematic. But if it des-ignates a mere agglomeration of cells with human DNA, then removing it poses no problem. Settling on a definition would help to resolve the dispute.

Ordinary language, as most of us are at least vaguely aware, serves various functions in our day-to-day lives. The twentieth-century philosopher Ludwig Wittgenstein thought the number of these functions to be virtually unlimited. Thus, among many other things, language is used to

ask questions tell jokes tell stories flirt with someone tell lies give directions guess at answers sing songs form hypotheses issue commands launch verbal assaults greet someone

For our purpose, two linguistic functions are par- A value claim is a claim that something is good,

ticularly important: (1) to convey information and bad, right, wrong, better, worse, more important, (2) to express or evoke feelings. Consider, for exam- or less important than some other thing. For exam-ple, the following statements: ple, the statement about the death penalty asserts

The death penalty, which is legal in twenty- the value claim that the death penalty is wrong or seven states, has been carried out most immoral. Such value claims are often the most often in Virginia; however, since 1977, Texas important part of the cognitive meaning of emotive holds the record for the greatest number statements. Thus, for the purposes of logic, we must of executions. be able to disengage the value claims of emotively

charged statements from the emotive meaning and

convey information about the death penalty, whereas The first statement is intended primarily to The death penalty is a cruel and inhuman treat these claims as separate statements. form of punishment in which hapless These observations suggest the reason that people prisoners are dragged from their cells and use emotive terminology as often as they do: Value summarily slaughtered only to satiate the claims as such normally require evidence to support bloodlust of a vengeful public. them. For example, the claim that the death penalty is immoral cannot simply stand by itself. It cries out for reasons to support it. But when value claims are couched in emotive terminology, the emotive �cloth-the second is intended to persuade us that the death ing� tends to obscure the fact that a value claim is penalty is bad. The second accomplishes this func-being made, and it simultaneously gives psycho-tion by engaging our feelings and not, as in an argu-logical momentum to that claim. As a result, read-ment, by establishing the truth of a claim. ers and listeners are inclined to swallow the value These statements accomplish their respective claim whole without any evidence. Furthermore, the functions through the distinct kinds of terminology mental laziness of many speakers and writers, com-in which they are phrased. Terminology that conveys bined with their inability to supply supporting rea-information is said to have cognitive meaning , sons for their value claims, reinforces the desirability and terminology that expresses or evokes feelings is of couching such claims in emotive terminology. said to have emotive meaning . Thus, in the first Many people, for example, will refer to someone statement the words �legal,� �twenty-seven,� �most as �crazy,� �stupid,� or �weird� when they want to often,� �Virginia,� �record,� and so on have primarily express the claim that what that person is doing is a cognitive meaning, while in the second statement bad or wrong and when they are unable or unwill-the words �cruel,� �inhuman,� �hapless,� �dragged,� ing to give reasons for this claim. Also, people �slaughtered,� �bloodlust,� and �vengeful� have a will use such words as �awesome� or �fantastic� to strong emotive meaning. Of course, these latter words express the claim that something is good. Those who have cognitive meaning as well. �Cruel� means tend-happen to be listening, especially if they are friendly ing to hurt others, �inhuman� means inappropriate with the speaker, will often accept these claims with-for humans, �hapless� means unfortunate, and so on. out hesitation. The emotively charged statement about the death The use of emotive terminology in advertising penalty illustrates two important points. The first is often results in huge profits for companies that spon-that statements of this sort usually have both cogni-sor the ads. The pharmaceutical industry is a prime tive meaning and emotive meaning. Therefore, since example. Nowadays, instead of pitching their ads to logic is concerned chiefly with cognitive meaning, we the doctors who prescribe the drugs, pharmaceutical must be able to distinguish and disengage the cogni-companies go directly to the patients, and they select tive meaning of such statements from the emotive names for their drugs that are easy to recall and convey meaning. The second point is that part of the cog-a positive, though often subtle, emotional message. nitive meaning of such statements is a value claim.

Section 2.1 Varieties of Meaning 71

For example, lunesta, a drug for insomnia, suggests a Now that we know that the rocks on the moon peaceful sleep bathed in moonlight (recall that luna are similar to those in our backyard and that means �moon�). Boniva, a drug for osteoporosis, sug- tadpoles can exist in a weightless environment, gests giving new life to one�s bones (�bone� + vita). and now that we have put the rest of the world Viagra suggests vigor and growth. Celebrex, a drug in order, can we concentrate on the problems for arthritis, suggests celebrating. Abilify, for depres- here at home? Like what makes people hungry sion, �abilifies� you to be happy and productive. and why is unemployment so elusive? Enablex, for overactive bladder, enables you to regain (Robert J. Boland)

control. Vuity, for blurry vision, improves your view. The conclusion of this argument is that our gov-Flonase, for allergies, gets your nasal passages flow-ernment should take money that has been spent ing. Aleve, for pain, alleviates it. With Tremfya, for on the space program and on international plaque psoriasis, you emerge tremfyant (triumphant). police actions and redirect it to solving domes-And Plenity, for weight loss, makes you feel that you tic problems. The author minimizes the impor-have plenty. These are just a few. tance of the space program by covertly sug-For another example of emotive terminology, con-gesting that it amounts to nothing more than sider the word �harvest.� This word evokes feelings work on ordinary rocks and tadpoles (which by associated with honest, hardworking farmers being themselves are relatively insignificant), and he rewarded for their labor in planting and tending their exaggerates the scope of the international effort by crops. To capitalize on this positive feeling, wood covertly suggesting that it has solved every prob-products companies speak of harvesting the trees in lem on earth but our own. Also, the phrase �put . . . 200-year-old forests, even though they had nothing to in order� suggests that the international effort has do with planting them, and surgeons speak of harvest-been no more important than restoring order to a ing the organs from the bodies of donors and the tissue room in one�s house. We might rephrase the argu-from aborted fetuses. In all of these cases, the use of the ment in emotively neutral language, making the word �harvest� is specifically calculated to elicit a favor-implicit suggestions and value claims explicit, as able or agreeable response from the listener. follows: Let us now consider emotive terminology as it occurs in arguments. In arguments, emotive termi- The space program has been confined to nology accomplishes basically the same function as it work on ordinary rocks and tadpoles. does in statements. It allows the arguer to make value Ordinary rocks and tadpoles are less important claims about the subject matter of the argument with- than domestic hunger and unemployment.

out providing evidence, and it gives the argument a Our international efforts have restored order

potential counterarguments before the reader or lis- These efforts have been directed to prob-tener has a chance to think of them. This steamroller kind of steamroller quality by which it tends to crush to every nation on earth but our own. quality also tends to paralyze the logical thought pro- lems that are less important than our own cesses of readers or listeners so that they are not able to domestic problems. see illogical arguments in their true light. These effects Therefore, our government should redirect of emotive terminology can be avoided if the reader funds that have been spent on these projects or listener will disengage the value claims and other to solving our own domestic problems. cognitive meanings from the emotive meaning of the By restructuring the argument in this way, we language and reexpress them as distinct premises. can more easily evaluate the degree to which the

Consider, for example, the following emotively premises support the conclusion. Inspection of

charged argument taken from the letters to the the premises reveals that the first, third, and pos-editor section of a newspaper: sibly fourth premises are false. Thus, the actual

72Chapter2 Language: Meaning and Definition

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). support provided by the premises is less than what one that can be interpreted as having more than one we might have first expected. If the argument were clearly distinct meaning in a given context. For exam-to be rephrased a second time so that the prem- ple, words such as �light,� �proper,� �critical,� �stress,� ises turned out true (for example, the first premise �mad,� �inflate,� �chest,� �bank,� �sound,� and �race� can might read �Part of the space program has been be used ambiguously. Thus, if one were to describe a devoted to research on ordinary rocks and tad- beer as a light pilsner, does this mean that the beer is poles�), the support given to the conclusion would light in color, light in calories, or light in taste? If one still be weaker than the author intended. were to describe an action as proper, does this mean

Now that we have distinguished emotive mean- proper in a moral sense or proper for achieving an

ing from cognitive meaning, let us explore some of intended result? Or if one were to describe a person as the ways that language having cognitive meaning critical, does this mean that the person is essential for a can be defective. Two of them are vagueness and certain task or that the person tends to criticize others? ambiguity. A vague expression is one that allows As is the case with vagueness, ambiguity can also for borderline cases in which it is impossible to tell affect entire statements. Such ambiguity often results if the expression applies or does not apply. Vague from the way in which certain words are combined. expressions often allow for a continuous range of For example, there was a newspaper headline that interpretations. The meaning is hazy, obscure, and read, �Tuna are biting off the Washington coast.� imprecise. For example, words such as �love,� �hap- Does this mean that the tuna are nibbling away at piness,� �peace,� �excessive,� �fresh,� �rich,� �poor,� the coastline or that fishermen are catching them off �normal,� �conservative,� and �polluted� are vague. the coast? Presumably it means the latter. Another We can rarely tell with any precision whether they headline read, �College students are turning to veg-apply to a given situation or not. How fresh does etables.� Does this mean that the students are meta-something have to be in order to be called �fresh�? morphosing into vegetables or that they are incor-

Vagueness can also affect entire statements. Such porating more vegetables into their diet? Again, the

vagueness may arise not so much from the individ- intended meaning is probably the latter. ual words as from the way in which the words are The difference between ambiguity and vague-combined. For example, suppose someone were to ness is that vague terminology allows for a rela-say, �Today our job situation is more transparent.� tively continuous range of interpretations, whereas First, what is the meaning of �job situation�? Does ambiguous terminology allows for multiple discrete it refer to finding a job, keeping a job, filling a job, interpretations. A vague expression creates a blur of completing a job, or bidding on a job? And what meaning, whereas an ambiguous expression mixes exactly does it mean for a job situation to be �trans- up otherwise clear meanings. However, many forms parent�? Does it mean that the job is more easily of expression are ambiguous in one context and perceived or comprehended? That the job is more vague in another. For example, the word �slow� in easily completed? That we can anticipate our future one context could mean either mentally challenged job needs more clearly? Or what else? or physically slow, but when the word refers to phys-

Not all cases of vagueness, however, are prob- ical slowness, it could be vague. How slow is slow?

lematic. To describe an acquaintance as �tall� or Similar remarks apply to �light,� �fast,� and �rich.�

�thin� often causes no trouble in ordinary conver- Ambiguity and vagueness are important in logic sation. Indeed, it may be burdensome to describe because there are countless occasions in which the this person in more precise language. Trouble arises evaluation of an argument leads to the observa-only when the language is not sufficiently precise for tion, �Well, that depends on what you mean by . . . .� what the situation demands. Certain phraseology in the argument is vague

The other way in which cognitive meanings can be or ambiguous, and its meaning must be clarified

defective is ambiguity. An ambiguous expression is before any evaluation can proceed. For example,

Section 2.1 Varieties of Meaning 73

Scientologists argue that their organization should in a court of law, and Smiley has not yet even be exempt from paying taxes because, they claim, been accused of anything.

Scientology is a religion. Evaluating their argument In this case the dispute arises over the ambiguity of requires that we clarify the meaning of �religion.� Pro-the word �guilty.� Brenda is using the word in the life advocates argue that abortion is wrong because moral sense. Given that Smiley has admitted to set-it results in the killing of human beings. But what is ting fire to the old schoolhouse, it is very likely that the meaning of �human being�? And some feminists he did indeed set fire to it and therefore is guilty of argue that leering glances constitute sexual harass-arson in the moral sense of the term. Warren, on ment. To evaluate their arguments we must clarify the the other hand, is using the word in the legal sense. meaning of �leering glances� and �sexual harassment.� Because Smiley has not been convicted in a court of The role of vagueness and ambiguity in arguments law, he is not legally guilty of anything. may be conveniently explored in the context of con-Disputes that arise over the meaning of language flicting arguments between individuals. Such con-are called verbal disputes. But not all disputes are of flicts are called disputes: this sort. Some disputes arise over a disagreement about facts, and these are called Amelia: Mrs. Wilson abuses her children. And factual disputes.

how do I know that? I saw her spank one of her Example: kids the other day after the kid misbehaved.

Javier: I know that Freddie stole a computer

Jane: Don�t be silly. Kids need discipline, and from the physics lab. Barbara told me that by disciplining her children, Mrs. Wilson is she saw Freddie do it. showing that she loves them.

Camille: That�s ridiculous! Freddie has never

Here, the problem surrounds the vagueness of the stolen anything in his life. Barbara hates words �abuse� and �discipline.� When does disci- Freddie, and she is trying to pin the theft on pline become abuse? The line separating the two is him only to shield her criminal�boyfriend.

sort will never be resolved. Here, the dispute centers on the factual issues of hazy at best, but unless it is clarified, disputes of this Another example: whether Barbara told the truth and whether Freddie stole the computer. Brenda: I�m afraid that Smiley is guilty of In dealing with disputes, the first question is arson. Last night he confided to me that he was

whether the dispute is factual, verbal, or some com-

the one who set fire to the old schoolhouse. bination of the two. If the dispute is verbal, then the Warren: No, you couldn�t be more mistaken. second question to be answered is whether the dis-In this country no one is guilty until proven so pute concerns ambiguity or vagueness.

Exercise 2.1

I. The following selection is taken from a speech delivered by George C. Wallace, former

governor of Alabama, on July 4, 1964. In this speech, Wallace attacked Lyndon John-son�s signing of the Civil Rights Act. The speech is liberally sprinkled with emotive ter-minology. Make a list of what you consider to be the twenty-five most highly charged words or phrases, and then indicate whether they are intended to evoke a favorable or an unfavorable attitude from the listener.

74Chapter2 Language: Meaning and Definition

We come here today in deference to the memory of those stalwart patriots who on July 4, 1776, pledged their lives, their fortunes, and their sacred honor to establish and defend the proposition that governments are created by the people, empowered by the people, derive their just powers from the consent of the people, and must forever remain subservient to the will of the people.

Today, 188 years later, we celebrate that occasion and find inspiration and determi-

nation and courage to preserve and protect the great principles of freedom enunci-ated in the Declaration of Independence.

It is therefore a cruel irony that the president of the United States has only yesterday

signed into law the most monstrous piece of legislation ever enacted by the United States Congress.

It is a fraud, a sham, and a hoax.

This bill will live in infamy. To sign it into law at any time is tragic. To do so upon the

eve of the celebration of our independence insults the intelligence of the American people.

It dishonors the memory of countless thousands of our dead who offered up their

very lives in defense of principles which this bill destroys.

Never before in the history of this nation have so many human and property rights

been destroyed by a single enactment of the Congress. It is an act of tyranny. It is the assassin�s knife stuck in the back of liberty.

With this assassin�s knife and a blackjack in the hand of the federal force-cult, the

left-wing liberals will try to force us back into bondage. Bondage to a tyranny more brutal than that imposed by the British monarchy which claimed power to rule over the lives of our forefathers under sanction of the omnipotent black-robed despots who sit on the bench of the United States Supreme Court.

This bill is fraudulent in intent, in design, and in execution.

It is misnamed. Each and every provision is mistitled. It was rammed through the

Congress on the wave of ballyhoo, promotions, and publicity stunts reminiscent of P. T. Barnum.

It was enacted in an atmosphere of pressure, intimidation, and even cowardice, as

demonstrated by the refusal of the United States Senate to adopt an amendment to submit the bill to a vote of the people.

To illustrate the fraud�it is not a civil rights bill. It is a federal penal code. It creates

federal crimes which would take volumes to list and years to tabulate because it affects the lives of 192 million American citizens. Every person in every walk and station of life and every aspect of our daily lives become subject to the criminal provisions of this bill.

It threatens our freedom of speech, of assembly, of association, and makes the exer-

cise of these freedoms a federal crime under certain conditions.

It affects our political rights, our right to trial by jury, our right to the full use and

enjoyment of our private property, the freedom from search and seizure of our pri-vate property and possessions, the freedom from harassment by federal police and, in short, all the rights of individuals inherent in a society of free men.

Ministers, lawyers, teachers, newspapers, and every private citizen must guard his

speech and watch his actions to avoid the deliberately imposed booby traps put into this bill. It is designed to make federal crimes of our customs, beliefs, and traditions. Therefore, under the fantastic powers of the federal judiciary to punish for contempt of court and under their fantastic powers to regulate our most intimate aspects of our lives by injunction, every American citizen is in jeopardy and must stand guard against these despots.

Section 2.1 Varieties of Meaning 75

II. The following selections were taken from the letters-to-the-editor section of a newspa-

per. Each can be interpreted as expressing one or more arguments. Begin by identifying the conclusion of each. Then disengage the covert assumptions, value claims, and other cognitive assertions from the emotive language and translate them into emotively neu-tral premises. Use the two examples in the text as models. Finally, evaluate the restruc-tured arguments. Some may turn out to be good ones.

★1. Why don�t animal lovers do something about these dog sled races? Have you

ever witnessed a race on television? Talk about torture. It�s sickening to watch the dogs, panting and their tongues hanging out, pull a heavily laden sled with a driver through snow and ice in bitter cold.

(Joe Shapiro)

2. How anyone who has seen even one photo of the fly-covered, starving children

in Somalia can still believe in a loving, everpresent, omnipotent God is beyond intelligent reasoning.

(William Blanchard)

3. The creationists have no right to impose their mistaken, ignorant, superstitious

beliefs on others. They claim the constitutional right to the free exercise of religion. How about the rights of the majority of people who want their children taught the scientific truth about evolution�not fallacious myths and superstitions from primitive societies.

(Andrew M. Underhill Jr.)

★4. God, guts, and guns made this great country of ours free, and you can bet your

buns it will take more of the same to keep it that way. One of the very last things in this world we need is handgun control.

(R. Kinzie)

5. The insanity plea should be done away with; criminals should lose this easy way

out. Killers can theoretically spend as little as six months in a mental hospital, then be released. It�s time to take a stand for safety and put psychotic killers in prison.

(Keith Aikens)

6. Until now, the protest against the holocaust in our own nation has been vocal

but far too small. The massacre of an unwanted generation through abortion and infanticide has sounded an alarm that should wake up every Christian. Helpless and guiltless little infants are mercilessly butchered daily in hospitals and clinics across our land. For the love of God, let us all urge the passage of the Human Life Bill, now before Congress.

(Jim Key)

★7. It�s time to challenge all this nonsense about the �celebration of diversity� in

our society. The more the schizophrenics preach the glories of diversity, the more we pull apart. This is not to deny appreciation of the ethnic roots, rituals, and foods, which add color to life. But to lay undue emphasis upon diversifica-tion results in destruction of the �social glue� that binds us together. Our fore-fathers framed one nation, indivisible. In the misguided effort to �celebrate�

76Chapter2 Language: Meaning and Definition

the uniqueness of every disparate culture and subculture, we betray our heri-tage and dilute our identities as Americans.

(Ruth M. Armstrong)

8. A kind and loving God surely favors the pro-choice attitude. He wants his

world inhabited by happy, well-fed children with parents who love and care for�them.

Our burgeoning population in Third World nations with constant famine

and disease, and many other human miseries, could be relieved if the Catholic Church were to adjust more of its ancient policies to our current civilization.

(Art Bates)

9. Thousands of years of organized religion have done nothing to solve any prob-

lems and have almost always exacerbated them by promoting fear, superstition, and irrational mythologies. Kneeling in prayer to some supernatural entity seeking �divine guidance� or, even more implausibly, �divine intervention,� is not only a waste of time, it is counterproductive because it lulls the supplicant into inactivity.

We must stand up, open our eyes, and face life�s challenges head-on in a

problem-solving approach that is reality-based, empirical, and above all, rational.

(James W. Baugh)

★10. Liberalism has turned our welfare system from a social safety net into a ham-

mock. We hand out money with few questions asked. When welfare recipients are asked for some contribution to our society in return, liberals scream that it�s unconstitutional.

Liberalism has transformed our criminal justice system into one that cares

more about the criminal�s past childhood problems than for the victim. Liberal-ism in its never-ending quest for �social justice� has sacrificed the rights of the majority while continuing to push the rights of a few to new limits.

Liberalism has turned our school system from one of excellence to one

where condoms and metal detectors are more important than prayer.

(Marc Sexton)

III. Visit a pharmacy and find the aisle that displays hair-care products (shampoos, condi-

tioners, etc.). Identify five items whose names have emotional meaning. Then, for each one, explain how its name is supposed to elicit a favorable response from the consumer.

IV. Determine whether the following disputes are verbal, factual, or some combination

of the two. If verbal, discuss whether the dispute arises from vagueness or ambiguity.

★ 1. Frank: Look at that huge tree that fell last night. It must have made a tremen-

dous crash when it came down.

Shirley: No, I�m afraid you�re quite wrong. Sound is a perception, and per-ceptions depend on a perceiver. Therefore, since nobody was around here last night, there was no crash.

2. Luna: Yesterday I visited the exhibition of the work of Jean Michel Basquiat at

the Central Gallery. What an interesting artist he is!

Carla: Don�t be ridiculous! That�s not art�it�s just graffiti.

Section 2.1 Varieties of Meaning 77

3. Phil: That was a great basketball game last night. Stephen Curry scored 37 points.

Arthur: Your statistics are all wet. Curry scored only 34 points.

★4. Roger: I think modern society is becoming more and more violent every day.

Just look at the increase in murder, rape, and robbery. Violence is clearly an evil that must be eradicated.

Mark: You might be right about the increase in crime, but the idea that violence is an evil is nonsense. Violence is quite natural. The universe was created in a tremendously violent big bang, the nuclear reactions that bring us sunlight are extremely violent, and insects and animals kill and devour one another all the time.

5. Kathy: I was saddened to hear about the death of your uncle. He was such a

wonderful man. You must be consoled knowing that he�s enjoying his heavenly reward.

Anne: Thanks, but I�m afraid I don�t know what you mean. If death is the end

of life, how could my uncle be alive right now in heaven?

6. Heidi: This morning I heard a lecture on the life of Jane Austen. She was such

a wonderfully educated woman.

David: That�s not true at all. Jane Austen dropped out of school when she was only eleven, and she never even attended high school, much less college or graduate school.

★7. Leslie: Your friend Paul told us that he would be visiting his parents in Knox-

ville this weekend. Therefore, he must not be at home.

Diana: I agree that Paul is probably not at home, but you didn�t hear him right. He said that his parents live in Nashville.

8. Karl: There�s a euthanasia measure on the ballot today, and I think I�ll vote

for it. It seems reasonable that terminally ill patients should be allowed to be disconnected from life-support systems so that they can die peacefully and naturally.

Sergio: You must be crazy! Euthanasia means giving people lethal injections, and that�s clearly murder.

9. Cheryl: Tomorrow I�m going to the Metallica concert. Their music is

fabulous.

Oliver: You call that music? Really it�s just noise, and incredibly loud noise at�that.

★10. Carol: Nelson could not have fought in the battle of Trafalgar, because that

battle occurred in 1806, and Nelson died in 1804.

Justin: Your knowledge of history is atrocious! Nelson did fight in Trafalgar, and the date was October 21, 1805.

11. Eric: I�ve just signed up for Philosophy 502�Dr. Peterson�s class in metaphys-

ics. I know I�m going to enjoy it because I�ve always been fascinated by magic and ghosts.

Leah: I�m afraid you�re in for a surprise.

78Chapter2 Language: Meaning and Definition

12. Harold: Professor Steinbeck is the most intelligent man I know. His lecture

series on matter and consciousness was simply brilliant.

Joyce: Steinbeck is actually an idiot. Yesterday, I watched while he tried to get his car started. When it wouldn�t start, he opened the hood, and he didn�t even notice that someone had stolen the battery.

★13. Thomas: George Marshall committed those crimes of child abuse through his

own free choice. Nobody put a gun to his head. Therefore, he should be pun-ished for them.

Emilie: That�s not true. It�s been established that George was severely abused himself when he was a child, and such children have an irresistible obsession to abuse others when they grow up.

14. Anthony: The sun is much smaller than the earth. You see, it�s just a small

thing up there in the sky. Therefore, since the sun�s gravitational attraction is proportional to its mass, the sun�s gravity is less than the earth�s.

Cindy: You are as stupid as they come. I agree the mass of the sun is less than that of the earth, but its volume is greater. Therefore, since gravitational attrac-tion is proportional to volume, the sun�s gravity is greater than the earth�s.

15. Tyrell: Blue Mango is a terrific three-piece band.

Kofi: That�s impossible. No band is terrific if it can only play three pieces.

★ 16. Fred: Today�s professional athletes are overpaid. Many of them make millions

of dollars a year.

Shawn: I don�t think they are overpaid at all. Just look at the owners of some of these teams. They make ten times as much as the athletes do.

17. Brian: All of us have a right to life, and that means to the things necessary to

sustain life. Therefore, that penniless, hungry person had a right to steal food from the grocery store.

Elaine: You�re wrong about that. Rights are created by laws, and there is no law that says we have a right to steal.

18. Penny: In my mind, the use of marijuana should be legalized everywhere.

After all, caffeine and alcohol are no less of a drug than marijuana, and it�s not illegal to enjoy a glass of beer or drink a cup of coffee.

Sam: Your conclusion is way off. Beer and coffee are not drugs; they�re foods.

★ 19. Jerry: In spite of the great strides technology has made in this country, poverty

remains a terrible problem. Why, some people earn less than $10,000 per year. The government should do something about it.

Frankie: I hardly think that $10,000 per year constitutes poverty. In many develop-ing nations the majority of inhabitants earn less than $1,000 per year.

20. Jamar: My friend Jarvis, who plays for the NFL, told me that he has never

taken any performance-enhancing drugs.

Kyrone: That can�t be true. I know Jarvis, too, and he told me that he has taken

Viagra for years.

Section 2.1 Varieties of Meaning 79

V. Create the following disputes:

1. A dispute focusing on the vagueness of the word �awesome�

2. A dispute focusing on the ambiguity of the word �tip�

3. A factual dispute focusing on the date of some historical event

2.2 The Intension and Extension of Terms

The main task of logic is the evaluation of arguments. Words that are not terms include verbs, nonsub-However, as we saw in the previous section, there are stantive adjectives, adverbs, prepositions, conjunc-countless arguments in which this task leads to the tions, and all nonsyntactic arrangements of words. observation, �Well, that depends on what you mean The following words or phrases are not terms; none

meaning of certain words in the argument is vague or dictatorial moreover ambiguous. Clearing up the problem often involves by . . . .� Such an observation usually indicates that the can serve as the subject of a statement: supplying a definition. Thus, the study of meaning and runs quickly craves definition is closely related to the main task of logic. In above and beyond cabbages into again this section we continue our inquiry into aspects of lin- the forest guistic meaning, and the results of this inquiry provide The last example is a nonsyntactic arrangement.

the basis for the theory of definition in the next section. At this point, it is important to distinguish the The basic units of any ordinary language are use of a word from the mention of a word. Without this distinction any word can be imagined to serve words. Our main concern in this chapter, however, is not with words in general but with terms. A as the subject of a statement and, therefore, to count term

as the subject of a statement. Terms consist of proper a term, but �wherever� (in quotes) can serve as the subject of a statement, such as � �Wherever� is an is any word or arrangement of words that may serve as a term. The word �wherever,� for example, is not names, common names, and descriptive phrases.

Here are some examples: eight-letter word.� But in this statement, it is not the Proper names word itself that is the subject but rather the quotedCommon names word. The word is said to be mentioned�not used.

Napoleon animal On the other hand, �wherever� is used in this state-North Dakota restitution ment: �I will follow you wherever you go.� In distin-Peloton house guishing terms from nonterms one must be sure that Toni Morrison activity the word or group of words can be used as the sub-

Robinson Crusoe ject of a statement. person

The previous section of this chapter explored

Descriptive phrases the cognitive meaning of language in general. The first president of the United States cognitive meaning of terms comprises two kinds: author of Hamlet intensional and extensional. The intensional books in my library meaning , or intension, consists of the quali-officers in the Swiss Navy ties or attributes that the term connotes, and the blue things extensional meaning, or extension, consists of

80Chapter2 Language: Meaning and Definition

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). the members of the class that the term denotes. For obnoxious and disgusting. To avoid this problem, example, the intensional meaning of the term �cat� we restrict the meaning of connotation to what is consists of the attributes of being furry, of having usually called the conventional connotation. The four legs, of moving in a certain way, of emitting conventional connotation of a term includes certain sounds, and so on, while the extensional the attributes that the term commonly calls forth in meaning consists of cats themselves�all the cats in the minds of competent speakers of the language. the universe. The term connotes the attributes and Under this interpretation, the connotation of a term denotes the cats. remains more or less the same from person to person

The intensional meaning of a term is otherwise and from time to time.

known as the connotation, and the extensional The denotation of a term also typically remains meaning is known as the denotation. Intension the same from person to person, but it may change and extension are roughly equivalent to the more over time. The denotation of �currently living cat,� modern terms sense and reference, respectively. for example, is constantly fluctuating as some Also, note that logic uses the terms connotation and cats die and others are born. The denotation of denotation differently from the way they are used the term �cat,� on the other hand, is presumably in grammar. In grammar, connotation refers to the constant because it denotes all cats�past, present, subtle nuances of a word, whereas denotation refers and future. to the word�s direct and specific meaning. Sometimes the denotation of a term can change

radically with the passage of time. The terms

�currently living dodo bird� and �current king of

Class members France,� for example, at one time denoted actu-Attributes (extension) (intension) ally existing entities, but today all such entities

Thomas Edison denotes have perished. Accordingly, these terms now have nnotes co Clever what is called empty extension . They are said Alexander Graham Bell Intuitive �Inventor� to denote the empty (or �null�) class, the class that Mary Anderson Creative has no members. Other terms with empty exten-Eli Whitney Imaginative

sion include �unicorn,� �leprechaun,� �gnome,�

�elf,� and �griffin.� While these terms have empty

extension, however, they do not have empty inten-

Exactly how a term connotes a set of attributes sion. �Currently living dodo bird� and �current

allows for at least two different interpretations. king of France,� as well as �unicorn,� �elf,� and Some philosophers take an objective approach �griffin,� connote a variety of intelligible attributes. and hold that a term connotes whatever attributes The fact that some terms have empty exten-something must have in order to be denoted by the sion leads us to an important connection between term. Others take what might be called a subjective extension and intension�namely, that intension approach and hold that a term connotes the attri- determines extension. The intensional mean-butes that occur in the minds of the people who use ing of a term serves as the criterion for decid-that term. This text takes the latter approach. ing what the extension consists of. Because we

In connection with this approach, however, we know the attributes connoted by the term �uni-

encounter the problem of terms connoting differ- corn,� for example, we know that the term has ent things to different people. Thus, to a cat lover empty extension. That is, we know that there are the term �cat� might connote the attributes of no four-legged mammals having a single straight being cuddly and adorable, while to someone who horn projecting from their forehead. Similarly, the hates cats it might connote the attributes of being intension of the word �cat� serves as the criterion

Section 2.2 The Intension and Extension of Terms 81

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). for determining what is and what is not a member A series of terms is in the order of increasing of the class of cats. extension when each term in the series (except

One kind of term that raises problems for the the first) denotes a class having more members than

intension-determines-extension rule is proper the class denoted by the term preceding it. In other names. For example, the name �David� might not words, the class size gets larger with each succes-appear to have any intension, but it denotes the sive term. Decreasing extension is, of course, the person who has this name. Although philosophers reverse of this order. Examples:

would not know what persons, if any, they denote. increasing extension: tiger, feline, mammal, One possible solution to this problem is that names animal are shorthand symbols for descriptions or bundles decreasing intension: tiger, feline, mammal, of descriptions. For example, �David� could be names must have some kind of intension or we increasing intension: animal, mammal, feline, tiger have disagreed about this, it would seem that proper shorthand for �the person who lives next door� or animal �the person who works at the corner store and who decreasing extension: animal, mammal, drives a green Chevy.� feline, tiger

Another possible solution to the problem of These examples illustrate a fact pertaining to most

proper names is that the intension of proper names such series: The order of increasing intension is consists of the causal chain of events leading from usually the same as that of decreasing extension. the point at which the name is first assigned to the Conversely, the order of decreasing intension is point at which a certain person learns about the usually the same as that of increasing extension. name. Thus, the first link in such a chain might be There are some exceptions, however. Consider the the baptismal event at which the name �David� is following series: given to a certain infant, the second link would be

the event in which a certain third party is informed unicorn; unicorn with blue eyes; unicorn of the first event, and so on. This entire chain of with blue eyes and green horn; unicorn events extending through the linguistic commu- with blue eyes, green horn, and a weight nity would then constitute the intension of �David.� of over 400 pounds

sion may be further illustrated by comparing The distinction between intension and exten- while the series exhibits the order of increasing intension, it does not exhibit the order of decreas-ing extension. Here is another, slightly different, the way in which these concepts can be used to proper names, intension determines extension. Each term in this series has empty extension; so, Thus, we conclude that for all terms, including give order to random sequences of terms. Terms example:

may be put in the order of increasing intension, living human being; living human being with increasing extension, decreasing intension, and a genetic code; living human being with a decreasing extension. A series of terms is in the genetic code and a brain; living human being order of increasing intension when each term with a genetic code, a brain, and a height of in the series (except the first) connotes more attri- less than 100 feet butes than the one preceding it. In other words,

each term in the series after the first is more spe- In this series, none of the terms has empty exten-cific than the one preceding it. (A term is specific sion, but each term has exactly the same extension to the degree that it connotes more attributes.) as the others. Thus, while the intension increases The order of decreasing intension is the reverse with each successive term, once again the extension of that of increasing intension. does not decrease.

82Chapter2 Language: Meaning and Definition

Exercise 2.2

I. The following exercises deal with words and terms.

1. Determine which of the following words or groups of words are terms and

which are nonterms.

extortion Thomas Jefferson laborious Empire State Building cunningly annoy practitioner render satisfactory seriousness graceful dancer forever wake up whoever studies not only interestingly impassive tallest man on the squad scarlet mountaintop reinvestment between therefore since

2. Name some of the attributes connoted by the following terms. Express your

answer with adjectives or adjectival phrases. Example: The term �elephant� connotes the attributes of being large, having tusks, and having a trunk.

drum wolf fanatic riot politician Mona Lisa carrot piano devil Statue of Liberty

3. In the following, name three items denoted by the terms in the left-hand

column and all items denoted by the terms in the right-hand column:

newspaper tallest mountain on earth scientist prime number less than 10 manufacturer governor of New York river language of Canada opera Scandinavian country

4. Put the following sequences of terms in the order of increasing intension:

★a. conifer, Sitka spruce, tree, spruce, plant

b. Italian sports car, car, vehicle, Maserati, sports car

c. doctor of medicine, person, brain surgeon, professional person, surgeon

d. wallaby, marsupial, mammal, animal, kangaroo

e. parallelogram, polygon, square, rectangle, quadrilateral

Section 2.2 The Intension and Extension of Terms 83

5. Construct a series of four terms that exhibits increasing intension but non-

decreasing extension.

II. Answer �true� or �false� to the following statements:

1. All words have an intensional meaning and an extensional meaning.

2. The intensional meaning of a term consists of the attributes connoted by

the�term.

3. The extensional meaning of a term consists of the members of the class

denoted by the term.

4. The extension of a term always remains the same with the passage of time.

5. Some terms have empty intension.

6. Some terms have empty extension.

7. The intension of a term determines the extension.

8. The intension of a term determines how specific the term is.

9. The order of increasing intension is always the same as that of decreasing

extension.

10. �Leprechaun� and �unicorn� have the same extension.

III. Create the following series of terms:

1. A series of four terms, relating to music, that are in the order of increasing

intension

2. A series of four terms, relating to beverages, that are in the order of increasing

extension

2.3 Definitions and Their Purposes

Over the years, philosophers have held various the definiens. The definiendum is the word or conflicting views about the purpose of defini- group of words that is supposed to be defined, and tions. For example, Plato claimed that defini- the definiens is the word or group of words that tions were intended to explicate the meaning of does the defining. For example, in the definition certain eternal essences or forms, such as justice, � �Tiger� means a large, striped, ferocious feline piety, and virtue. For most logicians today, how- indigenous to the jungles of India and Asia,� the ever, definitions are intended exclusively to expli- word �tiger� is the definiendum, and everything cate the meaning of words. In conformity with after the word �means� is the definiens. The defini-this latter position, we may define definition as ens is not itself the meaning of the definiendum; a group of words that assigns a meaning to some rather, it is the group of words that symbolizes (or word or group of words. Accordingly, every defi- that is supposed to symbolize) the same meaning nition consists of two parts: the definiendum and as the definiendum. Because we presumably know

84Chapter2 Language: Meaning and Definition

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). in advance what the definiens symbolizes, we are old word. The purpose of a stip ulative definition is led, via the definition, to understand what the usually to replace a more complex expression with definiendum symbolizes. It is in this way that the a simpler one. definition �assigns� a meaning to its definiendum. The need for a stipulative definition is often

occasioned by some new phenomenon or devel-

opment. For example, many years ago lions were

Definition crossbred with tigers. The word �tigon� was

selected to name the offspring of male tiger and a

female lion, and �liger� was selected to name the

offspring of a male lion and a female tiger. When

Definiendum = Definiens a zebra was crossbred with a donkey, the offspring

was called a �zeedonk.� Crossbreeding a lime

Word to be with a kumquat produced a fruit that was called Words that do a �limequat,� and crossbreeding a plum with an defined the defining apricot produced a fruit called a �plumcot� and a

�pluot.� All of these words were first assigned their

Once it has been decided that definitions expli- meanings through stipulative definitions.

cate the meaning of words, other disagreements Because people are continually coming up with emerge among the philosophers. Some argue that new creations, whether it be food concoctions, since a definition is merely a rule that allows one inventions, modes of behavior, or kinds of apparel, set of words (the definiens) to be used in place of stipulative definitions are continually being intro-another set (the definiendum), definitions com- duced to name them. The invention of computers municate no information at all about the subject provides a prime example. Today we have dozens matter of the definiendum. Others take the oppo- of new terms or new uses of old terms that did site tack and argue that since definitions result in not exist before the computer age: �cyberspace,� a clarification of language, they provide a means �e-mail,� �browser,� �hacker,� �dot-com,� �hard-for the discovery of deeper philosophical truths. It ware,� �software,� �download,� �website,� �webmas-seems, however, that neither of these approaches ter,� �server,� �boot,� �bar code,� �mouse,� �modem,� is able to make good sense of all the various kinds �cookies,� �spam,� �app,� �ego surfer,� �meme,� �QR of definitions that are actually employed in ordi- code,� �Android,� �Google,� �metaverse,� �tweet,� nary usage. As a result, instead of beginning their �texting,� and �sexting��to name just a few. Ear-analysis of definitions with a set of a priori cri- lier, in the area of biology, when a certain excre-teria, many logicians take a pragmatic approach tion of the pancreas was refined to its pure form, and begin with a survey of the various kinds of the word �insulin� was chosen to name it, and the

tions that they actually serve. This is the approach substance produced by certain Penicillium molds. 5 In mathematics, the symbol �10 definitions that are actually used and of the func- word �penicillin� was chosen for an antibacterial taken here. � was chosen as a

simple substitute for �10 3 10 3 10 3 10 3 10.�

Because a stipulative definition is a completely

Stipulative Definitions the first time, there can be no such thing as a �true� arbitrary assignment of a meaning to a word for A stipulative definition assigns a meaning to or �false� stipulative definition. Furthermore, for a word for the first time. This may involve either the same reason, a stipulative definition cannot coining a new word or giving a new meaning to an provide any new information about the subject

Section 2.3 Definitions and Their Purposes 85

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). matter of the definiendum. The fact that the ambiguity causes the most trouble. In many cases, word �tigon� was selected to replace �offspring the problem lies not with the obvious differences of a male tiger and a female lion� tells us nothing in meaning that words such as �light� and �bank� new about the nature of the animal in question. may have but with the subtle shadings of mean-One stipulative definition may, however, be more ing that are more likely to be confused with one or less convenient or more or less appropriate another. For example, if a woman is described as than another. �nice,� any number of things could be intended.

Stipulative definitions are misused in verbal She could be fastidious, refined, modest, pleas-

disputes when one person covertly uses a word in ant, attractive, or even lewd. A good lexical defi-a peculiar way and then proceeds to assume that nition will distinguish these various shadings everyone else uses that word in the same way. Under and thereby guard against the possibility that two these circumstances, that person is said to be using such meanings will be unconsciously jumbled the word �stipulatively.� In such cases, the assump- together. tion that other persons use the word in the same way

is rarely justified. Precising Definitions

Lexical Definitions The purpose of a precising definition is to reduce

the vagueness of a word. As we saw in the first sec-

A lexical definition is used to report the meaning tion of this chapter, an expression is vague if there that a word already has in a language. Dictionary are borderline cases in which it is impossible to tell definitions are all instances of lexical definitions. if the word applies or does not apply. Words such Thus, in contrast with a stipulative definition, as �fresh,� �wealthy,� and �high� are vague. Once which assigns a meaning to a word for the first the vagueness of such words is reduced by a precis-time, a lexical definition may be true or false ing definition, one can reach a decision as to the depending on whether it does or does not report applicability of the word to a specific situation. For the way a word is actually used. Because words are example, in relation to blood sugar, ��high� means frequently used in more than one way, lexical defi- that an A1C test reads above 5.7� is an example of a nitions have the further purpose of eliminating the precising definition. ambiguity that would otherwise arise if one of these Whenever words are taken from ordinary usage meanings were to be confused with another. Here and used in a highly systematic context, such as are two lexical definitions: science, mathematics, medicine, or law, they must

Light: (1) Something that makes things vis- always be clarified by means of a precising defini-ible; (2) electromagnetic radiation; (3) of little tion. The terms �force,� �energy,� �acid,� �element,� weight; (4) having fewer calories than the �number,� �equality,� �contract,� and �agent� have standard product. all been given precising definitions by specific

disciplines.

Bank: (1) The slope bordering a stream or Sometimes the substance of a court trial may river; (2) the inclination of an aircraft during a revolve around the precise usage of a term. A turn; (3) an institution for receiving, lending, harassment case before the U.S. Court of Appeals and safeguarding money. depended on whether a certain person qualified

Because a lexical definition lists the various as a supervisor. The court ruled that a supervisor

meanings that a word can have, a person who was one who had �the power to hire, fire, demote, consults such a definition is better prepared to promote, transfer, or discipline� an employee. This avoid ambiguous constructions of his or her ruling amounted to a precising definition of the own and to detect those of others. Undetected word �supervisor.�

86Chapter2 Language: Meaning and Definition

Prominent Peter Abelard 1079�1142Logicians

G enerally considered the greatest logician of the Middle Ages, Peter Abelard was born in the village of Le Pallet in the Brittany region of France. His parents were members of the French

nobility, and as their eldest son, Abelard was slated to

inherit substantial wealth and noble standing. However,

he gave up claim to this inheritance and the knighthood

that went with it, choosing instead the life of a scholar.

When he was only a teenager, Abelard went off to

Paris to study philosophy with William of Champeaux

at the cathedral school of Notre Dame. He proved to

be a brilliant student and arrogant to a fault. He openly

challenged the views of his teacher and seized on

every opportunity to debate William in public. Later,

he set up a rival school, describing its founder as the

�only remaining philosopher in the world.� Gradually,

he became renowned throughout all of Europe, and Hulton Archive/Getty Images he was eventually appointed to the faculty of Notre

Dame where he attracted hundreds of students eager supposed to marry. The couple decided to marry to learn from this illustrious master. secretly, and Heloise fled to a convent to shield her-

Around this time Abelard�s attentions were drawn self from the scandal mongers who persecuted her

to Heloise, the beautiful and brilliant young niece of for being ostensibly unwed. Meanwhile, a furious a prominent Parisian canon named Fulbert. Making Fulbert plotted to punish Abelard, and he hired a the acquaintance of Fulbert�s young prot�g� proved gang of marauders to break into Abelard�s lodgings a daunting task, since her uncle kept her closely in the middle of the night and castrate him. guarded. Nonetheless, Abelard persuaded Fulbert to After the castration, Abelard took refuge in one allow him to move into his house and tutor the gifted monastery after another. However, his arrogance niece, who, though only in her teens, had already mas- made him ill suited for monastic life, as he went out tered Greek and Hebrew. Fulbert saw this as a way of his way to provoke the other monks. Much later, he of providing Heloise with a first-rate higher education, returned to Paris where he taught until he was silenced but for Abelard it provided quite a different opportu- by the church for alleged heresy. At one point, he was nity. He later compared Fulbert�s credulity in allowing forced to burn one of his own books. Throughout all him access to his charge as akin to placing a lamb in of these calamities, Abelard remained devoted to his the care of a devouring wolf. scholarly endeavors. He developed a truth-functional

The tutoring sessions rapidly turned toward seduc- propositional logic and a theory of entailment, and he

tion, with Heloise a receptive student. Before long, wrote prolifically in the areas of metaphysics, ethics, Heloise became pregnant and gave birth to a son and philosophy of language. He is buried alongside whom she named Astrolabe (after the astronomical Heloise in the P�re-Lachaise Cemetery in Paris. Today, device). A public marriage mi ght have abated the their grave site is visited by people seeking solace ensuing scandal, but scholars and clerics were not from the frustrations of love.

Section 2.3 Definitions and Their Purposes 87

Another example involves the practice of surgical investigating the relationship between molecular

transplantation of vital organs. Before a heart trans- velocity and the phenomena of radiation, gas pres-plant can be conducted, the donor must be dead; sure, molecular elasticity, and molecular configura-otherwise, the surgeon will be accused of murder. tion. In short, this definition of �heat� provides the If the donor is dead for too long, however, the suc- impetus for an entire theory about heat. cess of the transplant will be imperiled. But exactly Other examples of theoretical definitions are the when is a person considered to be dead? Is it when the definition of �light� as a form of electromagnetic heart stops beating, when the person stops breathing, radiation and the definition of �force,� �mass,� and when rigor mortis sets in, or some other time? The �acceleration� in Newton�s second law of motion, as question involves the meaning of the term �moment expressed in the equation �F 5 MA.� The latter is a of death.� The courts have decided that �moment kind of contextual definition in which each term is of death� should be taken to mean the moment the defined in terms of the other two. Both definitions brain stops functioning, as measured by an electro- entail numerous deductive consequences about the encephalograph. This decision amounts to the accep- phenomena involved and suggest numerous avenues tance of a precising definition for �moment of death.� of experimental investigation.

A precising definition differs from a stipulative Not all theoretical definitions are associated with

definition in that the latter involves a purely arbi- science. Many terms in philosophy, such as �sub-trary assignment of meaning, whereas the assign- stance,� �form,� �cause,� �change,� �idea,� �good,� ment of meaning in a precising definition is not at �mind,� and �God,� have been given theoretical defi-all arbitrary. A great deal of care must be taken to nitions. In fact, most of the major philosophers in his-ensure that the assignment of meaning in a precis- tory have given these terms their own peculiar theo-ing definition is appropriate and legitimate for the retical definitions, and this fact accounts in part for context within which the term is to be employed. the unique character of their respective philosophies.

For example, Gottfried Wilhelm Leibniz�s definition

Theoretical Definitions of �substance� in terms of what he called �monads�

laid the foundation for his metaphysical theory, and

A theoretical definition assigns a meaning John Stuart Mill�s definition of �good� as the great-to a word by suggesting a theory that gives a cer- est happiness of the greatest number provided the tain characterization to the entities that the term underpinnings for his utilitarian theory of ethics. denotes. Such a definition provides a way of viewing Like stipulative definitions, theoretical definitions or conceiving these entities that suggests deductive are neither true nor false, strictly speaking. The reason consequences, further investigation (experimental is that theoretical definitions function as proposals to or otherwise), and whatever else would be entailed see or interpret some phenomenon in a certain way. by the acceptance of a theory governing these enti- Since proposals have no truth value, neither do theo-ties. The definition of the term �heat� found in texts retical definitions. They may, however, be more or less dealing with the kinetic theory of heat provides a interesting or more or less fruitful, depending on the good example: � �Heat� means the energy associ- deductive consequences they entail and on the out-ated with the random motion of the molecules of a come of the experiments they suggest. substance.� This definition does more than merely

conceiving the physical phenomenon that is heat. assign a meaning to a word; it provides a way of Persuasive Definitions In so doing, it suggests the deductive consequence The purpose of a persuasive definition is to that as the molecules of a substance speed up, the engender a favorable or unfavorable attitude toward temperature of the substance increases. In addition, what is denoted by the definiendum. This purpose is it suggests a number of experiments�experiments accomplished by assigning an emotionally charged

88Chapter2 Language: Meaning and Definition

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). or value-laden meaning to a word while making it �Capitalism� means the economic system in appear that the word really has (or ought to have) which individuals are afforded the God-given that meaning in the language in which it is used. freedom to own property and conduct busi-Thus, persuasive definitions amount to a certain ness as they choose. synthesis of stipulative, lexical, and, possibly, theo- �Capitalism� means the economic system in

to engender a certain attitude. As a result of this quest for money, and mutual understand-ing and respect are replaced by alienation, synthesis, a persuasive definition masquerades as retical definitions backed by the rhetorical motive which humanity is sacrificed to the wanton an honest assignment of meaning to a term while greed, and selfishness. condemning or blessing with approval the subject �Taxation� means the procedure by means of matter of the definiendum. Here are some examples which our commonwealth is preserved and of opposing pairs of persuasive definitions: sustained.

�Taxation� means the procedure used

�Abortion� means the ruthless murdering of

by bureaucrats to rip off the people who

innocent children.

elected them.

�Abortion� means a safe and established

surgical procedure whereby a woman is The objective of a persuasive definition is to influ-relieved of an unwanted burden. ence the attitudes of the reader or listener; thus, �Liberal� means a drippy-eyed do-gooder such definitions may be used with considerable obsessed with giving away other people�s effectiveness in political speeches and editorial money. columns. While persuasive definitions may, like �Liberal� means a genuine humanitarian lexical definitions, be evaluated as either true or committed to the goals of adequate housing false, the primary issue is neither truth nor falsity and health care and of equal opportunity for but the effectiveness of such definitions as instru-all of our citizens. ments of persuasion.

Exercise 2.3

I. Determine whether the following definitions are stipulative, lexical, precising, theoreti-

cal, or persuasive.

★1. �Blind� means, for federal income-tax purposes, either the inability to see

better than 20/200 in the better eye with glasses or having a field of vision of 20�degrees or less.

2. �Football� means a sport in which modern-day gladiators brutalize one another

while trying to move a ridiculously shaped �ball� from one end of the playing field to the other.

3. �Dogbomber� means an inconsiderate person who does not clean up the

messes his dog leaves on other people�s lawns.

★4. �Diffident� means lacking confidence in oneself; characterized by modest

reserve.

Section 2.3 Definitions and Their Purposes 89

5. �Magnetism� means a property of certain substances such as iron, cobalt, and

nickel that arises from the spin of the electrons in the unfilled inner shell of the atoms that compose the substance.

6. �Fiduciary� means having to do with a confidence or trust; a person who holds

something in trust.

★7. �Politician� means a person of unquestioned honesty and integrity whom the

people, in their collective wisdom, have duly elected to guide the ship of state and protect it from the reefs and shoals that threaten it on every side.

8. �Intoxicated,� for purposes of driving a car in many states, means having a

blood-alcohol content of 0.08 percent (.0008) or greater.

9. �Telepest� means an annoying telemarketer.

★10. �Sound� means a compression wave, in air or some other elastic medium,

having a frequency ranging (for humans) from 20 to 20,000 vibrations per second.

11. �Radioactive area� means, for purposes of the U.S. Nuclear Regulatory Com-

mission, any area accessible to individuals in which there exists radiation at such levels that a major portion of the body could receive in any one hour a dose in excess of 5 millirems or in any five consecutive days a dose in excess of 100 millirems.

12. �Neurosis� means a chronic emotional disturbance that arises from suppressed

or forgotten emotional stress (such as resentment, hostility, aggression, or guilt) experienced in early childhood.

★13. �Petrograb� means invading a country to steal its oil.

14. �Vaper� means a rude and disgusting individual who callously emits huge

clouds of smoke into the air, threatening the health and comfort of everyone in the vicinity.

15. �Diadem� means an ornamental headband worn as a badge of royalty; a crown.

★16. �Psychiatry� means the fortuitous melding of modern medicine with psychol-

ogy that promises relief to thousands of poor, desperate souls who suffer the pains of emotional disorder.

17. �Gene� means the hereditary unit that occupies a fixed chromosomal locus,

which through transcription has a specific effect on phenotype and which can mutate to various allelic forms.

18. �Ramster� means an animal produced by crossbreeding a rat with a hamster.

★19. �Intractable� means not easily governed; obstinate; unruly; not disposed to be

taught.

20. �Recession� means, for purposes of the National Bureau of Economic Research,

two consecutive quarters of negative growth in real GNP or in aggregate output for the entire economy.

90Chapter2 Language: Meaning and Definition

21. �Gravity� means a force that results from the universal attraction that every

particle of matter has for every other particle, and which varies directly with the mass of the particles and inversely with the square of the distance between them.

★22. �Assault� means, for legal purposes, an intentional and unprivileged act result-

ing in the apprehension of an immediate harmful or offensive contact.

23. �Cell phone� means the electronic device that keeps its owner transfixed and

detached while spending time with friends or relatives.

24. �Obelisk� means an upright, four-sided pillar that terminates in a pyramid; a

dagger.

★25. �Clonipus� means a cat that has been produced by cloning.

II. The following exercises involve constructing definitions:

1. Invent stipulative definitions for two new words that you wish to introduce

into the language for the first time.

2. Construct lexical definitions for �capital� and �depression,� and indicate two

different meanings for each.

3. Construct precising definitions for �middle-aged� and �alcoholic.� Interpret

both words as relating to people and specify the purpose for which the defini-tions are to be used.

4. Construct theoretical definitions for �energy� and �atom.�

5. Construct opposing pairs of persuasive definitions for �conservative� and

�socialism.�

III. Answer �true� or �false� to the following statements:

1. From the standpoint of logic, many definitions are concerned not with words

but with things.

2. The definiendum is the word or term that is supposed to be defined.

3. The definiens is the word or group of words that assigns a meaning to the word

being defined.

4. A stipulative definition is either true or false.

5. A lexical definition reports the way a word is actually used in a language.

6. One of the purposes of a lexical definition is to guard against the ambiguous

use of a word.

7. The meaning given to a word by a precising definition is completely arbitrary.

8. Theoretical definitions are either true or false, just as are lexical definitions.

9. Theoretical definitions provide a theoretical characterization of the entity or

entities denoted by the word being defined.

10. The purpose of a persuasive definition is to influence attitudes.

Section 2.3 Definitions and Their Purposes 91

IV. Create the following definitions:

1. A stipulative definition for a new word you have invented

2. A precising definition for the word �loud�

3. A persuasive definition for the word �terrorist�

4. A theoretical definition for the word �atom�

2.4 Definitional Techniques

In the previous section, we presented a survey of If you were attempting to teach a foreigner your own some of the kinds of definitions actu ally in use and native language, and neither of you understood a the functions they are intended to serve. In this sec- word of each other�s language, demonstrative defini-tion, we will investigate some of the techniques used tion would almost certainly be one of the methods to produce these definitions. These techniques may you would use. be classified in terms of the two kinds of meaning, Because demonstrative definitions are the most intensional and extensional, discussed in Section 2.2. primitive, they are also the most limited. In addition

to the limitations affecting all extensional definitions

Extensional (Denotative) Definitions limitation that the required objects be available for (which will be discussed shortly), there is the obvious An extensional (denotative) definition is one being pointed at. For example, if one wishes to define that assigns a meaning to a term by indicating the the word �sun� and it happens to be nighttime, or the members of the class that the definiendum denotes. word �dog� and none happens to be in the vicinity, There are at least three ways of indicating the mem- a demonstrative definition cannot be used. bers of a class: pointing to them, naming them Demonstrative definitions differ from the other individually, and naming them in groups. The three kinds of definitions in that the definiens is consti-kinds of definitions that result are called, respec- tuted at least in part by a gesture�the gesture of tively, demonstrative or ostensive definitions, enu- pointing. Since the definiens in any definition is a merative definitions, and definitions by subclass. group of words, however, a gesture, such as pointing,

Demonstrative (ostensive) definitions are must count as a word. While this conclusion may

probably the most primitive form of definition. All appear strange at first, it is supported by the fact that one need know to understand such a definition is the �words� in many sign languages consist exclu-the meaning of pointing. As the following examples sively of gestures. illustrate, such definitions may be either partial or Enumerative definitions assign a meaning to complete, depending on whether all or only some of a term by naming the members of the class the term the members of the class denoted by the definien- denotes. Like demonstrative definitions, they may dum are pointed to: also be either partial or complete. Examples:

�Chair� means this and this and this�as you �Actress� means a person such as Sasha point to several chairs, one after the other. Lane, Elle Fanning, and Joey King.

�Washington Monument� means that�as �Baltic state� means Estonia, Latvia, or you point to it. Lithuania.

92Chapter2 Language: Meaning and Definition

Complete enumerative definitions are usually members (or to the subclasses) of the class denoted

more satisfying than partial ones because they identify by the word being defined. Sometimes they even the definiendum with greater assurance. Relatively include a kind of demonstrative definition when few classes, however, can be completely enumer- they provide a picture of the object that the word ated. Many classes, such as the class of real numbers denotes. Not all lexical definitions have to occur in greater than 1 but less than 2, have an infinite number dictionaries, however. A lexical definition can just of members. Others, such as the class of stars and as well be spoken, as when one person attempts to the class of persons, while not infinite, have still too explain orally to another how a word is used in a many members to enumerate. Therefore, anything language. Such attempts, incidentally, often have approximating a complete enumerative definition of recourse to all three kinds of extensional definition. terms denoting these classes is virtually impossible. Stipulative definitions are used to assign a mean-Then there are others�the class of insects and the ing to a word for the first time. This task may be class of trees, for example�the vast majority of whose accomplished by all three kinds of extensional defi-members have no names. For terms that denote these nition. For example, a biologist engaged in naming classes, either a demonstrative definition or a defini- and classifying types of fish might assign names to tion by subclass is the more appropriate choice. the specific varieties by pointing to their respec-

A definition by subclass assigns a meaning to tive tanks (demonstrative definition), and then

a term by naming subclasses of the class denoted by she might assign a class name to the whole group the term. Such a definition, too, may be either partial by referring to the names of the specific varieties or complete, depending on whether the subclasses (definition by subclass). An astronomer might point named, when taken together, include all the mem- via his telescope to a newly discovered comet and bers of the class or only some of them. Examples: announce, �That comet will henceforth be known as

�Tree� means an oak, pine, elm, spruce, �Henderson�s Comet� � (demonstrative definition). maple, and the like. The organizer of a children�s game might make the �Flower� means a rose, lily, daisy, geranium, stipulation: �Kai, Daniela, and Martin will be called zinnia, and the like. �Buccaneers,� and Valeria, Tyrone, and Serenity will

be �Pirates� � (enumerative definition).

�Cetacean� means either a whale, a dolphin, Although it is conceivable that extensional defi-or a porpoise. nitions could also serve as techniques for theo-retical and persuasive definitions (though this �Fictional work� means either a poem, a play, a novel, or a short story. would be highly unusual), extensional definitions

The first two are partial, the second two complete. by themselves cannot properly serve as precising As with definitions by enumeration, complete defi- definitions for the following reason. The function nitions by subclass are more satisfying than partial of a precising definition is to clarify a vague word, ones; but because relatively few terms denote classes and vagueness is a problem affecting intensional that admit of a conveniently small number of sub- meaning. Because the intension is imprecise, the classes, complete definitions by subclass are often extension is indefinite. To attempt to render the difficult, if not impossible, to provide. intension precise by exactly specifying the exten-

Extensional definitions are chiefly used as sion (as with an extensional definition) would

techniques for producing lexical and stipulative be tantamount to having extension determine definitions. Lexical definitions are aimed at com- intension�which cannot be done. municating how a word is actually used, and one The principle that intension determines exten-of the ways of doing so is by identifying the mem- sion, whereas the converse is not true, underlies bers of the class that the word denotes. Dictionar- the fact that all extensional definitions suffer seri-ies frequently include references to the individual ous deficiencies. For example, in the case of the

Section 2.4 Definitional Techniques 93

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). demonstrative definition of the word �chair,� if all defined, a synonymous definition is a highly concise the chairs pointed to are made of wood, observ- way of assigning a meaning. Many words, however, ers might get the idea that �chair� means �wood� have subtle shades of meaning that are not connoted instead of something to sit on. Similarly, they by any other single word. For example, the word might get the idea that �Washington Monument� �wisdom� is not exactly synonymous with either means �tall� or �pointed� or any of a number of �knowledge,� �understanding,� or �sense�; and �envi-other things. From the definition of �actress,� read- ous� is not exactly synonymous with either �jealous� ers or listeners might think that �actress� means or �covetous.�

�woman��which would include countless indi- An etymological definition assigns a mean-viduals who have nothing to do with the stage or ing to a word by disclosing the word�s ancestry in screen. From the definition of �tree� they might get both its own language and other languages. Most the idea that �tree� means �firmly planted in the ordinary English words have ancestors either in ground,� which would also include the pilings of Old or Middle English or in some other language a building. And they might think that �cetacean� such as Greek, Latin, or French, and the current means �aquatic animal,� which includes salmon, English meaning (as well as spelling and pronun-tuna, squid, manatees, and so on. In other words, ciation) is often closely tied to the meaning (and it makes no difference how many individuals or spelling and pronunciation) of these ancestral subclasses are named in an extensional defini- words. For example, the English word �license� is tion, there is no assurance that listeners or readers derived from the Latin verb licere, which means will� get the intensional meaning. Extensions can to be permitted, and the English word �captain� suggest intensions, but they cannot determine them. derives from the Latin noun caput, which means

head.

Etymological definitions have special impor-

attributes that the word connotes. Because at least which all other associated meanings are derived. Unless one is familiar with this root meaning, one four strategies may be used to indicate the attri-often fails to place other meanings in their proper butes a word connotes, there are at least four kinds light or to grasp the meaning of the word when it is of intensional definitions: synonymous definition, used in its most proper sense. For example, the word etymological definition, operational definition, and �principle� derives from the Latin word principium, definition by genus and difference. An etymological definition of a word often conveys intensional definition is one that assigns a the word�s root meaning or seminal meaning from meaning to a word by indicating the qualities or Intensional (Connotative) Definitions tance for at least two reasons. The first is that the

definiens is a single word that connotes the same A which means beginning or source. Accordingly, the synonymous definition is one in which the �principles of physics� are those fundamental laws that provide the �source� of the science of physics. attributes as the definiendum. In other words, the The English word �efficient� derives from the Latin definiens is a synonym of the word being defined. verb efficere, which means to bring about. Thus, the Examples: �efficient cause� of something (such as the motion

�Physician� means doctor. of a car) is the agent that actually brings that thing �Intentional� means willful. about (the engine). �Voracious� means ravenous. The second reason for the importance of ety-

mological definitions is that if one is familiar with

�Observe� means see. the etymology of one English word, one often has

When a single word can be found that has access to the meaning of an entire constellation of

the same intensional meaning as the word being related words. For example, the word �orthodox�

94Chapter2 Language: Meaning and Definition

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). derives from the two Greek words ortho, mean- if and only if it has a pH of less than 7,� while good ing right or straight, and doxa, meaning belief or in other respects, is not an operational definition, opinion. From this, one might grasp that �ortho- because it prescribes no�operation. pedic� has to do with straight bones (originally Operational definitions were invented for the in children�pais in Greek means child), and that purpose of tying down relatively abstract scien-�orthodontic� has to do with straight teeth ( odon tific concepts to the solid ground of empirical in Greek means tooth). Similarly, if one is famil- reality. In this they succeed fairly well; yet, from iar with the etymological definition of �polygon� the standpoint of ordinary language usage, they (from the Greek words poly, meaning many, and involve certain deficiencies. One of these defi-ganos meaning angle), one might grasp the mean- ciencies concerns the fact that operational defini-ings of �polygamy� (from gamos, meaning mar- tions usually convey only part of the intensional riage) and �polygraph� (from graphein, meaning to meaning of a term. Certainly �brain activity� write). A polygraph is a lie detector that simulta- means more than oscillations on an electroen-neously records pulse rate, blood pressure, respira- cephalograph, just as �acid� means more than tion, and so on. litmus paper turning red. This deficiency becomes

An operational definition assigns a meaning more acute when one attempts to apply opera-

to a word by specifying certain experimental pro- tional definitions to terms outside the framework cedures that determine whether or not the word of science. For example, no adequate operational applies to a certain thing. Examples: definition could be given for such words as �love,�

One substance is �harder than� another if �respect,� �freedom,� and �dignity.� Within their and only if one scratches the other when the proper sphere, however, operational definitions two are rubbed together. are quite useful and important. Interestingly Ein-

stein developed his special theory of relativity in

�Brain activity� means that an electroenceph- partial response to the need for an operational

alograph shows oscillations when attached to definition of simultaneity. a patient�s head. A definition by genus and difference A �potential difference� exists between assigns a meaning to a term by identifying a genus two conductors if and only if a voltmeter term and one or more difference words that, when shows a reading when connected to the two combined, convey the meaning of the term being conductors. defined. Definition by genus and difference is more A solution is an �acid� if and only if litmus generally applicable and achieves more adequate paper turns red when dipped into it. results than any of the other kinds of intensional

definition. To explain how it works, we must first

Each of these definitions prescribes an operation explain the meanings of the terms genus, species, and to be performed. The first prescribes that the two specific difference. substances in question be rubbed together, the In logic, genus and species have a somewhat dif-second that the electro encephalograph be con- ferent meaning than they have in biology. In logic, nected to the patient�s head and observed for genus simply means a relatively larger class, and spe-oscillations, the third that the voltmeter be con- cies means a relatively smaller subclass of the genus. nected to the two conductors and observed for For example, we may speak of the genus animal and deflection, and the fourth that the litmus paper the species mammal, or of the genus mammal be placed in the solution and observed for color and the species feline, or of the genus feline and change. Unless it specifies such an operation, a the species tiger, or the genus tiger and the species definition cannot be an operational definition. Bengal tiger. In other words, genus and species are For example, the definition �A solution is an �acid� merely relative classifications.

Section 2.4 Definitional Techniques 95

The specific difference, or difference, is the attri- the five kinds of definition discussed in Section 2.3.

bute or attributes that distinguish the various spe- Stipulative, lexical, precising, theoretical, and per-cies within a genus. For example, the specific dif- suasive definitions can all be constructed accord-ference that distinguishes tigers from other species ing to the method of genus and difference. Lexical in the genus feline would include the attributes of definitions are typically definitions by genus and being large, striped, ferocious, and so on. Because difference, but they also often include etymological the specific difference is what distinguishes the definitions. Operational definition can serve as the species, when a genus is qualified by a specific dif- method for constructing stipulative, lexical, precis-ference, a species is identified. Definition by genus ing, and persuasive definitions, but because of the and difference is based on this fact. It consists of limitations we have noted, it typically could not be combining a term denoting a genus with a word or used to produce a complete lexical definition. Other group of words connoting a specific difference so techniques would have to be used in addition. Syn-that the combination identifies the meaning of the onymous definition may be used to produce only term denoting the species. lexical definitions. Since, in a synonymous defini-

Let us construct a definition by genus and differ- tion, the definiendum must have a meaning before

ence for the word �ice.� The first step is to identify a synonym can be found, this technique cannot be a genus of which ice is the species. The required used to produce stipulative definitions, and the fact genus is water. Next, we must identify a specific that the definiens of such a definition contains no difference (attribute) that makes ice a special form more information than the definiendum prohibits of water. The required difference is frozen. The its use in constructing precising, theoretical, and completed definition may now be written out: persuasive definitions.

Species This account of definitions is inevitably incom-Difference Genus

�Ice� means frozen water. plete. At the beginning of the chapter we saw that all

construct. Simply select a term that is more gen-A definition by genus and difference is easy to words�not just terms�stand in need of definitions, but the account given here is based on the intension and extension of terms . Nevertheless, many of the eral than the term to be defined, and then narrow techniques developed here can be applied to words it down so that it means the same thing as the term in general, and even to symbols. For example, Chap-being defined. Examples: ters 6 and 8 will present definitions of various sym-Species Difference Genus bols that are used in modern logic to connect one �Daughter� means female offspring. statement with another and to translate ordinary

�Husband� means married man. language statements into symbolic form. When logi-�Doe� means female deer. cians introduced these symbols many years ago, they �Fawn� means very young deer. did it through stipulative definitions. Also, as we will

Other examples are more complex: by certain tables, called truth tables, which establish each symbol�s meaning under all possible arrange-�Skyscraper� means see in Chapter 6, some of these symbols are defined very tall building.

�Tent� means a collapsible shelter made of ments of truth values. These definitions are probably canvas or other material that is stretched and best described as extensional, and they�are similar sustained by poles. in some ways to demonstrative definitions and enu-

�Tent� is the species, �shelter� is the genus, and �col- merative definitions. lapsible� and �made of canvas . . .� the difference. The applicability of the seven definitional tech-

Definition by genus and difference is the most niques in producing the five kinds of definition is

effective of the intensional definitions for producing summarized in Table 2.1.

96Chapter2 Language: Meaning and Definition

Table 2.1 Correlation of Definitional Techniques with Types of Definitions

Can Produce This Type of Definition

This TechniqueStipulativeLexicalPrecising TheoreticalPersuasive

Demonstrative yes yes no (unusual) (unusual)

Enumerative yes yes no (unusual) (unusual)

Subclass yes yes no (unusual) (unusual)

Synonymous no yes no no no

Etymological yes yes no no no

Operational (limited) yes yes (unusual) (unusual)

Genus and yes yes yes yes yes Difference

Exercise 2.4

I. Determine whether the following are demonstrative definitions, enumerative defini-

tions, definitions by subclass, synonymous definitions, etymological definitions, opera-tional definitions, or definitions by genus and difference.

★1. �Plant� means something such as a tree, a flower, a vine, or a cactus.

2. �Hammer� means a tool used for pounding.

3. A triangle is �equilateral� if and only if a compass, when placed sequentially on

two vertices and properly adjusted, strikes through the other two vertices.

★4. �State� means something such as Ohio, Arkansas, Minnesota, and Tennessee.

5. �Angel� is a word that originates from the Greek word angelos, which means

messenger.

6. �Newbie� means beginner.

★7. �House� means this:

8. �Painting� means something like da Vinci�s Mona Lisa, van Gogh�s Starry Night,

Botticelli�s Birth of Venus, or Picasso�s Guernica.

9. �Dessert� means something such as pie, cake, cookies, or ice-cream sundaes.

★ 10. �Ripe� means, for a watermelon, that it emits a deep, hollow sound when you

thump it with your hand.

11. �Universe� originates from the Latin word universus, which means whole or entire.

12. �Mountain� means something such as Everest, Rainier, Whitney, or McKinley.

★ 13. �Hurricane� means a storm having constant winds of at least 74 miles per hour

that originates at sea.

Section 2.4 Definitional Techniques 97

14. A substance is �translucent� if and only if when held up to a strong light some

of the light comes through.

15. �Insect� means something such as a fly, an ant, a wasp, or a caterpillar.

★16. �Poignant� is a word derived from the Latin word pungere, which means prick,

pierce, or sting.

17. �Facade� means face.

18. �Prime number� means a number greater than one that is divisible only by

itself and one.

★19. �Language� means something such as French, German, Spanish, or English.

20. �Tree� means this, and this, and this (as you point to several trees).

21. �Oak� means a tree that bears acorns.

★22. �Rapier� means sword.

23. An �electric current� flows in a circuit if and only if an ammeter connected in

series with the circuit shows a reading.

24. �Rapper� means someone such as Snoop Dogg, Eminem, Tupac Shakur, or Ice Cube.

★25. �Professional person� means a person such as a doctor, a lawyer, a professor, or

an architect.

26. �Error� means mistake.

27. �Tale� is a word that derives from the Old English word talu, which means talk.

★28. �Truck� means a vehicle used for hauling.

29. �Done� means, in reference to a baking cake, that a wooden toothpick poked

into the center comes out clean.

30. �Musical composition� means something such as a symphony, a concerto, a

sonata, or a toccata.

II. The following exercises involve constructing definitions:

1. Construct a partial enumerative definition for the following terms by naming

three members of the class the term denotes. Then find a nonsynonymous term that these members serve equally well to define. Example: �Coin� means some-thing such as a nickel, dime, or quarter. A nonsynonymous term is �round object.�

★a. skyscraper

b. corporation

c. island

d. composer

e. novel

2. Construct a complete enumerative definition for the following terms:

a. ocean

b. continent

98Chapter2 Language: Meaning and Definition

3. Construct a definition by subclass for the following terms by naming three

subclasses of the class the term denotes. Then find a nonsynonymous term that these subclasses serve equally well to define.

★a. animal

b. fish

c. vehicle

d. gemstone

e. polygon

4. Construct a complete definition by subclass for the following terms:

a. quadrilateral

b. circulating American coin

5. Construct synonymous definitions for the following terms:

★a. intersection

b. fabric

c. nucleus

d. abode

e. wedlock

f. cellar

g. summit

h. apparel

6. Construct operational definitions for the following words:

★a. genius

b. ferromagnetic

c. fluorescent

d. alkaline

e. polarized (light)

7. Construct definitions by genus and difference for the following terms. In each

definition, identify the genus term.

★a. drake

b. biologist

c. felony

d. widow

e. library

Section 2.4 Definitional Techniques 99

8. Consult a dictionary to find the etymological roots of the following words, and

then explain how they relate to the conventional meaning of these words.

★a. morphology

b. isomorphic

c. isotropic

d. phototropic

e. photography

f. lithography

g. lithology

h. psychology

III. Answer �true� or �false� to the following statements:

1. The technique of extensional definition may be used to produce precising

definitions.

2. The technique of extensional definition may be used to produce stipulative and

lexical definitions.

3. Most extensional definitions convey the precise intensional meaning of a term.

4. An intensional definition conveys the meaning of a term by indicating the

members of the class the term denotes.

5. In a synonymous definition, the definiens must be a single word.

6. The technique of synonymous definition may be used to construct precising

definitions.

7. Operational definitions typically convey the entire intensional meaning of a word.

8. The species is a subclass of the genus.

9. The specific difference is an attribute or set of attributes that identifies a species.

10. Definition by genus and difference may be used to produce stipulative, lexical,

precising, theoretical, and persuasive definitions.

IV. Create the following definitions:

1. A demonstrative definition for the word �sky�

2. An enumerative definition for the word �novel�

3. A synonymous definition for the word �bumbershoot�

4. A definition by subclass for the word �sport�

5. An etymological definition for the word �telephone�

6. An operational definition for the word �solid�

7. A definition by genus and difference for the word �fawn�

100Chapter2 Language: Meaning and Definition

2.5 Criteria for Lexical Definitions

Because the function of a lexical definition is to report essential meaning of �human� as the word is used in the way a word is actually used in a language, lexi- ordinary English. It says nothing about the impor-cal definitions are the ones we most often encounter tant attributes that distinguish humans from the and are what most people mean when they speak of other animals�namely, the capacity to reason and the �definition� of a word. Accordingly, it is appro- to use language on a sophisticated level. A more priate that we have a set of rules that we may use adequate definition would be � �Human� means the in constructing lexical definitions of our own and animal that has the capacity to reason and to speak.� in evaluating the lexical definitions of others. While If a lexical definition is to be given in terms some of these rules apply to the other kinds of defi- of an operational definition or in terms of any of nitions as well, the unique functions that are served the forms of extensional definition, it should usu-by stipulative, precising, theoretical, and persuasive ally be supplemented by one of the other forms definitions prescribe different sets of criteria. of intensional definition, preferably definition by

genus and difference. As noted, from the stand-

Rule 1: A Lexical Definition Should Conform point of ordinary language usage, an operational to the Standards of�Proper Grammar definition often conveys only part of the inten-

A definition, like any other form of expression, sional meaning of a word, and this part frequently

as follows: the essential meaning of a word; they cannot determine it precisely. As a result, no adequate Vacation is when you don�t have to go to lexical definition can consist exclusively of exten-work or school. sional definitions. definitions that are grammatically extensional definitions, at best they can only sug-incorrect are gest should be grammatically correct. Examples of misses the essential meaning altogether. As for

Furious means if you�re angry at someone.

Cardiac is like something to do with the heart. Rule 3: A Lexical Definition Should Be

The corrected versions are these: Neither Too Broad nor Too Narrow

If a definition is too broad, the definiens includes

�Vacation� means a period during which too much; if it is too narrow, the definiens includes activity is suspended from work or school. too little. If, for example, �bird� were defined as any warm-blooded animal having wings, the �Furious� means a condition of being very angry. definition would be too broad because it would �Cardiac� means pertaining to, situated near, include bats, and bats are not birds. If, on the

Technically the definiendum should be put in blooded, feathered animal that can fly, the defi-nition would be too narrow because it would or acting on the heart. other hand, �bird� were defined as any warm-

always followed. exclude ostriches and penguins, which cannot fly. quotation marks or italics, but this convention is not

The only types of lexical definitions that tend

to be susceptible to either of these defi ciencies are

Rule 2: A Lexical Definition Should synonymous definitions and definitions by genus Convey the Essential Meaning of the and difference. With synonymous definitions, one Word Being Defined must be careful that the definiens really is a syn-The word �human� is occasionally defined as feath- onym of the definiendum. For example, the defi-erless biped. Such a definition fails to convey the nition � �king� means ruler� is too broad because

Section 2.5 Criteria for Lexical Definitions 101

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). many rulers are not kings. �Ruler� is not genu- Arguably a person would have to know what �time� inely synonymous with �king.� As for definitions means before he or she could understand the pur-by genus and difference, one must ensure that the pose of a clock.

specific difference narrows the genus in exactly the Rule 5: A Lexical Definition Should Not Be right way. Both of the given definitions of �bird� Negative When It Can Be Affirmative are definitions by genus and difference in which the specific difference fails to restrict the genus in Of the following two definitions, the first is affirma-tive, the second negative: exactly the right manner.

�Concord� means harmony.

Rule 4: A Lexical Definition Should �Concord� means the absence of discord.

A definition is circular when the definiendum is intrinsically negative. For them, a negative defini-Avoid Circularity Some words, however, have meanings that are are

defined in terms of itself, or virtually in terms of tion is quite appropriate. Examples: itself. Sometimes the problem of circularity appears �Bald� means lacking hair. in connection with pairs of definitions. The follow-ing pair is circular: �Darkness� means the absence of light.

�Science� means the activity engaged in by Rule 6: A Lexical Definition Should scientists. Avoid Figurative, Obscure, Vague,�or �Scientist� means anyone who engages in Ambiguous Language

science. A definition is figurative if it involves metaphors

At other times, a definition may be intrinsically cir- or tends to paint a picture instead of exposing the

definition, the second a definition by genus and �Architecture� means frozen music. difference: cular. Of the following, the first is a synonymous essential meaning of a term. Examples:

�Camel� means a ship of the desert.

�Soporific� means soporiferous. A definition is obscure if its meaning is hidden

�Jewelers� rouge� means rouge used by a as a result of defective or inappropriate language. jeweler. One source of obscurity is overly technical language.

In the first example, the definiendum is virtually the Compare these two definitions:

same word as the definiens. As a result, anyone who �Bunny� means a mammalian of the family does not already know the meaning of �soporofic� Leporidae of the order Lagomorpha whose would probably not know the meaning of �sopo-young are born furless and blind. riferous,� either. In the second example, �jewelers� �Bunny� means a rabbit. rouge� is clearly defined in terms of itself. The cor-The problem lies not with technical language as rected definitions are as follows: such but with needlessly technical language. Because �Soporific� means tending to cause sleep. �bunny� is very much a nontechnical term, no tech-�Jewelers� rouge� means a very fine polishing nical definition is needed. On the other hand, some compound. words are intrinsically technical, and for them only a Certain operational definitions also run the risk of technical definition will suffice. Example:

circularity: �Neutrino� means a quasi-massless lepton

�Time� means whatever is measured by a obeying Fermi-Dirac statistics and having clock. one-half quantum unit of spin.

102Chapter2 Language: Meaning and Definition

A definition is vague if it lacks precision or if its visionaries in which the national wealth is

meaning is blurred�that is, if there is no way of tell- supposed to be held in common by the ing exactly what class of things the definiens refers people. to. Example: �Theism� means belief in that great Santa

�Democracy� means a kind of government Claus in the sky. where the people are in control. The second example also violates Rule 6 because it

than one distinct interpretation. Example: ambiguous if it lends itself to more Definiens Pertains This rule applies to any definition in which the �Triangle� means a figure composed of three context of the definiens is important to the mean-straight lines in which all the angles are equal ing of the definiendum. For example, the definition to two right angles. � �Deuce� means a tie in points toward a game or in Does this mean that each angle separately is equal in control, how they exercise their control, and what Rule 8: A Lexical Definition Should they are in control of. Indicate the Context to Which the A definition is This definition fails to identify the people who are contains a metaphor.

games toward a set� is practically meaningless with-

to two right angles or that the angles taken together out any reference to tennis. Whenever the definien-are equal to two right angles? Either interpretation dum is a word that means different things in differ-is possible given the ambiguous meaning of �all the ent contexts, a reference to the context is important. angles are equal to two right angles.� Examples:

�Strike� means (in baseball) a pitch at which

Rule 7: A Lexical Definition Should Avoid a batter swings and misses.

Affective Terminology �Strike� means (in bowling) the act of knock-

Affective terminology is any kind of word usage that ing down all the pins with the first ball of plays on the emotions of the reader or listener. It a frame. includes sarcastic and facetious language and any �Strike� means (in fishing) a pull on a line other kind of language that could influence atti- made by a fish in taking the bait.

tudes. Examples: It is not always necessary to make explicit reference

�Communism� means that �brilliant� inven- to the context, but at least the phraseology of the tion of Karl Marx and other foolish political definiens should indicate the context.

Exercise 2.5

I. Criticize the following definitions in light of the eight rules for lexical definitions:

★1. A sculpture is a three-dimensional image made of marble.

2. �Elusory� means elusive.

3. �Birdie� means sinking the ball in one stroke under par.

★4. A cynic is a person who knows the price of everything and the value of nothing.

(Oscar Wilde)

Section 2.5 Criteria for Lexical Definitions 103

5. �Semantics� is when somebody studies words.

6. �iPhone� means a handheld electronic device having an LCD screen and a single

toggle button at the bottom of the face.

★7. A theist is anyone who is not an atheist or an agnostic.

8. �Intelligence� means whatever is measured by an IQ test.

9. A symphony is a musical piece written for full orchestra.

★10. Feminism is a militant movement originated by a group of deviant women for

the purpose of undermining the natural distinction between the sexes.

11. �Wood� means fibrous, lignified cellulose.

12. Logic is the study of arguments including definitions.

★13. �Truculent� is if you�re cruel or fierce.

14. A house is a structure made of wood or stone intended for human habitation.

15. Satire is a kind of glass, wherein beholders do generally discover everybody�s

face but their own.

( Jonathan Swift)

★16. A carpenter�s square is a square used by a carpenter.

17. �Safety� means a play in which a player grounds the ball behind the player�s

own goal line when the ball was caused to cross the goal line by that player�s own team.

18. Puberty: the time in life in which the two sexes begin first to be acquainted.

( Johnson�s Dictionary)

★19. �Normal� means an attribute possessed by people who are able to get on in the

world.

20. An organic substance is any substance that is not inorganic.

21. Faith is the bird that sings when the dawn is still dark.

(Rabindranath Tagore)

★22. �Schooner� means sort of like a sailboat.

23. �Faith� means reason succumbing to insecurity.

24. �Gammon� means, in backgammon, a victory in which one player defeats

another before he or she can remove any of his or her men from the board.

★25. A cello is a stringed musical instrument played with a bow.

26. Tobacco is a plant grown in the southeastern United States that, when enjoyed

in the form of cigars and cigarettes, produces a most delightful and satisfying taste and aroma.

27. History is the unfolding of miscalculations.

(Barbara Tuchman)

104Chapter2 Language: Meaning and Definition

★28. �Clock� means a manufactured device featuring two pointers that rotate past a

set of numerals ranging from 1 to 12.

29. �Soap� means saponified glyceride.

30. Mackerel: a sea-fish.

( Johnson�s Dictionary)

★31. �Veganism� means a silly dietary practice that avoids beef and pork.

32. �Diet� means like when you cut back on your calories.

33. Animal: a living creature corporeal, distinct, on the one side, from pure spirit,

on the other, from pure matter.

(Johnson�s Dictionary)

★34. �Pen� means an instrument used for writing on paper.

35. Wine is an alcoholic beverage made from grapes.

II. Create the following defective lexical definitions:

1. A definition of �hexagon� that is too broad

2. A definition of �scarf� that is too narrow

3. A definition of �corporeal� that is negative

4. A definition of �butcher knife� that is circular

Summary

Linguistic expressions can have different kinds of meaning:

■ Cognitive meaning: Conveys information

■ Emotive meaning: Expresses or evokes feelings

Statements having emotive meaning often make value claims. When such statements occur in arguments, the value claims should be disengaged from the emotive terminol-ogy and expressed as separate premises.

Cognitive meanings can be defective in two ways:

■ Vagueness: The meaning is blurred.

■ Ambiguity: More than one clearly distinct meaning is possible.

A term is a word or phrase that can serve as the subject of a statement. Terms include:

■ Proper names (Napoleon, North Dakota, etc.)

■ Common names (animal, house, etc.)

■ Descriptive phrases (author of Hamlet, books in my library, etc.)

Summary105

Terms can have different kinds of meaning:

■ Intensional meaning (or intension): The attributes that the term connotes

■ Extensional meaning (or extension): The members of the class that the term denotes

Terms that refer to nonexistent things have empty extension.

A definition is a word or group of words that assigns a meaning to a word or group of words:

■ Definiendum: The word or group of words being defined

■ Definiens: The word or group of words that does the defining

Definitions can serve different purposes, so there are different kinds of definitions:

■ Stipulative definitions assign a meaning to a word when it first comes into use.

■ Lexical definitions report the meaning a word has within a community of users.

■ Precising definitions reduce the vagueness of a word.

■ Theoretical definitions appeal to a theory to characterize whatever the term denotes.

■ Persuasive definitions influence the attitudes of the community of users regarding whatever

the word denotes.

Intensional meaning and extensional meaning provide a basis for constructing definitions:

■ Extensional definitions assign a meaning by identifying the things the word denotes:

● Demonstrative definitions �point� to these things.

● Enumerative definitions name individuals that the word denotes.

● Definitions by subclass identify subclasses of these things.

■ Intensional definitions assign a meaning by identifying the attributes the word connotes:

● Synonymous definitions equate the word being defined with another word that connotes

the same attributes.

● Etymological definitions disclose the word�s ancestry.

● Operational definitions specify experimental procedures for determining whether the word

applies to a certain thing.

● Definitions by genus and difference identify a genus term and one or more difference words

that, when combined, convey the meaning of the definiendum.

Lexical definitions are governed by eight rules. They should:

■ Conform to the standards of proper grammar.

■ Convey the essential meaning of the word being defined.

■ Be neither too broad nor too narrow.

■ Avoid circularity.

■ Not be negative when they can be affirmative.

■ Avoid figurative, obscure, vague, or ambiguous language.

■ Avoid affective terminology.

■ Indicate the context to which the definiens pertains.

106Chapter2 Language: Meaning and Definition

3Informal Fallacies

3.1 Fallacies in General

3.2 Fallacies of Relevance

3.3 Fallacies of Weak Induction

3.4 Fallacies of Presumption, Ambiguity, and Illicit Transference

3.5 Fallacies in Ordinary Language

3.1 Fallacies in General

Learning about informal fallacies is empowering. Informal falla-cies are arguments that cheat, and they shouldn�t persuade anyone, even though they sometimes do. For example, if an arguer should reply to an opponent by calling that person an ignorant buffoon, then nobody will be persuaded. Also, the person attacked will get angry and the two parties will become further alienated. By learning the structure of informal fallacies, we refrain from committing them ourselves and we avoid being coerced and manipulated by others who use them.

Afallacy is a defect in an argument that arises from either a mistake in reason-ing or the creation of an illusion that makes a bad argument appear good. The fallacies that appear in this chapter involve errors that occur so often that they have been given specific names. The term non sequitur (�it does not follow�) is another name for fallacy. Both deductive and inductive arguments may contain fallacies; if they do, they are either unsound or uncogent, depending on the kind of argument. Conversely, if an argument is unsound or uncogent, it has one or more false premises or it contains a fallacy (or both).

Fallacies are usually divided into two groups: formal and informal. A formal

fallacy is one that may be identified by merely examining the form or structure of an argument. Fallacies of this kind are found only in deductive arguments that

107

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). have identifiable forms. Chapter 1 presented some of an improper arrangement of terms or statements, these forms: categorical syllogisms, disjunctive syl- it commits a formal fallacy. Section 1.5 investigated logisms, and hypothetical syllogisms. The following some of these forms and gave instruction on distin-categorical syllogism contains a formal fallacy: guishing the form from the content of an argument.

All bullfights are grotesque rituals. All of the exercises at the end of that section commit All executions are grotesque rituals. formal fallacies. Therefore, all bullfights are executions. Informal fallacies are those that can be detected

This argument has the following form: sider the following example: only by examining the content of an argument. Con-

All A are B. The Brooklyn Bridge is made of atoms. All C are B. Atoms are invisible. All A are C. Therefore, the Brooklyn Bridge is invisible.

By merely examining this form, one can see that it is To detect this fallacy one must know something invalid. The fact that A, B, and C stand respectively about bridges�namely, that they are large visible for �bullfights,� �grotesque rituals,� and �executions� objects, and even though their atomic components is irrelevant in detecting the fallacy. The problem are invisible, this does not mean that the bridges may be traced to the second premise. If the letters C themselves are invisible. and B are interchanged, the form becomes valid, and Or consider this example:

the original argument, with the same change intro- A chess player is a person. duced, also becomes valid (but unsound). Therefore, a bad chess player is a bad person.

Here is an example of a formal fallacy that occurs

in a hypothetical syllogism: To detect this fallacy one must know that the mean-

If apes are intelligent, then apes can solve ing of the word �bad� depends on what it modifies, puzzles. and that being a bad chess player is quite different Apes can solve puzzles. from being a bad�person. Therefore, apes are intelligent. The various informal fallacies accomplish their

purpose in so many different ways that no single

This argument has the following form: umbrella theory covers them all. Some fallacies work

If A then B. by getting the reader or listener to feel various emo-B. tions, such as fear, pity, or camaraderie, and then A. attaching a certain conclusion to those emotions.

Others attempt to discredit an opposing argument

In this case, if A and B are interchanged in the first by associating it with certain pejorative features of its premise, the form becomes valid, and the original author. And then there are those that appeal to vari-argument, with the same change, also becomes ous dispositions on the part of the reader or listener, valid. This fallacy and the one that precedes it will such as superstition or mental laziness, to get him or be discussed in later chapters. her to accept a conclusion. By studying the typical

In distinguishing formal from informal fallacies, ways in which arguers apply these techniques, one

remember that formal fallacies occur only in deduc- is less likely to be fooled by the fallacious arguments tive arguments. Thus, if a given argument is induc- posed by others or to stumble blindly into fallacies tive, it cannot contain a formal fallacy. Also, keep when constructing arguments for one�s own use. an eye out for standard deductive argument forms Since the time of Aristotle, logicians have such as categorical syllogisms and hypothetical syl- attempted to classify the various informal fallacies. logisms. If such an argument is invalid because of Aristotle himself identified thirteen and separated

108Chapter3 Informal Fallacies

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). them into two groups. The work of subsequent logi- of weak induction, fallacies of presumption, falla-cians has produced dozens more, rendering the task cies of ambiguity, and fallacies of illicit transference. of classifying them even more difficult. The presen- The final section of the chapter considers the related tation that follows divides twenty-two informal fal- topics of detecting and avoiding fallacies in the con-lacies into five groups: fallacies of relevance, fallacies text of ordinary language.

Exercise 3.1

I. Determine whether the fallacies committed by the following arguments are formal fal-

lacies or informal fallacies.

★1. If Rasputin was really mad, then he deceived Czar Nicholas II. Rasputin was

not really mad. Therefore, he did not deceive Czar Nicholas II.

2. Everything that runs has feet. The Columbia River runs very swiftly. Therefore,

the Columbia River has feet.

3. All people who believe we create our own reality are people who lack social

responsibility. All people governed by selfish motives are people who lack social responsibility. Therefore, all people who believe we create our own real-ity are people governed by selfish motives.

★4. The ship of state is like a ship at sea. No sailor is ever allowed to protest orders

from the captain. For the same reason, no citizen should ever be allowed to protest presidential policies.

5. Renowned violinist Pinchas Zukerman has said, �When it comes to vodka,

Smirnoff plays second fiddle to none.� We must therefore conclude that Smirnoff is the best vodka available.

6. If the Chinese government systematically kills its unwanted orphans, then the

Chinese government is immoral. The Chinese government is indeed immoral. Therefore, the Chinese government systematically kills its unwanted orphans.

★7. Mark Ruffalo, America Ferrera, and Ben Affleck are Democrats. Therefore, it

must be the case that all Hollywood stars are Democrats.

8. Mark Zuckerberg, cofounder of Facebook, argues in favor of opening up the

Arctic National Wildlife Refuge to oil drilling. But consider this: Zuckerberg is just a moneygrubbing capitalist who only cares about inflating his already bloated bank account. Clearly his arguments are ridiculous.

9. If plastic guns are sold to the public, then terrorists will carry them aboard

airliners undetected. If plastic guns are sold to the public, then airline hijack-ings will increase. Therefore, if terrorists carry plastic guns aboard airliners un detected, then airline hijackings will increase.

★10. Some corporate mergers are arrangements that produce layoffs. Some arrange-

ments that produce layoffs are social catastrophes. Therefore, some corporate mergers are social catastrophes.

Section 3.1 Fallacies in General 109

II. Create the following formal and informal fallacies:

1. A hypothetical syllogism that contains a formal fallacy (see the �apes� argu-

ment given earlier in this section)

2. An informal fallacy similar to the Brooklyn Bridge argument given earlier in

this section

3.2 Fallacies of Relevance

The fallacies of relevance share the common Child to playmate: Sesame Street is the best characteristic that the arguments in which they show on TV; and if you don�t believe it, I�m occur have premises that are logically irrelevant to going to call my big brother over here and the conclusion. Yet the premises may appear to be he�s going to beat you up.

psychologically relevant, so the conclusion may seem But it occurs among adults as well: to follow from the premises, even though it does not Lobbyist to senator: Senator Foxwell, of follow logically. In a good argument, the premises course you support our bill to reduce inheri-provide genuine evidence in support of the conclu-tance taxes. After all, you wouldn�t want the sion. In an argument that commits a fallacy of rel-press to find out about all the contributions evance, on the other hand, the connection between you receive from the Ku Klux Klan. premises and conclusion is emotional. To identify a fallacy of relevance, therefore, one must be able to The first example involves a physical threat, the distinguish genuine evidence from various forms of second a psychological one. While neither threat emotional appeal. provides any genuine evidence that the conclu-sion is true, both provide evidence that someone

might be injured. If the two types of evidence are

1. Appeal to Force ( confused with each other, both arguer and listener Argumentum

may be deluded into thinking that the conclusion is

ad Baculum: Appeal to the �Stick�) supported by evidence, when in fact it is not.

The fallacy of appeal to force occurs whenever an arguer presents a conclusion to another person

and tells that person either implicitly or explicitly Appeal to force that some harm will come to him or her if he or she

does not accept the conclusion. The fallacy always Threatens A R/L A = Arguer involves a threat by the arguer to the physical or A s psychological well-being of the listener or reader, ser R/L = Reader/ t s who may be either an individual or a group of Listener people. Obviously, such a threat is logically irrele- Conclusion vant to the subject matter of the conclusion, so any

argument based on such a procedure is fallacious.

The ad baculum fallacy often occurs when children The appeal to force fallacy usually accomplishes argue with one another: its purpose by psychologically impeding the reader

110Chapter3 Informal Fallacies

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). or listener from acknowledging a missing premise If the arguer succeeds in evoking pity from the lis-that, if acknowledged, would be seen to be false or tener or reader, the latter is likely to exercise his at least questionable. The two examples just given or her desire to help the arguer by accepting the can be interpreted as concealing the following argument. In this way the reader or listener may be premises, both of which are most likely false: fooled into accepting a conclusion that is not sup-

If my brother forces you to admit that ported by any evidence. The appeal to pity is quite Sesame Street is the best show on TV, then common and is often used by students on their Sesame Street is in fact the best show. instructors at exam time and by lawyers on behalf

of their clients before judges and juries.

If I succeed in threatening you, then you sup-

port the bill to reduce inheritance taxes.

The conclusion of the first argument is that Appeal to pity

Sesame Street is the best show on TV. But just

because someone is forced into saying that it is Evokes A R/L does not mean that such is the case. Similarly, the A = Arguer pity A conclusion of the second argument is that the sena-s ser R/L = Reader/ t tor supports the bill to reduce inheritance taxes. s Listener But just because the lobbyist succeeds in threaten-Conclusion ing the senator does not mean that the senator sup-ports the bill. Many of the other informal fallacies

can be interpreted as accomplishing their purpose Of course, some arguments that attempt to evoke in this way. sympathetic feelings from the reader or listener are arguments from not fallacious. We might call them compassion. Such arguments differ from the falla-2. Appeal to Pity ( Argumentum ad cious appeal to pity in that, in addition to evoking Misericordiam ) compassion on behalf of some person, they supply The appeal to pity fallacy occurs when an arguer information about why that person is genuinely attempts to support a conclusion by merely evok- deserving of help or special consideration. When-ing pity from the reader or listener. This pity may ever possible, these non fallacious arguments should be directed toward the arguer or toward some third show that the person in question is a victim of cir-party. Example: cumstances and not responsible for the dire straits

Taxpayer to judge: Your Honor, I admit that he finds himself in, that the recommended help or I declared thirteen children as dependents special consideration is not illegal or inappropriate, on my tax return, even though I have only and that it will genuinely help the person in ques-two. But if you find me guilty of tax evasion, tion. In contrast to such arguments, the appeal to my reputation will be ruined. I�ll probably pity proceeds by ignoring all of these considerations lose my job, my poor wife will not be able and attempts to support a conclusion by merely to have the operation that she desperately evoking pity from the reader or listener.

needs, and my kids will starve. Surely I am

not guilty. 3. Appeal to the People

not guilty.� Obviously, the conclusion is not The conclusion of this argument is �Surely I am (Argumentum ad Populum) logi-cally relevant to the arguer�s set of pathetic cir- Nearly everyone wants to be loved, esteemed, cumstances, although it is psychologically relevant. admired, valued, recognized, and accepted by others.

Section 3.2 Fallacies of Relevance 111

The appeal to the people uses these desires to Fear mongering has played at least a minor role get the reader or listener to accept a conclusion. Two in nearly every presidential election. A famous approaches are involved: one of them direct, the example was the so-called Daisy Commercial* used other indirect. in 1964 by the Lyndon Johnson campaign to defeat

The direct approach occurs when an arguer, Barry Goldwater. The commercial raised the spec-

addressing a large group of people, excites the emo- ter of Goldwater�s possible use of nuclear weapons tions and enthusiasm of the crowd to win acceptance in Vietnam if he were to become president. What if for his or her conclusion. The objective is to arouse Goldwater were elected? Could this lead to an all-a kind of mob mentality. This is the strategy used out nuclear confrontation with the Soviet Union? by nearly every propagandist and demagogue. Adolf No one could be sure of the answer. Even though the Hitler was a master of the technique, but speech commercial was pulled after its first airing, count-makers at Democratic and Republican national con- less news shows picked it up and repeated it over and ventions also use it with some measure of success. over again. The effect was to plant the fear of nuclear Waving flags and blaring music add to the overall annihilation in the minds of millions of voters, effect. Because the individuals in the audience want and in the end it swayed many of them to side with to share in the camaraderie, the euphoria, and the President Johnson. excitement, they find themselves accepting a variety President George W. Bush used fear to get the of conclusions with ever-increasing fervor. public to support the invasion of Iraq. He talked

An appeal to negative emotions can also generate the nation into believing that Saddam Hussein had

a mob mentality. The appeal to fear, also known weapons of mass destruction, and he could use them as fear mongering, is a variety of the direct form of against the United States, so it was essential that he the appeal to the people that occurs when an arguer be taken out. But no president in American history trumps up a fear of something in the mind of the has used the politics of fear more persistently than crowd and then uses that fear as a premise for some Donald Trump. Trump told the public that immi-conclusion. Of course many fears that we experience grants were to be feared because they were rapists in daily life are supported by solid evidence, such as and murderers, environmentalists were to be feared the fear of getting mugged in a dark alley when sev- because they weakened the nation, scientists were eral muggings have occurred there recently. In the to be feared because they told lies about every-appeal to fear fallacy, the fear is not supported by any thing, the news media were to be feared because solid evidence, and it usually rests on nothing more they created fake news, the Democrats were to be than irrational suspicion created by repeating a mes- feared because they want to avert climate change sage or rumor over and over again. As the message (which would weaken the economy), government gradually sinks in it causes the crowd to feel uneasy, officials were to be feared because they were part and this alone may be enough to get a large number of the deep state, and he himself was to be feared to accept the arguer�s conclusion. because he would unleash hell on anybody who

crossed him.

Appeal to Practically any social or political change is fertile the people ground for appeals to fear. When Darwin�s theory

of evolution began to be taught in the schools,

A Plays on need William Jennings Bryan argued that it would R/L A = Arguer for security, etc. increase the number of wars, undermine moral-A s ser R/L = Reader/ ity, convert love into hate, and destroy civilization. t s Listener

Conclusion

*This commercial, in its entirety, is readily available on the Internet.

112Chapter3 Informal Fallacies

His speeches were a driving force in the movement you ought to do it, too. If everyone believed that to ban the teaching of evolution. When the nation 2 + 2 = 5, that would not make it so. was debating whether women should be allowed to The appeal to vanity is another form of the vote, Elihu Root argued that suffrage would make indirect approach, and it often involves linking the women �hard, harsh, unlovable, repulsive.� When love, admiration, or approval of the crowd with some the Soviet Union developed the atomic bomb and famous figure who is loved, admired, or approved of. became a threat to world peace, Senator Joseph This version of the fallacy is often used by advertis-McCarthy argued that Soviet spies or sympathizers ers, parents, and people in general.

had infiltrated many branches of the government Of course you want to look as strong and including the State Department and the U.S. Army. beautiful as Lyzabeth Lopez. That means you The resulting atmosphere of fear wrecked the lives will want to buy her training app. of many innocent victims. Daniel Craig wears an Omega wristwatch. As these examples illustrate, the direct approach Thus, if you want to be like him, you will buy of appeal to the people is often accomplished through and wear an Omega watch, too. oral communication. However, writing can also pro-duce the same effect. By using such emotionally Mother to child: You want to grow up and be charged phrases as �champion of the free-enterprise just like Wonder Woman, don�t you? Then system,� �defender of the working man,� �bleeding- eat your liver and carrots.

heart liberal,� and �profligate spender,� polemicists The idea is that if you succeed in becoming like can awaken the same kind of mob mentality in their Lopez, Craig, or Wonder Woman, then you will win writings as they would if they were speaking. the love and approval of the crowd; but to become

In the indirect approach of appeal to the people, like them you must buy the advertised product or,

the arguer aims his or her appeal not at the crowd in the case of the little girl, eat the liver and carrots. as a whole but at one or more individuals sepa- Incidentally, these examples show how the appeal to rately, focusing on some aspect of those individu- vanity can overlap the false cause fallacy (presented als� relationship to the crowd. The indirect approach in Section 3.3), because they might be interpreted to includes such specific forms as the bandwagon argu- mean that using the training app will cause a person ment, the appeal to vanity, the appeal to snobbery, to look like Lyzabeth Lopez, and so on. Of course, and the appeal to tradition. any such causal connection is unlikely.

The bandwagon argument has this general In the appeal to snobbery the crowd that the

structure: Everybody believes such-and-such or arguer appeals to is a smaller group that is supposed to does such-and-such; therefore, you should believe be superior in some way�more wealthy, more power-or do such-and-such, too. Examples: ful, more culturally refined, more intelligent, and so on.

Everyone nowadays is on a keto diet. There- As the argument goes, if the listener wants to be part of fore, you should go on a keto diet, too. this group, then he or she will do a certain thing, think

in a certain way, or buy a certain product. Example:

Practically everybody believes in life after

death. Therefore, you should believe in life A Ferrari is not for everyone. Only those after death, too. with considerable means can acquire one.

The idea is that if you want to fit in with the crowd To show the world that you are among the and not stick out like a sore thumb, then you, too, select few, you will want to purchase and will go on the diet and believe in life after death. drive one of these distinguished sport cars. But of course the mere fact that a large group of Even if a group of snobs might happen to think or people happen to be doing something or believe feel something, this is not a logical reason for why in something is not, by itself, a logical reason why you should act in conformity.

Section 3.2 Fallacies of Relevance 113

The appeal to tradition is yet another variety In the direct approach the arousal of a mob men-

of the indirect appeal to the people. It occurs when tality produces an immediate feeling of belonging, an arguer cites the fact that something has become a even if it relates to something feared. Each person tradition as grounds for some conclusion. The claim feels united with the crowd, and this feeling evokes a that something is a tradition is basically synony- sense of strength and security. When the crowd roars mous with the claim that a lot of people have done it its approval of the conclusions that are then offered, that way for a long time. Examples: anyone who does not accept them automatically cuts

Traditionally, professional sporting events have himself or herself off from the crowd and risks the been preceded by the national anthem. There- loss of his or her security, strength, and acceptance. fore, professional sporting events should con- The same thing happens in the indirect approach, tinue to be preceded by the national anthem. but the context and technique are somewhat subtler.

long-standing tradition. Therefore, we should 4. Argument Against the Person Serving turkey on Thanksgiving Day is a

serve turkey next Thanksgiving Day. (Argumentum ad Hominem)

The mere fact that something has been done in a This fallacy always involves two arguers. One of certain way for a long time does not by itself justify them advances (either directly or implicitly) a cer-its being repeated in the future. Yet, there are some tain argument, and the other then responds by appeals to tradition that have conclusions that are directing his or her attention not to the first per-true for other reasons, and this may trick a reader son�s argument but to the first person himself. When or listener into thinking that the argument is a good this occurs, the second person is said to commit an one. Example: argument against the person.

Traditionally, guests have worn elegant cloth- The argument against the person occurs in three ing to Mrs. Channing�s annual cocktail party. forms: the ad hominem abusive, the ad hominem cir-Therefore, it would not be a good idea for cumstantial, and the tu quoque. In the ad hominem you to go naked to her party this year. abusive, the second person responds to the first per-

This argument is just as fallacious as the previous two. son�s argument by verbally abusing the first person. Example: The conclusion is clearly true, but the reason why it is true is not because of any tradition but because the Television entertainer Bill Maher argues purpose of a cocktail party is to foster a feeling of con- that religion is just a lot of foolish nonsense. viviality among the guests. One of the guests showing But Maher is an arrogant, shameless, self- up naked would threaten to destroy this purpose to righteous pig. Obviously his arguments are the detriment of the host and all the other guests. not worth listening to.

tradition with true conclusions applies to the other Incidentally, this final point about appeals to The author of this argument ignores the substance of Maher�s argument and instead attacks Maher him-forms of ad populum as well. If such arguments have self. However, because Maher�s personal attributes true conclusions, those conclusions are true for are irrelevant to whether the premises of his religion reasons other than the fact that the crowd believes argument support the conclusion, the argument something or feels something. attacking him is fallacious. Both the direct and indirect approaches of the ad populum fallacy have the same basic structure: Not all cases of the ad hominem abusive are so blunt, but they are just as fallacious. Example: You want to be accepted/included in the

group/loved/esteemed. . . . Therefore, you Dr. Phil argues that mutual self-esteem is should accept XYZ as true. essential to a good marriage. But Dr. Phil

114Chapter3 Informal Fallacies

is not terribly well educated, and he never that the Dalai Lama happens to be affected by these attended an Ivy League college. Thus, his circumstances is irrelevant to whether his premises arguments are worthless. support a conclusion. The ad hominem circum-

sive occurs when the responding arguer�s retort A very common form of the stantial is easy to recognize because it always takes ad hominem abu-this form: �Of course Mr. X argues this way; just look at the circumstances that affect him.� is to suggest that the opposing arguer consider going somewhere else�such as out of the coun-

try (�America�love it or leave it�), switching to a

different religion or political party, or doing some- Argument against thing ridiculous (such as jumping in a lake or flying the person

a kite). Example: Verbally A = Arguer 1 1 A 1 A 2 Billionaire Bill Gates argues that the govern-attacks P A = Arguer 2 re c t s 2 ment should pay more attention to climate s ents Reje change. Why doesn�t Mr. Gates just buy us all (A commits the 2

an electric vehicle? fallacy) Argument

This argument commits an ad hominem because instead of replying to Mr. Gates�s argument, the

second arguer directs his attention to Mr. Gates The tu quoque (�you too�) fallacy begins the himself and suggests that if he is so concerned about same way as the other two varieties of the ad climate change, he can reduce carbon emissions by hominem argument, except that the second arguer buying everyone an electric car. The aim is to show attempts to make the first appear to be hypocriti-that Mr. Gates is insincere. cal or arguing in bad faith. The second arguer

The ad hominemcircumstantial begins the usually accomplishes this by citing features in the

same way as the ad hominem abusive, but instead life or behavior of the first arguer that conflict of heaping verbal abuse on his or her opponent, with the latter�s conclusion. The fallacy often takes the respondent attempts to discredit the opponent�s the form, �How dare you argue that I should stop argument by alluding to certain circumstances that doing X; why, you do (or have done) X yourself.� affect the opponent. By doing so, the respondent Example:

hopes to show that the opponent is predisposed to Kim Kardashian argues that women should argue the way he or she does and should therefore not have children out of wedlock. But not be taken seriously. who is she to talk? She gave birth to her

Here is an example: daughter North without being married.

The Dalai Lama argues that China has no Clearly Kardashian�s argument is not worth business in Tibet and that the West should listening to.

do something about it. But the Dalai Lama Again, the fact that Kim Kardashian gave birth to a just wants the Chinese to leave so he can daughter out of wedlock is irrelevant to whether her return as leader. Naturally he argues this way. premises support her conclusion. Thus, the argu-Therefore, we should reject his arguments. ment is fallacious.

The author of this argument ignores the substance Keep in mind that the purpose of an ad hominem of the Dalai Lama�s argument and attempts to dis- argument is to discredit another person�s argument credit it by calling attention to certain circum- by placing its author in a bad light. Thus, for the stances that affect the Dalai Lama�namely, that he fallacy to be committed, there must always be two wants to return to Tibet as its leader. But the fact arguers (at least implicitly). If it should turn out that

Section 3.2 Fallacies of Relevance 115

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). the person being attacked is not an arguer, then the recently convicted on ten counts of perjury, personal comments made by the attacker may well and he hates Freddy and would love to see be relevant to the conclusion that is drawn. In gen- him sent to jail. Therefore, you should not eral, personal observations are relevant to conclu- believe Mickey�s testimony.

sions about what kind of person someone is (good, This argument commits no fallacy. The conclu-bad, stingy, trustworthy, and so forth) and whether sion is not that you should reject Mickey�s argument a person has done something. Example: but rather that you should reject his testimony.

North Korean dictator Kim Jong-un has mur- Testimony is not argument, and the fact that the dered political opponents and their entire witness is a known liar and has a motive to lie families, and he has committed other crimes now is relevant to whether we should believe him. including extermination, enslavement, Furthermore, note that the conclusion is not that torture, and rape. Also, he has threatened U. S. Mickey�s statement is literally false but rather that cities with nuclear annihilation. Kim Jong-un we should not believe the statement. It is quite pos-is therefore a thoroughly despicable and sible that Mickey really did see Freddy set fire to disgusting human being. the building and that Mickey�s statement to that

The conclusion is not that Kim Jong�s argument effect is true. But if our only reason for believing this statement is the mere fact that Mickey has is bad but that Kim Jong himself is bad. Because made it, then given the circumstances, we are not the premises give relevant support to this conclu-justified in that belief. Personal factors are never sion, the argument commits no fallacy. Another relevant to truth and falsity as such, but they are example: relevant to believability. Shakespeare cannot possibly have written Yet there is often a close connection between the thirty-six plays attributed to him, because truth and believability, and this provides one of the real Shakespeare was a two-bit country the reasons why ad hominem arguments are often businessman who barely finished the fourth effective. In evaluating any argument there are grade in school and who never left the con-always two issues to be considered: the quality of fines of his native England. the reasoning and the truth of the premises. As The conclusion is not that some argument of noted, both are irrelevant to the personal charac-Shakespeare�s is bad but that Shakespeare did not teristics of the arguer. But whether we accept the write certain plays. Again, since the premises are rel-premises as true may depend on the credibility evant to this conclusion, the argument commits no of the arguer. Knowing that the arguer is biased ad hominem fallacy. or has a motive to lie may provide good grounds Determining what kind of person someone for distrusting the premises. Another reason why is includes determining whether that person is ad hominem arguments are effective is that they trustworthy. Thus, personal comments are often engage the emotions of readers and listeners and relevant in evaluating whether a person�s procla- thereby motivate them to transfer their negative mations or statements, unsupported by evidence, feelings about the arguer onto the argument. warrant our belief. Examples of such statements

include promises to do something, testimony given

uct or service. Here is an example of an argument The fallacy of accident is committed when a general that discredits a witness: by a witness, and testimonials in support of a prod- 5. Accident

rule is applied to a specific case it was not intended

Mickey has testified that he saw Freddy to cover. Typically, the general rule is cited (either set fire to the building. But Mickey was directly or implicitly) in the premises and then

116Chapter3 Informal Fallacies

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). wrongly applied to the specific case mentioned in while Zoe was en route to meeting Ethan, someone the conclusion. Two examples: collapsed on the sidewalk right in front of her.

Freedom of speech is a constitutionally

guaranteed right. Therefore, John Q. Radical 6. Straw Man should not be arrested for his speech that The straw man fallacy is committed when an arguer incited the riot last week. distorts an opponent�s argument for the purpose of Zoe promised to meet Ethan for coffee, more easily attacking it, demolishes the distorted but while she was walking to the local argument, and then concludes that the opponent�s Starbucks, a pedestrian collapsed on the real argument has been demolished. By so doing, the sidewalk right in front of her and she arguer is said to have set up a straw man and knocked it stopped to administer CPR. Since people down, only to conclude that the real �man� (opposing are obligated to keep their promises, it was argument) has been knocked down as well.�Example: wrong for Zoe to miss her appointment with Ethan. Mr. Goldberg has argued against prayer in the

In the first example, the general rule is that free- public schools. Obviously Mr. Goldberg advo-dom of speech is normally guaranteed, and the spe- cates atheism. But atheism is what they used cific case is the speech made by John Q. Radical. to have in Russia. Atheism leads to the sup-Because the speech incited a riot, the rule does pression of all religions and the replacement not apply. In the second example, the general rule of God by an omnipotent state. Is that what is that people are obligated to keep their promises, we want for this country? I hardly think so.

Clearly Mr. Goldberg�s argument is nonsense.

and the specific case is the promise that Zoe made

to Ethan. The rule does not apply because saving a

human life is more important than keeping a coffee Straw man appointment.

Distorts

A Op A = Arguer

Accident A s ser Op = Opponent�s t s position

Conclusion

General rule

Straw man, along with argument against the person,

Misapplied are called refutational fallacies because they involve

one arguer refuting another. These are the only

fallacies presented in this chapter that always

Specific case involve two arguers. In the Goldberg argument,

Mr. Goldberg, who is the first arguer, has presented

an argument against prayer in the public schools.

The second arguer then attacks Goldberg�s argu-

The fallacy of accident gets its name from the fact ment by equating it with an argument for athe-

that one or more accidental features of the specific case ism. He then attacks atheism and concludes that make it an exception to the rule. In the first example, Goldberg�s argument is nonsense. Since Goldberg�s the accidental feature is that the speech incited a riot; argument had nothing to do with atheism, the in the second example, the accidental feature is that second argument commits the straw man fallacy.

Section 3.2 Fallacies of Relevance 117

As this example illustrates, the kind of distortion particular conclusion, but then a different conclusion,

the second arguer resorts to is often an attempt to often vaguely related to the correct conclusion, is drawn. exaggerate the first person�s argument or make it Whenever one suspects that such a fallacy is being look more extreme than it really is. Here are two committed, he or she should be able to identify the more examples: correct conclusion, the conclusion that the premises

The garment workers have signed a petition logically imply. This conclusion must be significantly arguing for better air circulation on the work different from the conclusion that is actually drawn. premises. Unfortunately, air-conditioning is Examples:

expensive. Air ducts would have to be run Crimes of theft and robbery have been throughout the factory, and a massive heat increasing at an alarming rate lately. The exchange unit installed on the roof. Also, conclusion is obvious: We must reinstate the the cost of operating such a system during death penalty immediately.

the summer would be astronomical. In view Abuse of the welfare system is rampant of these considerations, the petition must nowadays. Our only alternative is to abolish be rejected. the system altogether.

us with an argument favoring alcohol privi-The student status committee has presented At least two correct conclusions are implied by the leges on campus. What do the students want? premise of the first argument: either �We should Is it their intention to stay boozed up from the provide increased police protection in vulnerable day they enter as freshmen until the day they neighborhoods� or �We should initiate programs to graduate? Do they expect us to open a bar eliminate the causes of the crimes.� Reinstating for them? Or maybe a chain of bars all over the death penalty is not a logical conclusion at all. campus? Such a proposal is ridiculous! Among other things, theft and robbery are not capi-

In the first argument, the petition is merely for tal crimes. In the second argument the premises logically suggest some systematic effort to elimi-better air circulation in the factory�maybe a fan nate the cheaters rather than eliminating the system in the window during the summer. The arguer altogether. exaggerates this request to mean an elaborate air-conditioning system installed throughout the

building. He then points out that this is too expen-

sive and concludes by rejecting the petition. A sim- Missing the point ilar strategy is used in the second argument. The

arguer distorts the request for alcohol privileges to Actually entails mean a chain of bars all over campus. Such an idea Premises Conclusion �A� is so patently outlandish that no further argument is necessary.

7. Missing the Point Conclusion �B� (Ignoratio Elenchi)

All the fallacies we have discussed thus far have been

instances of cases where the premises of an argument

are irrelevant to the conclusion. Missing the pointIgnoratio elenchi means �ignorance of the proof.� illustrates a special form of irrelevance. This fallacy The arguer is ignorant of the logical implications occurs when the premises of an argument support one of his or her own premises and, as a result, draws

118Chapter3 Informal Fallacies

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). a conclusion that misses the point entirely. The fal- fruits and vegetables. But many of these lacy has a distinct structure all its own, but in some foods are essential to our health. Carrots are ways it serves as a catchall for arguments that are not an excellent source of vitamin A, broccoli is clear instances of one or more of the other fallacies. rich in iron, and oranges and grapefruit have An argument should not be identified as a case of lots of vitamin C. missing the point, however, if one of the other fal-

lacies fits.

Red herring

8. Red Herring Draws A R/L A = Arguer This fallacy is closely associated with missing the off track A s point ( ignoratio elenchi ). The red herring fallacy ser R/L = Reader/ t s Listener is committed when the arguer diverts the attention of the reader or listener by changing the subject Conclusion to a different but sometimes subtly related one.

He or she then finishes by either drawing a con-

clusion about this different issue or by merely pre- Both arguments commit the red herring fallacy. suming that some conclusion has been established. In the first, the original issue is whether nuclear By so doing, the arguer purports to have won the power is dangerous. The arguer changes this sub-argument. The fallacy gets its name from a proce- ject to the danger of electrocution and proceeds dure used to train hunting dogs to follow a scent. to draw a conclusion about that. The new subject A red herring (or bag of them) is dragged across is clearly different from the possibility of nuclear the trail with the aim of leading the dogs astray. explosion or meltdown, but the fact that both are Since red herrings have an especially potent scent related to electricity facilitates the arguer�s goal of (caused in part by the smoking process used to leading someone off the track. In the second argu-preserve them), only the best dogs will follow the ment, the original issue is pesticides, and the arguer original scent. changes it to the value of fruits and vegetables in

To use the red herring fallacy effectively, the one�s diet. Again, the fact that the second topic is

arguer must change the original subject of the argu- related to the first assists the arguer in committing ment without the reader or listener noticing it. One the fallacy. In neither case does the arguer draw a way of doing this is to change the subject to one that conclusion about the original topic, but by merely is subtly related to the original subject. Here are two diverting the attention of the reader or listener, the examples of this technique: arguer creates the presumption of having won the

Environmentalists are continually harp- argument. ing about the dangers of nuclear power. A second way of using the red herring effectively Unfortu nately, electricity is dangerous no is to change the subject to some flashy, eye-catch-matter where it comes from. Every year ing topic that is virtually guaranteed to distract the hundreds of people are electrocuted by listener�s attention. Topics of this sort include sex, accident. Since most of these accidents crime, scandal, immorality, death, and any other are caused by carelessness, they could topic that might serve as the subject of gossip. Here be avoided if people would just exercise is an example of this technique:

greater caution. Professor Conway complains of inadequate There is a good deal of talk these days about parking on our campus. But did you hear that the need to eliminate pesticides from our Conway�s eldest son, Jasper, was convicted a

Section 3.2 Fallacies of Relevance 119

year ago of embezzlement? Jasper managed Both the red herring and straw man fallacies to steal half a million dollars from the First are susceptible of being confused with missing the National Bank before officials figured out point, because all three involve a similar kind of what was going on. It�s really important that irrelevancy. To avoid this confusion, one should bank officials be constantly vigilant. Okay, note that both red herring and straw man proceed let�s move on to the next topic. by generating a new set of premises, whereas miss-

The red herring fallacy can be confused with the ing the point does not. Straw man draws a conclu-

straw man fallacy because both have the effect of draw- sion from new premises that are obtained by dis-ing the reader/listener off the track. This confusion can torting an earlier argument, and red herring, if it usually be avoided by remembering the unique ways in draws any conclusion at all, draws one from new which they accomplish this purpose. In the straw man, premises obtained by changing the subject. Miss-the arguer begins by distorting an opponent�s argument ing the point, however, draws a conclusion from and concludes by knocking down the distorted argu- the original premises. Also, in the red herring and ment. In the red herring, the arguer ignores the oppo- straw man, the conclusion, if there is one, is rel-nent�s argument (if there is one) and subtly changes evant to the premises from which it is drawn; but the subject. Thus, to distinguish the two fallacies, one in missing the point, the conclusion is irrelevant should attempt to determine whether the arguer has to the premises from which it is drawn. Finally, knocked down a distorted argument or simply changed remember that missing the point serves in part as the subject. Also keep in mind that straw man always a kind of catchall fallacy, and a fallacious argument involves two arguers, at least implicitly, whereas a red should not be identified as a case of missing the herring often does not. point if one of the other fallacies clearly fits.

Exercise 3.2

I. Identify the fallacies of relevance committed by the following arguments, giving a brief

explanation for your answer. If no fallacy is committed, write �no fallacy.�

★1. The position open in the accounting department should be given to Francine

Thompson. Francine has six hungry children to feed, and her husband desper-ately needs an operation to save his eyesight.

2. Erica Evans, who takes orders at the local Taco Bell, argues persuasively in favor

of increasing the minimum wage. But this is exactly what you would expect. Erica is paid the minimum wage, and if the minimum wage is increased, then her own salary will go up. Obviously Erica�s arguments are worthless.

3. The school board argues that our schools are in desperate need of repair. But

the real reason our students are falling behind is that they spend too much time on their smartphones. Becoming educated means a lot more than learning how to scroll up and down. The school board should send a letter to the parents urging them to confiscate their kids� smartphones.

★4. Whoever thrusts a knife into another person should be arrested. But surgeons

do precisely this when operating. Therefore, surgeons should be arrested.

5. You should read Irving Stone�s latest novel right away. It�s sold over a million copies,

and practically everyone in the Manhattan cocktail circuit is talking about it.

120Chapter3 Informal Fallacies

6. Friedrich Nietzsche�s philosophy is not worth the paper it�s printed on.

Nietzsche was an immoral reprobate who went completely insane from syphi-lis before he died.

★ 7. Surely you welcome the opportunity to join our protective organization. Think

of all the money you will lose from broken windows, overturned trucks, and damaged merchandise in the event of your not joining.

8. Senator Albright advocates increased Social Security benefits for people in the

lowest income bracket. It is regrettable that the senator finds it necessary to advocate socialism. Socialism defeats initiative, takes away promised rewards, and leads directly to inefficiency and big government. It was tried for years in Eastern Europe, and it failed miserably. Clearly, socialism is no good.

9. Something is seriously wrong with high school education these days. A recent

study showed that 70 percent of high school students find school boring, and many are tired and stressed. The obvious conclusion is that we should close the schools.

★10. Giving ridiculous reasons, the editors of the Daily Register have accused our

company of being one of the city�s worst water polluters. But the Daily Register is responsible for much more pollution than we are. After all, they own the West-ern Paper Company, and that company discharges tons of chemical residue into the city�s river every day.

11. If 20 percent of adult Americans are functionally illiterate, then it�s no wonder

that morons get elected to public office. In fact, 20 percent of adult Americans are functionally illiterate. Therefore, it�s no wonder that morons get elected to public office.

12. Ladies and gentlemen, today the lines of battle have been drawn. When the din

of clashing armor has finally died away, the Republican party will emerge victo-rious! We are the true party of the American people! We embody the values that all real Americans hold sacred! We cherish and protect our founding fathers� vision that gave birth to the Constitution! We stand for decency and righteous-ness; for self-determination and the liberty to conduct our affairs as each of us freely chooses! In the coming election, victory will be ours, so help us God!

★13. We�ve all heard the argument that too much time spent on video games is the

reason student grades are suffering. Yet, many of these video games are great. Super Smash Brothers is a fun party game, Guitar Hero teaches students about music, and World of Warcraft keeps students in touch with their friends. Today�s video games are wonderful!

14. Surely architect Norris is not responsible for the collapse of the Central Bank

Tower. Norris has had nothing but trouble lately. His daughter eloped with a child molester, his son committed suicide, and his alcoholic wife recently left for Las Vegas with his retirement savings.

15. The First Amendment to the Constitution prevents the government from

interfering with the free exercise of religion. The liturgical practice of the Reli-gion of Internal Enlightenment involves human sacrifice. Therefore, it would be wrong for the government to interfere with this religious practice.

Section 3.2 Fallacies of Relevance 121

★16. U.S. Senator Elizabeth Warren argues that big banks have been ripping off the

American consumer for years. It�s clear, however, that Warren says this only to scratch up more campaign contributions from those very consumers. Thus, you shouldn�t take her arguments seriously.

17. Professor Pearson�s arguments in favor of the theory of evolution should be

discounted. Pearson is a cocaine-snorting sex pervert and, according to some reports, a member of the Communist party.

18. Rudolf H�ss, commandant of the Auschwitz concentration camp, confessed

to having exterminated 1 million people, most of whom were Jews, in the Auschwitz gas chamber. We can only conclude that H�ss was either insane or an extremely evil person.

★19. TV commentator Maria Bartiromo argues that government should get off the back

of the American businessman. Obviously, Bartiromo wants to abolish government altogether. Yet without government there would be no defense, no judicial system, no Social Security, and no health and safety regulations. None of us wants to forgo these benefits. Thus, we can see that Bartiromo�s argument is absurd.

20. I know that some of you oppose the appointment of David Cole as the new

sales manager. On further consideration, however, I am confident you will find him well qualified for the job. If Cole is not appointed, it may become neces-sary to make severe personnel cutbacks in your department.

21. Animal rights activists say that animals are abused in biomedical research

labs. But consider this: Pets are abused by their owners every day. Probably 25 percent of pet owners should never get near animals. Some cases of abuse are enough to make you sick.

★22. Of course you want to buy a pair of Slinky fashion jeans. Slinky jeans really

show off your figure, and all the Hollywood starlets down on the Strip can be seen wearing them these days.

23. Pop music artist Marie Osmond says on TV that Nutrisystem is the best way to

get rid of belly fat. But this is exactly what you would expect given that Nutri-system pays her millions of dollars to make these ads. Thus, you shouldn�t take her testimonials too seriously.

24. Dr. Morrison has argued that eating too much sugar is responsible for the

majority of health problems in this country and that we should cut down on our sugar intake. Unfortunately, however, we must consign Dr. Morrison�s argument to the trash bin. Only yesterday I saw none other than Dr. Morrison himself eating a candy bar.

★25. Mr. Rhodes is suffering from amnesia and has no recollection whatever of the

events of the past two weeks. We can only conclude that he did not commit the crime of murdering his wife a week ago, as he has been accused of doing.

II. Answer �true� or �false� to the following statements:

1. In the appeal to force, the arguer physically attacks the listener.

2. In the direct variety of the appeal to the people, the arguer attempts to create a

kind of mob mentality.

122Chapter3 Informal Fallacies

3. If an arguer attempts to discredit courtroom testimony or a promise by pointing

out that the witness or the person making the promise is a liar, then the arguer commits an argumentum ad hominem (argument against the person) fallacy.

4. The argumentum ad hominem always involves two arguers.

5. In the argumentum ad hominem circumstantial, the circumstances cited by the

second arguer are intended precisely to malign the character of the first�arguer.

6. In the tu quoque fallacy, the arguer threatens the reader or listener.

7. In the fallacy of accident, a general rule is applied to a specific case where it

does not fit.

8. In the straw man fallacy, an arguer often distorts another person�s argument by

making it look more extreme than it really is.

9. Whenever one suspects that a missing the point fallacy is being committed, one

should be able to state the conclusion that is logically implied by the premises.

10. In the red herring fallacy, the arguer attempts to lead the reader or listener off

the track.

III. Identify the fallacies committed in the following dialogue. You should be able to find at

least one case of each fallacy presented in this section.

Food for Thought

�Let�s hit the produce section first,� Curtis says to his fianc�e, Talia, as they enter Pay-less grocery store.

�Okay,� she says.

�Oh, look,� says Curtis. �The corn is on sale. Let�s get some for dinner.�

�I don�t know,� says Talia. �Did you see that sign over the display? The corn is ge-

netically modified. I know we�ve never paid much attention to that sign in the past, but now I�m thinking that maybe we should.�

�Why�s that?� asks Curtis.

�I read an article the other day about foods containing genetically modified

organisms�they call them GMO foods�and now I know what�s behind this GMO business,� Talia replies.

�And what is behind it?� Curtis asks.

Talia picks up an ear and sniffs it. �For starters,� she says, �one of the reasons

they modify corn is to make it resistant to herbicides like Roundup�you know, the stuff you spray on the weeds in the garden. So with GMO corn the farmer can spray the whole field with Roundup, the weeds will die, and the cornstalks will be unaffected.�

�Sounds like a great way to grow corn,� Curtis says.

�Yes,� replies Talia, �but that means the corn contains a residue of Roundup. That�s

definitely not good. Roundup is a probable cause of cancer.�

�Good grief,� says Curtis. �If what you say is right, I think there is only one conclu-

sion: We must ban the sale of Roundup immediately.�

After pausing to scratch his head, he continues, �On the other hand, look at all

the people who are buying this corn. If everyone is buying it, then I think we should, too.�

Section 3.2 Fallacies of Relevance 123

�You�re right that everyone is buying it,� says Talia. �Nearly 90 percent of the corn

sold in this country is genetically modified. But that doesn�t mean that we should buy it. Look, there�s a small display of organic corn over there. Let�s get some of that.�

�Now wait just a minute,� says Curtis. �You better not be going organic on me. That

would be too much. You know I like to eat out at least once a week, and most res-taurants don�t serve organic food. If you insist on organic, then you will stay at home cooking your own food, while I go out.�

�Well, maybe I could eat conventional restaurant food once in a while,� says

Talia. �But organic food has become really appealing these days. That actor Jake Gyllenhaal promotes it to school kids. And some of my friends say that he is really sexy and that you look like him! Maybe you should think about switching to organic.�

�I look like Jake Gyllenhaal?� asks Curtis, looking flattered. �Wow! Maybe you�re

right. But now that you�ve raised the issue, that reminds me of something. Didn�t you tell me a while back that you had an uncle who grew organic food? If he makes a lot of money, you might inherit it. I bet that�s what�s behind this organic thing of yours.�

�Not at all,� says Talia. �But I�m glad you mentioned the farmers. Some of these

people have invested every cent they have in growing organic food. If consumers don�t buy it, these poor, hardworking farmers will all go broke. We really can�t let that happen. They�ve put their heart and soul into growing really healthy food for people like you and me. We can�t let them down.�

�And here�s another consideration,� continues Talia. �A basic principle of mo-

rality says that we should help others in need. The owner of this grocery store needs for people to buy his organic produce. Thus, I think we have an obligation to buy it.�

�Ha, ha,� Curtis laughs. �According to that argument we also have a moral obli-

gation to buy the GMO food. I don�t think we have a moral obligation to buy any-thing. But now that I think of it, you�ve always been a bit resistant to anything new. When smartphones came out, you didn�t want one, you didn�t want a flat-screen TV, and now you don�t want GMO corn. You should be more open to scientific developments.�

Talia smirks. �And what about you?� she asks. �You seem to think technology can

solve all our problems. By that line of reasoning, we should have robots serving our every need. Robot doctors, robot lawyers, robot cooks, robot grade-school teachers. But robots will never replace human beings. Human beings have feelings. They have hopes and fears and they love each other. Robots don�t love anything.�

�I know nothing about robots,� says Curtis, �but getting back to corn, just compare

this GMO corn with the organic corn in the other bin. The GMO corn looks larger and more appetizing than the organic. In fact, we do lots of things these days to make vegetables look more appetizing and grow better. For example, we add fertilizer to the soil. Nitrogen is an important ingredient in these fertilizers, and so is phosphorus and potassium.�

�I didn�t realize you knew so much about fertilizer,� Talia says. �I need some fertil-

izer for my flower garden. Could you recommend something?�

�Yes, I could. But in the meantime, shall we get some corn?� Curtis asks as he se-

lects two ears from the GMO bin.

Talia takes one ear from his hand and puts it back. She then crosses over to the

organic bin, tucks an ear under her arm, and smiles at Curtis. �One for you, one for me,� she says.

124Chapter3 Informal Fallacies

IV. Create the following informal fallacies:

1. An appeal to force dealing with an automobile collision

2. An appeal to pity about hiring someone for a job

3. An appeal to fear related to immigration

4. A bandwagon argument dealing with alcoholic beverages

5. An ad hominem abusive related to politics

6. An ad hominem circumstantial about gender identity

7. A tu quoque connected with diet and exercise

8. A red herring dealing with climate change

3.3 Fallacies of Weak Induction

The fallacies of weak induction occur not requisite ability to perceive or recall. The following because the premises are logically irrelevant to the examples illustrate these reasons: conclusion, as is the case with the eight fallacies

the premises provide at least a shred of evidence in ing a sustained nuclear fusion reaction at room temperature. In view of Dr. Bradshaw�s support of the conclusion, but the evidence is not expertise as a physician, we must conclude nearly good enough to cause a reasonable person to that this is indeed true. believe the conclusion. Like the fallacies of relevance, however, the fallacies of weak induction often involve This conclusion deals with nuclear physics, and the emotional grounds for believing the conclusion. premises and conclusion is not strong enough to sup- that the creation of muonic atoms of deu-terium and tritium hold the key to produc-port the conclusion. In each of the following fallacies, of relevance, but because the connection between Dr. Bradshaw, our family physician, has stated

authority is a family physician. Because it is unlikely that

a physician would be an expert in nuclear physics, the

9. Appeal to Unqualified Authority argument commits an appeal to unqualified authority.

(Argumentum ad Verecundiam) David Duke, former Grand Wizard of the Ku We saw in Chapter 1 that an argument from authority Klux Klan, has stated, �Jews are not good is an inductive argument in which an arguer cites the Americans. They have no understanding of authority or testimony of another person in support what America is.� On the basis of Duke�s of some conclusion. The appeal to unqualified authority, we must therefore conclude that authority fallacy is a variety of the argument from the Jews in this country are un-American.

authority and occurs when the cited authority or As an authority, David Duke is clearly biased, so his witness lacks credibility. There are several reasons statements cannot be trusted. why an authority or witness might lack credibility. The person might lack the requisite expertise, might Richard Gonzales, CEO of AbbVie pharma-be biased or prejudiced, might have a motive to lie ceuticals, testified before Congress that or disseminate �misinformation,� or might lack the AbbVie was not inflating the price of its

Section 3.3 Fallacies of Weak Induction 125

most popular drugs, and that the massive be weak regardless of the authority�s qualifica-price increases were justified by increased tions. Many questions in these areas are so hotly development costs. Therefore, we must contested that there is no conventional wisdom an believe what he says, when he spent authority can depend on. twenty times as much on self-enriching

stock buybacks. Appeal to unqualified

If Mr. Gonzales had admitted that the huge increases authority

in the price of its drugs were intended to fund stock Cites A Au A = Arguer buybacks, he would have opened the door to gov-A s ernment regulation. Thus, because Gonzales had a ser Au = Unqualified t s clear motive to lie, we should not believe his implau-authority sible statement to Congress. Conclusion

Old Mrs. Furguson (who is practically

dant stab the victim with a bayonet while 10. Appeal to Ignorance blind) has testified that she saw the defen-

she was standing in the twilight shadows (Argumentum ad Ignorantiam)

100�yards from the incident. Therefore, When the premises of an argument state that noth-members of the jury, you must find the ing has been proved one way or the other about defendant guilty. something, and the conclusion then makes a defi-

Here the witness lacks the ability to perceive what she nite assertion about that thing, the argument com-has testified to, so her testimony is untrustworthy. mits an appeal to ignorance. The issue usually

Of course if an authority is credible, the resulting involves something that is incapable of being proved

argument will contain no fallacy. Example: or something that has not yet been proved. Example:

The county tax collector issued a press release People have been trying for centuries to stating that property tax revenues are higher provide conclusive evidence for the claims of this year than last. Therefore, we conclude that astrology, and no one has ever succeeded. these revenues are indeed higher this year. Therefore, we must conclude that astrology

Normally a county tax collector would be consid- is a lot of nonsense. ered a qualified expert in the area of tax revenues, so Conversely, the following argument commits the assuming the tax collector has no reason to lie, this same fallacy.

argument is inductively strong. People have been trying for centuries to In deciding whether a person is a qualified disprove the claims of astrology, and no one authority, one should keep two important points has ever succeeded. Therefore, we must con-in mind. First, the person might be an author-clude that the claims of astrology are true. ity in more than one field. For example, a chem-The premises of an argument are supposed to ist might also be an authority in biology, or an provide positive evidence for the conclusion. The economist might also be an authority in law. The second point is that there are some areas in which premises of these arguments, however, tell us noth-ing about astrology; rather, they tell us about what practically no one can be considered an authority. certain unnamed and unidentified people have tried Such areas include politics, morals, and religion. unsuccessfully to do. This evidence may provide For example, if someone were to argue that abor-tion is immoral because a certain philosopher or some slight reason for believing the conclusion, but certainly not sufficient reason. religious leader has said so, the argument would

126Chapter3 Informal Fallacies

the mere ability to see and report what one sees is suf-

Appeal to ignorance ficient. Example:

No one has ever seen Mr. Andrews drink

Premise: a glass of wine, beer, or any other alco-

Nobody has proved holic beverage. Probably Mr. Andrews is a

that X is true.

nondrinker.

Because it is highly probable that if Mr. Andrews

were a drinker, somebody would have seen him

Conclusion: drinking, this argument is inductively strong. No

X is false. special qualifications are needed to be able to see

someone take a drink.

The second exception to the appeal to ignorance

of two important exceptions to the appeal to igno- cent until proven guilty. If the prosecutor in a criminal rance. The first stems from the fact that if quali-trial fails to prove the guilt of the defendant beyond fied researchers investigate a certain phenomenon reasonable doubt, counsel for the defense may justifi-These examples do, however, lead us to the first and a few other countries, a person is presumed inno-relates to courtroom procedure. In the United States

within their range of expertise and fail to turn up ably argue that his or her client is not guilty. Example: any evidence that the phenomenon exists, this Members of the jury, you have heard the pros-fruitless search by itself constitutes positive evi-ecution present its case against the defen-dence about the question. Consider, for example, dant. Nothing, however, has been proved the following argument: beyond a reasonable doubt. Therefore, under

Teams of scientists attempted over several the law, the defendant is not guilty.

decades to detect the existence of the lumi- This argument commits no fallacy because �not

The premises of this argument are true. Given the niferous aether, and all failed to do so. There- guilty� means, in the legal sense, that guilt beyond fore, the luminiferous aether does not exist. a reasonable doubt has not been proved. The defendant may indeed have committed the crime of which he or she is accused, but if the prosecutor circumstances, it is likely that the scientists in ques-fails to prove guilt beyond a reasonable doubt, the tion would have detected the aether if in fact it did defendant is considered �not guilty.� exist. Since they did not detect it, it probably does not exist. Thus, we can say that the given argument

is inductively strong (but not deductively valid). 11. Hasty Generalization As for the two arguments about astrology, if the (Converse Accident) attempts to prove or disprove the astrological claims had been done in a systematic way by qualified Hasty generalization is a fallacy that affects experts, the arguments would more likely be good. inductive generalizations. In Chapter 1, we saw Exactly what is required to qualify someone to inves- that an inductive generalization is an argument that tigate astrological claims is, of course, difficult to say. draws a conclusion about all members of a group But as these arguments stand, the premises state noth- from evidence that pertains to a selected sample. ing about the qualifications of the investigators, and so The fallacy occurs when there is a reasonable like-the arguments remain fallacious. lihood that the sample is not representative of the

It is not always necessary, however, that the investi- group. Such a likelihood may arise if the sample is

gators have special qualifications. The kinds of quali- either too small or not randomly selected. Here are fications needed depend on the situation. Sometimes two examples:

Section 3.3 Fallacies of Weak Induction 127

Attorney Michael Avenatti stole thousands On three separate occasions, I drank a of dollars from his client, Stormy Daniels, bottle of Figowitz beer and found it flat and and attorney Jason Kurland ripped off his bitter. Probably I would find every bottle of clients, who were lottery winners, for tens Figowitz beer flat and bitter. of millions. Also, attorney Thomas Giardi

stole millions of dollars in settlement funds Neither of these arguments commits the fallacy from his clients. The conclusion is obvious of hasty generalization, because in neither case is that today�s lawyers are just a bunch there any likelihood that the sample is atypical of of thieves. the group. In the first argument the fact that the

mice died in only two minutes suggests the exis-

The Tesla Model X Plaid electric vehicle tence of a causal connection between eating sub-costs more than $100,000, and so does the stance Z and death. If there is such a connection,

Audi RS e-tron and the Porsche Taycan GTS. it would hold for other mice as well. In the second It must be the case that all electric vehicles example, the fact that the taste of beer typically these days cost more than $100,000. remains constant from bottle to bottle causes the

In these arguments, a conclusion about a whole argument to be strong, even though only three bot-group is drawn from premises that mention only a tles were sampled. few instances. Because such small, atypical samples In the case of large samples, if the sample is not are not sufficient to support a general conclusion, random, it may not be typical of the larger group. each argument commits a hasty generalization. Example:

One hundred thousand voters from Orange

Hasty County, California, were surveyed on their generalization choice for governor, and 68 percent

said they intend to vote for the Republican

Specific case(s) candidate. Clearly the Republican candidate (not representative) will be elected.

Even though the sample cited in this argument is

Generalization large, the argument commits a hasty generalization.

The problem is that Orange County is overwhelm-

General rule ingly Republican, so the mere fact that 68 percent

intend to vote for the Republican candidate is no

The mere fact that a sample is small, however, other words, the survey was not conducted randomly, and for this reason the sample is biased. The need indication of how others in the state intend to vote. In

does not guarantee that it is typical. In the case of Chapter 12 of this book. Hasty generalization is otherwise called �converse small samples, various factors may intervene that accident� because it proceeds in a direction opposite render such a sample typical of the larger group. the other hand, the mere fact that a sample is large for randomness in samples is discussed further in does not necessarily mean that it is atypical. On

Examples: to that of accident. Whereas accident proceeds from the general to the particular, converse accident moves from the particular to the general. The premises cite Ten milligrams of substance Z was fed to four

mice, and within two minutes all four went some characteristic affecting one or more atypical into shock and died. Probably substance Z, in instances of a certain class, and the conclusion then this amount, is fatal to mice in general. applies that characteristic to all members of the class.

128Chapter3 Informal Fallacies

Prominent William of Ockham ca. 1285�1347Logicians

The English philosopher and theologian, William of Ockham, was born in or near the village of Ockham not far from London. Little is known about his childhood, and his biographers are not even certain about the year of his birth, with estimates run-

ning from 1280 to 1290. However, they are certain

that while Ockham was still a small boy, his parents

delivered him to the nearest Franciscan monastery to

be brought up in the monastic way of life. His par-

ents� intentions were realized when he entered the

Franciscan Order and was ordained in 1306.

Ockham studied theology at Oxford, possibly

under Duns Scotus, and he lectured there. He also

studied and taught at the University of Paris, where Ockham/Wikimedia Commons he wrote extensively on theology and philosophy.

Ockham�s theological views generated contro- pope was a heretic and had no business even being versy among theologians of the day, some of whom pope. This got the Avignon Franciscans into a great vehemently opposed him. In 1324, he was called to deal of trouble, and to extricate themselves they Avignon, then the location of the papal court, to purloined several horses and rode out of town in answer charges of�heresy. the middle of the night. Ludwig of Bavaria, the Holy

A panel of scholars had been appointed to Roman Emperor, gave them protection, and Ockham

review the charges made against Ockham, and he lived out the rest of his life in Munich. While there he was obliged to remain at a Franciscan house in Avi- turned his attention to politics and political philoso-gnon throughout the investigation, which lasted phy. He was a staunch advocate of the separation of four years. During this time, the Franciscan minister church and state, claiming that the pope had no right general, Michael of Cesena, was called to Avignon, to intervene in state affairs. The pope retaliated by because he had become embroiled in a controversy excommunicating him. with Pope John XXII over the issue of the poverty Ockham is best known for endorsing a prin-of Jesus and the apostles. Michael held that Jesus ciple of parsimony that has come to be called and the apostles did not own property but instead �Ockham�s razor.� This principle states that, among survived through goodwill offerings of people in the alternative explanations, the simplest one is the best. community. The Franciscans regarded themselves Ockham emphasized the importance of keeping as emulating the model set by Jesus and the apos- the number of entities hypothesized in an explana-tles, but the pope, who lived in luxury, obviously tion to an absolute minimum. In the area of logic, disagreed. he is known for his theory of truth conditions for cat-

Though Ockham had more than enough problems egorical propositions, for work in the foundations of

of his own, the minister general asked him to research inductive reasoning, for preliminary work on a three-the issue to see which position was right�the pope�s valued logic, and for developing a close approxima-or the minister general�s. Ockham ultimately came out tion to what would later come to be known as De on the side of the minister general, claiming that the Morgan�s rule.

Section 3.3 Fallacies of Weak Induction 129

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). between premises and conclusion depends on some and the mistake is based on something other than imagined causal connection that probably does not mere temporal succession. In reference to the exist. Whenever an argument is suspected of com-second argument, success as an executive causes mitting the false cause fallacy, the reader or listener increases in salary�not the other way around� should be able to say that the conclusion depends on The fallacy of variety is committed when what is taken to be the false cause occurs whenever the link cause of something is not really the cause at all 12. False Cause pro causa (�not the cause for the cause�). This

the supposition that so the argument mistakes the cause for the effect. X causes Y , whereas X probably In reference to the third argument, the increase in does not cause Y at all. Examples: crime is, for the most part, only coincidental with

During the past two months, every time that the increase in the number of laws. Obviously, the cheerleaders have worn blue ribbons the mere fact that one event is coincidental with in their hair, the basketball team has been another is not sufficient reason to think that one defeated. Therefore, to prevent defeats in caused the other. the future, the cheerleaders should get rid of A third variety of the false cause fallacy, and those blue ribbons. one that is probably committed more often than Successful business executives are paid either of the others in their pure form, is oversim-salaries in excess of $200,000. Therefore, plified cause. This variety occurs when a multitude the best way to ensure that Ferguson will of causes is responsible for a certain effect but the become a successful executive is to raise his arguer selects just one of these causes and repre-salary to at least $200,000. sents it as if it were the sole cause. Here are some There are more laws on the books today examples:

The first argument depends on the supposition that reduce crime, we must eliminate the laws. schools and high schools has been declining for years. Clearly, our teachers just aren�t doing their job these days. the blue ribbons caused the defeats, the second on committed than ever before. Therefore, to The quality of education in our grade than ever before, and more crimes are being the supposition that a high salary causes success, Today, all of us can look forward to a and the third on the supposition that laws cause longer life span than our parents and crime. In no case is it likely that any causal connec- grandparents. Obviously we owe our tion exists. thanks to the millions of dedicated

The first argument illustrates a variety of the false ensure our health. doctors who expend every effort to

cause fallacy called post hoc ergo propter hoc (�after this, therefore on account of this�). This variety of

the fallacy presupposes that just because one event False cause precedes another event, the first event causes the

second. Obviously, mere temporal succession is not

sufficient to establish a causal connection. Never- Premises theless, this kind of reasoning is quite common and

lies behind most forms of superstition. (Example:

�A black cat crossed my path and later I tripped and Depends on nonexistent or

are bad luck.�) Conclusion The second and third arguments illustrate a sprained my ankle. It must be that black cats really minor causal connection variety of the false cause fallacy called non causa

130Chapter3 Informal Fallacies

In reference to the first argument, the decline increase the likelihood that the next flip will come

in the quality of education is caused by many fac- up tails. tors, including lack of discipline in the home, lack For the gambler�s fallacy to be committed, the of parental involvement, too much time spent on events must be independent or nearly independent. social media and video games, and drug use by Such events include rolls of a pair of fair (unloaded) students. Poor teacher performance is only one dice, spins of a fair roulette wheel, and selections of these factors and probably a minor one at that. of lottery winning numbers. Events are not com-In the second argument, the efforts of doctors pletely independent whenever the skill of the gam-are only one among many factors responsible for bler affects the outcome. Thus, poker, blackjack, our longer life span. Other, more important fac- and horse-race betting provide less-than-perfect tors include a better diet, more exercise, reduced candidates for the gambler�s fallacy. smoking, safer highways, and more-stringent The false cause fallacy is often convincing occupational safety standards. because it is often difficult to determine whether

The oversimplified cause fallacy is usually moti- two phenomena are causally related. A lengthy

vated by self-serving interests. Sometimes the time lapse between the operation of the cause arguer wants to take undeserved credit for him- and the occurrence of the effect can exacerbate self or herself or give undeserved credit to some the problem. For example, the thirty-year inter-movement with which he or she is affiliated. At val between exposure to asbestos and the onset other times, the arguer wants to heap blame on of asbestosis impeded the recognition of a causal an opponent or shift blame from himself or her- connection. Also, when two events are causally self onto some convenient occurrence. Instances of related, determining the degree of relatedness the fallacy can resemble either the post hoc or the may be hard. Thus, there may be some connection non causa pro causa varieties in that the alleged between the electromagnetic field produced by cause can occur either prior to or concurrently high-voltage transmission lines and leukemia, but with the effect. It differs from the other varieties the connection may be extremely slight. Finally, of false cause fallacy in that the single factor when a causal connection is recognized, it may selected for credit or blame is often partly respon- be difficult to determine which is the cause and sible for the effect, but responsible to only a minor which is the effect. For example, an allergic reac-degree. tion may be connected with an episode of anxiety,

The last variety of false cause we will consider but it may be hard to tell if the reaction causes the

is called the gambler�s fallacy. This fallacy is com- anxiety or if the anxiety causes the reaction. mitted whenever the conclusion of an argument The realm of human action constitutes another depends on the supposition that independent area in which causal connections are notoriously events in a game of chance are causally related. difficult to establish. For example, the attorneys Here is an example: for accused murderer Dan White argued that

A fair coin was flipped five times in a row, Twinkies, Coke, and potato chips caused him to and each time it came up heads. Therefore, kill former San Francisco Mayor George Moscone. it is extremely likely that it will come up tails Other attorneys have blamed their clients� crimes on the next flip. on PMS, rap music, childhood abuse, mental dis-

ability, and hallucinations. The complex nature of

In fact, it is no more likely that the coin will come human motivation renders all such causal claims up tails on the next flip than it was on the first flip. difficult to evaluate. The situation may become Each flip is an independent event, so earlier flips even worse when a whole nation of people are have no causal influence on later ones. Thus, the involved. Thus, the recent drop in crime rates has fact that the earlier flips came up heads does not been attributed to �three strikes� laws, but it is

Section 3.3 Fallacies of Weak Induction 131

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). difficult to say whether this or some other factor is Attempts to outlaw pornography threaten really responsible. basic civil rights and should be summarily

One point that should be kept in mind when abandoned. If pornography is outlawed,

establishing causal connections is that statistical cor- censorship of newspapers and news relations by themselves often reveal little about what magazines is only a short step away. is actually going on. For example, if all that we knew After that there will be censorship of

frequently occur together, we might conclude any content of lectures delivered by university professors. Complete mind control by the number of things. We might conclude that both have about smoking and lung cancer was that the two textbooks, political speeches, and the a common cause, such as a genetic predisposition, central government will be the inevitable or we might conclude that lung cancer is a disease result.

contracted early in life and that it manifests itself in Both arguments attempt to persuade the reader or its early stages by a strong desire for tobacco. For-listener that the welfare of society rests on a �slip-tunately, in this case, we have more evidence than a pery slope� and that a single step in the wrong direc-mere statistical correlation. This additional evidence tion will result in an inevitable slide all the way to inclines us to believe that the smoking is a cause of the bottom. the cancer. The slippery slope fallacy can involve various kinds of causality. For example, someone might

false cause fallacy. It occurs when the conclusion down a tall tree would set off a cascade of fall-of an argument rests on an alleged chain reaction ing trees leading to the destruction of the forest. and there is not sufficient reason to think that the These arguments depend on pure physical causal-chain reaction will actually take place. Here is an ity. On the other hand, someone might argue that example: The fallacy of ing would set off a chain reaction leading to the slippery slope is a variety of the destruction of the building, or that chopping 13. Slippery Slope argue that removing a single brick from a build-

starting a rumor about the health of the economy

Immediate steps should be taken to outlaw would set off a chain reaction leading to the col-pornography once and for all. The contin- lapse of the stock market. Such an argument would ued manufacture and sale of pornographic depend on the kind of causality found in interper-material will almost certainly lead to an sonal communications. increase in sex-related crimes such as sexual The following example involves a chain reaction assault and incest. This in turn will gradually of mental causes where wanting one thing leads to erode the moral fabric of society and result wanting another. in an increase in crimes of all sorts. Eventu-

ally a complete disintegration of law and Professor Fallon has asked us to purchase order will occur, leading in the end to the a coffeemaker for her office. But we total collapse of civilization. shouldn�t do that because next she�ll

want a microwave, and then a convection

Because there is no good reason to think that the oven. Then she�ll want a full-sized refrig-mere failure to outlaw pornography will result in erator, a sink with hot and cold water, a all these dire consequences, this argument is fal- dishwasher, and a complete set of expen-lacious. An equally fallacious counterargument is sive china. This will exhaust the budget of as follows: our department.

132Chapter3 Informal Fallacies

Two points to keep in mind about slippery slope is allowed to get a foothold, death at the

are first, in real-life situations, the fallacy is almost hands of a physician will become com-always used to defend the status quo against some monplace. Physicians will think nothing of

listener fear change�to fear it so much that they will time patients will be killed against their will. For these reasons, physician-assisted cling rigidly to the prevailing state of affairs even if kind of attack. The strategy is to make the reader and administering a lethal injection, and in no it is thoroughly immoral. For example, prior to the suicide should never be allowed.

in favor of abolishing slavery. In a famous debate in chain of causes leading from the innocent first step to the final calamity is far from assured. If physician-Cincinnati in 1845, N. L. Rice, a Presbyterian min-assisted suicide were made lawful and the abuses ister, invited his listeners to imagine what would and over again to shut down opposing arguments This argument is probably fallacious because the Civil War slippery slope arguments were used over

happen if slavery were abolished forthright. In no envisioned by this argument began to occur, there is reason to believe that the medical community would time former slaves would be given the right to vote: step in and call for corrective measures. Also, as sus-

Then a colored man might be the next picious deaths began to mount up, law enforcement governor [of states having a black major- would step in. Thus, in critiquing slippery slope ity]; and colored men might constitute their arguments it is always important to examine care-Legislature, and sit on the bench as judges fully the chain of causes and look for weak links. in their courts. Thus the entire administra-

tion in those States would be placed in the

the principles of law and government. * Slippery slope hands of degraded men, wholly ignorant of

As a result, Mr. Rice wants us to believe, the govern- Innocent Disaster ment will collapse. first step The second point to keep in mind is that in every

case of a slippery slope fallacy the chain of causes is Chain reaction (not likely to occur) neither inevitable nor necessary. Somewhere along the line there are one or more links that are likely to fail.

In the case of the anti-abolitionist argument, one weak

link was the supposition that freed slaves, once given 14. Weak Analogy the right to vote, would actually be allowed to vote. As

it turned out, they were not. There were poll taxes, liter- This fallacy affects inductive arguments from acy tests, threats, and intimidation. Another weak link analogy. As we saw in Chapter 1, an argument was the supposition that slaves, once given the right to from analogy is an argument in which the conclu-vote, would run for public office. They did not. sion depends on the existence of an analogy, or

A similar critique applies to contemporary slip- similarity, between two things or situations. The

pery slope arguments used to crush support for fallacy of weak analogy is committed when the physician-assisted suicide. Example: analogy is not strong enough to support the con-

clusion that is drawn. Example:

Allowing a physician to assist a terminally ill

patient to end his life may seem innocent Amber�s dog is similar in many ways to Kyle�s enough to start with. But if this practice cat. Both like being petted, they enjoy being

around people, they beg for food at the

*N. L. Rice, A Debate on Slavery (New York, Wm. H. Moore & Co., dinner table, and they sleep with their owners. 1846), p. 33. Amber�s dog loves to romp on the beach with

Section 3.3 Fallacies of Weak Induction 133

Amber. Therefore, Kyle�s cat probably loves to As an illustration of when the requisite system-romp on the beach with Kyle. atic or causal relation does and does not exist, con-

In this argument the similarities cited between sider the following arguments:

Amber�s dog and Kyle�s cat probably have nothing The flow of electricity through a wire to do with the cat�s attitude toward romping on the is similar to the flow of water through beach. Thus, the argument is fallacious. a pipe. Obviously a large-diameter pipe The basic structure of an argument from anal-will carry a greater flow of water than a ogy is as follows: pipe of small diameter. Therefore, a large-

Entity A has attributes a, b, c, and z. of electricity than a small-diameter wire. diameter wire should carry a greater flow Entity B has attributes a, b, c.

Therefore, entity B probably has attribute The flow of electricity through a wire is z also. similar to the flow of water through a pipe.

Evaluating an argument having this form requires When water runs downhill through a pipe, a two-step procedure: (1) Identify the attributes the pressure at the bottom of the hill is a, b, c, greater than it is at the top. Thus, when . . . that the two entities A and B share, and (2) determine how the attribute electricity flows downhill through a wire, z , mentioned in

the conclusion, relates to the attributes a, b, c, . . . bottom of the hill than at the top. the voltage should be greater at the If some causal or systematic relation exists

between z and a, b, or c, the argument is strong; The first argument is good and the second is fal-otherwise, it is weak. In the given example, the lacious. Both arguments depend on the similarity two entities share the attributes of liking to be between water molecules flowing through a pipe petted, enjoying people, begging for food, and and electrons flow ing through a wire. In both sleeping with their owners. Because it is highly cases there is a systematic relation between the probable that none of these attributes is systemat- diameter of the pipe/wire and the amount of flow. ically or causally related to romping on the beach, In the first argument, this systematic relation pro-the argument is fallacious. vides a strong link between premises and conclu-

sion, and so the argument is a good one. But in

the second argument, a causal connection exists

Weak analogy between difference in elevation and increase in

pressure that holds for water but not for electric-

ity. Water molecules flowing through a pipe are

Premises significantly affected by gravity, but electrons

flowing through a wire are not. Thus, the second

argument is fallacious.

Depends on inadequate The theory and evaluation of arguments from analogy analogy is one of the most complex and elusive

Conclusion subjects in all of logic. Additional material on

arguments from analogy appears in Section 3.5

and Chapter 9 of this text.

134Chapter3 Informal Fallacies

Exercise 3.3

I. Identify the fallacies of weak induction committed by the following arguments, giving a

brief explanation for your answer. If no fallacy is committed, write �no fallacy.�

★1. The Daily News carried an article this morning about three local teenagers who

were arrested on charges of drug possession. Teenagers these days are nothing but a bunch of junkies.

2. If a car breaks down on the freeway, a passing mechanic is not obligated to

render emergency road service. For similar reasons, if a person suffers a heart attack on the street, a passing physician is not obligated to render emergency medical assistance.

3. There must be something to psychical research. Three famous physicists�

Oliver Lodge, James Jeans, and Arthur Stanley Eddington�took it seriously.

★4. The secretaries have asked us to provide lounge areas where they can spend

their coffee breaks. This request will have to be refused. If we give them lounge areas, next they�ll be asking for spas and swimming pools. Then it will be rac-quetball courts, tennis courts, and fitness centers. Expenditures for these facili-ties will drive us into bankruptcy.

5. The accumulation of pressure in a society is similar to the buildup of pres-

sure in a boiler. If the pressure in a boiler increases beyond a critical point, the boiler will explode. Accordingly, if a government represses its people beyond a certain point, the people will rise up in revolt.

6. A few minutes after Governor Harrison finished his speech on television, a

devastating earthquake struck southern Alaska. For the safety of the people up there, it is imperative that Governor Harrison make no more speeches.

★7. No one has ever been able to prove the existence of extrasensory perception.

We must therefore conclude that extrasensory perception is a myth.

8. Dr. Carlos Nobre, the world�s leading Amazon climatologist, says that Amazon

deforestation is an extremely urgent problem. Given Dr. Nobre�s status in the community of scientists, we must conclude that preserving the Amazon rain-forest is indeed an important goal.

9. Beyonce, Shawn Mendes, and Will Smith give loads of money to charitable

causes each year. It must be the case that every celebrity gives large amounts of money to charity every year.

★10. Vladimir Putin claims that the news media in Russia always report the news

accurately, and that there is no pressure by the government to present an offi-cial perspective. Given Putin�s intimate knowledge of the Russian government, we must conclude that news reporting in Russia is fair and objective.

11. Probably no life exists on Venus. Teams of scientists have conducted exhaustive

studies of the planet�s surface and atmosphere, and no living organisms have been found.

Section 3.3 Fallacies of Weak Induction 135

12. We don�t dare let the animal-rights activists get their foot in the door. If they

sell us on the idea that dogs, cats, and dolphins have rights, next it will be chick-ens and cows. That means no more chicken Kiev or beef Wellington. Next it will be worms and insects. This will lead to the decimation of our agricultural industry. The starvation of the human race will follow close behind.

★13. No one should buy clothing at a department store without first trying it on.

Accordingly, no one should buy food at a grocery store without first tasting it.

14. No one has proved conclusively that America�s nuclear power plants constitute

a danger to people living in their immediate vicinity. Therefore, it is perfectly safe to continue to build nuclear power plants near large metropolitan centers.

15. There are more churches in New York City than in any other city in the nation,

and more crimes are committed in New York City than anywhere else. So, if we are to eliminate crime, we must abolish the churches.

II. Answer �true� or �false� to the following statements:

1. If an arguer cites a statement by a recognized expert in support of a conclusion

and the statement falls within the expert�s range of expertise, then the arguer commits an appeal to unqualified authority.

2. If an arguer cites a statement in support of a conclusion and the statement

reflects the strong bias of its author, then the arguer commits an appeal to unqualified authority.

3. In the appeal to ignorance, the arguer accuses the reader or listener of being

ignorant.

4. If an attorney for the defense in an American or Canadian criminal trial argues

that the prosecution has proved nothing beyond a reasonable doubt about the guilt of the defendant, then the attorney commits an appeal to ignorance.

5. Hasty generalization always proceeds from the particular to the general.

6. The post hoc ergo propter hoc variety of the false cause fallacy presumes that X

causes Y merely because X happens before Y.

7. If an argument concludes that X causes Y simply because X and Y occur over

the same time interval, then the argument commits the non causa pro causa variety of the false cause fallacy.

8. If the conclusion of an argument depends on the occurrence of a chain

reaction of events, and there is good reason to believe that the chain reaction will actually occur, the argument commits a slippery slope fallacy.

9. The fallacy of weak analogy always depends on an alleged similarity between

two things or situations.

10. If an argument from analogy depends on a causal or systematic relationship

between certain attributes, and there is good reason to believe that this rela-tionship exists, then the argument commits no fallacy.

136Chapter3 Informal Fallacies

III. Identify the fallacies of relevance and weak induction committed by the following argu-

ments. If no fallacy is committed, write �no fallacy.�

★1. I have several friends who have cats for pets. Mitzi loves to attack its owner�s

ankles, Pouncer bites its owner whenever it gets petted, Luna loves to spring out of hiding places and get underfoot, Milo scratches its owner�s face, and Oliver hisses and growls whenever its owner tries to pick it up. Obviously, cats are very obnoxious creatures.

2. Tagging by graffiti artists has become a terrible problem in recent years. Obvi-

ously our schools are stifling the creative spirit of these young people.

3. These equal rights ordinances that supposedly protect gays and transexu-

als from discrimination are the work of the devil! Once these ordinances are enacted, sexual deviants, lurking in the shadows of public restrooms and locker rooms, will spring upon unsuspecting girls and women and perpe-trate all forms of molestation and assault. For the sake of your daughters, your wives, your mothers, and your grandmothers, never say yes to these innocent-sounding pieces of legislation.

★4. Todd Burpo, the best-selling author, argues in his book that heaven is real and

that his son Colton met Jesus and John the Baptist in a near-death experience. But it�s clear that Burpo says these things only to sell lots of books. Therefore, his arguments have no merit.

5. What the farmer sows in the spring he reaps in the fall. In the spring he sows

$8-per-bushel soybeans. Therefore, in the fall he will reap $8-per-bushel soybeans.

6. World-renowned physicist Stephen Hawking claimed that black holes do not

gobble up everything that falls into them without leaving a trace, but that some-thing is always left behind. Given Hawking�s stature as a scientist and the many years he worked on this problem, we should conclude that this is indeed the case.

★7. Emily has bought over 100 tickets on the weekly state lottery, and she has never

won anything. Therefore, the likelihood increases every week that she will win something if she continues to buy tickets.

8. Johnny, of course I deserve the use of your bicycle for the afternoon. After all, I�m

sure you wouldn�t want your mother to find out that you played hooky today.

9. Practically everyone has a fake ID card these days. Therefore, you should have

no qualms about getting one yourself.

★ 10. Ellen Quinn has argued that logic is not the most important thing in life.

Apparently Ellen advocates irrationality. It has taken 2 million years for the human race to achieve the position that it has, and Ellen would throw the whole thing into the garbage. What utter nonsense!

11. When water is poured on the top of a pile of rocks, it always trickles down to

the rocks on the bottom. Similarly, when rich people make lots of money, we can expect this money to trickle down to lower-income individuals.

12. Extensive laboratory tests have failed to prove any deleterious side effects of the new

painkiller lexaprine. We conclude that lexaprine is safe for human consumption.

Section 3.3 Fallacies of Weak Induction 137

★13. Environmentalists accuse us of blocking the plan to convert Antarctica into

a world park. In fact, nothing could be further from the truth. Antarctica is a huge continent teeming with life. It is the home of millions of penguins, seals, sea birds, and sea lions. Also, great schools of finfish and whales inhabit its coastal waters.

14. Michael Moore, the documentary filmmaker, argues that capitalist institu-

tions are ravaging America�s middle class and walking away with billions of dollars in their pockets. But Moore is just a big, fat, populist rabble-rouser, and a multi millionaire to boot. He should keep his mouth shut.

15. The physiology of dogs is very similar to that of humans. They have a liver, a

heart, a digestive system, kidneys, and a central nervous system. Exposure to lead affects the development of humans. Therefore, exposure to lead probably affects the development of dogs.

★16. Student loan debt is becoming a national crisis. Clearly, we should provide free

textbooks to all college students.

17. Kat Kerr, an evangelical preacher from Jacksonville, Florida, says that God told

her that Donald Trump would serve as president for eight consecutive years. Therefore, it follows that Trump did indeed win the 2020 election, just as he claims to have done.

18. Dear Internal Revenue Service: I received a notice that my taxes are being

audited for last year. But you have no right to do this. The deadline for filing a return was April 15, and I filed my tax return on April 12�a full three days before the deadline.

★19. To prevent dangerous weapons from being carried aboard airliners, those seek-

ing to board must pass through a magnetometer or X-ray device and submit to a possible pat-down search. Therefore, to prevent alcohol and drugs from being carried into rock concerts, those entering should submit to similar search procedures.

20. Mr. Flemming�s arguments against the rent-control initiative on the September

ballot should be taken with a grain of salt. As a landlord he would naturally be expected to oppose the initiative.

21. India is suffering a serious drought, thousands of children are dying of star-

vation in their mothers� arms, and homeless people line the streets of the major cities. Surely we must give these unfortunate downtrodden people the chance of bettering their condition in America, the land of wealth and opportunity.

★22. Members of the jury, you have heard Jack Fix testify that the defendant is not

in fact a drug dealer. But Mr. Fix is a known drug dealer himself, and a close friend of the defendant. Also, only a year ago, he was convicted of five counts of perjury. Therefore, you should discount Fix�s testimony.

23. It is ridiculous to hear that man from Peru complaining about America�s pov-

erty. Peru has twice as much poverty as America has ever had.

138Chapter3 Informal Fallacies

24. Folks who argue in favor of amnesty for undocumented immigrants are dead

wrong. Homeland security is of vital importance. National security is neces-sary for a healthy economy and the psychological well-being of all our citizens. We must work to secure our nation from all threats, domestic or foreign.

★25. Nobody has ever proved that immoral behavior by elected officials erodes

public morality. Therefore, we must conclude that such behavior does not erode public morality.

26. Freedom of speech is guaranteed by the First Amendment. Therefore, your

friend was acting within his rights when he shouted �Fire! Fire!� in that crowded theater, even though it was only a joke.

27. No one, on encountering a wristwatch lying on a forest trail, would expect that

it had simply appeared there without having been made by someone. For the same reason, no one should expect that the universe simply appeared without having been made by some intelligent being.

★28. On Monday I drank ten rum and Cokes, and the next morning I woke up with

a headache. On Wednesday I drank eight gin and Cokes, and the next morning I woke up with a headache. On Friday I drank nine bourbon and Cokes, and the next morning I woke up with a headache. Obviously, to prevent further headaches I must give up Coke.

29. Radio host Alex Jones says that the Sandy Hook school shooting, in which twenty-

six people, mostly children, were reportedly killed, was a hoax that was staged. Thus, it must be the case that the Sandy Hook shooting never actually occurred.

30. Some of the parents in our school district have asked that we provide bilingual

education in Spanish. This request will have to be denied. If we provide this service, then someone will ask for bilingual education in Greek. Then it will be German, French, and Hungarian. Polish, Russian, Chinese, Japanese, and Korean will follow close behind. We certainly can�t accommodate all of them.

IV. Identify the fallacies committed in the following dialogue. You should be able to find at

least one case of each fallacy presented in this section and one case of a fallacy presented in the previous section.

Gun Love

Holly and her friend Ben have just attended a panel discussion in response to a recent shooting on a college campus. As they leave the auditorium and head for the parking lot, Holly asks, �Do you agree with any of the panelists?�

�I agree with the panelist who said that guns should be banned on every campus,�

Ben replies. �And I would even go a step further and say that we need strict gun control throughout the United States. The simple fact is that guns are way too available in this country.�

�Sounds like you�re saying that the availability of guns leads to violent crime,� Holly

says. �Is that what you�re saying?�

�Yes, I do think guns lead to crime,� Ben replies.

�Well, let me ask you this,� Holly says. �Has anybody ever proved that? Has

anybody ever proved that the mere availability of guns leads to violent crime?

Section 3.3 Fallacies of Weak Induction 139

The answer is no. So until somebody proves that it does, we should conclude that gun availability does not lead to violence.�

�Well, you heard what one of the panelists said about England, didn�t you?� Ben

asks. �England has strict limitations on gun possession, and practically no gun-related crime. I think that proves that guns do cause crime. If the United States would adopt the English law, crime in this country would drop like a rock.�

�In fact,� Holly counters, �the England example proves nothing of the sort. There�s no

reason to think that what works in England would also work in the United States.�

�Why do you think that?� Ben asks.

�Well,� Holly responds, �you recall what the other panelist said about gun control

in this country. When Evanston, Illinois, banned guns altogether a few years back, that led to no reduction at all in violent crime. That shows that gun availability does not cause crime. If every city in the country followed Evanston�s example, they would get the same result.�

�Sounds like you�re pretty negative on gun control,� Ben observes.

�Yes, I am,� Holly replies. �As soon as we have gun registration, the next thing you

know we�ll need a license to buy a gun. And that will lead the government to impose a tax on gun ownership. Taxes always increase, and when gun owners are no longer able to pay the tax, that will lead to gun confiscation. Finally, when guns are completely removed from the hands of the public, there will be nothing to prevent some wild-eyed fanatic from taking over the government. Complete totalitarianism will follow in no time.�

�Do you really think that could happen in this country?� Ben asks.

�If it happened in Nazi Germany, it could happen here,� she replies. �And, as one of

the panelists said, it did happen in Nazi Germany, and Fascist Italy, too.�

�So, do you think you have a right to own a gun?� Ben asks.

�Absolutely,� Holly replies. �The current president of the NRA�his name escapes me

at the moment�says that the right to own a gun, even an assault weapon, is guaranteed by the Constitution. Given that he knows a lot about guns, what he says about them is certainly true.�

�I�ll withhold judgment on that,� says Ben. �But here�s what I think you should do.

You should take a couple deep breaths and calm down. You�ve turned into a gun nut. If people see what�s happened to you, they won�t take any of your arguments seriously.�

�Think what you want about me,� Holly says. �But the next time you�re robbed at

gunpoint, don�t come crying in my direction.�

Ben laughs. �Okay, I promise I won�t. Well, see you around . . . if I survive!�

�See you!� she says.

V. Create the following informal fallacies:

1. An appeal to unqualified authority relating to a psychic

2. An appeal to ignorance about life on other planets

3. A hasty generalization dealing with ageism

4. A post hoc ergo propter hoc connected with getting sick

5. A gambler�s fallacy about playing the lottery

6. A slippery slope fallacy dealing with credit card use

7. A weak analogy connected with caring for a pet

140Chapter3 Informal Fallacies

3.4 Fallacies of Presumption, Ambiguity,

and Illicit Transference

The fallacies of presumption include begging the that inadequate premises provide adequate sup-question, complex question, false dichotomy, and sup- port for the conclusion by leaving out a possibly pressed evidence. These fallacies arise not because the false (shaky) key premise, by restating a possibly premises are irrelevant to the conclusion or provide false premise as the conclusion, or by reasoning insufficient reason for believing the conclusion but in a circle. The Latin name for this fallacy, petitio because the premises presume what they purport to principii, means �request for the source.� The actual prove. Begging the question presumes that the premises source of support for the conclusion is not appar-provide adequate support for the conclusion when in ent, and so the argument is said to beg the ques-fact they do not, and complex question presumes that tion. After reading or hearing the argument, the a question can be answered by a simple �yes,� �no,� or observer is inclined to ask, �But how do you know other brief answer when a more sophisticated answer X?� where X is the needed support. is needed. False dichotomy presumes that an �either . . . or . . .� statement presents jointly exhaustive alternatives

when in fact it does not, and suppressed evidence pre- Begging the question sumes that no important evidence has been overlooked 1. 2. 3. by the premises when in fact it has. Premise Shaky premise Shaky premise

The fallacies of ambiguity include equivoca- Shaky key

tion and amphiboly. These fallacies arise from the premise Conclusion (missing) occurrence of some form of ambiguity in either the Conclusion premises or the conclusion (or both). As we saw in

Section 2.1, an expression is ambiguous if it is sus- Conclusion Conclusion (restates ceptible to different interpretations in a given con-premise) Conclusion text. The words �light� and �bank� are ambiguous, as is the statement �Tuna are biting off the Wash-

ington coast.� When the conclusion of an argument

depends on a shift in meaning of an ambiguous The first, and most common, way of committing word or phrase or on the wrong interpretation of this fallacy is by leaving a possibly false key premise an ambiguous statement, the argument commits a out of the argument while creating the illusion that fallacy of ambiguity. nothing more is needed to establish the conclusion.

The fallacies of illicit transference include Examples:

composition and division. Arguments that commit these fallacies involve the incorrect transference Murder is morally wrong. This being the case, of an attribute from the parts of something onto it follows that abortion is morally wrong.

the whole, or from the whole onto the parts. We know that humans are intended to eat lots

of fruit because the human hand and arm are

15. Begging the Question perfectly suited for picking fruit from a tree.

( It�s obvious that low-income individuals Petitio Principii )

should be given handouts from the govern-

The fallacy of begging the question is com- ment. After all, these people earn less than mitted whenever the arguer creates the illusion the average citizen.

Section 3.4 Fallacies of Presumption, Ambiguity, and Illicit Transference 141

Clearly, terminally ill patients have a right to have come from an intelligent creator?� Of course doctor-assisted suicide. After all, many of the claim that it did come from an intelligent cre-these people are unable to commit suicide ator may well be true, but the burden is on the by themselves. arguer to prove it. Without supporting reasons or

tion �How do you know that abortion is a form The first of these arguments begs the ques- evidence, the argument proves nothing. Yet most people who are predisposed to believe the conclu-sion are likely to accept the argument as a good of murder?� The second begs the question �Does one. The same can be said of most arguments that the structure and function of the human hand and beg the question, and this fact suggests another arm tell us what humans should eat?� And the reason why arguers resort to this fallacy: Such third and fourth beg the questions �Just because arguments tend to reinforce preexisting inclina-low-income individuals earn less than the aver-tions and beliefs. age citizen, does this imply that the government The second form of petitio principii occurs when should give them handouts?� and �Just because the conclusion of an argument merely restates a terminally ill patients cannot commit suicide by possibly false premise in slightly different lan-themselves, does it follow that they have a right to guage. In such an argument, the premise supports a doctor�s assistance?� the conclusion, and the conclusion tends to rein-These questions indicate that something has force the premise. Examples: been left out of the original arguments. Thus, the first argument is missing the premise �Abortion is Capital punishment is justified for the crimes a form of murder�; the�second is missing the prem- of murder and kidnapping because it is quite ise �The structure and function of the human hand legitimate and appropriate that someone and arm tell us what humans should eat�; and so be put to death for having committed such on. These premises are crucial for the soundness hateful and inhuman acts. of the arguments. If the arguer is unable to estab- Anyone who preaches revolution has a vision lish the truth of these premises, then the arguments of the future for the simple reason that if a prove nothing. However, in most cases of begging person has no vision of the future he could the question, this is precisely the reason why such not possibly preach revolution.

cal phraseology such as �of course,� �clearly,� �this ment is �justified� means the same thing as saying that it is �legitimate and appropriate,� and in the being the case,� and �after all,� the arguer hopes to second argument the premise and the conclusion create the illusion that the stated premise, by itself, to establish their truth, and by employing rhetori- In the first argument, saying that capital punish-premises are left unstated. The arguer is not able

provides adequate support for the conclusion when say exactly the same thing. However, by repeating the same thing in slightly different language, the in fact it does not. arguer creates the illusion that independent evi-The same form of begging the question often dence is being presented in support of the con-appears in arguments concerning religious topics to clusion, when in fact it is not. Both arguments justify conclusions about the existence of God, the contain rhetorical phraseology (�hateful and inhu-immortality of the soul, and so on. Example: man,� �simple reason,� and �could not possibly�)

The world in which we live displays an that help effect the illusion. The first argument amazing degree of organization. Obviously begs the question �How do you know that capital this world was created by an intelligent punishment really is legitimate and appropriate?� God. and the second begs the question �How do you

This argument begs the question �How do you know that people who preach revolution really do

know that the organization in the world could only have a vision of the future?�

142Chapter3 Informal Fallacies

The third form of petitio principii involves circu- This argument is valid, but it is unsound because it

lar reasoning in a chain of inferences having a first has a false premise. However, it commits no fallacy premise that is possibly false. Example: because, again, no illusion is created to cover any-

Verizon has the best wireless service. After thing up. Arguments having this form also appear all, their service has the clearest sound. in Chapter 7. And we know this is so because custom- As with these examples, arguments that beg the ers hear better with Verizon service. And question are normally valid. This is easy to see. this follows from the fact that Verizon has Any argument that includes the conclusion as one digital technology. But this is exactly what of the premises is clearly valid, and those forms you would expect given that Verizon has of the fallacy that leave a key premise out of the the best wireless service. argument become valid when that key premise

On encountering this argument, the attentive reader is introduced. The problem with arguments that beg the question is that they are usually unsound, is inclined to ask, �Where does this reasoning begin? or at least not clearly sound, because the prem-What is its source?� Since the argument goes in a ise needed to provide adequate support for the circle, it has no beginning or source, and as a result it conclusion is, at best, of uncertain truth value. proves nothing. Of course, in this example the circu-Because such arguments presume the truth of this larity is rather apparent, so the argument is not likely premise, begging the question is called a fallacy of to convince anyone. Cases in which circular reason-presumption. ing may convince involve long and complex argu-ments having premises that depend on one another

in subtle ways and a possibly false key premise that 16. Complex Question depends on the conclusion. The fallacy of complex question is commit-In all cases of begging the question, the arguer ted when two (or more) questions are asked in the uses some linguistic device to create the illusion that guise of a single question and a single answer is then inadequate premises provide adequate support for given to both of them. Every complex question pre-a conclusion. Without such an illusion, the fallacy sumes the existence of a certain condition. When the is not committed. Thus, the following arguments respondent�s answer is added to the complex ques-commit no fallacy: tion, an argument emerges that establishes the pre-No dogs are cats. sumed condition. Thus, although not an argument Therefore, no cats are dogs. as such, a complex question involves an implicit

London is in England and Paris is in France. argument. This argument is usually intended to trap Therefore, Paris is in France and London is in the respondent into acknowledging something that England. he or she might otherwise not want to acknowledge.

In both of these examples, the premise amounts to Examples: little more than a restatement of the conclusion. Yet Have you stopped cheating on exams?

both arguments are sound because they are valid Where did you hide the marijuana you were and have true premises. No fallacy is committed, smoking? because no illusion is created to make inadequate Let us suppose the respondent answers �yes� to the premises appear as adequate. We will study argu-first question and �under the bed� to the second. ments of this sort in Chapters 4 and 7. Here is another example: The following arguments emerge:

Rome is in Germany or Rome is in Germany. You were asked whether you have stopped Therefore, Rome is in Germany. cheating on exams. You answered, �Yes.�

Section 3.4 Fallacies of Presumption, Ambiguity, and Illicit Transference 143

Therefore, it follows that you have cheated The fallacy of complex question should be distin-in the past. guished from another kind of question known in law You were asked where you hid the marijuana as a leading question. A leading question is one in you were smoking. You replied, �Under the which the answer is in some way suggested in the bed.� It follows that you were in fact smok- question. Whether or not a question is a leading one ing marijuana. is important in the direct examination of a witness

by counsel. Example:

On the other hand, let us suppose that the respon-

dent answers �no� to the first question and Tell us, on April 9, did you see the defendant �nowhere� to the second. We then have the follow- shoot the deceased? ing arguments: (leading question)

Tell us, what did you see on April 9?

You were asked whether you have stopped (straight question) cheating on exams. You answered, �No.�

Therefore, you continue to cheat. Leading questions differ from complex questions You were asked where you hid the marijuana in that they involve no logical fallacies�that is, you were smoking. You answered, �Nowhere.� they do not attempt to trick the respondent into It follows that you must have smoked all admitting something he or she does not want to of it. admit. To distinguish between the two, however,

Obviously, each of the questions is really two questions have been asked. Here are some addi-one sometimes needs to know whether prior questions: tional examples of complex questions:

Did you cheat on exams in the past? If you did

Are you going to be a good little boy and eat

cheat in the past, have you stopped now?

your hamburger?

If respondents are not sophisticated enough to Were you smoking marijuana? If you were Is George Hendrix still telling lies? smoking it, where did you hide it? How long must I put up with your snotty behavior?

identify a complex question when one is put to When are you going to stop talking nonsense? them, they may answer quite innocently and be

trapped by a conclusion that is supported by no

evidence at all; or, they may be tricked into provid- 17. False Dichotomy

ing the evidence themselves. The correct response The fallacy of false dichotomy is committed when lies in resolving the complex question into its com- a disjunctive (�either . . . or . . .�) premise presents ponent questions and answering each separately. two unlikely alternatives as if they were the only

ones available, and the arguer then eliminates the

Complex undesirable alternative, leaving the desirable one as question the conclusion. Such an argument is clearly valid,

A Attempts to trap probably false, the argument is typically unsound. R/L A = Arguer by asking question The fallacy is often committed by children when but since the disjunctive premise is false, or at least

ds R/L = Reader/ arguing with their parents, by advertisers, and by

Respon Listener adults generally. Here are three examples:

Completed

argument Either you let me attend the Polo G concert

or I�ll be miserable for the rest of my life.

144Chapter3 Informal Fallacies

I know you don�t want me to be miserable and leaves it to the reader or listener to supply the for the rest of my life, so it follows that you�ll missing statements: let me attend the concert.

Either you buy me a new iPad, or I won�t be

Either you use Ultra Guard deodorant or you able to take class notes and I�ll flunk out of risk the chance of perspiration odor. Surely college. you don�t want to risk the chance of perspi-

ration odor. Therefore, you will want to use Either I continue to borrow your designer Ultra Guard deodorant. clothes, or I�ll have nothing to wear.

Either we adopt a one-world government , or The missing premise and conclusion are easily regional wars will continue forever. We cer- introduced. tainly can�t tolerate constant war. Therefore,

we must adopt a one-world government. 18. Suppressed Evidence

premise present the only alternatives available, but Chapter 1 explained that a cogent argument is In none of these arguments does the disjunctive in each case the arguer tries to convey that impres- an inductive argument with good reasoning and sion. For example, in the first argument, the arguer true premises. The requirement of true premises

goes to the concert or faces a lifetime of misery, and some important piece of evidence that outweighs the presented evidence and entails a very different tries to convey the impression that he or she either includes the proviso that the premises not ignore that no other alternatives are possible. Clearly, how-

ever, this is not the case. conclusion. If an inductive argument does indeed The fallacious nature of false dichotomy lies in the ignore such evidence, then the argument commits suppressed evidence the fallacy of

creation of an illusion that the disjunctive premise . Consider, for example, the following argument: presents jointly exhaustive alternatives. If it did, the premise would be true of necessity. For example, the Most dogs are friendly and pose no threat to statement �Either Reno is in Nevada, or it is not in people who pet them. Therefore, it would be Nevada� presents jointly exhaustive alternatives and safe to pet the little dog that is approaching is true of necessity. But in the fallacy of false dichot- us now.

omy, not only do the two alternatives fail to be jointly If the arguer ignores the fact that the little dog is exhaustive, but they are not even likely. As a result, the excited and foaming at the mouth (which suggests disjunctive premise is false, or at least probably false. rabies), then the argument commits a suppressed Thus, the fallacy amounts to making a false or prob-evidence fallacy. This fallacy is classified as a fallacy ably false premise appear true. of presumption because it works by creating the pre-Also, if one of the alternatives in the disjunctive sumption that the premises are both true and com-premise is factually true beyond any doubt, then the plete when in fact they are not. fallacy is not committed. For example, the following Perhaps the most common occurrence of the sup-argument is valid and sound: pressed evidence fallacy appears in inferences based on advertisements. Nearly every ad neglects to men-Either Seattle is in Washington, or it is in tion certain negative features of the product adver-Oregon.

Seattle is not in Oregon. tised. As a result, an observer who sees or hears an

False dichotomy is otherwise called �false bifurca- may commit the fallacy of suppressed evidence. For example, the ad for Carl�s Jr. burgers says: Therefore, Seattle is in Washington advertisement and then draws a conclusion from it

tion� and the �either-or fallacy.� Also, in most cases

the arguer expresses only the disjunctive premise �Feed your Happy.�

Section 3.4 Fallacies of Presumption, Ambiguity, and Illicit Transference 145

The ad implies that buying and eating Carl�s it did in 1940. Therefore, the U.S. military is a Jr. burgers will make you happy or increase your less effective fighting force today than it was happiness. What the ad fails to say is that the burg- in 1940.

ers, which are loaded with cholesterol and other The argument ignores the fact that battleships, unhealthy ingredients, will, along with other fast M1 rifles, and horse-drawn howitzers became foods, cause heart disease, stroke, type 2 diabetes, obsolete long ago and have been replaced by obesity, depression, digestive ailments, cancer, and more-effective ships and weaponry. Thus, it is not early death. By ignoring these facts, the argument true that the U.S. military is less effective today based on the ad is fallacious. than it was in 1940. Another rich source of the suppressed evidence fallacy is the biennial U.S. general election in which two candidates, usually a Republican and a Demo-

crat, compete with each other for a single office. Suppressed evidence During the campaign leading up to the election

countless speech makers (including the candidates

themselves) tear down the policies and accom- Ignores stronger evidence that Premises supports a different conclusion plishments of the candidate they oppose only to conclude that the candidate they favor should be elected. Example:

Ladies and gentlemen, Mr. Smith has sup-

ported policies that will bankrupt Medicare Conclusion and Social Security, his budgetary recom-

mendations increase the federal deficit,

and his approach to foreign policy will

destabilize the Middle East. Therefore, you Yet another form of suppressed evidence is com-should vote to elect Jones to this office. mitted by arguers who quote passages out of context

undisputed successes that Smith has achieved. Also, What the speaker fails to mention are the many from sources such as the Bible, the Constitution, and the Bill of Rights to support a conclusion that the passage was not intended to support. Consider, he fails to mention that Jones is even less qualified for example, the following argument against gun than Smith for the office, and that Jones�s policies control: bode even worse for Medicare, Social Security, the federal deficit, and the Middle East. When these The Second Amendment to the Constitution unmentioned facts are taken into account it becomes states that the right of the people to keep clear that it is actually Smith who should be elected. and bear arms shall not be infringed. But

Another way that an arguer can commit the sup- a law controlling handguns would infringe

pressed evidence fallacy is by ignoring important the right to keep and bear arms. There-events that have occurred with the passage of time fore, a law controlling handguns would be that render an inductive conclusion improbable. unconstitutional.

Romney used in a presidential debate with Barack The following argument resembles one that Mitt In fact, the Second Amendment reads, �A well regulated militia being necessary to the security Obama: of a free state, the right of the people to keep and

The U.S. military has fewer battleships, M1 bear arms shall not be infringed.� In other words, rifles, and horse-drawn howitzers today than the constitutional right to keep and bear arms

146Chapter3 Informal Fallacies

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). is in some way related to the preservation of a �right� in two senses. In the first premise �right� means well-regulated militia. Arguably a law controlling morally correct, but in the second it means a just claim handguns that is unrelated to the preservation of a or power. The fourth argument illustrates the ambigu-well-regulated militia could be constitutional. ous use of a relative word. The word �large� means dif-

The suppressed evidence fallacy is similar to the ferent things depending on the context. Other relative

form of begging the question in which the arguer words that are susceptible to this same kind of ambigu-leaves a key premise out of the argument. The differ- ity include �small,� �good,� �bad,� �light,� �heavy,� �dif-ence is that suppressed evidence leaves out a premise ficult,� �easy,� �tall,� and �short.� that requires a different conclusion, while that form of begging the question leaves out a premise that is

needed to support the Equivocationstated conclusion. However, because both fallacies proceed by leaving a premise

out of the argument, there are cases where the two Premises fallacies overlap.

19. Equivocation Word or phrase

used in two senses

The fallacy of equivocation occurs when the

conclusion of an argument depends on the fact Conclusion that a word or phrase is used, either explicitly

or implicitly, in two different senses in the argu-

ment. Such arguments are either invalid or have a

false premise, and in either case they are unsound. To be convincing, an argument that commits an Examples: equivocation must use the equivocal word in ways that

Some triangles are obtuse. Whatever is are subtly related. Of the examples just given, only the obtuse is ignorant. Therefore, some triangles third might fulfill this requirement. Since both uses are ignorant. of the word �right� are related to ethics, the unalert

observer may not notice the shift in meaning. Another

Any law can be repealed by the legislative technique is to spread the shift in meaning out over the authority. But the law of gravity is a law. course of a lengthy argument. Political speech makers Therefore, the law of gravity can be repealed often use phrases such as �equal opportunity,� �gun by the legislative authority. control,� �national security,� and �environmental pro-We have a duty to do what is right. We tection� in one way at the beginning of a speech and in have a right to speak out in defense of the quite another way at the end. A third technique con-innocent. Therefore, we have a duty to speak sists in using such phrases one way in a speech to one out in defense of the innocent. group and in a different way in a speech to an oppos-A mouse is an animal. Therefore, a large ing group. If the same people are not present at both mouse is a large animal. speeches, the equivocation is not detected.

In the first argument �obtuse� is used in two different

of angle, whereas in the second it means dull or stupid. senses. In the first premise it describes a certain kind 20. Amphiboly The second argument equivocates on the word �law.� The fallacy of amphiboly occurs when the arguer In the first premise it means statutory law, and in the misinterprets an ambiguous statement and then second it means law of nature. The third argument uses draws a conclusion based on this faulty interpretation.

Section 3.4 Fallacies of Presumption, Ambiguity, and Illicit Transference 147

The original statement is usually asserted by some- takes place in the biology lecture hall; is it the lec-one other than the arguer, and the ambiguity usually ture or the heart failures? The correct interpreta-arises from a mistake in grammar or punctuation�a tion is probably the former. The ambiguity can be missing comma, a dangling modifier, an ambigu- eliminated by inserting commas (�Professor John-ous antecedent of a pronoun, or some other careless son said that he will give a lecture, about heart arrangement of words. Because of this ambiguity, failure, in the biology lecture hall�) or by moving the statement may be understood in two clearly the ambiguous modifier (�Professor Johnson said distinguishable ways. The arguer typically selects the that he will give a lecture in the biology lecture unintended interpretation and proceeds to draw a hall about heart failure�). Ambiguities of this sort conclusion based on it. Here are some examples: are called syntactical ambiguities.

The tour guide said that standing in Greenwich Two areas where cases of amphiboly cause serious Village, the Empire State Building could problems involve contracts and wills. The drafters of easily be seen. It follows that the Empire these documents often express their intentions in State Building is in Greenwich Village. terms of ambiguous statements, and alternate inter-

pretations of these statements then lead to different

John told Henry that he had made a mistake. conclusions. Examples: It follows that John has at least the courage to admit his own mistakes. Mrs. Hart stated in her will, �I leave my

Professor Johnson said that he will give 500-carat diamond necklace and my pet a lecture about heart failure in the biol- chinchilla to Alice and Theresa.� Therefore, ogy lecture hall. It must be the case that we conclude that Alice gets the necklace and a number of heart failures have occurred Theresa gets the chinchilla. there recently. Mr. James signed a contract that reads, �In

The premise of the first argument contains a dan- exchange for painting my house, I promise gling modifier. Is it the observer or the Empire to pay David $5,000 and give him my new State Building that is supposed to be standing in Cadillac only if he finishes the job by May 1.� Greenwich Village? The factually correct inter- Therefore, since David did not finish until pretation is the former. In the second argument, May 10, it follows that he gets neither the

$5,000 nor the Cadillac.

the pronoun �he� has an ambiguous antecedent; it

can refer either to John or to Henry. Perhaps John In the first example, the conclusion obviously told Henry that Henry had made a mistake. In favors Alice. Theresa is almost certain to argue the third argument, the ambiguity concerns what that the gift of the necklace and chinchilla should

be shared equally by her and Alice. Mrs. Hart

could have avoided the dispute by adding either

Amphiboly �respectively� or �collectively� to the end of the

sentence. In the second example, the conclu-

sion favors Mr. James. David will argue that the

Premises Mentions ambiguous condition that he finish by May 1 affected only

statement the Cadillac and that he therefore is entitled to

the $5,000. The dispute could have been avoided

by properly inserting a comma in the language of

the promise.

Conclusion Misinterprets Amphiboly differs from equivocation in two

that statement important ways. First, equivocation is always

traced to an ambiguity in the meaning of a word or

148Chapter3 Informal Fallacies

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). phrase, whereas amphiboly involves a syntactical Sodium and chlorine, the atomic compo-ambiguity in a statement. The second difference nents of salt, are both deadly poisons. is that amphiboly usually involves a mistake made Therefore, salt is a deadly poison.

by the arguer in interpreting an ambiguous state- In these arguments, the attributes that are trans-ment made by someone else, whereas the ambi- ferred from the parts onto the whole are designated guity in equivocation is typically the arguer�s own by the words �Maria likes,� �excellent,� �invisible,� creation. If these distinctions are kept in mind, and �deadly poison,� respectively. In each case the it is usually easy to distinguish amphiboly from transference is illegitimate, and so the argument is equivocation. Occasionally, however, the two fal- fallacious. lacies occur together, as the following example Not every such transference is illegitimate, how-illustrates: ever. Consider the following arguments:

The Great Western Cookbook recommends Every atom in this teacup has mass. There-that we serve the oysters when thoroughly fore, this teacup has mass. stewed. Apparently the delicate flavor is

enhanced by the intoxicated condition of Every component in this picket fence is the diners. white. Therefore, the whole fence is white.

First, it is unclear whether �stewed� refers to the oys- In each case an attribute (having mass, being white) is transferred from the parts onto the ters or to the diners, and so the argument commits whole, but these transferences are quite legiti-an amphiboly. But if �stewed� refers to the oysters it mate. Indeed, the fact that the atoms have mass means �cooked,� and if it refers to the diners it means is the very reason why the teacup has mass. The �intoxicated.� Thus, the argument also involves an same reasoning extends to the fence. Thus, the equivocation. acceptability of these arguments is attributable, at

least in part, to the legitimate transference of an

21. Composition These examples illustrate the fact that the fal-attribute from parts onto the whole. The fallacy of lacy of composition is indeed an informal fallacy. composition is committed when the conclusion of an argument depends on the errone- It cannot be discovered by a mere inspection of ous transference of an attribute from the parts of the form of an argument�that is, by the mere something onto the whole. In other words, the fal- observation that an attribute is being transferred lacy occurs when it is argued that because the parts from parts onto the whole. In addition, detecting have a certain attribute, it follows that the whole has this fallacy requires a general knowledge of the that attribute, too, and the situation is such that the situation and of the nature of the attribute being attribute in question cannot be legitimately trans- transferred. The critic must be certain that, given ferred from parts to whole. Examples: the situation, the transference of this particular

Maria likes anchovies. She also likes choco- attribute is incorrect. late ice cream. Therefore, it is certain that Further caution is required by the fact that com-she would like a chocolate sundae topped position is sometimes confused with hasty gener-with anchovies. alization. The only time this confusion is possible

is when the �whole� is a class (such as the class of

Each player on this basketball team is an people in a city or the class of trees in a forest), and excellent athlete. Therefore, the team as a the �parts� are the members of the class. In such a whole is excellent. case composition proceeds from the members of the Each atom in this teacup is invisible. There- class to the class itself. Hasty generalization, on the fore, this teacup is invisible. other hand, proceeds from the specific to the general.

Section 3.4 Fallacies of Presumption, Ambiguity, and Illicit Transference 149

be followed. Examine the conclusion of the argu-

Composition ment. If the conclusion is a general statement�

that is, a statement in which an attribute is predi-

cated distributively to each and every member of

Parts a class�the fallacy committed is hasty generaliza-

tion. But if the conclusion is a class statement�

that is, a statement in which an attribute is predi-

cated collectively to a class as a whole�the fallacy

Attribute is is composition. Example: improperly transferred

Less fuel is consumed by a car than by a fire

truck. Therefore, less fuel is consumed in the

United States by cars than by fire trucks.

Whole At first sight this argument might appear to pro-

ceed from the specific to the general and, conse-

quently, to commit a hasty generalization. But in

fact the conclusion is not a general statement at all

Because it is sometimes easy to mistake a state- the whole class of cars uses less fuel than does the whole class of fire trucks (which is false, because but a class statement. The conclusion states that

position can be mistaken for hasty generalization. there are many more cars than fire trucks). Since ment about a class for a general statement, com-

Such a mistake can be avoided if one is careful the attribute of using less fuel is predicated collec-tively, the fallacy committed is composition. to keep in mind the distinction between these two kinds of statements. This distinction falls

back on the difference between the collective and the distributive predication of an attri- 22. Division

bute. Consider the following statements: The fallacy of division is the exact reverse of com-

Fleas are small. position. As composition goes from parts to whole, Fleas are numerous. division goes from whole to parts. The fallacy is

committed when the conclusion of an argument

The first statement is a general statement. The attri- depends on the erroneous transference of an attri-bute of being small is predicated distributively; that bute from a whole (or a class) onto its parts (or is, it is assigned (or distributed) to each and every members). Examples: flea in the class. Each and every flea in the class is

hand, is a statement about a class as a whole, or what its component elements, sodium and chlo-rine, are nonpoisonous. we will call a �class statement.� The attribute of being said to be small. The second statement, on the other Salt is a nonpoisonous compound. Therefore, numerous is predicated collectively; in other words, This airplane was made in Seattle. Therefore, it is assigned not to the individual fleas but to the every component part of this airplane was class of fleas. The meaning of the statement is not made in Seattle. that each and every flea is numerous but that the The Royal Society is over 300 years old. Pro-class of fleas is large. fessor Thompson is a member of the Royal

To distinguish composition from hasty general- Society. Therefore, Professor Thompson is

ization, therefore, the following procedure should over 300 years old.

150Chapter3 Informal Fallacies

In each case the attribute, designated respectively occur only when the �whole� is a class. In such by the words �nonpoisonous,� �made in Seattle,� a case, division proceeds from the class to the and �over 300 years old,� is illegitimately trans- members, whereas accident proceeds from the ferred from the whole or class onto the parts or general to the specific. Thus, if a class statement members. As with the fallacy of composition, is mistaken for a general statement, division may however, this kind of transference is sometimes be mistaken for accident. To avoid such a mistake, legitimate. The following arguments contain no one should analyze the premises of the argument. fallacy: If the premises contain a general statement, the

This teacup has mass. Therefore, the fallacy committed is accident; but if they contain atoms that compose this teacup have a class statement, the fallacy is division. Example:

mass. Stanley Steamers have almost disappeared. This field of poppies is uniformly orange. This car is a Stanley Steamer. Therefore, the individual poppies are Therefore, this car has almost disappeared. orange.

The first premise is not a general statement but a

class statement. The attribute of having almost dis-

Division appeared is predicated collectively. Accordingly,

the fallacy committed is division, not accident.

This example also illustrates how cases of divi-

sion that involve class statements can include a

Whole subtle form of equivocation. In the conclusion,

the word �disappeared� means fading from vision,

as when the lights are turned down; but in the first

Attribute is premise it means rarely seen. The equivocation is improperly transferred a kind of secondary fallacy that results from the

primary fallacy, which is division.

The next example shows how division turns up

in arguments dealing with averages.

Parts

The average American family has 2.5 children.

The Jones family is an average American

family.

Therefore, the Jones family has 2.5 children.

Obviously, one must be acquainted with the situa- The statement �The average American family has tion and the nature of the attribute being transferred 2.5 children� is not a general statement, but rather to decide whether the fallacy of division is actually a class statement. The sense of the statement is not committed. that each and every family has 2.5 children, but that

Just as composition can sometimes be con- the class of families having two parents is reducible

fused with hasty generalization (converse acci- to about 55 percent children and 45 percent adults. dent), division can sometimes be confused with Thus, once again, the fallacy is division, and not accident. As with composition, this confusion can accident.

Section 3.4 Fallacies of Presumption, Ambiguity, and Illicit Transference 151

Exercise 3.4

I. Identify the fallacies of presumption, ambiguity, and illicit transference committed by

the following arguments, giving a brief explanation for your answer. If no fallacy is committed, write �no fallacy.�

★1. Either we require forced sterilization of Third World peoples or the world pop-

ulation will explode and all of us will die. We certainly don�t want to die, so we must require forced sterilization.

2. Every sentence in this paragraph is well written. Therefore, the paragraph is

well written.

3. An athlete is a human being. Therefore, a good athlete is a good human being.

★4. James said that he saw a picture of a beautiful girl stashed in Stephen�s locker.

We can only conclude that Stephen has broken the rules, because girls are not allowed in the locker room.

5. Why is it so difficult for you to reach a decision?

6. Water will quench one�s thirst. Water is composed of hydrogen and oxygen.

Therefore, hydrogen will quench one�s thirst, and so will oxygen.

★7. People who lack humility have no sense of beauty, because everyone who has a

sense of beauty also has humility.

8. Butane is combustible. Therefore, it burns.

9. Twenty years ago, Kung Fong, the great sumo wrestler, could have yanked up

one of the fir trees in the new municipal arboretum with a single pull. There-fore, since Mr. Fong is as strong today as he was then, he could just as easily pull up one of those trees today.

★10. If Thomas gives Marie a ring, then Thomas and Marie will be engaged.

Thomas did give Marie a ring. In fact, he phoned her just the other night. Therefore, Thomas and Marie are engaged.

11. Alex, I heard your testimony in court earlier today. Tell me, why did you lie on

the witness stand?

12. Johnson is employed by the General Services Administration, and everyone

knows that the GSA is the most inefficient branch of the government. There-fore, Johnson must be an inefficient worker.

★13. All men are mortal. Therefore, someday man will disappear from the earth.

14. Each and every cell in this carrot is 90 percent water. Therefore, the entire

carrot is 90 percent water.

15. George said that he was interviewing for a job drilling oil wells in the supervi-

sor�s office. We can only conclude that the supervisor must have an awfully dirty office.

152Chapter3 Informal Fallacies

★16. During the fifty years that Mr. Jones worked, he contributed $90,000 to Social

Security. Now that he is retired, he stands to collect $200,000 from the system. Obviously he will collect a much greater monetary value than he contributed.

17. Either you major in business administration, or you have no chance at a suc-

cessful career. You told me that you aren�t majoring in business administration. It�s sad that you haven�t chosen a path that will lead to success.

18. Either James Franco or Miley Cyrus is a popular singer. James Franco is not a

popular singer. Therefore, Miley Cyrus is a popular singer.

★19. Switzerland is 48 percent Protestant. Heidi Gilsing is a Swiss. Therefore, Heidi

Gilsing is 48 percent Protestant.

20. Picasso is the greatest artist of the twentieth century. We know that this is so

because art critics have described him in these terms. These art critics are correct in their assessment because they have a more keenly developed sense of appreciation than the average person. This is true because it takes a more keenly developed sense of appreciation to realize that Picasso is the greatest artist of the twentieth century.

21. An atomic bomb causes more damage than a conventional bomb. Therefore,

during World War II more damage was caused by atomic bombs than by con-ventional bombs.

★22. Sylvia, I saw you shopping for wine the other day. Incidentally, are you still

drinking excessively?

23. The author warns about numerous computational errors in his accounting

text. Therefore, he must have written it very carelessly.

24. Emeralds are seldom found in this country, so you should be careful not to

misplace your emerald ring.

★ 25. Of course abortion is permissible. After all, a woman has a right to do as she

pleases with her own body.

II. Answer �true� or �false� to the following statements:

1. Arguments that commit the fallacy of begging the question are normally valid.

2. The effect of begging the question is to hide the fact that a premise may not

be�true.

3. The correct way of responding to a complex question is to divide the question

into its component questions and answer each separately.

4. False dichotomy always involves an �either . . . or . . .� statement, at least

implicitly.

5. The fallacy of equivocation arises from a syntactical defect in a statement.

6. The fallacy of amphiboly usually involves the ambiguous use of a single word.

7. Amphiboly usually arises from the arguer�s misinterpreting a statement made

by someone else.

Section 3.4 Fallacies of Presumption, Ambiguity, and Illicit Transference 153

8. The fallacy of composition always proceeds from whole to parts.

9. The fallacy of division always proceeds from parts to whole.

10. A general statement makes an assertion about each and every member of

a class.

11. A class statement makes an assertion about a class as a whole.

12. In the statement �Divorces are increasing,� an attribute is predicated distributively.

13. In the statement �Waistlines are increasing,� an attribute is predicated

distributively.

14. Composition and division involve the distributive predication of an attribute.

15. Equivocation and amphiboly are classified as fallacies of ambiguity.

III. Examine the following arguments in light of the fallacies presented in this chapter,

giving a brief explanation for your answer. If no fallacy is committed, write �no fallacy.�

★1. In his History of the American Civil War, Jeffry Noland argues that the war had

little to do with slavery. However, as a historian from Alabama, Noland could not possibly present an accurate account. Therefore, his arguments should be discounted.

2. Mr. Wilson said that on July 4 he went out on the veranda and watched the

fireworks go up in his pajamas. We conclude that Mr. Wilson must have had an exciting evening.

3. Donald Trump says that the best way of preventing future school shootings is

by arming the teachers. Thus, to prevent future tragedies of this sort, we must make sure that every teacher in the United States is armed with a gun.

★4. A crust of bread is better than nothing. Nothing is better than true love. There-

fore, a crust of bread is better than true love.

5. Every member of the Delta Club is over seventy years old. Therefore, the Delta

Club must be over seventy years old.

6. Of course you should eat Wheaties. Wheaties is the breakfast of champions,

you know.

★7. Surely it�s morally permissible to kill animals for food. If God didn�t want us to

eat animals, he wouldn�t have made them out of meat.

8. If you pursue a career in music, you are likely to end up wealthy. Just look at

Rihanna. She�s worth $600 million, and so is Julio Iglesias. Jay Z is worth $1 billion, and Paul McCartney is worth $1.2 billion.

9. No one has ever proved that the human fetus is not a person with rights. There-

fore, abortion is morally wrong.

★10. Giant pandas are rapidly disappearing. This animal is a giant panda. Therefore,

this animal should disappear any minute now.

154Chapter3 Informal Fallacies

11. When a car breaks down so often that repairs become pointless, the car is

thrown on the junk heap. Similarly, when a person becomes old and diseased, he or she should be mercifully put to death.

12. The twenty-story Carson Building is constructed of concrete blocks. Each and

every concrete block in the structure can withstand an earthquake of 9.5 on the Richter scale. Therefore, the building can withstand an earthquake of 9.5 on the Richter scale.

★13. Childhood obesity is a major problem these days. Obviously our public health

officials have not been doing their job.

14. This administration is not anti-German, as it has been alleged. Germany is

a great country. It has contributed immensely to the world�s artistic treasury. Goethe and Schiller made magnificent contributions to literature, and Bach, Beethoven, Wagner, and Brahms did the same in music.

15. Paul, it was great to see you at the party the other night. Everyone there was

doing crack. Incidentally, how long have you been dealing that stuff?

★16. Mike Sommers, president and CEO of the American Petroleum Institute, says

that natural-gas fracking is perfectly safe and poses no threat to the nation�s drinking water. Therefore we must conclude that fracking is just as safe as he claims it is.

17. Senator Chris Murphy�s arguments for the protection of wilderness areas

should be ignored. Murphy is just another one of those tree-hugging liber-als who supports such legislation only to please the environmental nuts in his home state of Connecticut.

18. Professor Andrews, surely I deserve a B in logic. I know that I have gotten F�s

on all the tests, but if you give me an F for my final grade, I will lose my schol-arship. That will force me to drop out of school, and my poor, aged parents, who yearn to see me graduate, will be grief stricken for the rest of their lives.

★ 19. Molecules are in constant random motion. The Statue of Liberty is composed

of molecules. Therefore, the Statue of Liberty is in constant random motion.

20. Either we have prayer in our public schools or the moral fabric of society will

disintegrate. The choice should be obvious.

21. White sheep eat more than black sheep (because there are more of them).

Therefore, this white sheep eats more than that black sheep.

★ 22. If someone rents a piece of land and plants crops on it, the landlord is never

permitted to come and take those crops for himself when harvest time arrives. Similarly, if couples enlist the services of a surrogate mother to provide them with a baby, the mother should never be allowed to renege on the deal and keep the baby for herself once it is born.

23. Motives and desires exert forces on people, causing them to choose one thing

over another. But force is a physical quantity, governed by the laws of physics. Therefore, human choices are governed by the laws of physics.

Section 3.4 Fallacies of Presumption, Ambiguity, and Illicit Transference 155

24. Each and every brick in the completely brick-faced Wainright Building has a

reddish-brown color. Therefore, the Wainright Building has a reddish-brown color.

★25. Humanitarian groups have argued in favor of housing for the poor. Unfor-

tunately, these high-density projects have been tried in the past and have failed. In no time they turn into ghettos with astronomical rates of crime and delinquency. Clearly, these humanitarian arguments are not what they seem.

26. Pauline said that after she had removed her new mink coat from the shipping

carton she threw it into the trash. We conclude that Pauline has no apprecia-tion for fine furs.

27. We know that induction will provide dependable results in the future because

it has always worked in the past. Whatever has consistently worked in the past will continue to work in the future, and we know that this is true because it has been established by induction.

★28. What goes up must come down. The price of food has been going up for years.

Therefore, it will surely come down soon.

29. Mr. Prime Minister, I am certain you will want to release the members of our

National Liberation Group whom you currently hold in prison. After all, I�m sure you will want to avoid having car bombs go off in the centers of your most heavily populated cities.

30. Recent studies have shown that conventional food has the same vitamins, min-

erals, proteins, and other nutrients as organic food. Therefore, it�s just as good to eat conventional food as organic food.

★31. We�ve all heard the complaint that millions of Americans are without adequate

health care. But America�s doctors, nurses, and hospitals are among the best in the world. Thousands of people come from abroad every year to be treated here. Clearly there is nothing wrong with our health-care system.

32. Former President Donald Trump has argued that good management is essen-

tial to any business. But who is he to talk? Trump�s own mismanagement drove his casinos into bankruptcy five times in twenty-three years.

33. The farmers of our state have asked that we introduce legislation to pro-

vide subsidies for soybeans. Unfortunately, we will have to turn down their request. If we give subsidies to the soybean farmers, then the corn and wheat growers will ask for the same thing. Then it will be the cotton grow-ers, citrus growers, truck farmers, and cattle raisers. In the end, the cost will be astronomical.

★34. The travel brochure states that walking up O�Connell Street, the statue of Par-

nell comes into view. Apparently that statue has no trouble getting around.

35. Criminals are basically stupid, because anyone who isn�t basically stupid

wouldn�t be a criminal.

156Chapter3 Informal Fallacies

36. Professor Glazebrooks�s theory about the origin of the Martian craters is

undoubtedly true. Rudolph Orkin, the great concert pianist, announced his support of the theory in this morning�s newspaper.

★37. Mr. Franklin has lost at the craps table for the last ten throws of the dice. There-

fore, it is extremely likely that he will win on the next throw.

38. Raising a child is like growing a tree. Sometimes violent things, such as cutting

off branches, have to be done to force the tree to grow straight. Similarly, cor-poral punishment must sometimes be inflicted on children to force them to develop properly.

39. Good steaks are rare these days, so don�t order yours well done.

★40. The Book of Mormon is true because it was written by Joseph Smith. Joseph

Smith wrote the truth because he was divinely inspired. We know that Joseph Smith was divinely inspired because the Book of Mormon says that he was, and the Book of Mormon is true.

41. The students attending Bradford College come from every one of the fifty

states. Michelle attends Bradford College. Therefore, Michelle comes from every one of the fifty states.

42. Rhubarb pie is a dessert. Therefore, whoever eats rhubarb pie eats a dessert.

★ 43. The vast majority of car accidents occur within twenty miles of one�s home.

Apparently it is much more dangerous to drive close to home than far away from home.

44. Either you buy me a new BMW, or I won�t be able to get back and forth to

school. I know you want me to go to school, so the choice is clear.

45. Nobody has ever proved that using cell phones causes brain tumors. Therefore,

using cell phones does not cause brain tumors.

★ 46. When I visited Dr. Ames about my cholesterol, she insisted that I go on a statin

drug. She says everybody should be on a statin. And when I saw Dr. Collins for depression, he prescribed Prozac. And when the Prozac gave me nausea, he prescribed Zofran to stop the nausea. Doctors are all the same. They all take their orders from the pharmaceutical industry.

47. Dozens of species of plants and animals are being wiped out every year, even

though we have laws to prevent it. Clearly, we should repeal the Endangered Species Act.

48. People are driving their cars like maniacs tonight. There must be a full moon.

★ 49. A line is composed of points. Points have no length. Therefore, a line has no

length.

50. Are you in favor of the ruinous economic policy of the Democratic Platform

Committee?

IV. Identify the fallacies committed in the following dialogue. You should be able to find

at least one case of each fallacy presented in this section and a few of the fallacies pre-sented in the two previous sections.

Section 3.4 Fallacies of Presumption, Ambiguity, and Illicit Transference 157

Personal Paper Mill

Carly catches sight of Brad as the two head for their class in sociology. �Hi!� she says as she joins him. �Did you get your paper finished?�

�Yes,� he replies. �And given that first-rate work deserves an A, I deserve an A for

this paper.�

�That�s great!� she says.

�But just between you and me, I had my girlfriend write my paper.�

�Don�t tell me that.� Carly looks disgusted. �Do you think that�s moral?�

�Beats me. Why do you always insist on giving moral advice to everyone?�

�The real question is, why do you always ignore moral advice?� she asks. �Well, the way I see it,� Brad says, �there�s really nothing wrong with plagiarism.

After all, students who plagiarize their work usually get better grades. Anyway, I�m just following in the footsteps of most U.S. presidents. They have speech-writers who write most of their speeches. Why shouldn�t I get someone to write my papers?�

�You are so stupid!� Carly replies. �Honestly, I really think everyone should stop

listening to you.�

�Okay, how about this?� he asks. �Either I get someone to write this paper for

me, or I�ll get an F in the class. I can�t afford to get an F; thus, I get someone to write the paper for me.�

�That�s not much better,� Carly replies. �But you should realize that plagiarism is

really dangerous. If you get caught, it could destroy your reputation.�

�I doubt many students would think badly of me,� he says. �Practically everyone

plagiarizes their work nowadays, and if they do it, so can I.�

�It�s not true that all students plagiarize their work,� Carly replies. �In fact, I am

quite certain that none of them do. I certainly don�t, and all of the students I know don�t.�

�Ah, but you do plagiarize your work,� Brad insists.

�What do you mean?� Carly asks.

�Well, every word of your paper appears in the dictionary, so it�s clear that you

plagiarized it from the dictionary.�

�Very funny,� Carly says. �But here�s what I think. I think you should ask Professor

Halvorson for an extension on your paper and write it yourself. She really takes pla-giarism seriously, and she insists that all work submitted to her be original.�

�But this paper is original work,� Brad insists, as they enter the social sciences

building. �My girlfriend has never written a paper like this before, and she didn�t copy it from anyone.�

Carly laughs. �I don�t think that�s what she means by �original.� Incidentally, did

you remember to attach a copy of the integrity pledge to the front of your paper? The academic integrity policy says that all instructors must direct their students to sign their full name on the integrity pledge, which asserts that the work being sub-mitted has not been plagiarized.�

�Yes, I have attached a copy of the pledge document to the paper, and I signed it

�Professor Judy Halvorson,� just as the rule says.�

�How dense can you get?� Carly asks. �You�re not supposed to sign the profes-

sor�s name! You�re supposed to sign your own name.�

�Oh,� says Brad. For the first time he looks concerned. �But that means I would

be lying.�

158Chapter3 Informal Fallacies

�Yes, it would,� Carly agrees, �but I don�t see why that should bother you. Still, I

can assure you of this: If you hand in the paper, you won�t graduate.�

�You�re wrong about that,� Brad counters. �Consider this. I�m a member of the

senior class, and the senior class graduates in May. That means I�ll graduate in May.�

�I wouldn�t bet on it,� Carly says, as they approach their classroom. �You forget

that I�m on the Academic Integrity Committee, and I�ve sworn to uphold the policy on plagiarism.�

�Uh oh.� Brad looks concerned again. �That had slipped my mind. But you won�t

rat on me, will you? Please, please don�t. All my life I have struggled with my sense of self-esteem, and if I get an F on this paper, I will feel absolutely crushed.�

�Rat on you? Well, that depends,� says Carly.

They enter the classroom, and Brad gives Carly a tense smile. �On what?� Carly smiles grimly back. �On whether you hand in the paper your girlfriend wrote.�

V. Create the following informal fallacies:

1. A begging the question connected with religion

2. A complex question dealing with cyberstalking

3. A false dichotomy related to a Tiffany watch

4. An equivocation involving the word �fan�

5. An amphiboly depending on the ambiguous use of a pronoun

6. A composition fallacy dealing with a church choir

3.5 Fallacies in Ordinary Language

This section addresses two topics. The first concerns and the strands of reasoning may have to be dis-the challenge of detecting the fallacies of others in entangled before the fallacies can be named. Yet ordinary language, and the second relates to the goal another problem arises from the fact that argu-of avoiding fallacies in one�s own arguments. ments in ordinary language are rarely presented in

complete form. A premise or conclusion often is

Detecting Fallacies left unexpressed, which may obscure the nature of

the evidence that is presented or the strength of the

Most of the informal fallacies that we have seen link between premises and conclusion. thus far have been clear-cut, easily recognizable Consider, for example, the following letter that instances of a specific mistake. When fallacies occur appeared in a newspaper: in ordinary usage, however, they are often neither

clear-cut nor easily recognizable. The reason is God, I am sick of �women�s rights�! Every that there are innumerable ways of making mis- time one turns on the news we hear about takes in arguing, and variations inevitably occur some form of discrimination against some that may not be exact instances of any specifically poor female who wants to be a fireman�or named fallacy. In addition, one fallacious mode of some �remark� that suggests or implies arguing may be mixed with one or more others, women are inferior to men.

Section 3.5 Fallacies in Ordinary Language 159

I, for one, do not want to be rescued by and mother is quite a different issue. Also, the ref-a �woman fireman,� especially if I am a erence to men hunting for food suggests a possible 6-foot-2 male and she is a 5-foot-6 female. begging of the question: Are we still locked into a Why is it that women find their �role� so �hunter-gatherer� social structure? degrading? What is wrong with being a wife The paragraph about whether women have and mother, staying home while the male proved themselves to be as inventive, capable, and goes out and �hunts for food� and brings it courageous as men begs yet another question: home to his family? Assuming, for the sake of argument, that this is true, I don�t think women have proven them- have women been allowed to occupy roles in soci-selves to be as inventive, as capable (on the ety where such inventiveness, capability, and cou-average) of world leadership, as physically rageousness can be demonstrated? Furthermore, capable, or as �courageous� as men. Most this paragraph commits a red herring fallacy and/ of us have yet to fight a war (the average or misses the point: Even if women have not proved American woman) and let�s face it ladies, this, what does that have to do with the issue? Most who wants to? jobs do not require any high degree of inventiveness Whether a person is female, black, white, or courage or a capacity for world leadership. handicapped�whatever�ability is The paragraph about ability begs yet another what counts in the final analysis. Women question: Is it in fact the case that women have less cannot demand �equality��no one can� ability? I am not aware that anything of the sort has unless it is earned. ever been proved. Finally, the last paragraph throws When push comes to shove and a damsel out another red herring. What does moving pianos is in distress, she is hard-pressed to protect and beating off rapists have to do with most jobs or herself and usually has to be rescued by a the question of equal pay for equal work? man. Until I can move a piano, beat off a Probably the single most important requirement potential robber or rapist, or fight a war, I am for detecting fallacies in ordinary language is alert-quite content to be a woman, thank you. ness. The reader or listener must pay close attention

(Patricia Kelley) to what the arguer is saying. What is the conclusion?

What are the reasons given in support of the con-

This letter presents numerous fallacies. The phrase clusion? Are the reasons relevant to the conclusion? �poor female who wants to be a fireman� suggests Do the reasons support the conclusion? If the reader a mild or listener is half asleep or lounging in that passive, ad hominem abusive, and equating women�s rights in general with the right to be a firefighter drugged-out state that attends much television suggests a straw man. The second paragraph com- viewing, then none of these questions will receive mits another straw man fallacy by supposing that answers. Under those circumstances the reader or the job of firefighter inevitably entails such activities listener will never be able to detect informal falla-as climbing up ladders and rescuing people. Surely cies, and he or she will accept even the worst reason-there are many male firefighters who cannot do this. ing without the slightest hesitation. The same paragraph also can be interpreted as beg-

fighters want the specific job of rescuing tall men? ging the question: Do women who want to be fire- Avoiding Fallacies

The third paragraph throws out a red herring. Why do people commit informal fallacies? Unfortu-

The issue is whether women have the right to be nately, this question allows for no simple, straight-considered for a job of their choice and whether they forward answer. The reasons underlying the com-must be paid as much as a man in the same situation. mission of fallacies are complex and interconnected. Whether there is something wrong with being a wife However, we can identify three factors that lead to

160Chapter3 Informal Fallacies

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). most of the informal mistakes in reasoning. The first combined with an emotional disposition favoring or is intent. Many fallacies are committed intention- opposing some person or thing. The careless mental ally. The arguer may know full well that his or her posture opens the door, so to speak, to fallacious reasoning is defective but goes ahead with it anyway reasoning, and the emotional disposition pushes because of some benefit for himself or herself or the arguer through it. Even people who are thor-some other person. All of the informal fallacies oughly versed in the informal fallacies occasionally we have studied can be used for that purpose, but succumb to the deadly combination of mental care-some of them are particularly well suited to it. These lessness and emotional impetus. For example, argu-include the appeal to force, appeal to pity, appeal to ments such as the following ad hominem abusive can the people, straw man, ad hominem, complex ques- sometimes be heard in the halls of university phi-tion, false dichotomy, and suppressed evidence. losophy departments:

Here is such a case of appeal to force: Professor Ballard�s argument in favor of

I deserve a chocolate sundae for dessert, and restructuring our course offering isn�t worth if you don�t buy me one right now, I�ll start a hoot. But what would you expect from screaming and embarrass you in front of all someone who publishes in such mediocre of the people in this restaurant. journals? And did you hear Ballard�s recent

And here is a case of false dichotomy that conveys lecture on Aristotle? It was total nonsense.

the appearance of being intentionally committed: When people who should know better are con-

Either you take me on a Caribbean cruise, or fronted with the fact that their argument commits I�ll have a nervous breakdown. It�s up to you. a common fallacy, they often admit with embar-

The key to avoiding fallacies that are intentionally rassment that they have not been thinking and then

committed probably lies in some form of moral edu- revise their argument according to logical princi-cation. The arguer must come to realize that using ples. In contrast, people who are not familiar with intellectually dishonest means in order to acquire the distinction between good and fallacious reason-something he or she does not deserve is just another ing will likely deny that there is anything wrong with form of cheating. their argument. Thus, the key to avoiding fallacies

The situation becomes more complicated, how- that arise from mental carelessness lies in develop-

ever, when the sought-after goal is morally justified. ing a thorough familiarity with the informal falla-Arguers sometimes use fallacious reasoning inten- cies, combined with a habitual realization of how tionally to trick a person into doing something that emotions affect people�s reasoning. Everyone should is really for that person�s own good. Here is a false realize that unchecked emotions are an open invi-dichotomy of that sort: tation to illogical reasoning, and they can lead a

person to commit quite blindly every one of the fal-

Given the beneficial consequences of controlled Either you control your eating and get regu- lacies we have studied thus far. lar exercise, or you�ll have a heart attack and The third factor that leads to the commission of die. The choice is yours. informal fallacies is far more difficult to contend with than the first two. It consists in the influence of eating and regular exercise, some moral philoso-what we might call the �worldview� of the arguer. By phers will find nothing wrong with this argument. worldview we mean a cognitive network of beliefs, Others will contend that manipulating someone into attitudes, habits, memories, values, and other ele-doing something violates human dignity. In either ments that conditions and renders meaningful the case, such arguments are logically unacceptable. world in which we live. Beginning in infancy, our The second factor that leads to the commission worldview emerges quietly and unconsciously from of informal fallacies is a careless mental posture enveloping influences�culture, language, gender,

Section 3.5 Fallacies in Ordinary Language 161

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). religion, politics, and social and economic status. as advocates, and when they enter a courtroom they As we grow older, it continues to develop through see themselves going into battle for their clients. In the shaping forces of education and experience. any battle, winning is the most important objec-Once it has taken root, our worldview determines tive. But this viewpoint presupposes that truth and how each of us sizes up the world in which we live. justice are either unattainable in the courtroom or Given a set of circumstances, it indicates what is of secondary importance. Thus, while many law-reasonable to believe and what is unreasonable. yers would evaluate this argument as nonfallacious,

In connection with the construction and evalua- many nonlawyers would reject it as a weak analogy.

tion of arguments, an arguer�s worldview determines For another example, consider the following the answer to questions about importance, relevance, causal inference:

whether a sample is typical or atypical of a group, what After enslaving most of Eastern Europe for nearly fifty years, the evil Soviet empire causal connections, the qualifications of authorities,

can and cannot be taken for granted, and other fac- finally collapsed. Obviously God listened to tors. However, because these determinations inevita-our prayers. bly involve unexamined presuppositions, the arguer�s

worldview can lead to the commission of informal This argument reflects the worldview of many the-

are likely candidates, but the ones especially suscep- tens to prayers, that God is influenced by prayers, tible are appeal to pity, straw man, missing the point, fallacies. All of the fallacies we have studied so far ists. It presupposes that there is a God, that God lis-

that God has the power to alter the course of his-

appeal to unqualified authority, hasty generaliza- tory, and that God does alter the course of history.

ging the question, false dichotomy, and suppressed a good one, the atheist will reject it as a blatant case evidence. tion, false cause, slippery slope, weak analogy, beg- While the theist is likely to consider this argument

of false cause.

Thus, a person with a victim mentality may think To avoid fallacies that arise from the influence of

favorable treatment; an uncritical conservative may tique his or her presuppositions. Doing so inclines cite with complete confidence the authority of con-that his pathetic circumstances really justify some worldviews, the arguer must acknowledge and cri-

the arguer to couch his or her arguments in language

spiracy theorist Alex Jones, a person with a racist that takes those presuppositions into account. The

a handful of Asian, Black, or Hispanic people really intelligently crafted and, it is hoped, more persua-is typical of the larger class; a person with a liberal worldview may conclude that the errant behavior of result is nearly always an argument that is more

sive. However, the task of recognizing and critiquing

worldview may quite innocently distort an oppo- one�s presuppositions is not easy. Presuppositions

life arguer may consider it obvious that the fetus is people are not even aware that they have a world-a person with rights, while a pro-choice arguer may nent�s argument by equating it with fascism; a pro- are intrinsically linked to one�s worldview, and many

view. The reason is that worldviews are formed

take it for granted that the fetus is not a person with through a process that is largely unconscious. Thus, rights, and so on. Consider, for example, the follow- the arguer must first recognize that he or she has a ing argument from analogy: worldview and must then exercise constant vigilance

A court trial is like a professional football over the presuppositions it comprises. game. In a professional football game, the Even after one�s presuppositions have been most important thing is winning. Similarly, in exposed and thoroughly critiqued, however, there a trial, the most important thing is winning. is no guarantee that one�s arguments will agree with

the arguments of others who have critiqued their

This argument is consistent with the worldview worldviews. This is because a person�s worldview

of many, if not most, lawyers. Lawyers are trained reflects the unique perspective that person has on

162Chapter3 Informal Fallacies

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). the world. No two people share exactly the same Now, besides lung cancer and other nasty busi-perspective. Nevertheless, disclosing and critiqu- ness, secondhand smoke causes deafness and ing the presuppositions in one�s worldview lays a impotence. Was secondhand smoke a problem foundation for meaningful communication with when people heated their homes solely by other reasonable arguers, and it provides a context fireplaces? How about those romantic teepees of reasonableness for working out disagreements. with the smoke hole at the top? And what

responsible for most informal fallacies are intent, research about the problems caused by these as well as barbecues. A little cancer with your mental carelessness combined with emotional dis-In summary, the three factors that are probably about fireplaces in new homes? Let�s have some positions, and unexamined presuppositions in the hot dog, anyone? arguer�s worldview. However, these factors rarely (Pat Sharp) occur in isolation. In the vast majority of cases, two This argument seems to commit the fallacy of or all three conspire to produce fallacious reasoning. equivocation. The arguer begins by using �sec-This fact exacerbates the difficulty in avoiding infor- ondhand smoke� to refer to the smoke from burn-mal fallacies in one�s own arguments and in detect- ing tobacco, and then uses the term to refer to the ing fallacies in the arguments of others. smoke from fireplaces, teepee fires, and barbe-

Now let us consider some cases of real-life argu- cues. Smoke from burning tobacco is clearly not

ments in light of the factors we have just discussed. the same thing as smoke from burning wood or All are taken from letters-to-the-editor columns of charcoal. Alternately, the argument might be seen newspapers and magazines. The first relates to affir- to beg the question: �But do people burn tobacco mative action programs: in their fireplaces and barbecues?� These falla-

I�m a nonracist, nonsexist, white male born in cies probably arise either from the intentions of 1969, who has never owned a slave, treated the author or from carelessness in failing to distin-anyone as inferior because of his or her race, or guish the two kinds of secondhand smoke. In either sexually harassed a female co-worker. In other event, the author is probably hostile to government words, I don�t owe women or minorities a thing. efforts to control secondhand tobacco smoke in con-Since when are people required to pay for the fined areas. sins of their predecessors simply because they The next argument deals with gun control:

belong to the same race or gender? Detroit, the seventh largest city and one with (Ben Gibbons) strict gun laws, had 596 homicides last year. In

The author of this argument presupposes that the same year Phoenix, the ninth largest city

efited him in any way. Among other things, he owners to be licensed, had 136 homicides. Criminals don�t fear the toothless criminal-jus-presupposes that his white ancestors in no way racist and sexist patterns in society have not ben- and one that at the time did not require gun benefited from their being white and that none of tice system, but they do fear armed citizens. these benefits passed down to him. On the other (Paul M. Berardi) hand, given that he has received such benefits, he This argument commits a false cause fallacy. The may presuppose that he is not obligated to pay any author presupposes that the availability of guns of them back. Of course, none of these things may caused Phoenix to have a lower homicide rate than have occurred, but the author should at least address Detroit. The arguer also presupposes that Detroit these issues. Because he does not address them, the and Phoenix are comparable as to homicide rate argument begs the question. merely because they are roughly the same size. As

The next argument relates to secondhand smoke a result, the argument involves a weak analogy and

from cigarettes: also begs the question. The additional factors of

Section 3.5 Fallacies in Ordinary Language 163

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). emotion and intent may also be present. The arguer reflect an emotional disposition in favor of religious probably hates the prospect of gun control, and fundamentalism. he may be fully aware of the fact that Phoenix and The last argument we will consider relates to Eng-Detroit are not comparable for his purpose, but he lish as the official U.S. language:

went ahead with the comparison anyway. This great country has been held together The next argument deals with religious for more than 200 years because of one fundamentalism: simple thing: the English language. There

If we compromise God�s word, we compro- are two things we must do: Make English the mise the truth. To say that the fundamental- official language of the United States and do ist is a loud shrill voice drowning out religious away with bilingual education. moderation implies that diluted truth is (David Moisan)

better than absolute truth. This argument misses the point. The arguer pre-

(Gerald Gleason) supposes that making English the official language

This argument begs the question. The arguer pre- would guarantee that all citizens speak it and that supposes that there is a God, that God has spoken, doing away with bilingual education would accel-that God has revealed his intentions to fundamen- erate the learning process of immigrant children. talist preachers, and that those preachers accurately The argument may also reflect a fear of changes in report the word of God. The argument also seems to U.S. society as a result of recent immigration.

Exercise 3.5

I. Most of the following selections were taken from letters-to-the-editor columns of news-

papers and magazines. Identify any fallacies that may be committed, giving a brief explanation for your answer. Then, if a fallacy is identified, discuss the possible factors that led the arguer to commit the fallacy.

★1. Exporting cigarettes [to Asia] is good business for America; there is no

reason we should be prohibited from doing so. Asians have been smoking for decades; we are only offering variety in their habit. If the Asians made tobacco smoking illegal, that would be a different situation. But as long as it is legal, the decision is up to the smokers. The Asians are just afraid of American supremacy in the tobacco industries.

(Pat Monohan)

2. When will these upper-crust intellectuals realize that the masses of work-

ing people are not in cozy, cushy, interesting, challenging, well-paying jobs, professions and businesses? My husband is now fifty-one; for most of the last thirty-three years he has worked in the same factory job, and only the thought of retiring at sixty-two has sustained him. When he reaches that age in eleven years, who will tell him that his aging and physically wracked body must keep going another two years? My heart cries out for all the poor souls who man the assembly lines, ride the trucks or work in the fields or mines, or in the poorly ventilated, hot-in-summer, cold-in-winter factories and garages. Many

164Chapter3 Informal Fallacies

cannot afford to retire at sixty-two, sixty-five, or even later. Never, never let them extend the retirement age. It�s a matter of survival to so many.

(Isabel Fierman)

3. Women in military combat is insane. No society in its right mind would have

such a policy. The military needs only young people and that means the only women who go are those in their childbearing years. Kill them off and society will not be able to perpetuate itself.

(Jack Carman)

★ 4. Dear Ann: I�ve read that one aspirin taken every other day will reduce the risk

of heart attack. Why not take two and double the protection?

(Boston)

5. The American Civil Liberties Union did a study that found that in the last

eighty years it believes twenty-five innocent people have been executed in the United States. This is unfortunate. But, there are innocent people who die each year in highway accidents. Out of 40,000 deaths, how many deaths are related to driving while intoxicated? How many more thousands are injured and incur financial ruin or are invalids and handicapped for the remainder of their lives?

(Mahlon R. Braden)

6. Mexico�s president expresses legitimate concern when he questions supplying

oil to Americans who are unwilling to apply �discipline� in oil consumption. In view of the fact that his country�s population is expected to double in only twenty-two years, isn�t it legitimate for us to ask when Mexicans will apply the discipline necessary to control population growth and quit dumping their excess millions over our borders?

(Wayne R. Bartz)

★ 7. A parent would never give a ten-year-old the car keys, fix him or her a martini,

or let him or her wander at night through a dangerous part of town. The same holds true of the Internet. Watch what children access, but leave the Net alone. Regulation is no substitute for responsibility.

(Bobby Dunning)

8. How would you feel to see your children starving and have all doors slammed

in your face? Isn�t it time that all of us who believe in freedom and human rights stop thinking in terms of color and national boundaries? We should open our arms and hearts to those less fortunate and remember that a time could come when we might be in a similar situation.

(Lorna Doyle)

9. A capital gains tax [reduction] benefits everyone, not just the �rich,� because

everyone will have more money to invest or spend in the private economy, resulting in more jobs and increasing prosperity for all. This is certainly better than paying high taxes to a corrupt, self-serving, and incompetent government that squanders our earnings on wasteful and useless programs.

(David Miller)

Section 3.5 Fallacies in Ordinary Language 165

★10. After reading �Homosexuals in the Churches,� I�d like to point out that I don�t

know any serious, capable exegetes who stumble over Saint Paul�s denunciation of homosexuality. Only a fool (and there seem to be more and more these days) can fail to understand the plain words of Romans, Chapter one. God did not make anyone �gay.� Paul tells us in Romans 1 that homosexuals become that way because of their own lusts.

(LeRoy J. Hopper)

11. When will they ever learn�that the Republican party is not for the people who

voted for it?

(Alton L. Stafford)

12. Before I came to the United States, I was in Berlin where I visited the famous

zoo. In one of the large cages were a lion and a tiger. Both respected each oth-er�s strength. It occurred to me that it was a good illustration of �balance of power.� Each beast followed the other and watched each other�s moves. When one moved, the other did. When one stopped, the other stopped.

In today�s world, big powers or groups of powers are trying to maintain the

status quo, trying to be as strong as or stronger than the other. They realize a conflict may result in mutual destruction. As long as the countries believe there is a balance of power we may hope for peace.

(Emilie Lackow)

★13. Doctors say the birth of a baby is a high point of being a doctor. Yet a medical

survey shows one out of every nine obstetricians in America has stopped deliv-ering babies.

Expectant mothers have had to find new doctors. In some rural areas,

women have had to travel elsewhere to give birth.

How did this happen? It�s part of the price of the lawsuit crisis.

The number of lawsuits Americans file each year is on the rise. Obstetri-

cians are among the hardest hit�almost three out of four have faced a mal-practice claim. Many have decided it isn�t worth the risk.

(Magazine ad by the Insurance Information Institute)

14. The conservative diatribe found in campus journalism comes from the mouths

of a handful of affluent brats who were spoon-fed through their high school years. Put them on an ethnically more diverse campus, rather than a Princeton or a Dartmouth, and then let us see how long their newspapers survive.

(David Simons)

15. I see that our courts are being asked to rule on the propriety of outlawing video

games as a �waste of time and money.�

It seems that we may be onto something here. A favorable ruling would open the door to new laws eliminating show business, spectator sports, cock-tail lounges, the state of Nevada, public education and, of course, the entire federal bureaucracy.

(A. G. Dobrin)

★16. The death penalty is the punishment for murder. Just as we have long jail terms

for armed robbery, assault and battery, fraud, contempt of court, fines for

166Chapter3 Informal Fallacies

speeding, reckless driving and other numerous traffic violations, so must we have a punishment for murder. Yes, the death penalty will not deter murders any more than a speeding ticket will deter violating speed laws again, but it is the punishment for such violation!

(Lawrence J. Barstow)

17. Would you rather invest in our nation�s children or Pentagon waste? The choice

is yours.

(Political ad)

18. If the advocates of prayers in public schools win on this issue, just where

will it end? Perhaps next they will ask for prayers on public transportation? Prayers by government workers before they start their job each day? Or maybe, mandatory prayers in public restaurants before starting each meal might be a good idea.

(Leonard Mendelson)

★19. Pigeons are forced to leave our city to battle for life. Their struggle is an endless

search for food. What manner of person would watch these hungry creatures suffer from want of food and deny them their survival? These helpless birds are too often ignored by the people of our city, with not the least bit of compassion shown to them. Pigeons are God�s creatures just as the so-called human race is. They need help.

(Leslie Ann Price)

20. You take half of the American population every night and set them down in

front of a screen watching people getting stabbed, shot and blown away. And then you expect them to go out into the streets hugging each other?

(Mark Hustad)

21. The fact is that the hype over �acid rain� and �global warming� is just that:

hype. Take, for example, Stephen Schneider, author of Global Warming. In his current �study� he discusses a �greenhouse effect of catastrophic proportions,� yet twenty years ago Schneider was a vocal proponent of the theory of a �new ice age.�

(Urs Furrer)

★22. Just as our parents did for us, my husband and I rely solely on Christian Science

for all the health needs of our four sons and find it invaluable for the quick cure of whatever ailments and contagions they are subject to. One particular heal-ing that comes to mind happened several years ago when our youngest was a toddler. He had a flu-type illness that suddenly became quite serious. We called a Christian Science practitioner for treatment and he was completely well the next morning.

(Ellen Austin)

23. As somebody who has experienced the tragedy of miscarriage�or sponta-

neous abortion�at eight weeks, I greatly resent the position that a fetus is not a baby. I went through the grief of losing a baby, and no one should tell me�otherwise.

(Ann Fons)

Section 3.5 Fallacies in Ordinary Language 167

24. How can we pledge allegiance to the flag of the United States of America and

not establish laws to punish people who burn the flag to make a statement? We are a people who punish an individual who libels another person but will not seek redress from an individual who insults every citizen of this great country by desecrating the flag.

(William D. Lankford)

★25. The notion of �buying American� is as misguided as the notion of buying

Wisconsin, or Oshkosh, Wisconsin, or South Oshkosh, Wisconsin. For the same reasons that Wisconsin increases its standard of living by trading with the rest of the nation, America increases its standard of living by trading with the rest of the world.

(Phillip Smith)

26. We�ve often heard the saying, �Far better to let 100 guilty men go free than

to condemn one innocent man.� What happens then if we apply the logic of this argument to the question, �Is a fetus an unborn human being?� Then is it not better to let 100 fetuses be born rather than to mistakenly kill one unborn human being? This line of reasoning is a strictly humanist argument against abortion.

(James Sebastian)

27. In our society it is generally considered improper for a man to sleep, shower,

and dress amid a group of women to whom he normally would be sexually attracted. It seems to me, then, to be equally unacceptable that a gay man sleep, shower, and dress in a company of men to whom, we assume, he would be no less sexually attracted.

(Mark O. Temple)

★28. I say �bravo� and �right on!� Now we have some real-life humane heroes to

look up to! These brave people [a group of animal liberators] went up against the insensitive bureaucratic technology, and won, saving former pet animals from senseless torture.

If researchers want to experiment, let them use computers, or themselves�

but not former pet animals! I know it�s bad enough they use monkeys and rats, but if those animals are bred knowing nothing else but these Frankensteins abusing them it�s different (but not better) than dogs or cats that have been loved and petted all their lives to suddenly be tortured and mutilated in the name of science. End all animal research! Free all research animals!

Right on, animal liberators!

(Linda Magee)

29. Dear Ann: Recently I was shopping downtown in 20-below-zero weather. A

stranger walked up to me and said, �I wonder how many beautiful rabbits died so you could have that coat?� I noticed she was wearing a down coat, so I asked if the geese they got the down from to make her coat were still alive. She looked surprised. Obviously she had never given it a thought.

If people are so upset about cruelty to animals, why don�t they go after the

folks who refuse to spend the money to have their pets neutered and spayed?

168Chapter3 Informal Fallacies

Thousands of dogs are put to death every year because the animal pounds can�t feed and house them. Talk about cruelty to animals�that�s the best example there is.

(�Baby It�s Cold Outside�)

30. I prayed for the U.S. Senate to defeat the prayer amendment�and it did. There

is a God.

(Richard Carr)

★31. People of the Philippines, I have returned! The hour of your redemption is

here! Rally to me! Let the indomitable spirit of Bataan and Corregidor lead on! As the lines of battle roll forward to bring you within the zone of operations, rise and strike! For future generations of your sons and daughters, strike! Let no heart be faint! Let every arm be steeled! The guidance of divine God points the way! Follow in his name to the Holy Grail of righteous victory!

(General Douglas MacArthur)

32. As the oldest of eleven children (all married), I�d like to point out our com-

bined family numbers more than 100 who vote only for pro-life candidates. Pro-lifers have children, pro-choicers do not.

(Mrs. Kitty Reickenback)

33. I am twelve years old. My class had a discussion on whether police used unnec-

essary force when arresting the people from Operation Rescue.

My teacher is an ex-cop, and he demonstrated police holds to us. They don�t

hurt at all unless the person is struggling or trying to pull away. If anybody was hurt when they were arrested, then they must have been struggling with the officers trying to arrest them.

(Ben Torre-Bueno)

★34. As corporate farms continue to gobble up smaller family farms, they control a

larger percentage of the grain and produce raised in the United States. Some have already reached a point in size where, if they should decide to withhold their grain and produce from the marketplace, spot shortages could occur and higher prices would result. The choice is to pay us family farmers now or pay the corporations later.

(Delwin Yost)

35. If you buy our airline ticket now you can save 60 percent, and that means

60 percent more vacation for you.

(Radio ad)

36. Why all the flap about atomic bombs? The potential for death is always with

us. Of course, if you just want something to worry about, go ahead. Franklin D. Roosevelt said it: �The only thing we have to fear is fear itself.�

(Lee Flemming Reese)

★37. September 17 marked the anniversary of the signing of the U.S. Constitution.

How well have we, the people, protected our rights? Consider what has hap-pened to our private-property rights.

Section 3.5 Fallacies in Ordinary Language 169

�Property has divine rights, and the moment the idea is admitted into soci-

ety that property is not as sacred as the laws of God, anarchy and tyranny begin.� John Quincy Adams, 1767�1848, Sixth President of the United States.

Taxes and regulations are the two-edged sword which gravely threatens the

fabric of our capitalistic republic. The tyranny of which Adams speaks is with us today in the form of government regulators and regulations which have all but destroyed the right to own property. Can anarchy be far behind?

(Timothy R. Binder)

38. Evolution would have been dealt serious setbacks if environmentalists had

been around over the eons trying to save endangered species.

Species are endangered because they just do not fit the bigger picture any

more as the world changes. That�s not bad. It�s just life.

In most cases we have seen the �endangered species� argument is just a ruse;

much deeper motives usually exist, and they are almost always selfish and personal.

(Tom Gable)

39. The problem that I have with the pro-choice supporters� argument is that they

make �choice� the ultimate issue. Let�s face facts. No one has absolute free-dom of choice sanctioned by the law. One can choose to rob a bank, but it�s not lawful. Others can choose to kill their one-year-old child, but it is not legal. Why then should a woman have the legal right to take the life of her unborn child?

(Loretta S. Horn)

★40. If a car or truck kills a person, do politicians call for car control or truck con-

trol? And call in all cars/trucks?

If a child burns down a house do we have match control or child control and

call in all of each?

Gun control and confiscation is equally as pathetic a thought process in an

age of supposed intelligence.

(Pete Hawes)

41. I was incensed to read in your article about the return of anti-Semitism that

New York City Moral Majority Leader Rev. Dan C. Fore actually said that �Jews have a God-given ability to make money, almost a supernatural ability . . .� I find it incredibly ironic that he and other Moral Majority types conve-niently overlook the fact that they, too, pack away a pretty tidy sum themselves through their fund-raising efforts. It is sad that anti-Semitism exists, but to have this prejudice voiced by leaders of religious organizations is deplorable. These people are in for quite a surprise come Judgment Day.

(John R. Murks)

42. I couldn�t help but compare the ducks in Fairbanks, Alaska, who refuse to

fly south for the winter, to our welfare system. Instead of going south, four ducks have set up housekeeping there because people keep feeding them. The biologist dealing with the ducks said that while it�s nice that people are concerned, in the long run it isn�t good for the ducks because it conditions them to stay when they need to get to warmer temperatures. Our welfare system, in

170Chapter3 Informal Fallacies

many instances, conditions people to stay on welfare. Compassion is a wonderful virtue, and we need to help people who need help, but is it compassionate to keep people on welfare who should be developing and using their God-given abilities to do for themselves what the government is doing for them?

(Donna Symes)

★43. Why are people so shocked that Susan Smith apparently chose to kill her chil-

dren because they had become an inconvenience? Doesn�t this occur every day in abortion clinics across the country? We suspect Smith heard very clearly the message many feminists have been trying to deliver about the expendable nature of our children.

(Kevin and Diana Cogan)

44. What�s wrong with kids today? Answer: nothing, for the majority of them.

They are great.

Witness the action of two San Diego teenage boys recently, when the Normal

Heights fire was at its worst. They took a garden hose to the roof of a threat-ened house�a house belonging to four elderly sisters, people they didn�t even know. They saved the house, while neighboring houses burned to the ground.

In the Baldwin Hills fire, two teenage girls rescued a blind, retired Navy

man from sure death when they braved the flames to find him, confused, out-side his burning house. He would probably have perished if they hadn�t run a distance to rescue him.

(Theodore H. Wickham)

45. Now that Big Brother has decided that I must wear a seatbelt when I ride in a

car, how long will it take before I have to wear an inner tube when I swim in my pool, a safety harness when I climb a ladder, and shoes with steel- reinforced toecaps when I carry out the garbage?

(G. R. Turgeon)

★46. Dear Ann: I was disappointed in your response to the girl whose mother used

the strap on her. The gym teacher noticed the bruises on her legs and backside and called it �child abuse.� Why are you against strapping a child when the Bible tells us in plain language that this is what parents should do?

The Book of Proverbs mentions many times that the rod must be used.

Proverbs 23:13 says: �Withhold not correction from the child for if thou beat-est him with the rod he shall not die.� Proverbs 23:14 says: �Thou shalt beat him with the rod and shalt deliver his soul from death.�

There is no substitute for a good whipping. I have seen the results of trying

to reason with kids. They are arrogant, disrespectful and mouthy. Parents may wish for a more �humane� way, but there is none. Beating children is God�s way of getting parents to gain control over their children.

(Davisville, W.Va.)

47. The Fourth Amendment guarantees our right to freedom from unreasonable

search and seizure. It does not prohibit reasonable search and seizure. The matter of sobriety roadblocks to stop drunk drivers boils down to this: Are such roadblocks reasonable or unreasonable? The majority of people answer:

Section 3.5 Fallacies in Ordinary Language 171

�Reasonable.� Therefore, sobriety roadblocks should not be considered to be unconstitutional.

(Haskell Collier)

48. The Supreme Court recently ruled that a police department in Florida did not

violate any rights of privacy when a police helicopter flew over the backyard of a suspected drug dealer and noticed marijuana growing on his property. Many people, including groups like the Anti-Common Logic Union, felt that the sus-pect�s right to privacy outweighed the police department�s need to protect the public at large.

The simple idea of sacrificing a right to serve a greater good should be allowed in certain cases. In this particular case the danger to the public wasn�t extremely large; marijuana is probably less dangerous than regular beer. But anything could have been in that backyard�a load of cocaine, an illegal stock-pile of weapons, or other major threats to society.

(Matt Cookson)

★49. The mainstream press finds itself left behind by talk radio, so they try to minimize

its importance. Americans are finding the true spirit of democracy in community and national debate. Why should we be told what to believe by a news weekly or the nightly news when we can follow public debate as it unfolds on talk radio?

(Adam Abbott)

50. The issue is not whether we should subsidize the arts, but whether anyone should

be able to force someone else to subsidize the arts. You and I are free to give any amount of our money to any artistic endeavor we wish to support. When the government gets involved, however, a group of bureaucrats is given the power to take our money and give it to the arts they wish to support. We are not consulted. That is not a way to promote a responsible culture. That is tyranny.

(Jerry Harben)

51. Who are these Supreme Court justices who have the guts to OK the burning of

our flag?

If the wife or daughter of these so-called justices were raped, could the rapist

be exonerated because he took the First Amendment? That he was just express-ing himself? How about murder in the same situation?

(Robert A. Lewis)

★52. I have one question for those bleeding hearts who say we should not have used

the atomic bomb: If the nation responsible for the Rape of Nanking, the Man-churian atrocities, Pearl Harbor, and the Bataan Death March had invented the bomb first, don�t you think they would have used it? So do I.

(Bill Blair)

53. There is something very wrong about the custom of tipping. When we go to a

store, we don�t decide what a product is worth and pay what we please; we pay the price or we leave. Prices in coffee bars and restaurants should be raised, waiters should be paid a decent wage, and the words �no tipping� should be clearly visible on menus and at counters.

(George Jochnowitz)

172Chapter3 Informal Fallacies

54. Most Americans do not favor gun control. They know that their well-being

depends on their own ability to protect themselves. So-called �assault rifles� are used in few crimes. They are not the weapon of choice of criminals, but they are for people trying to protect themselves from government troops.

(Larry Herron)

★55. Holding a gun, a thief robs John Q. Public of thousands of dollars. Holding

a baby, an unmarried mother robs taxpayers of thousands of dollars. If one behavior is considered a crime, then so should the other.

(Louis R. Ward)

II. Turn to the pages of a newspaper, magazine, or electronic publication and find an

instance of a fallacious argument in the editorials or letters to the editor. Identify the premises and conclusion of the argument and write an analysis at least one paragraph in length, identifying the fallacy or fallacies committed and the factors that may have led the arguer to commit them.

III. Select a topic that you are familiar with, and write a brief argument about that topic that

contains at least three fallacies.

Summary

Fallacy: A defect in an argument that arises from a mistake in reasoning or the creation of an illu-sion that makes a bad argument appear good. There are two kinds of fallacy:

■ Formal fallacy: Detectable by analyzing the form of an argument

■ Informal fallacy: Detectable only by analyzing the content of an argument

Fallacies of Relevance: The premises are not relevant to the conclusion:

■ Appeal to force: Arguer threatens the reader/listener.

■ Appeal to pity: Arguer elicits pity from the reader/listener.

■ Appeal to the people: Arguer incites a mob mentality (direct form) or appeals to our desire for

security, love, or respect (indirect form). This fallacy includes appeal to fear, the bandwagon argument, appeal to vanity, appeal to snobbery, and appeal to tradition.

■ Argument against the person: Arguer personally attacks an opposing arguer by verbally

abusing the opponent (ad hominem abusive), presenting the opponent as predisposed to argue as he or she does (ad hominen circumstantial), or by presenting the opponent as a hypocrite (tu quoque).

Note: For this fallacy to occur, there must be two arguers.

■ Accident: A general rule is applied to a specific case it was not intended to cover.

■ Straw man: Arguer distorts an opponent�s argument and then attacks the distorted argument.

Note: For this fallacy to occur, there must be two arguers.

Summary173

■ Missing the point: Arguer draws a conclusion different from the one supported by the

premises.

Note: Do not cite this fallacy if another fallacy fits.

■ Red herring: Arguer leads the reader/listener off the track.

Fallacies of Weak Induction: The premises may be relevant to the conclusion, but they supply insufficient support for the conclusion:

■ Appeal to unqualified authority: Arguer cites an untrustworthy authority.

■ Appeal to ignorance: Premises report that nothing is known or proved about some subject,

and then a conclusion is drawn about that subject.

■ Hasty generalization: A general conclusion is drawn from an atypical sample.

■ False cause: Conclusion depends on a nonexistent or minor causal connection. This fallacy

has four forms: post hoc ergo propter hoc, non causa pro causa, oversimplified cause, and the gambler�s fallacy.

■ Slippery slope: Conclusion depends on an unlikely chain reaction of causes.

■ Weak analogy: Conclusion depends on a defective analogy (similarity).

Fallacies of Presumption: The premises presume what they purport to prove:

■ Begging the question: Arguer creates the illusion that inadequate premises are adequate by

leaving out a key premise, restating the conclusion as a premise, or reasoning in a circle.

■ Complex question: Multiple questions are concealed in a single question.

■ False dichotomy: An �either . . . or . . . � premise presents two unlikely alternatives as if they

were the only ones available.

■ Suppressed evidence: Arguer ignores important evidence that requires a different

conclusion.

Fallacies of Ambiguity: An ambiguous word, phrase, or statement leads to an incorrect conclusion:

■ Equivocation: Conclusion depends on a shift in meaning of a word or phrase.

■ Amphiboly: Conclusion depends on an incorrect interpretation of an ambiguous statement

made by someone other than the arguer.

Fallacies of Illicit Transference: An attribute is incorrectly transferred from the parts of something onto the whole or from the whole onto the parts:

■ Composition: An attribute is incorrectly transferred from the parts to the whole.

■ Division: An attribute is incorrectly transferred from the whole to the parts.

Fallacies that occur in real-life argumentation may be hard to detect:

■ They may not exactly fit the pattern of the named fallacies.

■ They may involve two or more fallacies woven together in a single passage.

Three factors underlie the commission of fallacies in real-life argumentation:

■ The intent of the arguer (the arguer may intend to mislead someone).

■ Mental carelessness combined with unchecked emotions.

■ Unexamined presuppositions in the arguer�s worldview.

174Chapter3 Informal Fallacies

4Categorical Propositions

4.1 The Components of Categorical Propositions

4.2 Quality, Quantity, and Distribution

4.3 Venn Diagrams and the Modern Square of Opposition

4.4 Conversion, Obversion, and Contraposition

4.5 The Traditional Square of Opposition

4.6 Venn Diagrams and the Traditional Standpoint

4.7 Translating Ordinary Language Statements into Categorical Form

4.1 The Components of�Categorical

Propositions

Learning about categorical propositions is empowering. Suppose someone with whom you are in a relationship questions your com-mitment. You reply, �I do love you. Therefore, it�s not the case that I don�t love you.� Your argument is clearly valid, but as stated, there is no simple way to prove it valid. However, if it is translated into two categorical propositions, the proof is easy. Most single-premise arguments are like this one in regard to proof of validity.

In Chapter 1 we saw that a proposition (or statement�here we are ignoring the distinction) is a sentence that is either true or false. A proposition that relates two classes, or categories, is called a categorical proposition . The classes in question are denoted respectively by the subject term and the predicate term , and the proposition asserts that either all or part of the class denoted by the subject term is included in or excluded from the class denoted by the predicate term. Here are some examples of categorical propositions:

175

Reality TV stars hope for recognition. Many categorical propositions, of course, are not Junk foods do not belong in school cafeterias. in standard form because, among other things, Many of today�s high-tech devices involve they do not begin with the words �all,� �no,� or the metaverse. �some.� In the final section of this chapter we will

develop techniques for translating categorical

Not all romances have a happy ending. propositions into standard form, but for now we Oprah Winfrey publishes magazines. may restrict our attention to those that are already

The first statement asserts that the entire class of in standard form. The words �all,� �no,� and �some� are called reality TV stars is included in the class of people who quantifiers because they specify how much of hope for recognition, the second that the entire the subject class is included in or excluded from class of junk foods is excluded from the class the predicate class. The first form asserts that the of things that belong in school cafeterias, and whole subject class is included in the predicate the third that part of the class of today�s high-class, the second that the whole subject class is tech devices is included in the class of things excluded from the predicate class, and so on. (Inci-that involve the metaverse. The fourth state-dentally, in formal deductive logic the word �some� ment asserts that part of the class of romances always means at least one.) The letters S and P is excluded from the class of things that have a stand, respectively, for the subject and predicate happy ending, and the last statement asserts that terms, and the words �are� and �are not� are called the class that has Oprah Winfrey as its single the copula because they link (or �couple�) the sub- member is included in the class of people who ject term with the predicate term. publish magazines. Consider the following example: Since any categorical proposition asserts that

either all or part of the class denoted by the sub- All members of the American Medical ject term is included in or excluded from the class Association are people holding degrees denoted by the predicate term, it follows that there from recognized academic institutions. are exactly four types of categorical propositions:

(1) those that assert that the whole subject class This standard-form categorical proposition is ana-is included in the predicate class, (2) those that lyzed as follows: assert that part of the subject class is included in

the predicate class, (3) those that assert that the quantifier: all whole subject class is excluded from the predi- subject term: members of the American cate class, and (4) those that assert that part of Medical Association

class. A categorical prop osition that expresses predicate term: people holding degrees from recognized academic these relations with complete clarity is called a the subject class is excluded from the predicate copula: are standard-form categorical proposition institutions . A

categorical proposition is in standard form if and In resolving standard-form categorical propo-only if it is a substitution instance of one of the sitions into their four components, one must keep following four forms: these components separate. They do not overlap.

All S are P. In this regard, note that �subject term� and �predi-No S are P. cate term� do not mean the same thing in logic Some S are P. that � subject� and �predicate� mean in grammar. Some S are not P. The subject of the example statement includes the

176Chapter4 Categorical Propositions

Prominent Alice Ambrose 1906�2001Logicians

Alice Ambrose was born in 1906 in Lexington, Illinois. She was orphaned at age 13, but still managed to attend Millikin University in Decatur and graduate with a major in mathematics and philosophy. After earning a PhD in philosophy

from the University of Wisconsin, where she worked

on Whitehead and Russell�s Principia Mathematica,

she entered a postdoctoral program at Cambridge

University. There she studied under, and became a

close disciple of, the famous philosopher Ludwig

Wittgenstein. (She received a second PhD from that

university in 1938.) In 1937 she accepted a teaching

position in philosophy at Smith College, where she Courtesy Pat Safford remained until her retirement in 1972.

Within a year after arriving at Smith, Ambrose met insight, won widespread recognition by philosophers

and married the philosopher Morris Lazerowitz, with and logicians. whom she coauthored several books and articles. From 1975 to 1976, Ambrose served as president One was a textbook in symbolic logic, commonly of the American Philosophical Association (Easter n called �Ambrose and Lazerowitz� that was used by Division). Interestingly, in that office she was imme-a generation of young philosophers. Other subjects diately preceded by John Rawls, and immediately on which Ambrose did important work include the succeeded by Hillary Putnam. Ambrose was also a foundations of mathematics, finitism in mathematics, dedicated supporter of peace and social justice, and logical impossibility, the jus tification of induction, and she remained active as a speaker and writer until her Wittgenstein�s the ory of proof. Ambrose was a par- death in 2001 at the age of 94. Today, Smith College ticularly lucid writer, and this, combined with her keen sponsors an annual address in her honor.

quantifier �all,� but the subject term does not. Sim- For the sake of uniformity, our discussion will ilarly, the predicate includes the copula �are,� but always restrict the copula to �are� and �are not.� the predicate term does not. The last section of this chapter describes tech-

Two additional points should be noted about niques for translating these alternate forms into

standard-form categorical propositions. The first the two accepted ones. is that the form �All S are not P� is not a standard Originated by Aristotle, the theory of categorical form. This form is ambiguous and can be ren- propositions has constituted one of the core topics in dered as either �No S are P� or �Some S are not logic for over 2,000 years. It remains important even P,� depending on the content. The second point today because many of the statements we make in is that there are exactly three forms of quantifiers ordinary discourse are either categorical propositions and two forms of copulas. Other texts allow the as they stand or are readily translatable into them. various forms of the verb �to be� (such as �is,� �is Standard-form categorical propositions represent not,� �will,� and �will not�) to serve as the copula. an ideal of clarity in language, and a familiarity with

Section 4.1 The Components of�Categorical Propositions 177

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). the relationships that prevail among them provides a may be combined to produce categorical syllogisms, backdrop of precision for all kinds of linguistic usage. a kind of argumentation that is closely related to the In Chapter 5 we will see how categorical propositions most basic forms of human reasoning.

Exercise 4.1

I. In the following categorical propositions identify the quantifier, subject term, copula,

and predicate term.

★1. Some executive-pay packages are insults to ordinary workers.

2. No stressful jobs are occupations conducive to a healthy lifestyle.

3. All oil-based paints are products that contribute to photochemical smog.

★4. Some preachers who are intolerant of others� beliefs are not television

evangelists.

5. All trials in which a coerced confession is read to the jury are trials in which a

guilty verdict can be reversed.

6. Some artificial hearts are mechanisms that are prone to failure.

★7. No sex-education courses that are taught competently are programs that are

currently eroding public morals.

8. Some universities that emphasize research are not institutions that neglect

undergraduate education.

9. All occasions of civil disobedience are protests that should be tolerated.

★10. No fast-food restaurants are establishments that promote public health.

11. Some cases of road rage are episodes that end in tragedy.

12. Some asylum seekers are not travelers who will reach their destinations.

★13. All animal-rights activists are individuals motivated by empathy.

14. No foods that are loaded with antibiotics are meals suitable for humans.

15. Some school uniforms are useful tools for social conditioning.

★16. Some presidential candidates are not threats to democracy.

II. Create two new examples of each of the four kinds of standard-form categorical

propositions. Make sure each has a quantifier, subject term, copula, and predicate term.

178Chapter4 Categorical Propositions

4.2 Quality, Quantity, and Distribution

Quality and quantity are attributes of categorical But categorical propositions have no �qualifier.� In propositions. In order to see how these attributes universal propositions the quality is determined by pertain, it is useful to rephrase the meaning of the quantifier, and in particular propositions it is categorical propositions in class terminology: determined by the copula.

Proposition Particular propositions mean no more and no Meaning in class

All S are P. tion. The statement �Some S are P� does not imply Every member of the S that some notation less than the meaning assigned to them in class nota-class is a member of S are not P, and the statement �Some S are the P class; that is, the not P� does not imply that some S are P. It often hap-S class is included in pens , of course, that substitution instances of these the P class. statement forms are both true. For example, �Some

No S are P. apples are red� is true, as is �Some apples are not red.� No member of the S class

is a member of the P But the fact that one is true does not necessitate that class; that is, the S class is the other be true. �Some zebras are animals� is true excluded from the P class. (because at least one zebra is an animal), but �Some

Some S are P. zebras are not animals� is false. Similarly, �Some tur- At least one member of

the S class is a member keys are not fish� is true, but �Some turkeys are fish� of of the P class. is false. Thus, the fact that one of these statement

affirmative or negative depending on whether it affirms Some S are forms is true does not logically imply that the other At least one member is true, as these substitution instances clearly prove. not P. of the S class is not Since the early Middle Ages the four kinds of cate-a member of of the gorical propositions have commonly been designated P class. by letter names corresponding to the first four vowels The quality of a categorical proposition is either of the Roman alphabet: A , E , I , O . The universal affirmative is called an A proposition , the universal or denies class membership. Accordingly, �All S are P � negative an E proposition , the particular affirma-and �Some S are P � have affirmative quality, and �No tive an I proposition , and the particular negative S are P � and �Some S are not P � have negative qual-an O proposition . Tradition has it that these letters ity. These are called affirmative propositions and were derived from the first two vowels in the Latin negative propositions , respectively. words affirmo (�I affirm�) and nego (�I deny�), thus: The quantity of a categorical proposition is either

ment makes a claim about every member or just someUniversalAE member of the class denoted by the subject term. �All universal or particular, depending on whether the state- n S are P� and �No S are P� each assert something about f every member of the f g S class and thus are universal propositions . �Some S are P� and �Some S are not P� ParticularIO

assert something about one or more members of the S r class and hence are particular propositions. m

Note that the quantity of a categorical proposi- o

tion may be determined through mere inspection

of the quantifier. �All� and �no� immediately imply The material presented thus far in this section universal quantity, while �some� implies particular. may be summarized as follows:

Section 4.2 Quality, Quantity, and Distribution 179

Proposition Letter name Quantity Quality All S are P. A universal affirmative No S are P. E universal negative Some S are P. I particular affirmative Some S are not P. O particular negative

Unlike quality and quantity, which are attri- �distributed term� given above, S is distributed

butes of propositions,distribution is an attribute and P is not. In other words, for any universal of the terms (subject and predicate) of proposi- affirmative (A) proposition, the subject term, tions. A term is said to be distributed if the prop- whatever it may be, is distributed, and the predi-osition makes an assertion about every member cate term is undistributed. of the class denoted by the term; otherwise, it is Let us now consider the universal negative undistributed. Stated another way, a term is dis- (E) proposition. �No S are P� states that the S and tributed if and only if the statement assigns (or P classes are separate, which may be represented as distributes) an attribute to every member of the follows: class denoted by the term. Thus, if a statement

asserts something about every member of the S class, then S is distributed; if it asserts something about every member of the P class, then P is dis-tributed; otherwise SPS and P are undistributed.

Let us imagine that the members of the classes

denoted by the subject and predicate terms of a This statement makes a claim about every member categorical proposition are contained respectively of S and every member of P. It asserts that every in circles marked with the letters �S� and �P.� The member of S is separate from every member of P, meaning of the statement form �All and also that every member of P S are P � may is separate from then be represented by the following diagram: every member of S. Accordingly, by our defini-

tion, both the subject and predicate terms of uni-

versal negative (E) propositions are distributed.

The particular affirmative (I) proposition states

S that at least one member of S is a member of P.

P If we represent this one member of S that we are

certain about by an asterisk, the resulting diagram

The S circle is contained in the P circle, which looks like this: represents the fact that every member of S� is a member of P. (Of course, should S and P repre-sent terms denoting identical classes, the two cir- *S cles would overlap exactly.) As the diagram shows,

�All S are P� makes a claim about every member P of the S class, since the statement says that every member of S is in the P class. But the statement Since the asterisk is inside the P class, it repre-does not make a claim about every member of the sents something that is simultaneously an S and P class, since there may be some members of P a P; in other words, it represents a member of the that are outside of S. Thus, by the definition of S class that is also a member of the P class. Thus,

180Chapter4 Categorical Propositions

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). the statement �Some S are P � makes a claim about subject and predicate terms of a categorical prop-one member (at least) of S and also one member osition may or may not possess, depending on the (at least) of P, but not about all members of either kind of proposition. If the proposition in question class. Hence, by the definition of distribution, nei- is an A type, then the subject term, whatever it ther S nor P is distributed. may be, is distributed. If it is an E type, then both

The particular negative (O) proposition asserts terms are distributed; if an I type, then neither;

that at least one member of S is not a member of P. If and if an O type, then the predicate term. If a we once again represent this one member of S by an certain term is distributed in a proposition, this asterisk, the resulting diagram is as follows: simply means that the proposition says something

about every member of the class that the term

*S denotes. If a term is undistributed, the proposition

does not say something about every member of

the class.

P An easy way to remember the rule for distribu-

tion is to keep in mind that universal (A and E)

Since the other members of S may or may not be statements distribute their subject terms and outside of P, it is clear that the statement �Some negative (E and O) statements distribute their S are not P � does not make a claim about every predicate terms. As an aid to remembering this member of S, so S is not distributed. But, as may arrangement, the following mnemonic may be seen from the diagram, the statement does be useful: �Unprepared Students Never Pass.� assert that every member of P is separate and dis- Attending to the first letter in these words may tinct from this one member of S that is outside the help one recall that Universals distribute Subjects, P circle. Thus, in the particular negative (O) prop- and Negatives distribute Predicates. Another osition, P is distributed and S is undistributed. mnemonic that accomplishes the same purpose is

At this point the notion of distribution may be �Any Student Earning B�s Is Not On Probation.� In

somewhat vague and elusive. Unfortunately, there this mnemonic the first letters may help one recall is no simple and easy way to make the idea graphi- that A statements distribute the Subject, E state-cally clear. The best that can be done is to repeat some ments distribute Both terms, I statements distrib-of the things that have already been said. First of ute Neither term, and O statements distribute the all, distribution is an attribute or quality that the Predicate.

Two mnemonic devices for distribution

�Unprepared Students Never Pass� �Any Student Earning B�s Is Not On Probation�

Universals distribute Subjects. A distributes Subject.

Negatives distribute Predicates. E distributes Both.

I distributes Neither.

O distributes Predicate.

Finally, note that the attribute of distribution, developments in this chapter, is essential to the eval-

while not particularly important to subsequent uation of syllogisms in the next chapter.

Section 4.2 Quality, Quantity, and Distribution 181

The material of this section may now be summarized as follows:

Proposition Letter name Quantity Quality Terms distributed All S are P. A universal affirmative S No S are P. E universal negative S and P Some S are P. I particular affirmative none Some S are not P. O particular negative P

Exercise 4.2

I. For each of the following categorical propositions identify the letter name, quantity,

and quality. Then state whether the subject and predicate terms are distributed or undistributed.

★1. No vampire movies are films without blood.

2. All governments that bargain with terrorists are governments that encourage

terrorism.

3. Some symphony orchestras are organizations on the brink of bankruptcy.

★4. Some Chinese leaders are not thoroughgoing opponents of capitalist economics.

5. All human contacts with benzene are potential causes of cancer.

6. No labor strikes are events welcomed by management.

★7. Some hospitals are organizations that overcharge the Medicare program.

8. Some affirmative action plans are not programs that result in reverse

discrimination.

II. Change the quality but not the quantity of the following categorical propositions:

★1. All drunk drivers are threats to others on the highway.

2. No wildlife refuges are locations suitable for condominium developments.

3. Some slumlords are people who eventually wind up in jail.

★4. Some CIA operatives are not champions of human rights.

5. All identity thieves are criminals who deserve jail time.

6. No college athletes are students who can be paid by equipment manufacturers.

★7. Some stay-at-home parents are workers who deserve Social Security.

8. Some news media are not disseminators of accurate information.

III. Change the quantity but not the quality of the following categorical propositions:

★ 1. All owners of pit bull terriers are people who can expect expensive lawsuits.

2. No tax proposals that favor the rich are fair proposals.

182Chapter4 Categorical Propositions

3. Some grade-school administrators are people who choke the educational

process.

★ 4. Some residents of Manhattan are not people who can afford to live there.

5. All hate crimes are offenses deserving enhanced sentencing.

6. No suicide bombers are individuals with moral sensibilities.

★7. Some prescription medicines are substances hazardous to human health.

8. Some shelter animals are not good prospects for adoption.

IV. Change both the quality and the quantity of the following categorical propositions:

★1. All oil spills are events catastrophic to the environment.

2. No foods containing glyphosate are healthy snacks.

3. Some spring breaks are episodes that end with hangovers.

★4. Some corporate lawyers are not people with a social conscience.

5. All coal-fired electric plants are major sources of pollution.

6. No fashion shows are celebrations of normal clothing.

★7. Some economic forums are venues for airing financial fears.

8. Some paparazzi are not welcome guests at celebrity functions.

V. Create four categorical propositions, having different letter names, about your favorite

food, and identify their quality, quantity, and distributed terms.

4.3 Venn Diagrams and the Modern

Square of Opposition

Existential Import All Tom Cruise�s movies are hits.

The primary goal of our inquiry into categorical prop- All unicorns are one-horned animals. ositions is to disclose the role that such propositions In ordinary conversation the first proposition implies play in the formation of arguments. However, it turns that Tom Cruise has indeed made some movies. In out that we can interpret universal (A and E) prop- other words, the statement has existential import. It ositions in two different ways, and according to one implies that one or more things denoted by the sub-of these interpretations an argument might be valid, ject term actually exist. On the other hand, no such while according to the other it might be invalid. Thus, implication is made by the statement about unicorns. before turning to the evaluation of arguments, we must The statement might be said to be true, because uni-explore the two possible interpretations of universal corns, by definition, have a single horn. But the state-propositions. Our investigation will focus on what is ment does not imply that unicorns actually exist. called existential import. To illustrate this concept, Thus, the question arises: Should universal proposi-consider the following pair of propositions: tions be interpreted as implying that the things talked

Section 4.3 Venn Diagrams and the Modern Square of Opposition 183

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). about actually exist? Or should they be interpreted as other words, existence counts for something. On the implying no such thing? In response to this question, other hand, the Boolean standpoint is �closed� (unre-logicians have taken two different approaches. Aristotle ceptive) to existence. When things exist, the Boolean held that universal propositions about existing things standpoint does not recognize their existence, and have existential import. In other words, such state- universal statements about those things have no exis-ments imply the existence of the things talked about: tential import. Conversely, when things do not exist,

neither the Aristotelian nor the Boolean standpoint

Aristotelian standpoint recognizes any existence. Under these circumstances,

All pheasants Implies the existence the Aristotelian standpoint merges with the Boolean.*

No pine trees ean standpoint only with regard to universal (A and E) Implies the existence propositions. The two standpoints are identical with are birds. of pheasants. The Aristotelian standpoint differs from the Bool-are maples. of pine trees. regard to particular (I and O) propositions. Both the All satyrs are Does not imply the Aristotelian and the Boolean standpoints recognize vile creatures. existence of satyrs. that particular propositions make a positive assertion

about existence. For example, from both standpoints,

The first two statements have existential import the statement �Some cats are animals� asserts that at because their subject terms denote actually existing least one cat exists that is an animal. Also, from both things. The third statement has no existential import, standpoints, �Some fish are not mammals� asserts that because satyrs do not exist. at least one fish exists that is not a mammal. Thus, from

On the other hand, according to the theory devel- both standpoints, the word �some� implies existence.�

oped by the nineteenth-century logician George Adopting either the Aristotelian or the Boolean Boole (as perfected by John Venn), no universal standpoint amounts to accepting a set of ground propositions have existential import. Such statements rules for interpreting the meaning of universal prop-never imply the existence of the things talked about: ositions. Either standpoint can be adopted for any Boolean standpoint categorical proposition or any argument composed

of categorical propositions. Taking the Aristotelian

All trucks Does not imply the standpoint amounts to recognizing that universal are vehicles. existence of trucks. statements about existing things convey evidence

No roses about existence. Conversely, for a statement to Does not imply the

are daisies. convey such evidence, the Aristotelian standpoint existence of roses.

must be taken and the subject of the statement must

We might summarize these results by saying that the All werewolves Does not imply the denote actually existing things. Taking the Boolean are monsters. existence of werewolves. standpoint, on the other hand, amounts to ignoring any evidence about existence that universal state-ments might convey. Aristotelian standpoint is �open� (or receptive) to existence.* When things exist, the Aristotelian stand-

point recognizes their existence, and universal state- *For a more complete account of the relation between the Aristotelian ments about those things have existential import. In and the Boolean standpoints, see �Existential Import: Historical Back-

ground,� by Patrick J. Hurley. This article is available on MindTap.

� In ordinary language, the word �some� occasionally implies something less than actual existence. For example, the statement �Some unicorns

*In general, we interpret this openness to existence as extending to the are tenderhearted� does not seem to suggest that unicorns actually subject class, the predicate class, and the complements of these classes. exist, but merely that among the group of imaginary things called �uni-However, in the present account we confine our attention to the sub-corns,� there is a subclass of tenderhearted ones. In the vast majority ject class. The concept of class complement is discussed in Section 4.4 of cases, however, �some� in ordinary language implies existence. The (Obversion).logical �some� conforms to these latter uses.

184Chapter4 Categorical Propositions

Prominent George Boole 1815�1864Logicians

T he English mathematician and philosopher George Boole is known primarily for the devel-opment of Boolean algebra�a type of logic based on the three fundamental operations of and, or, and not. The American philosopher and logician

Charles Sanders Peirce was captivated by Boole�s

ideas, and he saw a possible application in the area of

electrical circuitry. Fifty years later, the American math-

ematician and engineer Claude Shannon put theory

to practice by showing how Boole�s system could be

used in designing telephone routing switches. This

innovation subsequently led to the development of

electronic digital computers.

Boole�s early years were marked by struggle. His Library of Congress Prints and Photographs Division [LC-USZ62-61664]

father, John, was a cobbler, and his mother, Mary Ann,

a lady�s maid. They could afford only the most basic of Lei bniz�s earlier spec ulations on the relationship education for their son, which John supplemented by between mathematics and logic. It also showed how teaching mathematics and science to young George the symbolism of mathematics could be imported into and by hiring a Latin tutor for him. Boole taught him- logic. This work won him a professorship at Queens self Greek, French, and German. His father, a leading College, in Cork, Ireland, where he remained for the member of the Mechanics Institute, was intrigued by rest of his life. Seven years later he published a much the application of mathematics in making instruments, larger and more mature work on the same subject, An and he passed this interest to his son. Though pov- Investigation of the Laws of Thought, on Which Are erty limited the resources available to him, Boole used Founded the Mathematical Theories of Logic and mathematics journals borrowed from the institute to Probabilities. This later work presented a complete further his mathematics education on his own. system of symbolic reasoning.

When George was only sixteen, his father�s shoe- Boole married Mary Everest (the niece of Sir

making business folded, and it fell to him to support George Everest, after whom Mt. Everest is named). the family by working as an assistant teacher. When He met her when she came to visit her famous uncle he was twenty-two, he took over the operation of a in Cork, and the relationship developed through boarding school after its former director had died, and his giving her lessons on differential equations. The his whole family assisted him in running it. Throughout couple had five daughters, but when Boole was only this period, Boole continued his study of mathematics, forty-nine his life was cut short from what was prob-and at age twenty-nine he published a paper on the ably pneumonia. Boole had walked two miles in the use of algebraic methods in solving differential equa- pouring rain to lecture at Queens, and he delivered tions. In recognition of this work he received the Royal the lecture in wet clothing. After he developed a Society Me dal, which brought him considerable fame high fever and became desperately ill, his wife, in the mathematical world. thinking that a good cure always mirrors the cause,

Three years later Boole published The Mathemati- poured cold water on him as he lay in bed. He died

cal Analysis of Logic, which brought to fruition some shortly thereafter.

Section 4.3 Venn Diagrams and the Modern Square of Opposition 185

Because the Boolean standpoint is closed to The members of the class denoted by each term

existence, it is simpler than the Aristotelian stand- should be thought of as situated inside the corre-point, which recognizes existential implications. sponding circle. Thus, the members of the S class For this reason, we will direct our attention first (if any such members exist) are situated inside the to arguments considered from the Boolean stand- S circle, and the members of the P class (if any such point. Later, in Section 4.5, we will extend our members exist) are situated inside the P circle. If any treatment to the Aristotelian standpoint. members are situated inside the area where the two

circles overlap, then such members belong to both

Venn Diagrams the S class and the P class. Finally, if any members

From the Boolean standpoint, the four kinds of cat- are situated outside both circles, they are members of neither S nor P. egorical propositions have the ]following meaning. Suppose, for example, that the S class is the class Notice that the first two (universal) propositions of Americans and the P class is the class of farmers. imply nothing about the existence of the things Then, if we use numerals to identify the four pos-denoted by S: sible areas, the diagram looks like this:

All S are P.  No members of S are outside P. No S are P.  No members of S are inside P. Some S are P.  At least one S exists that is a P. Some S are not P. 1 2 3 4  At least one S exists that

is not a P.

Adopting this interpretation of categorical AmericansFarmers

propositions, the nineteenth-century logician John

Venn developed a system of diagrams to represent Anything in the area marked �1� is an American the information they express. These diagrams have but not a farmer, anything in the area marked �2� is come to be known as Venn diagrams . both an American and a farmer, and anything in the A Venn diagram is an arrangement of overlap-area marked �3� is a farmer but not an American. ping circles in which each circle represents the class The area marked �4� is the area outside both circles; denoted by a term in a categorical proposition. thus, anything in this area is neither a farmer nor an Because every categorical proposition has exactly American. two terms, the Venn diagram for a single categori-We can now use Venn diagrams to represent the cal proposition consists of two overlapping cir-information expressed by the four kinds of categor-cles. Each circle is labeled so that it represents one ical proposition. To do this we make a certain kind of the terms in the proposition. Unless otherwise of mark in a diagram. Two kinds of marks are used: required, we adopt the convention that the left-hand shading an area and placing an X in an area. Shad-circle represents the subject term, and the right-ing an area means that the shaded area is empty,* hand circle the predicate term. Such a diagram looks and placing an X in an area means that at least one like this: thing exists in that area. The X may be thought of

as representing that one thing. If no mark appears

*In many mathematics texts, shading an area of a Venn diagram indi-

SPcates that the area is not empty. The significance of shading in logic is

exactly the opposite.

186Chapter4 Categorical Propositions

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). in an area, this means that nothing is known about that is both an S and a P. Finally, the O proposi-that area; it may contain members or it may be tion asserts that at least one S exists, and that S is empty. Shading is always used to represent the con- not a P. This is represented by placing an X in the tent of universal (A and E) propositions, and plac- part of the S circle that lies outside the P circle. ing an X in an area is always used to represent the This X represents an existing thing that is an S but content of particular (I and O) propositions. The not a P. content of the four kinds of categorical proposi- Because there is no X in the diagrams that rep-tions is represented as follows: resent the universal propositions, these diagrams

say nothing about existence. For example, the

diagram for the A proposition merely asserts that nothing exists in the part of the S circle that lies outside the P circle. The area where the two circles

A: All S are P. overlap and the part of the P circle that lies outside

the S circle contain no marks at all. This means that

SP something might exist in these areas, or they might

be completely empty. Similarly, in the diagram for

the E proposition, no marks appear in the left-hand

E: No S are P. part of the S circle and the right-hand part of the

P circle. This means that these two areas might

SP contain something or, on the other hand, they

might not.

I: Some S are P. X

SP The Modern Square of Opposition

Let us compare the diagram for the A proposition with the diagram for the O proposition. The dia-

O: Some S are not P. X gram for the A proposition asserts that the left-hand

part of the S circle is empty, whereas the diagram

SP for the O proposition asserts that this same area

is not empty. These two diagrams make assertions

Shading = emptiness that are the exact opposite of each other. As a result,

X = existence their corresponding statements are said to contra-

dict each other. Analogously, the diagram for the

E proposition asserts that the area where the two

Recall that the A proposition asserts that no circles overlap is empty, whereas the diagram for

members of S are outside P. This is represented the I proposition asserts that the area where the two by shading the part of the S circle that lies out- circles overlap is not empty. Accordingly, their cor-side the P circle. The E proposition asserts that no responding propositions are also said to contradict members of S are inside P. This is represented by each other. This relationship of mutually contradic-shading the part of the S circle that lies inside the tory pairs of propositions is represented in a dia-P circle. The I proposition asserts that at least one gram called the modern square of opposition. S�exists and that S is also a P. This is represented This diagram, which arises from the modern (or by placing an X in the area where the S and P�cir- Boolean) interpretation of categorical propositions, cles overlap. This X represents an existing thing is represented as follows:

Section 4.3 Venn Diagrams and the Modern Square of Opposition 187

Logically undetermined� relation between the A

A undetermined and the E because it results not from any E

relation between these propositions but

undetermined Logically tra tor dic Logically truth values of the I and the O. As the dic tra tory Con y The same can be said about the false from the fact the subject class is empty. Con undetermined contradictories of the A and E proposi-

tions, these propositions can be said to

be vacuously false.

Logically Testing Immediate Inferences

undetermined

Since the modern square of opposition

If two propositions are related by the contra- provides logically necessary results, we

dictory relation , they necessarily have opposite can use it to test certain arguments for validity. We

given as true, the corresponding O proposition the pertinent truth value in the square. We then use I must be false. Similarly, if a certain truth value. Thus, if a certain A proposition is begin by assuming the premise is true, and we enter

proposition the square to compute the truth value of the con-

is given as false, the corresponding E proposition clusion. If the square indicates that the conclusion

sible. In particular, given the truth value of an is invalid. Here is an example: A or must be true. But no other inferences are pos- is true, the argument is valid; if not, the argument O proposition, nothing can be determined

propositions. These propositions are said to have Therefore, it is false that all trade spies are masters at bribery. logically undetermined truth value about the truth value of the corresponding Some trade spies are not masters at bribery. E or I

. Like

all propositions, they do have a truth value, but Arguments of this sort are called immedi-logic alone cannot determine what it is. Similarly, ate inferences because they have only one given the truth value of an E or I proposition, premise. Instead of reasoning from one prem-nothing can be determined about the truth value ise to the next, and then to the conclusion, we of the corresponding A or O propositions. They, proceed immediately from the single prem-too, are said to have logically undetermined truth ise to the conclusion. To test this argument for value. validity, we begin by assuming that the prem-

Finally, suppose we should be given a set of ise, which is an O proposition, is true, and

four propositions about nonexisting things, we enter this truth value in the square of opposi-such as flying zebras. The A proposition, which tion. We then use the square to compute the truth asserts that there are no flying zebras outside value of the corresponding A proposition. By the the predicate class will be true because there contradictory relation, the A proposition is false. are no flying zebras at all, and the E proposi- Since the conclusion claims that the A proposition tion, which asserts that there are no flying zebras is false, the conclusion is true, and therefore the inside the predicate class will be true for the argument is valid. Arguments that are valid from same reason. These propositions are said to be the Boolean standpoint are said to be uncondi-vacuously true, because their truth value results tionally valid because they are valid regardless of solely from the fact that the subject class is empty, whether their terms refer to existing things. or void of members. Furthermore, the truth value Note that the conclusion of this argument has of these propositions does not violate the �logically the form �It is false that all S are P.� Technically,

188Chapter4 Categorical Propositions

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). statements of this type are not standard-form cat- an X in an area is the opposite of shading an egorical propositions because, among other things, area: they do not begin with a quantifier. To remedy this

difficulty, we adopt the convention that statements

having this form are equivalent to � �All S are P � is false.� Analogous remarks apply to the negations of the It is false that all A are B. X E, I, and O statements.

Here is another example: AB

It is false that all meteor showers are

common spectacles. Any statement that is diagrammed by entering Therefore, no meteor showers are common an X in an area is a particular proposition. Thus, spectacles. as the diagram shows, �It is false that all A are B�

We begin by assuming that the premise is true. Since is actually a particular proposition. By similar rea-the premise claims that an soning, �It is false that no A are B� is also a particu-A proposition is false, we enter �false� into the square of opposition. We then lar proposition. use the square to compute the truth value of the cor- To diagram �It is false that some A are B,� we responding E do the exact opposite of what we would do to proposition. Since there is no relation that links the diagram �Some A are B.� For �Some A are B,� we A and E propositions, the E proposition has undetermined truth value. Thus, the conclusion would enter an X in the overlap area. Thus, to dia-of the argument has undetermined truth value, and gram �It is false that some A are B,� we shade the the argument is invalid. overlap area:

We can also use Venn diagrams to test imme-

diate inferences for validity. However, using this

technique often requires that we diagram statements

beginning with the phrase �It is false that.� Let us It is false that some A are B. begin by showing how to diagram such statements.

Here are two examples: AB

It is false that all A are B.

It is false that some A are B.

The first statement claims that �All Any statement that is diagrammed by shading an A are B � is

of what we would do to diagram �All some A are B� is actually a universal proposition. By A are B .� To similar reasoning, �It is false that some false. Thus, to diagram it, we do the exact opposite area is a universal proposition. Thus, �It is false that

diagram �All A are not B� A are B ,� we shade the left-hand part of is also a universal proposition. the A circle: Now let us use Venn diagrams to test an imme-

diate inference. To do so we begin by using letters

to represent the terms, and we then draw Venn

All A are B. diagrams for the premise and conclusion. If the

To diagram �It is false that all A information expressed by the conclusion diagram B is contained in the premise diagram, the argument is valid; if not, it is invalid. Here is the symbolized A are B ,� we enter form of the trade spies inference that we tested an X in the left-hand part of the A circle. Entering earlier.

Section 4.3 Venn Diagrams and the Modern Square of Opposition 189

Some T are not M. To diagram the premise, we enter an X in the left-Therefore, it is false that all T are M. hand part of the M circle, and for the conclusion we

shade the overlap area:

The next step is to draw two Venn diagrams, one

for the premise and the other for the conclusion. For

the premise, we enter an X in the left-hand part of

the It is false that all M are C. X T circle, and for the conclusion, as we have just seen, we enter an X in the left-hand part of the T circle: MC

No M are C.

Some T are not M. X

TM Here, the conclusion diagram asserts that the

overlap area is empty. Since this information is not

contained in the premise diagram, the inference is

It is false that all T are M. X invalid.

We conclude with a special kind of inference:

All smartphones are wireless devices.

Therefore, some smartphones are wireless

To evaluate the inference, we look to see devices.

whether the information expressed by the con- The completed Venn diagrams are as follows: clusion diagram is also expressed by the prem-ise diagram. The conclusion diagram asserts

that something exists in the left-hand part of the

T circle. Since this information is also expressed by the premise diagram, the inference is valid. All S are W.

In this case, the diagram for the conclusion is SW identical to the diagram for the premise, so it is clear that premise and conclusion assert exactly the same thing. However, as we will see in

Sections 4.5 and 4.6, for an immediate inference Some S are W. X to be valid, it is not necessary that premise and

conclusion assert exactly the same thing. It is SW

only necessary that the premise assert at least as The information of the conclusion diagram is not much as the conclusion. contained in the premise diagram, so the inference Here is the symbolized version of the meteor is invalid. However, if the premise were interpreted shower inference evaluated earlier: as having existential import, then the S circle in the

premise diagram would not be empty. Specifically,

It is false that all M are C. there would be members in the overlap area. This Therefore, no M are C. would make the inference valid.

190Chapter4 Categorical Propositions

Arguments of this sort are said to commit the Finally, while all of these forms proceed from a uni-

existential fallacy. From the Boolean standpoint, the versal premise to a particular conclusion, it is important existential fallacy is a formal fallacy that occurs to see that not every inference having a universal prem-whenever an argument is invalid merely because the ise and a particular conclusion commits the existential premise lacks existential import. Such arguments fallacy. For example, the inference �All A are B; there-always have a universal premise and a particular con- fore, some A are not B� does not commit this fallacy. clusion. The fallacy consists in attempting to derive a This inference is invalid because the conclusion contra-conclusion having existential import from a premise dicts the premise. Thus, to detect the existential fallacy, that lacks it. one must ensure that the invalidity results merely from

The existential fallacy is easy to detect. Just look the fact that the premise lacks existential import. This

for a pair of diagrams in which the premise diagram can easily be done by constructing a Venn diagram. contains shading and the conclusion diagram con-

tains an X. If the X in the conclusion diagram is in Note on Boole, Venn, the same part of the left-hand circle that is unshaded and Existential Import in the premise diagram, then the inference commits the existential fallacy. In the smartphone example Students occasionally ask about the motivation we just considered, the premise diagram contains behind Boole�s elimination of existential import shading, and the conclusion diagram contains an X. from universal propositions. What is to be gained Also, the X in the conclusion diagram is in the over- by this move, they ask? What�s the point? To answer lap area, and this area is unshaded in the premise this question, we begin with a clarification: It was not diagram. Thus, the inference commits the existential Boole who took this step, it was Venn. That Boole fallacy. interpreted universal statements as having existential

There are exactly eight inference forms that import is clear from the fact that he allows for infer-

commit the existential fallacy. Four of them are as fol- ences that move from the universal to the particu-lows. (The other four are left for an exercise.) Among lar. One of these inferences is what is called conver-these forms, recall that any proposition asserting that sion by limitation: �All A are B� implies �Some B are a particular (I or O) proposition is false is a universal A.�1 The other (which will be treated in Section 4.5 proposition, and any proposition asserting that a of this chapter) is called subalternation: �All A are B� universal (A or E) proposition is false is a particular implies �Some A are B.�2 The only way these infer-proposition. With this in mind, you can see that all ences can be valid is if the class A is considered to of these forms proceed from a universal premise to a have at least one member. particular conclusion. The first logician who confronted head-on the

issue of the existential import of universal propo-

sitions was John Venn. In his Symbolic Logic, Venn

Existential fallacy defended the logic of Boole, but at the same time

All A are B. he insisted that universal propositions should be Therefore, some A are B. interpreted as having no existential import. This

modification of Boole�s logic came to be accepted

It is false that some A are not B. as the official Boolean position. Thus, when we Therefore, it is false that no A are B.

No A are B.

Therefore, it is false that all A are B. 1G. Boole, The Mathematical Analysis of Logic (Cambridge: Macmillan,

It is false that some A are B. 2Ibid, 28. Boole reiterates both of these inferences in The Laws of Therefore, some A are not B. Barclay, and Macmillan, 1847), 27�28 .

Thought (London: Macmillan, 1854), 229.

Section 4.3 Venn Diagrams and the Modern Square of Opposition 191

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). speak today of the Boolean standpoint or the who is 20 feet tall. Again, most logicians would Boolean interpretation of universal propositions, reject this outcome. And the problem is exacer-what we mean is Boole�s position as amended by bated as the complexity of the terms increases. Venn. In some cases it may be very difficult to see

Venn�s work can be considered as launching a exactly what happens when the statement is

two-pronged attack on existential import. Accord- given existential import. ing to the first prong, insisting that all universal The second prong of Venn�s attack consists in propositions have existential import results in resolutely adhering to Boole�s interpretation of �numerous perplexities.� One of these perplexities the meaning of universal propositions. According arises from contraposing A type propositions (see to this interpretation, universal propositions say Section 4.4 of this chapter). Suppose, for exam- nothing about where class members exist. They ple, we are given the true statement �All A are B.� simply say where class members do not exist. Venn According to the position Venn is attacking, both invites us to consider this pair of propositions: �All A and B represent non-empty classes. When we A are B� and �No A are B.�7 These propositions are contrapose this statement, we get the true state- said to be contraries, and if they are interpreted ment �All non-B are non-A,� where both non-B as having existential import, they cannot both be and non-A represent non-empty classes.3 Now, true (see Section 4.5 of this chapter). But if they are most of the time this operation is unproblematic, given the Boolean interpretation, there is no reason but suppose you are given the presumably true to think they cannot both be true. According to the statement �All measurable things are finite things.� first, nothing exists in the part of A that lies outside When we contrapose it, we get the true statement B, and according to the second, nothing exists in �All infinite things are immeasurable things,� and the part of A that lies inside B. Now there is nothing this implies that there is at least one infinite thing inconsistent at all about these two interpretations. (God?). Many logicians would reject this result. Clearly they can both be true. But, taken together,

Another perplexity relates to complex groups they imply that there is nothing in A at all. Thus,

of statements and single statements having the Boolean interpretation of universal proposi-complex terms. In the case of complex groups of tions, pure and simple, without anything else being statements, Venn shows how existential import added, implies there are propositions that have no can result in inconsistencies, where not all the existential import. Boole himself apparently did members of the group can be true.4 For single not notice this. statements having complex terms, the compo- In summary, eliminating existential import nents can combine in ways that render exis- does away immediately with the perplexities that tential import problematic.5 For example, the Venn identifies. Without existential import, we statement �All A who are B are C� is generally need not be concerned whether there are infinite acceptable. But if A, B, and C are replaced by things that actually exist, or whether there really �basketball players,� �20 feet tall,� and �formi- are such things as basketball players who are dable athletes,�6 insisting on existential import 20 feet tall. Also, we need not worry whether exis-implies there is at least one basketball player tential import causes inconsistencies in complex

groups of statements. Finally, abandoning exis-

tential import harmonizes perfectly with Boole�s

3J. Venn, Symbolic Logic (London: Macmillan, 1881), 137�138. interpretation of universal statements whereby 4The example Venn uses is: All x is either y and z, or not-y; All xy is such statements merely say where class members w or not z; No wx is zx. J. Venn, Symbolic Logic(London: Macmillan, 1881), 139�140, 147.

5Ibid., 139.

6 7 This example, and the �infinite things� example are mine .J. Venn, Symbolic Logic(London: Macmillan, 1881), 145�146.

192Chapter4 Categorical Propositions

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). do not exist, and not where they do exist. The logical system having existential import is more result is a system of logic that is simpler and widely applicable to real-life situations than one more elegant than one that allows for existential that lacks it. We will explore this idea further in import. However, from a practical standpoint, a Section 4.5 of this chapter.

Exercise 4.3

I. Draw Venn diagrams for the following propositions.

★1. No life decisions are happenings based solely on logic.

2. All electric motors are machines that depend on magnetism.

3. Some political campaigns are mere attempts to discredit opponents.

★4. Some folk-music lovers are not fans of Jay-Z.

5. All redistricting plans are sources of controversy.

6. No tax audits are pleasant experiences for cheaters.

★7. Some housing developments are complexes that exclude children.

8. Some cruise ships are not steam-driven vessels.

II. Use the modern square of opposition to determine whether the following immediate

inferences are valid or invalid from the Boolean standpoint.

★1. No sculptures by Rodin are boring creations.

Therefore, all sculptures by Rodin are boring creations.

2. It is false that some lunar craters are volcanic formations.

Therefore, no lunar craters are volcanic formations.

3. All trial lawyers are people with stressful jobs.

Therefore, some trial lawyers are people with stressful jobs.

★4. All dry martinis are dangerous concoctions.

Therefore, it is false that some dry martinis are not dangerous concoctions.

5. It is false that no jazz musicians are natives of New Orleans.

Therefore, some jazz musicians are not natives of New Orleans.

6. Some country doctors are altruistic healers.

Therefore, some country doctors are not altruistic healers.

★7. No fertility drugs are solutions to every problem.

Therefore, it is false that all fertility drugs are solutions to every problem.

8. It is false that no credit cards are things that contain holograms.

Therefore, some credit cards are things that contain holograms.

9. It is false that some stunt pilots are not colorful daredevils.

Therefore, it is false that some stunt pilots are colorful daredevils.

Section 4.3 Venn Diagrams and the Modern Square of Opposition 193

★10. No vampires are avid connoisseurs of garlic bread.

Therefore, it is false that some vampires are avid connoisseurs of garlic bread.

11. No talk radio shows are accurate sources of information.

Therefore, some talk radio shows are not accurate sources of information.

12. Some stellar constellations are spiral-shaped objects.

Therefore, no stellar constellations are spiral-shaped objects.

★13. It is false that some soap bubbles are not occasions of glee.

Therefore, some soap bubbles are occasions of glee.

14. It is false that all weddings are light-hearted celebrations.

Therefore, some weddings are not light-hearted celebrations.

15. It is false that some chocolate souffl�s are desserts containing olives.

Therefore, it is false that all chocolate souffl�s are desserts containing olives.

III. Use Venn diagrams to evaluate the immediate inferences in Part II of this exercise.

Identify any that commit the existential fallacy.

IV. This section of Chapter 4 identified four forms of the existential fallacy. Use Venn dia-

grams to identify the other four. In doing so, keep in mind that all forms of this fallacy have a universal premise and a particular conclusion, that �It is false that some A are B� and �It is false that some A are not B� are universal propositions, and �It is false that all A are B� and �It is false that no A are B� are particular.

V. Create five new immediate inferences composed of standard-form categorical proposi-

tions. Then test each for validity.

4.4 Conversion, Obversion,

and Contraposition

For a preliminary glimpse into the content of this �Some dogs are non-retrievers� claims the same section, consider the statement �No dogs are cats.� thing, so again, the two statements have the same This statement claims that the class of dogs is meaning and the same truth value. separated from the class of cats. But the statement Conversion, obversion, and contraposition are �No cats are dogs� claims the same thing. Thus, operations that can be performed on a categori-the two statements have the same meaning and cal proposition, resulting in a new statement that the same truth value. For another example, con- may or may not have the same meaning and truth sider the statement �Some dogs are not retriev- value as the original statement. Venn diagrams are ers.� This statement claims there is at least one dog used to determine how the two statements relate outside the class of retrievers. But the statement to each other.

194Chapter4 Categorical Propositions

The simplest of the three operations is statement is called the converse of the given state-conversion , ment. To see how the four types of categorical prop-Conversion statement is �No hedgehogs are foxes.� This new

and it consists of switching the subject term with ositions relate to their converse, compare the follow-the predicate term. For example, if the statement ing sets of Venn diagrams: �No foxes are hedgehogs� is converted, the resulting

Given statement formConverse

All A are B. All B are A.

ABAB

No A are B. No B are A.

ABAB

Some A are B. X Some B are A. X

ABAB

Some A are X Some B are X not B. not A.

ABAB

If we examine the diagram for the E statement, On the other hand, the diagram for the A

we see that it is identical to that of its converse. Also, statement is clearly not identical to the diagram for its the diagram for the I statement is identical to that converse, and the diagram for the O statement is not of its converse. This means that the E statement identical to the diagram for its converse. Also, these and its converse are logically equivalent, and the I pairs of diagrams are not the exact opposite of each statement and its converse are logically equivalent. other, as is the case with contradictory statements. Two statements are said to be logically equivalent

statements when they necessarily have the same Conversion truth value (as we will see again in Chapter 6). Thus, Switch converting an E or I statement gives a new statement that always has the same truth value (and the same meaning) as the given statement. These equivalences SP are strictly proved by the Venn diagrams for the E (quantifier) (copula) and I statements.

Section 4.4 Conversion, Obversion, and Contraposition 195

This means that an A statement and its converse are predicate with its term complement. The first part logically unrelated as to truth value, and an O state- of this operation was treated in Exercise 4.2. It con-ment and its converse are logically unrelated as to sists in changing �No S are P� to �All S are P� and vice truth value. In other words, converting an A or O versa, and changing �Some S are P� to �Some S are not statement gives a new statement whose truth value is P� and vice versa. logically undetermined in relation to the given state- The second step requires understanding the ment. The converse of an A or O statement does have concept of class complement. The complement of a a truth value, of course, but logic alone cannot tell us class is the group consisting of everything outside what it is. the class. For example, the complement of the class

Because conversion yields necessarily deter- of dogs is the group that includes everything that

mined results for E and I statements, it can be is not a dog (cats, fish, trees, and so on). The term used as the basis for immediate inferences having complement is the word or group of words that these types of statements as premises. The follow- denotes the class complement. For terms consist-ing inference forms are valid: ing of a single word, the term complement is usu-

No A are B. ally formed by simply attaching the prefix �non� to Therefore, no B are A. the term. Thus, the complement of the term �dog�

is �non-dog,� the complement of the term �book� is

Some A are B. �non-book,� and so on. Therefore, some B are A. The relationship between a term and its com-

Since the conclusion of each inference form nec- plement can be illustrated by a Venn diagram. For example, if a single circle is allowed to repre-essarily has the same truth value as the premise, if sent the class of dogs, then everything outside the the premise is assumed true, it follows necessarily circle represents the class of non-dogs: that the conclusion is true. On the other hand, the next two inference forms are invalid. Each com-mits the fallacy of illicit conversion:

All A are B.

Therefore, all B are A. non-dogs dogs

Some A are not B.

Therefore, some B are not A.

Here are two examples of inferences that commit the

fallacy of illicit conversion: We now have everything we need to form

All cats are animals. (True) the obverse of categorical propositions. First we Therefore, all animals are cats. (False) change the quality (without changing the quan-

tity), and then we replace the predicate term with

Some animals are not dogs. (True) its term complement. For example, if we are given Therefore, some dogs are not animals. (False) the statement �All horses are animals,� then the

obverse is �No horses are non-animals�; and if we

Obversion are given the statement �Some trees are maples,�

then the obverse is �Some trees are not non-

More complicated than conversion, obversion maples.� To see how the four types of categorical requires two steps: (1) changing the quality (with- propositions relate to their obverse, compare the out changing the quantity), and (2) replacing the following sets of Venn diagrams:

196Chapter4 Categorical Propositions

Given statement formObverse

All A are B. No A are

non-B.

ABAB

No A are B. All A are

non-B.

ABAB

Some A are B. X Some A are X

not non-B.

ABAB

Some A are X Some A are X not B. non-B.

ABAB

To see how the obverse diagrams are drawn, keep It is easy to see that if a statement is obverted and

in mind that �non-B� designates the area outside the then obverted again, the resulting statement will be B circle. Thus, �No A are non-B� asserts that the area identical to the original statement. For example, the where A overlaps non-B is empty. This is represented obverse of �All horses are animals� is �No horses by shading the left-hand part of the A circle. �All A are non-animals.� To obvert the latter statement we are non-B� asserts that all members of A are outside again change the quality (�no� switches to �all�) and B . This means that no members of A are inside B, so replace �non-animals� with its term complement. the area where A overlaps B is shaded. �Some A are The term complement is produced by simply delet-not non-B� asserts that at least one member of A is ing the prefix �non.� Thus, the obverse of the obverse not outside B. This means that at least one member is �All horses are animals.� of A is inside B, so an X is placed in the area where When a term consists of more than a single word, A and B overlap. Finally, �Some A are non-B� asserts more ingenuity is required to form its term comple-that at least one member of A is outside B, so an X is ment. For example, if we are given the term �ani-placed in the left-hand part of the A circle. mals that are not native to America,� it would not be

If we examine these pairs of diagrams, we see that

the diagram for each given statement form is identical Obversion

obverse. Thus, if we obvert an r A an (for I and OA statement that hap-(fo) pens to be true, the resulting statement will be true; if of the four types of categorical proposition is logi- quality term complement ) d E cally equivalent to (and has the same meaning as) its to the diagram for its obverse. This means that each Change Replace with SP

we obvert an O statement that happens to be false, the (quantifier) (copula) resulting statement will be false, and so on.

Section 4.4 Conversion, Obversion, and Contraposition 197

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). appropriate to form the term complement by writing Some A are B. Some A are not B. �non-animals that are not native to America.� Clearly Therefore, some Therefore, some it would be better to write �animals native to America.� A are not non-B. A are non-B.

Even though this is technically not the complement of Because the conclusion of each inference form nec-the given term, the procedure is justified if we allow a essarily has the same truth value as its premise, if reduction in the scope of discourse. This can be seen the premise is assumed true, it follows necessarily as follows. Technically the term complement of �ani-that the conclusion is true. mals that are not native to America� denotes all kinds of things such as ripe tomatoes, battleships, gold rings,

about animals (that is, we reduce the scope of discourse Like obversion, contraposition requires two to animals), then the complement of this term is �ani-and so on. But if we suppose that we are talking only Contraposition

used to supply the link between the premise and the As is the case with conversion, obversion can be with their term complements. For example, if the statement �All goats are animals� is contraposed, conclusion of immediate inferences. The following the resulting statement is �All non-animals are non-mals native to America.� and (2) replacing the subject and predicate terms steps: (1) switching the subject and predicate terms inference forms are valid: goats.� This new statement is called the contraposi-

All A are B. No A are B. tive of the given statement. To see how all four types Therefore, no Therefore, all of categorical propositions relate to their contrapos-A are non-B. A are non-B. itive, compare the following sets of diagrams:

Given statement formContrapositive

All A are B. All non-B are

non-A.

ABAB

No A are B. No non-B

are non-A.

ABAB

Some A are B. X Some non-B

are non-A.

ABAB

Some A are X Some non-B X not B. are not non-A.

ABAB

198Chapter4 Categorical Propositions

As with conversion and obversion, contraposi-

Contraposition tion may provide the link between the premise and

Switch a the conclusion of an immediate inference. The fol-nd re pl a c e w i th lowing inference forms are valid: co m ple t e r m men t

All A are B.

SP Therefore, all non-B are non-A.

(quantifier) (copula)

Some A are not B.

Therefore, some non-B are not non-A.

On the other hand, the following inference

To see how the first diagram on the right is forms are invalid. Each commits the fallacy of illicit

drawn, remember that �non-A� designates the contraposition: area outside A. Thus, �All non-B are non-A� asserts that all members of non-B are outside A. Some A are B. This means that no members of non-B are inside Therefore, some non-B are non-A.

A. Thus, we shade the area where non-B over- No A are B. laps A . �No non-B are non-A � asserts that the Therefore, no non-B are non-A. area where non-B overlaps non-A is empty. Since non-B is the area outside the B circle and non-A is Here are two examples of inferences that commit the area outside the A circle, the place where these the fallacy of illicit contraposition: two areas overlap is the area outside both circles. B Thus, we shade this area. �Some non- are non-A � No dogs are cats. (True) asserts that something exists in the area where Therefore, no non-cats are non-dogs. (False) non-B overlaps non-A . Again, this is the area out-side both circles, so we place an X in this area. Some animals are non-cats. (True) Finally, �Some non-B are not non-A� asserts that Therefore, some cats are non-animals. (False) at least one member of non-B is outside non-A. This means that at least one member of non- In regard to the first inference, an example of some-B is inside thing that is both a non-cat and a non-dog is a pig. A , so we place an X in the area where non-B overlaps A. Thus, the conclusion implies that no pigs are pigs,

Now, inspection of the diagrams for the A which is false. In regard to the second inference,

and O if both premise and conclusion are obverted, the statements reveals that they are identical to the diagrams of their contrapositive. Thus, premise becomes �Some animals are not cats,� which the A statement and its contrapositive are logi- is true, and the conclusion becomes �Some cats are cally equivalent (and have the same meaning), not animals,� which is false. and the O statement and its contrapositive are logically equivalent (and have the same meaning).

statements are neither identical to nor the exact equivalent results, note the second and third vowels in the words. Conversion works for E and I propositions, contraposition works for opposite of the diagrams of their contrapositives. On the other hand, the diagrams of the E and I To help remember when conversion and contraposition yield logically This means that contraposing an A and O propositions. E or I statement gives a new statement whose truth value is logi- C O N V E R S I O N cally undetermined in relation to the given C O N T R A P O S I T I O N statement.

Section 4.4 Conversion, Obversion, and Contraposition 199

Both illicit conversion and illicit contraposition are unconditional, and they hold true regardless of

are formal fallacies: They can be detected through whether the terms in the propositions denote actu-mere examination of the form of an argument. ally existing things. Thus, they hold for propositions

Finally, note that the Boolean interpretation of about unicorns and leprechauns just as they do for

categorical propositions has prevailed throughout propositions about dogs and animals. These results this section. This means that the results obtained are summarized in the following table.

Conversion: Switch Subject and Predicate Terms

Given StatementConverseTruth Value

E: No A are B. No B are A. Same truth value as

I given statement : Some A are B. Some B are A .

A: All A are B. All B are A. Undetermined truth O value : Some A are not B . Some B are not A .

Obversion: Change Quality; Replace Predicate with Term Complement

Given StatementObverseTruth Value

A: All A are B. No A are non-B.

E: No A are B. All A are non-B. Same truth value as I: Some A are B. Some A are not non-B. given statement

O: Some A are not B. Some A are non-B.

Contraposition: Switch Subject and Predicate Terms; Replace Each with Its Term Complement

Given StatementContrapositiveTruth Value

A: All A are B. All non-B are non-A. Same truth value as O given statement : Some A are not B . Some non-B are not non-A .

E: No A are B. No non-B are non-A. Undetermined truth I: Some A are B. Some non-B are non-A. value

Exercise 4.4

I. Exercises 1 through 6 provide a statement, its truth value in parentheses, and an operation

to be performed on that statement. Supply the new statement and the truth value of the new statement. (Hint: For number 1, first obtain the converse of �No A are non-B�; then, since the given statement is a true E proposition, use this information to calculate the truth value of the converse.)

Exercises 7 through 12 provide a statement, its truth value in parentheses, and a

new statement. Determine how the new statement was derived from the given statement and supply the truth value of the new statement. (Hint: For number 7, first compare the

200Chapter4 Categorical Propositions

terms in the given statement with the terms in the new statement. This will tell you the operation. Then, since the given statement is a false E proposition, use this information together with the operation to calculate the truth value of the new statement.)

Truth

Given statement Operation New statement value

★ 1. No A are non-B. (T) conv.

2. Some A are B. (T) contrap.

3. All A are non-B. (F) obv.

★ 4. All non-A are B. (F) contrap.

5. Some non-A are not B. (T) conv.

6. Some non-A are non-B. (T) obv.

★ 7. No non-A are non-B. (F) No B are A.

8. Some A are not non-B. (T) Some A are B.

9. All A are non-B. (F) All non-B are A.

★10. No non-A are B. (F) All non-A are non-B.

11. Some non-A are not B. (T) Some non-B are not A.

12. Some A are non-B. (F) Some non-B are A.

II. Perform the operations of conversion, obversion, and contraposition as indicated.

1. Convert the following propositions and state whether the converse is logically

equivalent or not logically equivalent to the given proposition.

★a. All hurricanes are storms intensified by global warming.

b. No cosmetic surgeries are completely successful procedures.

c. Some murals by Diego Rivera are works that celebrate the revolutionary

spirit.

d. Some forms of carbon are not substances with a crystalline structure.

2. Obvert the following propositions and state whether the obverse is logically

equivalent or not logically equivalent to the given proposition.

★a. All radically egalitarian societies are societies that do not preserve individ-

ual liberties.

b. No cult leaders are people who fail to brainwash their followers.

c. Some college football coaches are people who do not slip money to their

players.

d. Some budgetary cutbacks are not actions fair to the poor.

3. Contrapose the following propositions and state whether the contrapositive is

logically equivalent or not logically equivalent to the given proposition.

★a. All physicians whose licenses have been revoked are physicians ineligible to

practice.

b. No unpersecuted migrants are migrants granted asylum.

Section 4.4 Conversion, Obversion, and Contraposition 201

c. Some politicians who do not defend Social Security are politicans who do

not want to increase taxes.

d. Some opponents of gay marriage are not opponents of civil unions.

III. Use conversion, obversion, and contraposition to determine whether the following argu-

ments are valid or invalid. For those that are invalid, name the fallacy committed.

★1. All commodity traders are gamblers who risk sudden disaster.

Therefore, all gamblers who risk sudden disaster are commodity traders.

2. No child abusers are people who belong in day-care centers.

Therefore, all child abusers are people who do not belong in day-care centers.

3. Some cryptocurrencies are not traceable media of exchange.

Therefore, some untraceable media of exchange are not conventional currencies.

★4. Some irresponsible people are illogical people.

Therefore, some logical people are responsible people.

5. Some organ transplants are not sensible operations.

Therefore, some organ transplants are senseless operations.

6. No individuals who laugh all the time are people with a true sense of humor.

Therefore, no people with a true sense of humor are individuals who laugh all the time.

★7. All periods when interest rates are high are times when businesses tend not

to expand.

Therefore, all times when businesses tend to expand are periods when interest rates are low.

8. Some swimsuits are not garments intended for the water.

Therefore, some garments intended for the water are not swimsuits.

9. No promises made under duress are enforceable contracts.

Therefore, no unenforceable contracts are promises made in the absence of duress.

★10. All Russian oligarchs are thieves who hide their assets.

Therefore, no Russian oligarchs are thieves who fail to hide their assets.

11. Some graffiti writers are artists relieving pent-up frustrations.

Therefore, some artists relieving pent-up frustrations are graffiti writers.

12. Some peaceful revolutions are episodes that erupt in violence.

Therefore, some episodes that do not erupt in violence are non-peaceful revolutions.

★13. Some insurance companies are not humanitarian organizations.

Therefore, some humanitarian organizations are not insurance companies.

14. Some fossil fuels are unrenewable energy sources.

Therefore, some fossil fuels are not renewable energy sources.

15. All hired killers are criminals who deserve the death penalty.

Therefore, all criminals who deserve the death penalty are hired killers.

202Chapter4 Categorical Propositions

★16. No nonprescription drugs are medicines without adverse effects.

Therefore, no medicines with adverse effects are prescription drugs.

17. All fire-breathing dragons are lizards that languish in soggy climates.

Therefore, no fire-breathing dragons are lizards that flourish in soggy climates.

18. Some distant galaxies are not structures visible to the naked eye.

Therefore, some structures visible to the naked eye are not distant galaxies.

★19. All unpleasant experiences are things we do not like to remember.

Therefore, all things we like to remember are pleasant experiences.

20. Some pro-lifers are not people concerned with child welfare.

Therefore, some pro-lifers are people unconcerned with child welfare.

IV. Create a new categorical proposition for each of the four forms (A, E, I, O). Then con-

vert, obvert, and contrapose each one, and state whether the resulting proposition is logically equivalent to the original proposition.

4.5 The Traditional Square of Opposition

In Section 4.3 we adopted the Bool- Contrary AE ean standpoint, and we saw how T the modern square of opposition T Con applies regardless of whether the y tra tor propositions refer to actually exist-dic dic Subalternation Subalternation ing things. In this section, we adopt tra tor y Con the Aristotelian standpoint, which recognizes that universal proposi-F F tions about existing things have exis-IO tential import. For such propositions Subcontrary the traditional square of opposition becomes applicable. Like the modern square, the The four relations in the traditional square of traditional square of opposition is an opposition may be characterized as follows: arrangement of lines that illustrates logically Contradictory  opposite truth value necessary relations among the four kinds of cat- Contrary  at least one is false (not egorical propositions. However, because the Aris- both true) totelian standpoint recognizes the additional Subcontrary  at least one is true (not factor of existential import, the traditional square both false) supports more inferences than does the modern Subalternation  truth flows downward, square. It is represented as follows: falsity flows upward

Section 4.5 The Traditional Square of Opposition 203

The contradictory relation is the same as that not animals,� the corresponding I proposition �Some

found in the modern square. Thus, if a certain A propo- cats are animals� is true. Thus, the I and O proposi-sition is given as true, the corresponding O proposition tions cannot both be false, but they can both be true. is false, and vice versa, and if a certain A proposition is �Some animals are cats� and �Some animals are not given as false, the corresponding O proposition is true, cats� are both true. and vice versa. The same relation holds between the The subalternation relation is represented E and I propositions. The contradictory relation thus by two arrows: a downward arrow marked with expresses complete opposition between propositions. the letter T (true), and an upward arrow marked

The contrary relation differs from the contradic- with an F (false). These arrows can be thought of

tory in that it expresses only partial opposition. Thus, as pipelines through which truth values �flow.� The if a certain A proposition is given as true, the corre- downward arrow �transmits� only truth, and the sponding E proposition is false (because at least one upward arrow only falsity. Thus, if an A proposition must be false), and if an E proposition is given as true, is given as true, the corresponding I proposition is the corresponding A proposition is false. But if an A true also, and if an I proposition is given as false, proposition is given as false, the corresponding E prop- the corresponding A proposition is false. But if an A osition could be either true or false without violating proposition is given as false, this truth value cannot the �at least one is false� rule. In this case, the E proposi- be transmitted downward, so the corresponding I tion has logically undetermined truth value. Similarly, proposition will have logically undetermined truth if an E proposition is given as false, the corresponding value. Conversely, if an I proposition is given as true, A proposition has logically undetermined truth value. this truth value cannot be transmitted upward, so

These results are borne out in ordinary language. the corresponding A proposition will have logically

Thus, if we are given the actually true A proposi- undetermined truth value. Analogous reasoning tion �All cats are animals,� the corresponding E prevails for the subalternation relation between the proposition �No cats are animals� is false, and if we E and O propositions. To remember the direction are given the actually true E proposition �No cats of the arrows for subalternation, imagine that truth are dogs,� the corresponding A proposition �All �trickles down,� and falsity �floats� up. cats are dogs� is false. Thus, the A and E proposi- Now that we have examined these four relations tions cannot both be true. However, they can both individually, let us see how they can be used together to be false. �All animals are cats� and �No animals are determine the truth values of corresponding proposi-cats� are both false. tions. The first rule of thumb that we should keep in

The subcontrary relation also expresses a kind of mind when using the square to compute more than

partial opposition. If a certain I proposition is given as one truth value is always to use contradiction first. false, the corresponding O proposition is true (because Now, let us suppose that we are told that the nonsensi-at least one must be true), and if an O proposition is cal proposition �All adlers are bobkins� is true. Suppose given as false, the corresponding I proposition is true. further that adlers actually exist, so we are justified in But if either an I or an O proposition is given as true, using the traditional square of opposition. By the con-then the corresponding proposition could be either tradictory relation, �Some adlers are not bobkins� is true or false without violating the �at least one is true� false. Then, by either the contrary or the subalterna-rule. Thus, in this case the corresponding proposition tion relation, �No adlers are bobkins� is false. Finally, would have logically undetermined truth value. by either contradictory, subalternation, or subcontrary,

Again, these results are borne out in ordinary lan- �Some adlers are bobkins� is true.

guage. If we are given the actually false I proposition Next, let us see what happens if we assume that �Some cats are dogs,� the corresponding O proposi- �All adlers are bobkins� is false. By the contradictory tion �Some cats are not dogs� is true, and if we are relation, �Some adlers are not bobkins� is true, but given the actually false O proposition �Some cats are nothing more can be determined. In other words,

204Chapter4 Categorical Propositions

alternation yield undetermined results, and given a It is false that all A are B. true O proposition (the one whose truth value we Therefore, no A are B. just determined), subcontrary and subalternation

yield undetermined results. Thus, the corresponding It is false that no A are B.E Therefore, all A are B. and I propositions have logically undetermined truth value. This result illustrates two more rules of

thumb. Assuming that we always use the contradic- Illicit subcontrary tory relation first, if one of the remaining relations Some A are B.

others will as well. The other rule is that whenever Some A are not B. one statement turns out to have logically undeter-yields a logically undetermined truth value, the Therefore, it is false that some A are not B. mined truth value, its contradictory will also. Thus, Therefore, some A are B.

statements having logically undetermined truth Illicit subalternation value will always occur in pairs, at opposite ends of Some A are not B. diagonals on the square. Therefore, no A are B.

Testing Immediate Inferences It is false that all A are B. Next, let us see how we can use the traditional square Therefore, it is false that some A are B. of opposition to test immediate inferences for valid-

ity. Here is an example: Cases of the incorrect application of the contradic-

All Rolex watches are works of art. tory relation are so infrequent that an �illicit contra-Therefore, it is false that no Rolex watches dictory� fallacy is not usually recognized. are works of art. As we saw at the beginning of this section, for the

Since the premise is an A proposition, by the con- ositions to which it is applied must assert some-trary relation the corresponding E proposition is thing about actually existing things. The question false. But this is exactly what the conclusion says, so We begin, as usual, by assuming the premise is true. totelian standpoint must be adopted, and the prop-traditional square of opposition to apply, the Aris-

the argument is valid. totelian standpoint is adopted but the propositions may now be asked, What happens when the Aris-

Here is another example: are about things that do not exist? The answer is

Some viruses are structures that attack T cells. that under these conditions the traditional square Therefore, some viruses are not structures reverts to the modern square (see Section 4.3). that attack T cells. Inferences that are based on a correct application of

Here the premise and conclusion are linked by the the contradictory relation are valid, but inferences subcontrary relation. According to that relation, if that are based on an otherwise correct application the premise is assumed true, the conclusion has logi- of the other three relations are invalid and commit cally undetermined truth value, and so the inference the existential fallacy. is invalid. It commits the formal fallacy of illicit sub- The reason for this result is easy to see. The modern contrary . Analogously, inferences that depend on an square of opposition rests on the Boolean standpoint, incorrect application of the contrary relation commit and the traditional square rests on the Aristotelian the formal fallacy of illicit contrary, and inferences standpoint. But the Aristotelian standpoint dif-that depend on an illicit application of subalternation fers from the Boolean only with respect to universal commit the formal fallacy of illicit subalternation. propositions about existing things. The Aristotelian Some forms of these fallacies are as follows: standpoint interprets these propositions as having

Section 4.5 The Traditional Square of Opposition 205

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). existential import, while the Boolean standpoint does is false, is actually a particular proposition. Thus, this not. But universal propositions about things that do example, like the first one, proceeds from the univer-not exist have no existential import from the Aristo- sal to the particular. telian standpoint. In other words, the Aris-

totelian standpoint interprets these propo-

sitions in exactly the same way as does the Existential fallacy examples: Two standpoints Boolean. Thus, for universal propositions

about things that do not exist, the traditional All cats are animals. Boolean: Invalid, existential fallacy Some cats are animals. Aristotelian: Valid square reverts to the modern square.*

Accordingly, the existential fallacy is All unicorns are animals. Boolean: Invalid, existential fallacy

committed from the Aristotelian stand- Some unicorns are animals. Aristotelian: Invalid, existential fallacy point when and only when contrary, sub-

contrary, and subalternation are used (in an

otherwise correct way) to draw a conclusion The phrase conditionally valid applies to an from a premise about things that do not exist. All inference after the Aristotelian standpoint has been such inferences begin with a universal proposition, adopted and we are not certain if the subject term which has no existential import, and they conclude of the premise denotes actually existing things. For with a particular proposition, which has existential example, the following inference is conditionally valid:

import. The existential fallacy is never committed All students who failed the exam are stu-

it committed in connection with conversion, obver- Therefore, some students who failed the sion, or contraposition, all of which hold regardless in connection with the contradictory relation, nor is dents on probation. of existence. The following inferences commit the exam are students on probation. existential fallacy from the Aristotelian standpoint: The validity of this inference rests on whether there

All witches who fly on broomsticks are fear- were in fact any students who failed the exam. The less women. inference is either valid or invalid, but we lack suf-Therefore, some witches who fly on broom- ficient information about the meaning of the premise sticks are fearless women. to tell which is the case. Once it becomes known that No wizards with magical powers are malevo- there are indeed some students who failed the exam, lent beings. we can assert that the inference is valid from the Aris-Therefore, it is false that all wizards with totelian standpoint. But if there are no students who magical powers are malevolent beings. failed the exam, the inference is invalid because it

commits the exis tential fallacy.

The first depends on an otherwise correct use of Similarly, all inference forms that depend on valid

the subalternation relation, and the second on an oth- applications of contrary, subcontrary, and subalter-erwise correct use of the contrary relation. If flying nation are conditionally valid because we do not witches and magical wizards actually existed, both know if the letters in the propositions denote actu-arguments would be valid. But since they do not exist, ally existing things. For example, the following infer-both arguments are invalid and commit the existen- ence form, which depends on the contrary relation, tial fallacy. In regard to the second example, recall that is conditionally valid:

the conclusion, which asserts that an A proposition All A are B.

Therefore, it is false that no A are B.

are X� is vacuously true, as is �No unicorns are X.� Thus, their contra- and B, respectively, the resulting inference is valid. dictories, �Some unicorns are not X � and �Some unicorns are X � are But if �unicorns� and �animals� are substituted, the *Following on this result, from the Aristotelian standpoint �All unicorns If �dogs� and �animals� are substituted in place of Avacuously false. It makes no difference what X stands for.

206Chapter4 Categorical Propositions

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). resulting inference is invalid because it commits the Now that we have seen how the traditional square existential fallacy. In Section 4.3, we noted that all of opposition, by itself, is used to test inferences for inferences (and inference forms) that are valid from validity and soundness, let us see how it can be used the Boolean standpoint are unconditionally valid. together with the operations of conversion, obver-They are valid regardless of whether their terms sion, and contraposition to prove the validity of infer-denote actually existing things. ences that are given as valid. Suppose we are given the

In testing an inference for validity, we are never following valid inference:

Regardless of whether the premise is actually true or All inappropriate remarks are faux pas. Therefore, some faux pas are not appropri-concerned with the actual truth of the premise.

false, we always begin by assuming it to be true, and ate remarks. then we determine how this assumption bears on the

truth or falsity of the conclusion. The actual truth of To prove this inference valid, we select letters to the premise affects only the soundness of the argu-represent the terms, and then we use some combi-ment. So let us now turn to the question of sound-nation of conversion, obversion, and contraposi-ness. Recall from Section 1.4 that a sound argument tion together with the traditional square to find the is one that is valid and has all true premises, and intermediate links between premise and conclusion: consider the following example:

All non-A are F. (assumed true)

All cats are dogs.

Some non-A are F. (true by subalternation)

Therefore, some cats are dogs.

Some F are non-A. (true by conversion)

The premise is obviously false; but if we assume it Therefore, some F (true by obversion) to be true, then it follows necessarily by subalterna- are not A.

inference is unsound. succeeding step is validly derived from the one preced-ing it by the relation written in parentheses at the right. Here is another example: Since the conclusion (which is the last step) follows by a No rabbits are toads. is valid. However, because the premise is false, the The premise is the first line in this proof, and each tion that the conclusion is true. Thus, the inference

series of three necessary inferences, the inference is valid.

Therefore, it is false that all rabbits are toads. Various strategies can be used to construct proofs

This inference is sound. By the contrary relation it is such as this, but one useful procedure is first to con-valid, and it also has a true premise. centrate on obtaining the individual terms as they

Here is a final example: appear in the conclusion, then to attend to the order

Some unicorns are not gazelles. of the terms, and finally to use the square of opposition Therefore, it is false that all unicorns are to adjust quality and quantity. As the faux pas proof

ise asserts the existence of something that does not actu- A,� while �non-A� appears in the prem-ise, leads us to think of obversion. But using obversion ally exist (namely, unicorns). In other words, the premise to change �non-A � into � A � requires that the �non-A � seems to be self-contradictory. Nevertheless, the infer-in the premise be moved into the predicate position via This inference differs from the others in that the prem- sometimes necessary. The fact that the predicate of the conclusion is � gazelles. illustrates, however, variations on this procedure are

ence can be evaluated in the usual way. If the premise is conversion. The latter operation, however, is valid only assumed true, then it necessarily follows that the conclu-on E and I statements, and the premise is an A state-sion is true by the contradictory relation. Thus, the infer-ment. The fact that the conclusion is a particular state-ence is valid. But the inference is unsound because it has a ment suggests subalternation as an intermediate step, false premise. The premise asserts the existence of some-thus yielding an I statement that can be converted. thing that does not actually exist.

Section 4.5 The Traditional Square of Opposition 207

Exercise 4.5

I. Use the traditional square of opposition to find the answers to these problems. When a

statement is given as false, simply enter an �F� into the square of opposition and com-pute (if possible) the other truth values.

★1. If �All fashion fads are products of commercial brainwashing� is true, what is

the truth value of the following statements?

a. No fashion fads are products of commercial brainwashing.

b. Some fashion fads are products of commercial brainwashing.

c. Some fashion fads are not products of commercial brainwashing.

2. If �All fashion fads are products of commercial brainwashing� is false, what is

the truth value of the following statements?

a. No fashion fads are products of commercial brainwashing.

b. Some fashion fads are products of commercial brainwashing.

c. Some fashion fads are not products of commercial brainwashing.

3. If �No sting operations are cases of entrapment� is true, what is the truth value

of the following statements?

a. All sting operations are cases of entrapment.

b. Some sting operations are cases of entrapment.

c. Some sting operations are not cases of entrapment.

★4. If �No sting operations are cases of entrapment� is false, what is the truth value

of the following statements?

a. All sting operations are cases of entrapment.

b. Some sting operations are cases of entrapment.

c. Some sting operations are not cases of entrapment.

5. If �Some assassinations are morally justifiable actions� is true, what is the truth

value of the following statements?

a. All assassinations are morally justifiable actions.

b. No assassinations are morally justifiable actions.

c. Some assassinations are not morally justifiable actions.

6. If �Some assassinations are morally justifiable actions� is false, what is the truth

value of the following statements?

a. All assassinations are morally justifiable actions.

b. No assassinations are morally justifiable actions.

c. Some assassinations are not morally justifiable actions.

★7. If �Some obsessive-compulsive behaviors are not curable diseases� is true, what

is the truth value of the following statements?

a. All obsessive-compulsive behaviors are curable diseases.

b. No obsessive-compulsive behaviors are curable diseases.

c. Some obsessive-compulsive behaviors are curable diseases.

8. If �Some obsessive-compulsive behaviors are not curable diseases� is false,

what is the truth value of the following statements?

208Chapter4 Categorical Propositions

a. All obsessive-compulsive behaviors are curable diseases.

b. No obsessive-compulsive behaviors are curable diseases.

c. Some obsessive-compulsive behaviors are curable diseases.

II. Use the traditional square of opposition to determine whether the following imme diate

inferences are valid or invalid. Name any fallacies that are committed.

★ 1. All advocates of school prayer are individuals who insist on imposing their

views on others.

Therefore, some advocates of school prayer are individuals who insist on imposing their views on others.

2. It is false that no jailhouse informants are people who can be trusted.

Therefore, some jailhouse informants are not people who can be trusted.

3. All homemakers are people with real jobs.

Therefore, it is false that no homemakers are people with real jobs.

★4. It is false that some trolls are not creatures who live under bridges.

Therefore, it is false that no trolls are creatures who live under bridges.

5. Some campus romances are episodes plagued by violence.

Therefore, some campus romances are not episodes plagued by violence.

6. Some pornographic publications are materials protected by the First Amendment.

Therefore, it is false that no pornographic publications are materials protected by the First Amendment.

★7. It is false that all mainstream conservatives are people who support free legal

services for the indigent.

Therefore, no mainstream conservatives are people who support free legal ser-vices for the indigent.

8. It is false that some forms of human creativity are activities amenable to math-

ematical analysis.

Therefore, it is false that all forms of human creativity are activities amenable to mathematical analysis.

9. It is false that some tooth fairies are daytime visitors.

Therefore, some tooth fairies are not daytime visitors.

★ 10. It is false that some orthodox psychoanalysts are not individuals driven by a

religious fervor.

Therefore, it is false that some orthodox psychoanalysts are individuals driven by a religious fervor.

11. Some school buses manufactured on the moon are not plasma-powered vehicles.

Therefore, it is false that all school buses manufactured on the moon are plasma-powered vehicles.

12. It is false that some network news programs are exercises in mediocrity.

Therefore, it is false that no network news programs are exercises in mediocrity.

★ 13. No flying reindeer are animals who get lost in the fog.

Therefore, it is false that all flying reindeer are animals who get lost in the fog.

Section 4.5 The Traditional Square of Opposition 209

14. It is false that no leveraged buyouts are deals unfair to workers.

Therefore, all leveraged buyouts are deals unfair to workers.

15. It is false that some wood ticks are not carriers of Lyme disease.

Therefore, some wood ticks are carriers of Lyme disease.

III. Use the traditional square of opposition to determine whether the following immediate

inferences are valid or invalid and sound or unsound. Name any fallacies that are committed.

★1. All dolphins are polar bears.

Therefore, it is false that no dolphins are polar bears.

2. It is false that some recessions are not periods of economic decline.

Therefore, it is false that no recessions are periods of economic decline.

3. It is false that some suicide survivors are comeback kids.

Therefore, some suicide survivors are not comeback kids.

★4. It is false that some ruby earrings are not pieces of jewelry.

Therefore, some ruby earrings are pieces of jewelry.

5. It is false that all visitors to Rio are carnival addicts.

Therefore, no visitors to Rio are carnival addicts.

6. Some tax cheats are not honest citizens.

Therefore, no tax cheats are honest citizens.

★7. All truthful lies are curious assertions.

Therefore, some truthful lies are curious assertions.

8. It is false that no bankrupt hair salons are thriving enterprises.

Therefore, all bankrupt hair salons are thriving enterprises.

9. It is false that some functional skateboards are not devices equipped with wheels.

Therefore, all functional skateboards are devices equipped with wheels.

★10. Some film directors are artistic visionaries.

Therefore, some film directors are not artistic visionaries.

IV. Exercises 1 through 10 provide a statement, its truth value in parentheses, and an operation

to be performed on that statement. Supply the new statement and the truth value of the new statement. Exercises 11 through 20 provide a statement, its truth value in parentheses, and a new statement. Determine how the new statement was derived from the given statement and supply the truth value of the new statement. Take the Aristotelian standpoint in work-ing these exercises and assume that the terms refer to actually existing things.

Operation/ Truth

Given statement relation New statement value

★1. All non-A are B. (T) contrap.

2. Some A are non-B. (F) subalt.

3. No A are non-B. (T) obv.

★4. Some non-A are not B. (T) subcon.

5. No A are non-B. (F) contradic.

210Chapter4 Categorical Propositions

6. No A are B. (T) contrap.

★ 7. All non-A are B. (T) contrary

8. Some A are not non-B. (F) obv.

9. No A are non-B. (F) conv.

★10. Some non-A are non-B. (F) subcon.

11. Some non-A are not B. (T) All non-A are B.

12. Some A are non-B. (T) Some non-B are A.

★13. All non-A are B. (F) No non-A are non-B.

14. Some non-A are not B. (T) No non-A are B.

15. All A are non-B. (F) All non-B are A.

★16. Some non-A are non-B. (F) No non-A are non-B.

17. Some A are not non-B. (T) Some B are not non-A.

18. No non-A are B. (T) Some non-A are not B.

★19. No A are non-B. (F) All A are non-B.

20. Some non-A are B. (F) Some non-A are not B.

V. Use either the traditional square of opposition or conversion, obversion, or contraposi-

tion to determine whether the following immediate inferences are valid or invalid. For those that are invalid, name the fallacy committed.

★ 1. It is false that some jogging events are not aerobic activities.

Therefore, it is false that no jogging events are aerobic activities.

2. No meat-eating vegetarians are individuals with a high-protein diet.

Therefore, no individuals with a high-protein diet are meat-eating vegetarians.

3. Some jobs in health care are not glamorous occupations.

Therefore, some jobs in health care are glamorous occupations.

★4. Some terminally ill patients are patients who do not want to live.

Therefore, some patients who want to live are recovering patients.

5. All Barbie dolls are toys that engender a false sense of values.

Therefore, no Barbie dolls are toys that engender a true sense of values.

6. All flying elephants are jolly pachyderms.

Therefore, some flying elephants are jolly pachyderms.

★7. It is false that some international terrorists are political moderates.

Therefore, some international terrorists are not political moderates.

8. No pet hamsters are animals that need much attention.

Therefore, it is false that all pet hamsters are animals that need much attention.

9. Some hedge-fund managers are not responsible investors.

Therefore, some responsible investors are not hedge-fund managers.

Section 4.5 The Traditional Square of Opposition 211

★10. It is false that all substances that control cell growth are hormones.

Therefore, no substances that control cell growth are hormones.

11. Some cases of whistle-blowing are actions disloyal to employers.

Therefore, some cases of whistle-blowing are not actions loyal to employers.

12. No stolen computer chips are easy items to trace.

Therefore, no difficult items to trace are computer chips that are not stolen.

★13. Some economists are followers of Ayn Rand.

Therefore, some economists are not followers of Ayn Rand.

14. All porcelain figurines are fragile artifacts.

Therefore, it is false that some porcelain figurines are not fragile artifacts.

15. Some pleasant recollections are not missed opportunities.

Therefore, some availed opportunities are not unpleasant recollections.

VI. Use the traditional square of opposition together with conversion, obversion, and con-

traposition to prove that the following immediate inferences are valid. Show each inter-mediate step in the deduction.

★1. All insurance policies are cryptically written documents.

Therefore, some cryptically written documents are insurance policies.

2. No gemstones that do not contain chromium are emeralds.

Therefore, some stones that are not emeralds are not gemstones that contain chromium.

3. It is false that some Ficus benjaminas are untemperamental house plants.

Therefore, all Ficus benjaminas are temperamental house plants.

★4. All exogenous morphines are addictive substances.

Therefore, it is false that all addictive substances are endogenous morphines.

5. No people who do not advocate free-enterprise economics are fundamentalist

Christians.

Therefore, it is false that some fundamentalist Christians are not people who advocate free-enterprise economics.

6. It is false that some Gothic cathedrals are buildings that do not feature pointed arches.

Therefore, some buildings that feature pointed arches are Gothic cathedrals.

★7. Some people who recognize paranormal events are not non-scientists.

Therefore, it is false that no scientists are people who recognize paranormal events.

8. It is false that no unhealthy things to ingest are food additives.

Therefore, some food additives are not healthy things to ingest.

9. It is false that some illegal searches are not sobriety checkpoints.

Therefore, some sobriety checkpoints are not legal searches.

★10. It is false that some feminists are not advocates of equal pay for equal work.

Therefore, it is false that all advocates of equal pay for equal work are non-feminists.

VII. Create a new categorical proposition for each of the four forms (A, E, I, O). Then, for

each one, write out the three categorical propositions related to it by the traditional

212Chapter4 Categorical Propositions

square of opposition (similar to Part I of this exercise). Finally, assuming the original propositions are true, use the traditional square to determine the truth value of the other three propositions.

4.6 Venn Diagrams and�the� Traditional

Standpoint

Earlier in this chapter we saw how Venn diagrams

can be used to represent the content of categorical A: All S are P X propositions from the Boolean standpoint. With a slight modification they can also be used to repre-SP sent the content of categorical propositions from the traditional, or Aristotelian, standpoint. These modi-

fied Venn diagrams can then be used to prove the SE: No are P X relationships of the traditional square of opposition,

and also to test the validity of immediate inferences SP from the traditional standpoint. The difference between the Boolean standpoint and the Aristotelian standpoint concerns only uni- In the diagram for the A statement, the left-hand versal (A and E) propositions. From the Boolean part of the S circle is shaded, so if there are any standpoint, universal propositions have no existen- members of S, they must be in the area where the tial import, but from the Aristotelian standpoint they two circles overlap. Thus, a circled X is placed in do have existential import when their subject terms the overlap area. In the diagram for the E state-refer to actually existing things. For example, from the ment, the overlap area is shaded, so if there are Boolean standpoint the statement �All raccoons are any members of S they must be in the left-hand pests� does not imply the existence of anything, but part of the S circle. Thus, a circled X is placed in from the Aristotelian standpoint it implies the exis- this area. tence of raccoons. Thus, if we are to construct a Venn The diagrams for the I and O statements are the diagram to represent such a statement from the Aris- same from the Aristotelian standpoint as they are totelian standpoint, we need to use some symbol that from the Boolean: represents this implication of existence.

The symbol that we will use for this purpose is an

X surrounded by a circle. Like the X�s that we have

used up until now, this circled X signifies that some- I: Some S are P. X thing exists in the area in which it is placed. However,

the two symbols differ in that the uncircled X repre- SP sents the positive claim of existence made by particular

(I and O) propositions, whereas the circled X represents an implication of existence made by universal proposi- O: Some S are not P. X tions about actually existing things. For the purpose at

hand, a circled X is placed inside the S circle as follows: SP

Section 4.6 Venn Diagrams and�the� Traditional Standpoint 213

Proving the Traditional

Square of Opposition X X

We can now use this modified Venn SPSP diagram technique to prove the rela-Contrary tions of the traditional square of opposi-AE tion.* Having such a proof is important T T because up until now these relations Con y have only been illustrated with various tra tor dic examples; they have not been proved. dic Subalternation Subalternation tra tor The accompanying figure reproduces y Con the traditional square of opposition F together with Venn diagrams that rep-F

resent the Aristotelian interpretation IO Subcontrary of the four standard-form categorical propositions.

Let us begin with the contradictory

relation. If the X X A statement is given

as true, then the left-hand part of the SPSPS circle is empty. This makes the O

statement false, because it claims that

the left-hand part of the S circle is not

empty. And if the O statement is given Next, we turn to the contrary relation. If the A as true, then the left-hand part of the S circle is statement is given as true, then the overlap area is not empty, which makes the A statement false. On not empty, which makes the E statement false. By the other hand, if the O statement is given as false, analogous reasoning, if the E statement is given then the left-hand part of the S circle is empty. as true, the overlap area is empty, which makes However, given that some members of S exist, the A statement false. However, if the A statement they must be in the overlap area. This double out- is given as false (making the O statement true), come makes the A statement true. Also, if the A then the E statement could be either true or false statement is given as false, then either the left- depending on whether or not the overlap area is hand part of the S circle is not empty, or the over- empty. Thus, in this case the E statement would lap area is empty (or both). If the left-hand part have logically undetermined truth value. By anal-of the S circle is not empty, then the O statement ogous reasoning, if the E statement is given as is true. Alternately, if the overlap area is empty, false (making the I statement true), the A state-then, given that some members of S exist, they ment could be either true or false depending on must be in the left-hand part of the S circle, and, whether or not the left-hand part of the S circle is once again, the O statement is true. Analogous empty. Thus, the A statement would have logically reasoning applies for the relation between the E undetermined truth value. and I statements. Turning next to the subcontrary relation, if the

I statement is given as false, then the area where

*The modified Venn diagram technique can also be used to prove the the S and P circles overlap is empty. Given that at

validity of conversion, obversion, and contraposition from the Aris- least one S exists, there must be something in the totelian standpoint, but to do so a circled X must be entered in the left-hand part of the S circle, which makes the Ounshaded part of both the S and P circles and also in the unshaded area statement true. By analogous reasoning, if the outside both circles.O

214Chapter4 Categorical Propositions

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). statement is given as false, there must be something standpoint, testing the inference from the Bool-in the overlap area, making the I statement true. ean standpoint is often simpler. If the inference But if the I statement is given as true, then the O is valid, then it is valid from both standpoints. statement could be either true or false depending But if the inference is invalid from the Boolean on whether something exists in the left-hand part standpoint and has a particular conclusion, then of the S circle. Thus, the O statement would have it may be useful to test it from the Aristotelian undetermined truth value. Similarly, if the O state- standpoint. Let us begin by testing an inference

ment would have undetermined truth value. I state- Therefore, some A are B. First, we draw Venn diagrams from the Boolean Finally, we consider subalternation. If the A state-standpoint for the premise and conclusion: ment is given as true, then something exists in the either true or false depending on whether some- All A are B. thing exists in the overlap area. Thus, the ment is given as true, then the I statement could be form for validity: area where the S and P circles overlap, which makes

given as false, then the overlap area is empty, making All A are B. the the I statement true as well. And if the I statement is A statement false. But if the A statement is given

as false (making the O statement true), then the IAB statement could be either true or false depending on whether something exists in the overlap area. Thus, the I statement would have logically undetermined truth value. And if the I Some A are B. X statement is given as true,

then the A statement could be either true or false AB depending on whether or not the left-hand part of

the S circle is empty. Thus, the A statement would The information of the conclusion diagram is not have logically undetermined truth value. Analogous represented in the premise diagram, so the infer-reasoning applies for the subalternation relation ence form is not valid from the Boolean standpoint. between the E and O statements. Thus, noting that the conclusion is particular, we

adopt the Aristotelian standpoint and assume for

Testing Immediate Inferences the moment that the subject of the premise (A)

denotes at least one existing thing. This thing is

From the Aristotelian standpoint, the modified represented by placing a circled X in the open area Venn diagram technique involving circled X�s of that circle: can be used to test immediate inferences. The

only requirement is that the subject and predicate

terms of the conclusion be the same as those of the

premise. Such inferences depend on the square of All A are B. X opposition and do not involve the operations of

conversion, obversion, and contraposition. Venn AB diagrams can also be used to test inferences involv-

ing these latter operations, but a further modifica-

tion must be introduced. Some A are B. X

Since any inference that is valid from the Bool-

ean standpoint is also valid from the Aristotelian AB

Section 4.6 Venn Diagrams and�the� Traditional Standpoint 215

Now the information of the conclusion diagram is The term in question is �penguins,� and at least one represented in the premise diagram. Thus, the infer- penguin actually exists. Thus, the condition is ful-ence form is conditionally valid from the Aristo- filled, and the inference is valid from the Aristotelian telian standpoint. It is valid on condition that the standpoint. circled X represents at least one existing thing. Another example:

its form. Here is an example: To test a complete inference we begin by testing All sugarplum fairies are delicate creatures. Therefore, some sugarplum fairies are deli-

No penguins are birds that can fly. cate creatures.

Therefore, it is false that all penguins are This immediate inference has the same form as the birds that can fly. first one we tested. The form is not valid from the

First, we reduce the immediate inference to its form Boolean standpoint, but it is conditionally valid and test it from the Boolean stand-point: from the Aristotelian standpoint:

No P are B. All S are D. X

PBSD

It is false that

all X Some S are D. X P are B.

PBSD

Since the inference form is not valid from the Boolean The final step is to see if the circled X represents at standpoint, we adopt the Aristotelian standpoint and least one existing thing. The circled X is in the S circle assume for the sake of this test that the subject of the and S stands for �sugarplum fairies,� which do not premise (P) denotes at least one existing thing: exist. Thus, the requisite condition is not fulfilled,

and the inference is not valid from the Aristotelian

standpoint. The inference commits the existential

No P are B. X fallacy from the Aristotelian standpoint.

The steps involved in testing an immediate infer-

PB ence from the Aristotelian standpoint may now be

summarized:

It is false that 1. Reduce the inference to its form and test it from X all P are B. the Boolean standpoint. If the form is valid,

P proceed no further. The inference is valid from B both stand points.

The Venn diagrams show that the inference form is 2. If the inference form is invalid from the Boolean conditionally valid from the Aristotelian standpoint. standpoint and has a particular conclusion, then It is valid on condition that the circled X represents at adopt the Aristotelian standpoint and look to see least one existing thing. Since the circled X is in the P if the left-hand premise circle is partly shaded. If it circle, the final step is to see if the term in the inference is, enter a circled X in the unshaded part and retest corresponding to P denotes something that exists. the form.

216Chapter4 Categorical Propositions

3. If the inference form is conditionally valid, the Aristotelian standpoint. If it does not,

determine if the circled X represents some- the inference is invalid, and it commits the thing that exists. If it does, the condition existential fallacy from the Aristotelian is fulfilled, and the inference is valid from standpoint.

Exercise 4.6

I. Use the modified Venn diagram technique to determine if the following immediate

inference forms are valid from the Boolean standpoint, conditionally valid from the Aristotelian standpoint, or invalid.

★1. Some A are not B.

Therefore, no A are B.

2. It is false that some A are B.

Therefore, it is false that all A are B.

3. It is false that no A are B.

Therefore, some A are B.

★4. All A are B.

Therefore, it is false that no A are B.

5. Some A are B.

Therefore, it is false that some A are not B.

6. Some A are not B.

Therefore, it is false that all A are B.

★7. It is false that some A are B.

Therefore, no A are B.

8. It is false that some A are not B.

Therefore, some A are B.

9. It is false that all A are B.

Therefore, no A are B.

★10. No A are B.

Therefore, some A are not B.

II. Use the modified Venn diagram technique to determine if the following immediate

inferences are valid from the Boolean standpoint, valid from the Aristotelian stand-point, or invalid. Identify any inferences that commit the existential fallacy from either standpoint.

★1. No summer romances are banal pastimes.

Therefore, it is false that some summer romances are banal pastimes.

Section 4.6 Venn Diagrams and�the� Traditional Standpoint 217

2. It is false that some people who hunger for wealth are not victims of their

obsession.

Therefore, some people who hunger for wealth are victims of their obsession.

3. No lamps containing genies are ordinary sources of light.

Therefore, some lamps containing genies are not ordinary sources of light.

★4. It is false that some duck hunters are animal-rights activists.

Therefore, some duck hunters are not animal-rights activists.

5. All repressive political regimes are insults to human dignity.

Therefore, no repressive political regimes are insults to human dignity.

6. It is false that all skating rinks are playgrounds for amateurs.

Therefore, some skating rinks are not playgrounds for amateurs.

★7. All pixies who slide down moonbeams are fun-loving daredevils.

Therefore, it is false that no pixies who slide down moonbeams are fun-lov-ing daredevils.

8. It is false that some graduate teaching assistants are not underpaid laborers.

Therefore, it is false that no graduate teaching assistants are underpaid laborers.

9. Some housing projects are developments riddled with crime.

Therefore, it is false that no housing projects are developments riddled with crime.

★ 10. It is false that some thunderstorms are quiescent phenomena.

Therefore, all thunderstorms are quiescent phenomena.

11. No flower gardens are creations that feature skunkweed.

Therefore, it is false that all flower gardens are creations that feature skunkweed.

12. It is false that no incendiary devices are contraptions that misfire.

Therefore, some incendiary devices are not contraptions that misfire.

★13. It is false that some pet lovers are people who think that animals are mere

machines.

Therefore, it is false that all pet lovers are people who think that animals are mere machines.

14. No werewolves are creatures who lurk about in the daytime.

Therefore, it is false that all werewolves are creatures who lurk about in the daytime.

15. Some soccer games are not thrilling events to watch.

Therefore, no soccer games are thrilling events to watch.

III. Create five new immediate inferences composed of standard-form categorical propo-

sitions. Then use the modified Venn diagram technique to test each for�validity.

218Chapter4 Categorical Propositions

4.7 Translating Ordinary Language

Statements into Categorical Form

Although few statements that occur in ordinary writ- Some roses Some roses are red

in standard form, many of them can be translated All tigers are All tigers are into standard-form propositions. Such translation ten and oral expression are categorical propositions are red. flowers. has two chief benefits. The first is that the operations carnivorous. carnivorous animals.

cal propositions (contrary, subcontrary, etc.) become and inferences pertinent to standard-form categori- 2. Nonstandard Verbs applicable to these statements. The second is that such According to the position adopted earlier in this statements, once translated, are completely clear and chapter, the only copulas that are allowed in stan-unambiguous as to their meaning. Many statements dard-form categorical propositions are �are� and in ordinary language are susceptible to multiple inter- �are not.� Statements in ordinary usage, however, pretations, and each interpretation represents one often incorporate other forms of the verb �to be.� possible mode of translation. The effort to translate Such statements may be translated as the following

tions and thus helps prevent misunderstanding and Some college stu- Some college students are confusion. such statements discloses the various interpreta- examples illustrate:

Translating statements into categorical form is like dents will become people who will become

any other kind of translation in that no set of specific educated. educated. rules will cover every possible form of phraseology. Some dogs would Some dogs are animals Yet, one general rule always applies: Understand the rather bark than that would rather bark meaning of the given statement, and then reexpress bite. than bite. it in a new statement that has a quantifier, subject

term, copula, and predicate term. Some of the forms In other statements no form of the verb �to be� of phraseology that are typically encountered are occurs at all. These may be translated as the follow-terms without nouns, nonstandard verbs, singular ing examples indicate:

propositions, adverbs and pronouns, unexpressed Some birds fly Some birds are animals and nonstandard quantifiers, conditional state- south during the that fly south during the ments, exclusive propositions, �the only,� and excep- winter. winter. tive propositions.

All ducks swim. All ducks are swimmers.

1. Terms Without Nouns All ducks are animals that

The subject and predicate terms of a categorical swim.

pronoun that serves to denote the class indicated by proposition must contain either a plural noun or a 3. Singular Propositions the term. Nouns and pronouns denote classes, while A singular proposition (statement) is a prop-adjectives (and participles) connote attributes. If a osition that makes an assertion about a specific term consists of only an adjective, a plural noun or person, place, thing, or time. Singular propositions pronoun should be introduced to make the term are typically translated into universals by means of genuinely denotative. Examples: a parameter. A parameter is a phrase that, when

Section 4.7 Translating Ordinary Language Statements into Categorical Form 219

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). introduced into a statement, affects the form but not I hate gin. All people identical to the meaning. Some parameters that may be used to me are people who translate singular propositions are these: hate gin.

people identical to All things identical to places identical to gin are things that I things identical to hate.

cases identical to In translating singular statements, note that the param-times identical to eter �people identical to� is not the same as �people

For example, the statement �Socrates is mortal� like similar to� or �people like.� There may be many people may be translated as �All people identical to Socrates. Also note that parameters should Socrates, but there is only one person identical to Socrates are people who are mortal.� Because when the term in question already has a plural noun not be used

only one person is identical to Socrates, namely (or pronoun) that denotes the intended class. Such use Socrates himself, the term �people identical to is not wrong, technically, but it is redundant. Example: Socrates� denotes the class that has Socrates as its

only member. In other words, it simply denotes Diamonds Correct: All diamonds Socrates. Such a translation admittedly leaves are carbon are carbon

singular statements usually have existential Redundant: All things identi-import, whereas universal statements do not�at some of the original information behind, because allotropes. allotropes least from the Boolean standpoint. But if such cal to diamonds translations are interpreted from the Aristotelian are things identi-standpoint, the existential import is preserved. cal to carbon

allotropes.

Here are some examples:

George went home. All people identical 4. Adverbs and Pronouns

to George are people When a statement contains a spatial adverb such who went home. as �where,� �wherever,� �anywhere,� �everywhere,�

Sandra did not go or �nowhere,� or a temporal adverb such as �when,� No people identical

shopping. �whenever,� �anytime,� �always,� or �never,� it may to Sandra are people

who went shopping. be translated in terms of �places� or �times,� respec-

There is a TV in the tively. Statements containing pronouns such as All places identical �who,� �whoever,� �anyone,� �what,� �whatever,� or bedroom. to the bedroom are �anything� may be translated in terms of �people� or places there is a TV. �things,� respectively. Examples: or

in the bedroom. He always wears a All times he goes to Some TVs are things

suit to work. work are times he

The moon is full All things identical to wears a suit. tonight. the moon are things

that are full tonight. He is always clean All times are times he

or shaven. is clean shaven.

All times identical to She never brings No times she goes to tonight are times the her lunch to school. school are times she moon is full. brings her lunch.

220Chapter4 Categorical Propositions

Nowhere on earth No places on earth A fish is not a No fish are mammals. are there any are places there are mammal.

unicorns. unicorns. A tiger roared. Some tigers are ani-Whoever works All people who work mals that roared.

hard will succeed. hard are people who Children are human All children are

will succeed. beings. human beings.

Whenever he wins All times he wins are Children live next Some children are he celebrates. times he celebrates. door. people who live She goes where All places she next door. she chooses. chooses to go are

places she goes.

She does what All things she wants 6. Nonstandard Quantifiers she wants. to do are things she

does. In some ordinary language statements, the quan-

tity is indicated by words other than the three

Notice the order of the subject and predicate terms standard-form quantifiers. Such words include in the last four examples. When translating state- �few,� �a few,� �not every,� �anyone,� and various ments such as these it is often easy to confuse the other forms. Another problem occurs when the subject term with the predicate term. However, quantifier �all� is combined with the copula �are since these statements are all translated as A type not.� As we have already seen, statements of the categorical propositions, such a mix-up amounts to form �All S are not P� are not standard-form cat-committing the fallacy of illicit conversion. To pre- egorical propo sitions. Depending on their mean-vent it from happening, keep this rule in mind: For ing, they should be translated as either �No S �W� words (�who,� �what,� �when,� �where,� �who- are P� or �Some S are not P.� When the intended ever,� �whatever,� �whenever,� �wherever�), the lan- meaning is �Some S are not P,� the meaning may guage following the �W� word goes into the subject be indicated by placing oral emphasis on the word term of the categorical proposition. �all.� For example, �All athletes are not superstars�

means �Some athletes are not superstars.� Here are

some additional examples:

5. Unexpressed Quantifiers A few soldiers are Some soldiers are Many statements in ordinary usage have quan- heroes. heroes.

tifiers that are implied but not expressed. In Anyone who votes is All voters are citizens.

by the most probable meaning of the statement. Not everyone who Some voters are not Examples: introducing the quantifiers one must be guided a citizen.

votes is a Democrat. Democrats.

Emeralds are green Not a single dog is No dogs are cats. All emeralds are gems. a cat. green gems.

There are lions in All newborns are not No newborns are Some lions are ani-the zoo. able to talk. people able to talk. mals in the zoo.

A tiger is a mammal. All tigers are All prisoners are not Some prisoners are

mammals. violent. not violent people.

Section 4.7 Translating Ordinary Language Statements into Categorical Form 221

Many entertainers Some entertainers Conditional statements having a negated conse-are comedians are comedians. quent are usually best translated as E propositions.

Several demonstra- Examples: Some demonstrators

tors were arrested. are people who were If it�s a turkey, then it�s No turkeys are

arrested. not a mammal. mammals.

Few sailors entered Some sailors are If an animal has four legs, No four-legged the regatta. people who entered then it is not a bird. animals are birds.

sailors are not people A knife will cut only if it No knives that cut the regatta and some who entered the isn�t dull. are dull knives.

Notice that this last statement beginning with �few� following the word �if� goes in the subject, state-ments containing �unless� are translated as cat-regatta. The word �unless� means �if not.� Since language

tion. Such statements (and some beginning with �a egorical propositions having negated subject terms. cannot be translated as a single categorical proposi-few�) must be translated as a compound arrangement Examples:

of an I proposition and an O proposition. Statements Tomatoes are edible All unspoiled toma-beginning with �almost all� and �not quite all� must unless they are spoiled. toes are edible

occur in arguments, the arguments must be treated in Unless a boy mis- All boys who do the same way as those containing exceptive proposi-be handled in the same way. When these statements tomatoes. tions, which will be discussed shortly. behaves he will be not misbehave are

treated decently. boys who will be

treated decently.

7. Conditional Statements

When the antecedent and consequent of a condi- 8. Exclusive Propositions

tional statement refer to the same class of things, Many propositions that involve the words �only,� �none the statement can usually be translated into cat- but,� �none except,� and �no . . . except� are exclusive egorical form. Such statements are always trans- propositions. Efforts to translate them into categorical lated as universals. Language following the word propositions often lead to confusing the subject term �if � goes in the subject term of the categorical with the predicate term. To avoid such confusion keep proposition, and language following �only if � goes in mind that language following �only,� �none but,� in the predicate term. Examples: �none except,� and �no . . . except� goes in the predicate

term of the categorical proposition. For example, the

If it�s a mouse, then All mice are statement �Only executives can use the silver elevator� it�s a mammal. mammals. is translated �All people who can use the silver elevator If a bear is hungry, All hungry bears are are executives.� If it were translated �All executives are then it is dangerous. dangerous animals. people who can use the silver elevator,� the translation

Jewelry is expensive would be incorrect. Examples: All pieces of jewelry

if it is made of gold. made of gold are Only elected offi- All people who will

expensive things. cials will attend the attend the conven-

A car is a Camry only All Camrys are convention. tion are elected if it�s a Toyota. Toyotas. officials.

222Chapter4 Categorical Propositions

None but the brave All people who 9. �The Only�

deserve the fair. deserve the fair are Statements beginning with the words �the only�

brave people. are translated differently from those beginning

No birds except All birds proud with �only.� For example, the statement �The only peacocks are proud of their tails are cars that are available are Chevrolets� means �If of their tails. peacocks. a car is available, then it is a Chevrolet.� This in He owns only blue- All stocks he owns turn is translated as �All cars that are available are chip stocks. are blue-chip stocks. Chevrolets.� In other words, language following

She invited only �the only� goes in the subject term of the categori-All people she wealthy socialites. cal proposition. Examples: invited are wealthy

socialites.

The only animals that All animals that live live in this canyon are in this canyon are

For a statement involving �only,� �none but,� skunks. skunks.

�none except,� and �no . . . except� to be a genu- Accountants are the All those who inely exclusive proposition, the word that follows only ones who will be will be hired are these words must be a plural noun or pronoun. hired. accountants. If the word that follows �only,� �none but,� or

the like designates an individual, the statement Statements involving �the only� are similar to

ment �Only Megan painted a picture� asserts that the statement is about an individual, two state-Megan painted a picture really asserts two things. For example, the state- those involving �only� in this one respect: When

and that no other person ments are needed to translate it. For example,

painted a picture. Thus it would be translated as �The only person who painted a picture is Megan�

people who painted a picture, and all people who person painted a picture. The statement is equiv-painted a picture are people identical to Megan.� two statements: �All people identical to Megan are means that Megan painted a picture, and no other

alent in meaning to �Only Megan painted a pic-

This section of the text will ignore cases where the ture.� Thus, it is translated �All people identical to

ignates an individual. people who painted a picture are people identical Also note that many English statements con-word following �only,� �none but,� or the like des- Megan are people who painted a picture, and all

to Megan.� Statements involving �the only� that

taining �only� are ambiguous because �only� can refer to individuals are ignored throughout the be interpreted as modifying alternate words in the remainder of this chapter. statement. Consider, for example, the statement

�He only jogs after sunset.� Does this mean �He

jogs and does not walk after sunset� or �The only is the only person who jogs after sunset� or �He 10. Exceptive Propositions time he jogs is after sunset�? If the statement�s Propositions of the form �All except S are P� and �All context does not provide an answer, the translator but S are P� are exceptive propositions. They must be is free to pick any of these senses for translation. translated not as single categorical propositions but This same ambiguity, incidentally, affects the last as pairs of conjoined categorical propositions. State-two examples in the earlier list. Accordingly, they ments that include the phrase �none except,� on the might also be translated �All things he owns are other hand, are exclusive (not exceptive) proposi-blue-chip stocks� and �All socialites she invited tions. �None except� is synonymous with �none but.� are wealthy people.� Here are some examples of exceptive propositions:

Section 4.7 Translating Ordinary Language Statements into Categorical Form 223

All except stu- No students are invited Because exceptive propositions cannot be trans-dents are invited. people, and all nonstu- lated into single categorical propo sitions, many of

dents are invited people. the simple inferences and operations pertinent to

All but managers No managers are categorical propositions cannot be applied to them. must report to people who must report Arguments that contain exceptive propositions the president. to the president, and as premises or conclusion can be evaluated only

all nonmanagers are through the application of extended techniques. people who must report This topic is taken up in the next chapter. to the president.

Key word (to be eliminated)Translation hint

whoever, wherever, always, anyone, use �all� together with people,

never, etc. places, times

a few, several, many use �some�

if . . . then use �all� or �no�

unless use �if not�

only, none but, none except, use �all�

no . . . except

the only use �all�

all but, all except, few two statements required

not every, not all use �some . . . are not�

there is, there are use �some�

Rule for A propositions

Language following these words goes in the subject term: �if,� �the only,� and �W�

words (�who,� �what,� �when,� �where,� �whoever,� �whatever,� �whenever,�

�wherever�).

Language following these words goes in the predicate term: �only if,� �only,� �none

but,� �none except,� and �no . . . except.�

Exercise 4.7

I. Translate the following into standard-form categorical propositions.

★1. Any bank that makes too many risky loans will fail.

2. Temporary workers are not eligible for fringe benefits.

3. Terrorist attacks succeed whenever security measures are lax.

224Chapter4 Categorical Propositions

★ 4. Bromine is extractable from seawater.

5. Not all guilt feelings are psychological aberrations.

6. Every jazz fan admires Duke Ellington.

★7. If it�s a halogen, then it isn�t chemically inert.

8. A television show that depicts violence incites violence.

9. Manipulators do not make good marriage partners.

★10. None but pirate ships fly the Jolly Roger.

11. She oversleeps whenever she�s depressed.

12. She�s depressed whenever she oversleeps.

★ 13. A man is a bachelor only if he is unmarried.

14. Warmth always relieves pain.

15. Joseph J. Thomson discovered the electron.

★ 16. A few organic silicones are used as lubricants.

17. Only nuclear-powered vehicles are suitable for deep-space exploration.

18. Comets are the only heavenly bodies with tails.

★ 19. There is a giant star in the Tarantula Nebula.

20. If an activity emits methane, then it contributes to climate change.

21. No shellfish except oysters make pearls.

★22. Only diabetics require insulin treatments.

23. The electroscope is a device for detecting static electricity.

24. Occasionally there are concerts in Central Park.

★25. Berlin was the setting for the 1936 Olympic Games.

26. The Kentucky Derby is never run in January.

27. The only way to get rid of a temptation is to yield to it.

★28. Where there�s smoke, there�s fire.

29. Lunar eclipses do not occur unless the moon is full.

30. Radio transmissions are disrupted whenever sunspot activity increases.

★31. If an ore isn�t radioactive, then it isn�t pitchblende.

32. All but the rats left the sinking ship.

33. A pesticide is dangerous if it contains DDT.

★34. John Grisham writes only novels about lawyers.

35. He who hesitates is lost.

36. Modern corporations are all run in the interest of their managers.

Section 4.7 Translating Ordinary Language Statements into Categorical Form 225

★37. Unless the sun is shining, a rainbow cannot occur.

38. Whoever suffers allergic reactions has a weakened immune system.

39. All fruits except pineapples ripen after they are picked.

★40. Few corporate raiders are known for their integrity.

41. Monkeys are found in the jungles of Guatemala.

42. Monkeys are mammals.

★43. I like strawberries.

44. All passengers are not allowed to smoke on board the aircraft.

45. All flowers are not fragrant.

★46. Talisha travels where she wants.

47. Bats are the only true flying mammals.

48. Not every river runs to the sea.

★49. Physicists do not understand the operation of superconductors.

50. Many apartment dwellers are victimized by noise.

51. There are forced labor camps in China.

★52. Whatever increases efficiency improves profitability.

53. Dolphins are swimming between the breakers.

54. Feathers are not heavy.

★55. Few picnics are entirely free of ants.

56. A civil right is unalienable if it is a human right.

57. She says what she pleases.

★58. Several contestants won prizes.

59. An animal is a feline only if it is a cat.

60. Renee does whatever she is told to do.

II. The following exercises contain typical mistakes that students make in attempting to

translate statements into standard form. Correct the errors and redundancies in these attempted translations.

★ 1. Some of the figure-skating finalists are performers who are athletes that may

win medals.

2. All cars identical to BMWs are the only cars that young lawyers drive.

3. All vertebrates except cartilaginous fishes are animals with a bony skeleton.

★ 4. No downhill skiers are effective competitors if they suffer from altitude sickness.

5. All substances like cobalt are things that are substances identical to ferromag-

netic metals.

226Chapter4 Categorical Propositions

6. No people identical to nuclear pacifists are people who believe a just war is

possible.

★ 7. All people identical to matadors are not performers who succumb easily to fear.

8. All companies identical to Google are looking forward to a bright future.

9. No toxic dumps are ecological catastrophes unless they leak.

★10. All crocodiles are things identical to dangerous animals when they are hungry.

III. Translate the premise and conclusion of the following immediate inferences into stan-

dard-form categorical propositions. Then use conversion, obversion, contraposition, or the traditional square of opposition to determine whether each is valid or invalid. For those that are invalid, name the fallacy committed.

★1. Flu vaccines are never completely effective.

Therefore, not every flu vaccine is completely effective.

2. Only economic sanctions threaten a military superpower.

Therefore, every economic sanction threatens a military superpower.

3. A child destined to succeed in life has high self-esteem.

Therefore, a child with low self-esteem is not destined to succeed in life.

★4. If a date is a good dinner companion, then she does not spend every minute

texting. Therefore, if a date does not spend every minute texting, then she is not a bad dinner companion.

5. There are banks that engage in money laundering.

Therefore, there are banks that do not engage in money laundering.

6. Not a single major storm fails to cause flight cancellations.

Therefore, every major storm causes flight cancellations.

★7. Tax cuts never please everyone.

Therefore, every tax cut pleases everyone.

8. Military exercises are sometimes condemned by opponents.

Therefore, any military exercise is condemned by opponents.

9. It is false that not a single charter school is an academic underperformer.

Therefore, several charter schools are academic underperformers.

★10. Fake artworks are sold on eBay.

Therefore, a few items sold on eBay are fake artworks.

IV. Create a total of fifteen new categorical propositions (not in standard form) that may be

described as follows:

1. One contains a term without a noun.

2. One contains the nonstandard verb �will.�

3. One begins with the name �Margie.�

4. One contains the adverb �never.�

Section 4.7 Translating Ordinary Language Statements into Categorical Form 227

5. One contains the pronoun �whoever.�

6. One contains an unexpressed quantifier.

7. One begins with the words �not everyone.�

8. One begins with the word �few.�

9. One begins with the words �a few.�

10. One is a conditional (if . . . then) statement.

11. One contains the word �unless.�

12. One begins with the word �only.�

13. One begins with the words �no . . . except.�

14. One begins with the words �the only.�

15. One begins with the words �all except.�

Finally, translate each of these new statements into one or more standard-form categorical propositions.

Summary

Categorical Proposition: A proposition that relates two classes (or categories). Standard-form cat-egorical propositions occur in four forms and are identified by letter names:

■ A: All S are P.

■ E: No S are P.

■ I: Some S are P.

■ O: Some S are not P.

Every standard-form categorical proposition has four components:

■ Quantifier (�all,� �no,� �some�)

■ Subject term

■ Copula (�are,� �are not�)

■ Predicate term

The quality of a categorical proposition:

■ Affirmative (All S are P, Some S are P.)

■ Negative (No S are P, Some S are not P.)

The quantity of a categorical proposition:

■ Universal (All S are P, No S are P.)

■ Particular (Some S are P, Some P are not P.)

228Chapter4 Categorical Propositions

The subject and predicate terms are distributed if the proposition makes an assertion about every member of the class denoted by the term; otherwise, undistributed:

■ A: Subject term is distributed.

■ E: Subject and predicate terms are distributed.

■ I: Neither term is distributed.

■ O: Predicate term is distributed.

Universal (A and E) propositions allow for two different interpretations:

■ Aristotelian: Universal propositions about existing things have existential import.

■ Boolean: Universal propositions have no existential import.

The modern square of opposition is a diagram that represents necessary inferences from the Boolean standpoint:

■ A and O propositions contradict each other.

■ E and I propositions contradict each other.

The content of categorical propositions may be represented by two-circle Venn diagrams:

■ Shading an area indicates that the area is empty.

■ Entering an X in an area means that the area is not empty.

Using Venn diagrams to test an immediate inference:

■ Enter the content of the premise and conclusion in separate Venn diagrams.

■ See if the content of the conclusion diagram is contained in the premise diagram.

Three operations that sometimes yield logically equivalent results:

■ Conversion: Switch S and P. Logically equivalent results for E, I.

■ Obversion: Change the quality, replace P with its term complement. Logically equivalent

results for A, E, I, O.

■ Contraposition: Switch S and P, replace S and P with term complements. Logically

equivalent results for A, O.

Two formal fallacies may occur when these operations are used to derive conclusions:

■ Illicit conversion: Performing conversion on an A or O premise.

■ Illicit contraposition: Performing contraposition on an E or I premise.

The traditional square of opposition applies to categorical propositions when the Aristotelian standpoint is adopted and the subject term refers to existing things:

■ Contrary: Holds between A and E. At least one is false.

■ Subcontrary: Holds between I and O. At least one is true.

■ Subalternation: Holds between A and I and between E and O. Truth flows downward and

falsity flows upward.

■ Contradiction: Holds as in the modern square.

Three formal fallacies may occur when the traditional square is used to derive conclusions:

■ Illicit contrary: Results from an incorrect application of Contrary.

Summary229

■ Illicit subcontrary: Results from an incorrect application of Subcontrary.

■ Illicit subalternation: Results from an incorrect application of Subalternation.

Existential fallacy: Occurs when Contrary, Subcontrary, or Subalternation are used on premises whose subject terms refer to nonexistent things.

Venn diagrams may be modified to apply to the Aristotelian standpoint:

■ For A and E: Enter a circled X in the unshaded part of the subject circle.

■ The circled X represents the temporary assumption of existence.

■ May be used to prove the traditional square and test immediate inferences.

Translation: Propositions not in standard form may be put into standard form.

■ Translation must have a proper quantifier, subject term, copula, predicate term.

■ Translate singular propositions by using a parameter.

■ Translate adverbs and pronouns by using �persons,� �places,� �things,� �times.�

■ For A propositions:

● Language following �if,� �the only,� and �W� words goes in the subject term.

● Language following �only if,� �only,� �none but,� �none except,� and �no . . . except� goes

in the predicate term.

230Chapter4 Categorical Propositions

5Categorical Syllogisms

5.1 Standard Form, Mood, and Figure

5.2 Venn Diagrams

5.3 Rules and Fallacies

5.4 Reducing the Number of Terms

5.5 Ordinary Language Arguments

5.6 Enthymemes

5.7 Sorites

5.1 Standard Form, Mood, and Figure

Learning about categorical syllogisms is empowering. Suppose that a student in your class on the romantic novel presents this argument related to Emily Bront�s Wuthering Heights: �Anyone for whom love is an all-consuming passion invites suffering, and Catherine�s love for Heathcliff was precisely that. Thus, in loving Heathcliff in that way, Catherine invited suffering.� This argu-ment, which comprises two premises, can be proven valid by translating it into a categorical syllogism.

In the general sense of the term, a syllogism is a deductive argument consisting of two premises and one conclusion. Provisionally we will define a categorical syllogism as a syllogism consisting of three categorical propositions and contain-ing a total of three different terms, each of which appears twice in distinct proposi-tions. (We will give a more precise definition shortly.) The following argument is a categorical syllogism:

All soldiers are patriots.

No traitors are patriots.

Therefore, no traitors are soldiers.

231

Each of the three terms in a categorical syllogism the following syllogism is not in standard form,

has its own name depending on its position in the because the fourth condition is violated: argument. The major term, by definition, is the

premises, is the one that occurs once in each prem- To put this syllogism into standard form the ise and does not occur in the conclusion. Thus, for order of the premises must be reversed. The major the argument just given, the major term is �soldiers,� premise (the one containing �masterpieces,� which is the minor term is �traitors,� and the middle term is the predicate term in the conclusion) must be listed �patriots.� first, and the minor premise (the one containing The premises of a categorical syllogism also have �paintings,� which is the subject term in the conclu-is the subject of the conclusion. The Some watercolors are masterpieces. middle term , Therefore, some paintings are masterpieces. which provides the middle ground between the two predicate of the conclusion, and the All watercolors are paintings. minor term

their own names. The major premise, by defini- sion) must be listed second. tion, is the one that contains the major term, and the Now that we have a definition of standard-form minor premise is the one that contains the minor categorical syllogism, we can give a more precise term. Thus, in the syllogism just given the major definition of categorical syllogism. A categorical premise is �All soldiers are patriots,� and the minor syllogism is a deductive argument consisting premise is �No traitors are patriots.� Now that we of three categorical propositions that is capable are supplied with these definitions, we may proceed of being translated into standard form. For an to the idea of standard form. A standard-form argument to qualify as a categorical syllogism, all categorical syllogism is one that meets the three statements need not be standard-form cat- following four conditions: egorical propositions; but if they are, the analy-1. All three statements are standard-form categori-sis is greatly simplified. For this reason, all of the cal propositions. syllogisms presented in the first four sections of this chapter will consist of statements that are in 2. The two occurrences of each term are identical. standard form. In later sections, techniques will 3. Each term is used in the same sense throughout be developed for translating non-standard-form the argument. syllogisms into equivalent arguments that are in 4. The major premise is listed first, the minor standard form. premise second, and the conclusion last.

The first condition requires that each statement

have a proper quantifier, subject term, copula,

and predicate term. The second condition is clear. Standard form of a syllogism

The third rules out the possibility of equivoca- Major premise 1. Quantifier copula (contains major term) tion. For example, if a syllogism containing the

word �men� used that term in the sense of human Minor premise 2. Quantifier copula (contains minor term) beings in one statement and in the sense of male 3. Quantifier copula Conclusion human beings in another statement, the syllo-Minor Major gism would really contain more than three terms term term and would therefore not be in standard form.

Finally, the fourth condition merely requires that

the three statements be listed in the right order.

The syllogism about soldiers is in standard form After a categorical syllogism has been put

because all four conditions are fulfilled. However, into standard form, its validity or invalidity may

232Chapter5 Categorical Syllogisms

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). be determined through mere inspection of the No painters are sculptors. form. The individual form of a syllogism consists Some sculptors are artists. of two factors: mood and figure. The mood of a Therefore, some artists are not painters.

categorical syllogism consists of the letter names This syllogism is in standard form. The mood is of the propositions that make it up. For example, if EIO and the figure is four. The form of the syllo-the major premise is an A proposition, the minor gism is therefore designated as EIO-4. premise an O proposition, and the conclusion To remember how the four figures are defined, an E proposition, the mood is AOE . To determine imagine the four possible arrangements of the the mood of a categorical syllogism, one must middle term as depicting the outline of a shirt first put the syllogism into standard form; the collar: letter name of the statements may then be noted to the side of each. The mood of the syl-

logism is then designated by the order

major premise first, the letter for the minor 1 4 X X premise second, and the letter for the con-of these letters, reading the letter for the X X clusion last.

The figure of a categorical syllogism is 2 3

determined by the location of the two occur-

rences of the middle term in the premises.

Four different arrangements are possible. If we let The only problem with this device is that it may S represent the subject of the conclusion (minor lead you to confuse the second figure with the term), P the predicate of the conclusion (major third. To avoid this confusion, keep in mind that term), and M the middle term, and leave out the for these two figures the S and P terms go on the quantifiers and copulas, the four possible arrange- same �collar flap� as the middle term. These terms ments may be illustrated as follows: are represented by the X�s in the diagram. Thus,

for Figure 2, the occurrences of the middle term

are on the right side of the syllogism, and for

Figure 1Figure 2 Figure 3 they are on the left. MPPM Since there are four kinds of categorical prop-

S tions in a categorical syllogism, there are 64 pos-PSP sible moods (4 S ositions and there are three categorical proposi-MSM � 4 � 4 = 64). And since there

are four different figures, there are 256 different

forms of categorical syllogisms (4 � 64 = 256).

Figure 3Figure 4 Once the mood and figure of a syllogism is known, MPPM the validity of the syllogism can be determined by MSMS checking the mood and figure against a list of valid

S syllogistic forms. To do this, first adopt the Boolean PSP standpoint and see if the syllogism�s form appears in the following table of unconditionally valid forms. If

In the first figure the middle term is top left, bottom it does, the syllogism is valid from the Boolean stand-point. In other words, it is valid regardless of whether right; in the second, top right, bottom right, and so its terms denote actually existing things. on. Example:

Section 5.1 Standard Form, Mood, and Figure 233

Unconditionally Valid Forms

Figure 1Figure 2Figure 3Figure 4

AAA s Conditionally EAE IAI AEE rm valid EAE AEE AI I IAI fo

AI I 15 EIO OAO EIO

EIO AOO EIO

Aristotelian Unconditionally

If the syllogism does not appear on the list of Number of valid valid Boolean

unconditionally valid forms, then adopt the

Aristotelian standpoint and see if the syllogism�s

form appears in the following table of condi- Information conveyed by premises tionally valid forms. If it does, the syllogism is

valid from the Aristotelian standpoint on con-

dition that a certain term (the �critical� term) The graph shows that when the premises of a syl-denotes actually existing things. The required logistic form are recognized as conveying infor-condition is stated in the last column. mation about existence, an additional nine forms

become valid.

Interestingly, during the Middle Ages logic stu-

Conditionally Valid Forms dents used to memorize a little poem that served

Figure 1 Figure 2 Figure 3 Figure 4 Required as a rule of thumb for distinguishing valid from

Condition invalid syllogisms. The vowels in the words identi-

AAI fied the mood, and the words �prioris,� �secundae,� AEO AEO S exists

EAO and so on the figure.* EAO

AAI EAO M exists Barbara, Celarent, Darii, Ferioque prioris;

EAO Cesare, Camestres, Festino, Baroco

AAI secundae; P exists

Tertia, Darapti, Disamis, Datisi, Felapton,

Bocardo, Ferison habet: quarta insuper

For example, the addit AAI-1 is valid from the Aristotelian standpoint if the subject of the conclusion (the minor Bramantip, Camenes, Dimaris, Fesapo, term) denotes actually existing things. The Fresison. EAO-3 is valid if the middle term denotes actually existing things. For example, the �Barbara� syllogism (this desig-Thus, if we are given an AAI-1 syllogism and the minor nation is still encountered today) is AAA-1, �Celar-term is �cats,� then the syllogism is valid from the Aris- EAE ent� is -1, and so on. This poem conforms totelian standpoint. But if the minor term is �unicorns,� substantially to the two tables given earlier, except then the syllogism is invalid. On the other hand, if the that five forms have been left out. The reason these minor term is �students who failed the exam� and we forms were left out is that the logicians of that time are not certain if there are any such students, then the

syllogism is conditionally valid.

The relationship between the Aristotelian stand- *This poem appears for the first time in William of Sherwood�s Intro-

point and the Boolean standpoint is illustrated in the ductiones in Logicam, written ca. 1260, but it was probably not following bar graph: originally composed by him. See William Kneale and Martha Kneale,

Development of Logic (Oxford: Clarendon Press, 1962), 232.

234Chapter5 Categorical Syllogisms

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). considered them weak: They draw a particular con- Then use the figure to determine the arrangement of

ger) universal conclusion. For example, the weaker E No ______ are M. AAI clusion from premises that would support a (stron- the middle terms:

-1 is left out in favor of the stronger AAA-1.

Needless to say, few students today depend on this I Some M are ______.

obtain the mood and figure. But sometimes we need to We have seen how, given the syllogism, we can Finally, supply the major and minor terms, using poem to distinguish valid from invalid syllogisms. O Some ______ are not ______.

to the syllogistic form. Suppose we are given the form conclusion is always repeated in the first premise, EIO-4. To reconstruct the syllogistic form is easy. First and the subject of the conclusion is repeated in use the mood to determine the skeleton of the form: the second premise: go in the reverse direction: from the mood and figure predicate of the conclusion. The predicate of the the letters S and P to designate the subject and

E No ______ are ______. E No P are M.

I Some ______ are ______. I Some M are S.

O Some ______ are not ______. O Some S are not P.

Exercise 5.1

I. The following syllogisms are in standard form. Identify the major, minor, and middle

terms, as well as the mood and figure of each. Then use the two lists of valid syllogis-tic forms to determine whether each is valid from the Boolean standpoint, valid from the Aristotelian standpoint, or invalid.

★1. All neutron stars are things that produce intense gravity.

All neutron stars are extremely dense objects.

Therefore, all extremely dense objects are things that produce intense gravity.

2. No insects that eat mosquitoes are insects that should be killed.

All dragonflies are insects that eat mosquitoes.

Therefore, no dragonflies are insects that should be killed.

3. No environmentally produced diseases are inherited afflictions.

Some psychological disorders are not inherited afflictions.

Therefore, some psychological disorders are environmentally produced diseases.

★4. No people who mix fact with fantasy are good witnesses.

Some hypnotized people are people who mix fact with fantasy.

Therefore, some hypnotized people are not good witnesses.

Section 5.1 Standard Form, Mood, and Figure 235

5. All ozone molecules are good absorbers of ultraviolet rays.

All ozone molecules are things destroyed by chlorine.

Therefore, some things destroyed by chlorine are good absorbers of ultra-violet rays.

II. Put the following syllogisms into standard form, using letters to represent the terms,

and name the mood and figure. Then use the two lists of valid syllogistic forms to deter-mine whether each is valid from the Boolean standpoint, valid from the Aristotelian standpoint, or invalid.

★1. No Republicans are Democrats, so no Republicans are big spenders, since

all big spenders are Democrats.

2. Some latchkey children are not kids who can stay out of trouble, for some

youngsters prone to boredom are latchkey children, and no kids who can stay out of trouble are youngsters prone to boredom.

3. No rent-control proposals are regulations welcomed by landlords, and all

regulations welcomed by landlords are measures that allow a free hand in raising rents. Therefore, some rent-control proposals are measures that allow a free hand in raising rents.

★4. Some insects that feed on milkweed are not foods suitable for birds, inas-

much as no monarch butterflies are foods suitable for birds and all mon-arch butterflies are insects that feed on milkweed.

5. No friendships based on texting are relationships likely to last, and some

budding romances are friendships based on texting. Thus, some relation-ships likely to last are budding romances.

6. Some African nations are not countries deserving military aid, because

some African nations are not upholders of human rights, and all countries deserving military aid are upholders of human rights.

★7. All pranksters are exasperating individuals, consequently some leprechauns

are exasperating individuals, since all leprechauns are pranksters.

8. Some racists are not people suited to be immigration officials, given that some

humanitarians are not people suited to be immigration officials, and no humanitarians are racists.

9. No people who respect human life are terrorists, and all airline hijackers

are terrorists. Hence, no airline hijackers are people who respect human life.

★10. Some silicates are crystalline substances, because all silicates are oxygen

compounds, and some oxygen compounds are not crystalline substances.

III. Reconstruct the syllogistic forms from the following combinations of mood

and figure:

★1. OAE-3

2. EIA-4

236Chapter5 Categorical Syllogisms

3. AII-3

★ 4. IAE-1

5. AOO-2

6. EAO-4

★7. AAA-1

8. EAO-2

9. OEI-3

★ 10. OEA-4

IV. Construct the following syllogisms.

★1. An EIO-2 syllogism with these terms� major�dogmatists; minor�

theologians; middle�scholars who encourage freethinking.

2. An unconditionally valid syllogism in the first figure with a particular affir-

mative conclusion and these terms: major�people incapable of objectivity; minor �Supreme Court justices; middle�lockstep ideologues.

3. An unconditionally valid syllogism in the fourth figure having two univer-

sal premises and these terms: major�teenage suicides; minor�heroic epi-sodes; middle�tragic occurrences.

★4. A valid syllogism having mood OAO and these terms: major�things

capable of replicating by themselves; minor�structures that invade cells; middle�viruses.

5. A valid syllogism in the first figure having a universal negative conclusion

and these terms: major�guarantees of marital happiness; minor�prenup-tial agreements; middle�legally enforceable documents.

V. Answer �true� or �false� to the following statements:

1. Every syllogism is a categorical syllogism.

2. Some categorical syllogisms cannot be put into standard form.

3. The statements in a categorical syllogism need not be expressed in standard

form.

4. The statements in a standard-form categorical syllogism need not be

expressed in standard form.

5. In a standard-form categorical syllogism the two occurrences of each term

must be identical.

6. The major premise of a standard-form categorical syllogism contains the

subject of the conclusion.

7. To determine its mood and figure, a categorical syllogism must first be put

into standard form.

Section 5.1 Standard Form, Mood, and Figure 237

8. In a standard-form syllogism having Figure 2, the two occurrences of the

middle term are on the right.

9. The unconditionally valid syllogistic forms are valid from both the Boolean

and the Aristotelian standpoints.

10. The conditionally valid syllogistic forms are invalid if the requisite condition

is not fulfilled.

5.2 Venn Diagrams

Venn diagrams provide the most intuitively evi- M dent and, in the long run, easiest to remember

technique for testing the validity of categorical

syllogisms. The technique is basically an exten- 1

sion of the one developed in Chapter 4 to rep- 2 4 3 resent the informational content of categorical

propositions. Because syllogisms contain three 5 6 7 terms, whereas propositions contain only two,

requires three overlapping circles. Anything in the area marked �1� is an M but nei-These circles should be drawn so that seven the application of Venn diagrams to syllogisms SP

areas are clearly distinguishable within the dia- is both an ther an S nor a P, anything in the area marked �2�

subject of the conclusion, the lower-right circle diagram and then inspecting the diagram to to the predicate of the conclusion, and the top see whether it necessarily implies the truth of the circle to the middle term. This convention is easy conclusion. If the information in the diagram does do to remember because it conforms to the arrange-this, the argument is valid; otherwise it is invalid. ment of the terms in a standard-form syllogism: The use of Venn diagrams to evaluate syllogisms The subject of the conclusion is on the lower left, usually requires a little practice at first. Perhaps the the predicate of the conclusion is on the lower best way of learning the technique is through illus-right, and the middle term is in the premises, trative examples, but a few pointers are needed above the conclusion.* first: for each term. The precise order of the labeling so on. is not critical, but we will adopt the convention The test procedure consists of transferring of always assigning the lower-left circle to the the information content of the premises to the gram. The second step is to label the circles, one area marked �3� is a member of all three classes, and S and an M but not a P, anything in the

1. Marks (shading or placing an X) are entered

*Some textbooks present three-circle Venn diagrams with two cir- only for the premises. No marks are made for cles on the top and one on the bottom. In our discussions we will the conclusion.

present them the way John Venn himself drew them, with two 2. If the argument contains one universal premise, (London: Macmillan, 1881), 105, 114, 116. this premise should be entered first in the diagram. circles on the bottom and one on the top. See his Symbolic Logic

238Chapter5 Categorical Syllogisms

If there are two universal premises, either one can This means that the X may be in either (or be done first. both) of the two areas�but it is not known

3. When entering the information contained which one.

in a premise, one should concentrate on the 7. An X should never be placed in such a way that

circles corresponding to the two terms in the it dangles outside of the diagram, and it should statement. While the third circle cannot be never be placed on the intersection of two lines. ignored altogether, it should be given only

minimal attention.

4. When inspecting a completed diagram to Wrong:Wrong:

see whether it supports a particular conclu-

sion, one should remember that particular

statements assert two things. �Some S are P� means �At least one S exists and that S is a P�; �Some S are not P� means �At least one S exists and that S is not a P.�

5. When shading an area, one must be careful

to shade all of the area in question. Examples:

Boolean Standpoint

Right: Because the Boolean standpoint does not rec-

ognize universal premises as having existen-

tial import, its approach to testing syllogisms is

simpler and more general than that of the

Aristotelian standpoint. Hence, we will begin by test-

Wrong: ing syllogisms from that standpoint and later proceed

to the Aristotelian standpoint. Here is an example:

6. The area where an X goes is always initially 1. No P are M.EAE-2 divided into two parts. If one of these parts has All S are M. already been shaded, the X goes in the unshaded No S are P. part. Examples: Since both premises are universal, it makes no difference

which premise we enter first in the diagram. To enter the

major premise, we concentrate our attention on the M

Right: and P circles, which are highlighted with color:

If one of the two parts is not shaded, the X goes

on the line separating the two parts. Examples:

No P are M.

Right:

Section 5.2 Venn Diagrams 239

Prominent 1834�1923John Venn Logicians

John Venn is known mainly for his circle dia-grams, which have contributed to work in many areas of mathematics and logic, including com-puter science, set theory, and statistics. His book, The Logic of Chance (1866), advanced probability

theory by introducing the relative frequency inter-

pretation of probability; it significantly influenced

later developments in statistical theory as well. In

Symbolic Logic (1881) he defended George Boole

against various critics and rendered the new logic

intelligible to nonmathematical thinkers. Finally,

in The Principles of Empirical and Inductive Logic

(1889), he criticized Mill�s methods of induction as

being of limited application as an engine of discov- Reproduced by permission of the Master and Fellows of Gonville and Caius College, Cambridge, UK ery in�science.

John Venn was born in Hull, England, the son of Venn became a fel low of the Royal Society and recei-

Henry Venn, the Drypool parish rector, and Martha ved the degree of Doctor of Science. Sykes Venn, who died when John was a child. The The greater part of Venn�s life cen tered com-Venns were prominent members of the evangelical pletely on his association with Cambridge. In 1857 movement within the Church of England. John Venn�s he became a fellow of Caius, and he remained grandfather had been an evangelical leader, as was a member of the college foundation for sixty-his father, whom his contemporaries regarded as the six years, until his death. During the last twenty head of the evangelical movement. His father served years of his life he served as college president, for many years in an administrative capacity for the he wrote a history of the college. Also, in collab-Church Missionary Society, and John was expected to oration with� his son, he completed Part I of the follow in the family tradition. In 1858, after graduating massive Alumni Cantabrigienses, which contains from Gonville and Caius (pronounced �keys�) College, short biographies of 76,000 graduates and office-Cambridge, he was ordained and served for a time as holders ranging from the university�s earliest days a curate in parishes near London. through 1751.

Perhaps owing to his contact with Henry Sidgwick John Venn�s son said of his father that he was a �fine

and other Cambridge agnostics, Venn�s confidence in walker and mountain climber.� Also, in keeping with the Thirty-Nine Articles of the Church of England be gan his view that abstract subjects such as logic and math-to erode. Also, as a result of his reading the works of ematics ought to serve practical utility, Venn loved to De Morgan, Boole, and Mill, his interest shifted almost use this knowledge to build machines. He invented totally from theological matters to issues related to logic. a cricket bowling machine that was used against the At age twenty-eight, Venn returned to Cambridge to best batsman of an Australian team. The machine become a lecturer in logic and probability theory. Five �clean bowled� this batsman four times. Today, Venn years later, he married Susanna Carnegie Edmonstone, is memorialized by a stained-glass window in the the daughter of an Anglican cleric, and they had one dining hall of Caius College that contains a represen-child, John Archibald Venn. In 1883, at age forty-nine, tation of a Venn diagram.

240Chapter5 Categorical Syllogisms

We now complete the diagram by entering the M minor premise. In doing so, we concentrate our

attention on the S and M circles, which are high-lighted with color:

No S are M.

All Again, the conclusion states that the area where the S are M.S and P circles overlap is shaded. Inspection of the

diagram reveals that only part of this area is shaded,

SP so the syllogistic form is invalid.

Another example:

The conclusion states that the area where the S and 3. Some P are M. IAI-4 P circles overlap is shaded. Inspection of the diagram reveals that this area is indeed shaded, so All M are S. the syllogistic form is valid. Because the form is Some S are P. valid from the Boolean standpoint, it is uncondi- We enter the universal premise first. To do so, we tionally valid. In other words, it is valid regardless concentrate our attention on the M and S circles: of whether its premises are recognized as having

existential import. M

Here is another example:

2. All M are P. AEE-1

No S are M. All M are S.

No S are P.

Again, both premises are universal, so it makes no SP difference which premise we enter first in the dia-

gram. To enter the major premise, we concentrate To enter the particular premise, we concentrate our our attention on the attention on the M and P circles. This premise tells us to M and P circles:

place an X in the area where the M and P circles overlap.

M Because part of this area is shaded, we place the X in

the remaining area:

All M are P.

SP Some P are M. X

To enter the minor premise, we concentrate our

attention on the M and S circles: SP

Section 5.2 Venn Diagrams 241

The conclusion states that there is an X in the area M where the S and P circles overlap. Inspection of the 6. All M are P . AOO-1 diagram reveals that there is indeed an X in this area, Some S are not M. so the syllogistic form is valid. Some S are not P .

The examples that follow are done in a single step.

1 2 X

4. All P are M. AOO-2 SP

Some S are not M.

Some S are not P.

In entering the particular premise, we concentrate

our attention on the part of the S circle that lies

X outside the M circle (colored area). Because this

S area is divided into two parts (the areas marked P

�1� and �2�), we place the X on the line (arc of

The universal premise is entered first. The par- the P circle) separating the two areas. The conclu-ticular premise tells us to place an X in the part of sion states that there is an X that is inside the S the S circle that lies outside the M circle. Because circle but outside the P circle. There is an X in the part of this area is shaded, we place the X in the S circle, but we do not know whether it is inside remaining area. The conclusion states that there or outside the P circle. Hence, the syllogistic form is an X that is inside the S circle but outside the P is invalid. circle. Inspection of the diagram reveals that there

is indeed an X in this area, so the syllogistic form M is valid. 7. All M are P . AAA-1

All S are M.

M All S are P .

5. Some M are P . IAI-1

All S are M.

Some S are P .

X 2

1 SP

SP This is the �Barbara� syllogism. The conclusion

states that the part of the S circle that is outside the

As usual, we enter the universal premise first. In P circle is empty. Inspection of the diagram reveals entering the particular premise, we concentrate on that this area is indeed empty. Thus, the syllogistic the area where the form is valid. M and P circles overlap. (For emphasis, this area is colored in the diagram.)

Because this overlap area is divided into two parts M (the areas marked �1� and �2�), we place the X on 8. Some M are not P . OIO-1

parts. The conclusion states that there is an X in the Some S are not P . X 1 area where the the line (arc of the S circle) that separates the two Some S are M. 2

S and P circles overlap. Inspection of a X b

the diagram reveals that the single X is dangling out-

side of this overlap area. We do not know if it is in or

out. Thus, the syllogistic form is invalid. SP

242Chapter5 Categorical Syllogisms

In this diagram no areas have been shaded, so there there is an X that is inside the S circle but outside are two possible areas for each of the two X�s. The X the P circle. We have no certainty that the X from from the first premise goes on the line (arc of the S the first premise is inside the S circle, and while the circle) separating areas 1 and 2, and the X from the X from the second premise is inside the S circle, second premise goes on the line (arc of the P circle) we have no certainty that it is outside the P circle. separating areas a and b. The conclusion states that Hence, the syllogistic form is invalid.

When entering an X . . .

Look at the general area where the X goes. If neither of the sub-areas is

shaded, put the X on the line (arc of circle) separating the two sub-areas.

We have yet to explain the rationale for placing an M, but it is a P.� Again, the diagram says more

the X on the boundary separating two areas when than the premise says, and so it is incorrect. In neither of the areas is shaded. Consider this syllo- the third diagram, which is done correctly, the X gistic form: is placed on the boundary between the two areas.

No S is not an M, and it may or may P are M . not be a This diagram asserts: �At least one

Some P.� In other words, noth-S are not M . ing at� all is said about P , and so Some S are P . the diagram represents exactly the

Wrong:Wrong:Right: content of the second premise.

MMM

Aristotelian Standpoint

For the syllogistic forms tested thus

far, we have adopted the Boolean

S X X standpoint, which does not recog-X nize universal premises as having PSPSP existential import. We now shift to

the Aristotelian standpoint, where

existential import can make a difference to validity.

In each of the three diagrams the content of the To test a syllogism from the Aristotelian standpoint, first premise is represented correctly. The prob- we follow basically the same procedure we followed lem concerns placing the X from the second in Section 4.6 to test immediate inferences: premise. In the first diagram the X is placed inside

the 1. Reduce the syllogism to its form and test it S circle but outside both the M circle and the P circle. This diagram asserts: �At least one from the Boolean standpoint. If the form is S is not an valid, proceed no further. The syllogism is M and it is also not a P .� Clearly the diagram says more than the premise does, and so it is incor- valid from both standpoints. rect. In the second diagram the X is placed inside 2. If the syllogistic form is invalid from the the S circle, outside the M circle, and inside the P Boolean standpoint and has universal prem-circle. This diagram asserts: �At least one S is not ises and a particular conclusion, then adopt the

Section 5.2 Venn Diagrams 243

Aristotelian standpoint and look to see if there is The conclusion asserts that there is an X that is inside a Venn circle that is completely shaded except for the C circle but outside the T circle. Inspection of one area. If there is, enter a circled X in that area the diagram reveals no X�s at all, so the syllogism is and retest the form. invalid from the Boolean standpoint. Proceeding

3. If the syllogistic form is conditionally valid, to step 2, we adopt the Aristotelian standpoint and,

determine if the circled X represents some- noting that the conclusion is particular and that the thing that exists. If it does, the condition is ful- F circle is all shaded except for one area, we enter a filled, and the syllogism is valid from the Aris- circled X in that area:

totelian standpoint. F

In regard to step 2, if the diagram contains no

Venn circle completely shaded except for one area,

then the syllogism is invalid from the Aristotelian X standpoint. However, if it does contain such a Venn

circle and the syllogism has a particular conclusion,

then we place a circled X in the one unshaded area. CT This circled X represents the temporary assump-tion that the Venn circle in question is not empty. The diagram now indicates that the syllogism is con-In regard to step 3, if the circled X does not ditionally valid, so we proceed to step 3 and deter-represent something that exists, then the syllo-mine whether the circled X represents something gism is invalid. As we will see in Section 5.3, such that actually exists. Since the circled X represents an syllogisms commit the existential fallacy from the F , and since F stands for fighter pilots, the circled X Aristotelian standpoint. does represent something that exists. Thus, the con-The table of conditionally valid syllogistic dition is fulfilled, and the syllogism is valid from the forms presented in Section 5.1 names nine forms Aristotelian standpoint. that are valid from the Aristotelian standpoint if The F circle in this diagram represents the critical a certain condition is fulfilled. The following syl-term, first mentioned in connection with the table logism has one of those forms: of conditionally valid forms in Section 5.1. The criti-cal term is the term listed in the farthest right-hand 9. No fighter pilots are tank commanders. column of that table. As the diagram shows, if the All fighter pilots are courageous circle for the critical term has at least one member, individuals. then this member is necessarily also in the circle Therefore, some courageous individuals representing the subject of the conclusion (in this are not tank commanders. case C ). This fact provides the basis for a definition First, we replace the terms with letters and test the of critical term . The critical term is the term in a syllogism from the Boolean standpoint: categorical syllogism which, when it denotes at least

one existing thing, guarantees that the subject of the

No F conclusion denotes at least one existing thing. F are T . EAO-3 Here is another example: All F are C .

Some C are not T . 10. All reptiles are scaly animals.

All currently living tyrannosaurs are

reptiles.

Therefore, some currently living tyranno-

CT saurs are scaly animals.

244Chapter5 Categorical Syllogisms

First we test the syllogism from the Boolean standpoint: The diagram now indicates that the syllogism is

R conditionally valid, so we proceed to the third

All R are S. AAI-1 step and determine whether the circled X repre-All C are R. sents something that actually exists. Since the cir-Some C are S. cled X represents a C, and C stands for currently

living tyrannosaurs, the circled X does not repre-

sent something that actually exists. Thus, the con-

dition is not fulfilled, and the syllogism is invalid.

CS As we will see in the next section of this chapter,

The conclusion asserts that there is an X in the area the syllogism commits the existential fallacy from the Aristotelian standpoint. where the C and S circles overlap. Since the diagram In determining whether the circled X stands contains no X�s at all, the syllogism is invalid from the for something that exists, we always look to the Boolean standpoint. Proceeding to step 2, we adopt the Venn circle that is all shaded except for one area. Aristotelian standpoint. Then, after noticing that the If the term corresponding to that circle denotes conclusion is particular and that the C circle is all shaded existing things, then the circled X represents one except for one area, we enter a circled X in that area: of those things. In some diagrams, however, there R may be two Venn circles that are all shaded except

for one area, and each may contain a circled X in

the unshaded area. In these cases we direct our

attention only to the circled X needed to draw

X the conclusion. If that circled X stands for some-

thing that exists, the argument is valid; if not, it

C is�invalid. S

Exercise 5.2

I. Use Venn diagrams to determine whether the following standard-form categorical

syllogisms are valid from the Boolean standpoint, valid from the Aristotelian stand-point, or invalid. Then, identify the mood and figure, and cross-check your answers with the tables of valid syllogisms found in Section 5.1.

★ 1. All corporations that overcharge their customers are unethical businesses.

Some unethical businesses are investor-owned utilities.

Therefore, some investor-owned utilities are corporations that overcharge their customers.

2. No fossil fuels are renewable forms of energy.

Some combustible gases are fossil fuels.

Therefore, some combustible gases are not renewable forms of energy.

3. No individuals truly concerned with the plight of suffering humanity are

people motivated primarily by self-interest.

All television evangelists are people motivated primarily by self-interest.

Section 5.2 Venn Diagrams 245

Therefore, some television evangelists are not individuals truly concerned with the plight of suffering humanity.

★4. All high-fat diets are diets high in cholesterol.

Some diets high in cholesterol are not healthy food programs.

Therefore, some healthy food programs are not high-fat diets.

5. No engineering majors are candidates for nighttime hookups.

No candidates for nighttime hookups are deeply emotional individuals. Therefore, no deeply emotional individuals are engineering majors.

6. All impulse buyers are consumers with credit cards.

All shopaholics are impulse buyers.

Therefore, all shopaholics are consumers with credit cards.

★ 7. No pediatricians are individuals who jeopardize the health of children.

All faith healers are individuals who jeopardize the health of children. Therefore, no faith healers are pediatricians.

8. Some individuals prone to violence are not men who treat others humanely.

Some police officers are individuals prone to violence.

Therefore, some police officers are not men who treat others humanely.

9. Some ATM locations are places criminals lurk.

All places criminals lurk are places to avoid at night.

Therefore, some places to avoid at night are ATM locations.

★10. No corporations that defraud the government are organizations the govern-

ment should deal with.

Some defense contractors are not organizations the government should deal�with.

Therefore, some defense contractors are not corporations that defraud the�government.

11. All circular triangles are plane figures.

All circular triangles are three-sided figures.

Therefore, some three-sided figures are plane figures.

12. All supernovas are objects that emit massive amounts of energy.

All quasars are objects that emit massive amounts of energy.

Therefore, all quasars are supernovas.

★13. No people who profit from the illegality of their activities are people who want

their activities legalized.

All drug dealers are people who profit from the illegality of their activities. Therefore, no drug dealers are people who want their activities legalized.

14. Some individuals who risk heart disease are people who will die young.

Some vapers are individuals who risk heart disease.

Therefore, some vapers are people who will die young.

15. Some communications satellites are rocket-launched failures.

All communications satellites are devices with antennas.

Therefore, some devices with antennas are rocket-launched failures.

246Chapter5 Categorical Syllogisms

★16. All currently living dinosaurs are giant reptiles.

All giant reptiles are ectothermic animals.

Therefore, some ectothermic animals are currently living dinosaurs.

17. All survivalists are people who enjoy simulated war games.

No people who enjoy simulated war games are soldiers who have tasted the agony of real war.

Therefore, all soldiers who have tasted the agony of real war are survivalists.

18. No spurned lovers are Valentine�s Day fanatics.

Some moonstruck romantics are Valentine�s Day fanatics.

Therefore, some moonstruck romantics are not spurned lovers.

★19. No theocracies are regimes open to change.

All theocracies are governments that rule by force.

Therefore, some governments that rule by force are not regimes open to�change.

20. Some snowflakes are not uniform solids.

All snowflakes are six-pointed crystals.

Therefore, some six-pointed crystals are not uniform solids.

II. Use Venn diagrams to obtain the conclusion that is validly implied by each of the fol-

lowing sets of premises. If no conclusion can be validly drawn, write �no conclusion.�

★ 1. No P are M. 6. No M are P.

All S are M. Some S are not M.

2. Some P are not M. ★7. All M are P.

Some M are S. All S are M.

3. Some M are P.8. All P are M.

All S are M. All S are M.

★4. Some M are not P.9. No P are M.

All M are S. Some M are S.

5. Some P are M. ★10. No P are M.

All M are S. No M are S.

III. Answer �true� or �false� to the following statements:

1. In the use of Venn diagrams to test the validity of syllogisms, marks are some-

times entered in the diagram for the conclusion.

2. When an X is placed on the arc of a circle, it means that the X could be in either

(or both) of the two areas that the arc separates.

3. If an X lies on the arc of a circle, the argument cannot be valid.

4. When representing a universal statement in a Venn diagram, one always shades

two of the seven areas in the diagram (unless one of these areas is already shaded).

5. If a completed diagram contains two X�s, the argument cannot be valid.

6. If the conclusion asserts that a certain area is shaded, and inspection of the

diagram reveals that only half that area is shaded, the argument is valid.

Section 5.2 Venn Diagrams 247

7. If the conclusion asserts that a certain area contains an X and inspection

of the diagram reveals that only half an X appears in that area, the argu-ment is valid.

8. If the conclusion is in the form �All S are P,� and inspection of the diagram

reveals that the part of the S circle that is outside the P circle is shaded, then the argument is valid.

9. If, in a completed diagram, three areas of a single circle are shaded, and placing

a circled X in the one remaining area would make the conclusion true, then the argument is valid from the Aristotelian standpoint but not from the Boolean�standpoint.

10. If, in a completed diagram, three areas of a single circle are shaded, but the

argument is not valid from the Boolean standpoint, then it must be valid from the Aristotelian standpoint.

IV. Create a standard-form categorical syllogism relating to each of the following topics:

clean energy, motion pictures, military drones, immigration, money in professional sports. Then, test each of your syllogisms with a Venn diagram.

5.3 Rules and Fallacies

The idea that valid syllogisms conform to certain Boolean Standpoint

but logicians of today generally settle on five or two depend on the concept of distribution, the second two on the concept of quality, and the last six.* If any one of these rules is violated, a spe-on the concept of quantity. In applying the first two cific formal fallacy is committed and, accordingly, rules are discussed in Aristotle�s own account, Of the five rules presented in this section, the first rules was first expressed by Aristotle. Many such

the syllogism is invalid. Conversely, if none of the rules, you may want to recall either of the two mne-monic devices presented in Chapter 4: �Unprepared rules is broken, the syllogism is valid. These rules Students Never Pass� and �Any Student Earning B�s may be used as a convenient cross-check against Is Not On Probation.� These mnemonics help one the method of Venn diagrams. We will first con-remember that the four categorical propositions sider the rules as they apply from the Boolean distribute their terms as follows: standpoint, and then shift to the Aristotelian standpoint. Statement type Terms distributed

A subject E subject, predicate

*Some texts include a rule stating that the three terms of a categorical I none syllogism must be used in the same sense throughout the argument. In

our discussion, this requirement is included as part of the definition O predicate of standard-form categorical syllogism and is subsequently incorporated into the definition of categorical syllogism. See Section 5.1. Here is the first rule.

248Chapter5 Categorical Syllogisms

Rule 1: The Middle Term Must Be All tigers are mammals. Distributed at Least Once. All mammals are animals.

All animals are tigers.

Fallacy: Undistributed middle

In this standard-form categorical syllogism the Example: In the first example the major term, �animals,� is All sharks are fish. distributed in the conclusion but not in the major All salmon are fish. premise, so the syllogism commits the fallacy of All salmon are sharks. illicit major, or, more precisely, �illicit process of the major term.� In the second example the minor term, �animals,� is distributed in the conclusion but not in middle term is �fish.� In both premises �fish� occurs the minor premise. The second example therefore as the predicate of an A proposition and therefore commits the fallacy of illicit minor, or �illicit process it is not distributed in either premise. Thus, the syl-of the minor term.� logism commits the fallacy of undistributed middle In applying this rule, one must always examine and is invalid. If the major premise were rewritten the conclusion first. If no terms are distributed in to read �All fish are sharks,� then �fish� would be the conclusion, Rule 2 cannot be violated. If one or distributed in that premise and the syllogism would both terms in the conclusion are distributed, then be valid. But, of course, it would still be unsound the appropriate premise must be examined. If the because the rewritten premise would be false. term distributed in the conclusion is also distributed The logic behind Rule 1 may be explained by in the premise, then the rule is not violated. But, if recounting how the middle term accomplishes its the term is not distributed in the premise, the rule intended purpose, which is to provide a common is violated and the syllogism is invalid. In applying ground between the subject and predicate terms of Rule 2 (and also Rule 1), you may find it helpful to the conclusion. Let us designate the minor, major, and begin by marking all the distributed terms in the middle terms by the letters S , P , and M , respectively, syllogism�either by circling them or by labeling and let us suppose that M is distributed in the major them with a superscript letter �d.� premise. By definition, P is related to the whole of The logic behind Rule 2 is easy to understand. the M class. Then, when the M class is related either Let us once again designate the minor, major, and in whole or in part to S, S and P necessarily become middle terms by the letters S , P , and M , respectively, related. Analogous reasoning prevails if we suppose and let us suppose that a certain syllogism commits that M is distributed in the minor premise. But if M is the fallacy of illicit major. The conclusion of that syl-undistributed in both premises, S and P may be related logism then makes an assertion about every member to different parts of the M class, in which case there is of the P class, but the major premise makes an no common ground for relating S and P . This is exactly assertion about only some members of the P class. what happens in our fish example. The terms �salmon� Because the minor premise, by itself, says nothing at and �sharks� are related to different parts of the fish all about the P class, the conclusion clearly contains class, so no common ground exists for relating them. information not contained in the premises, and the

syllogism is therefore invalid. Analogous reasoning

Rule 2: If a Term Is Distributed in the applies to the fallacy of illicit minor. Conclusion, Then It Must Be Distributed Rule 2 becomes intuitively plausible when it is rec-in a Premise. ognized that distribution is a positive attribute. Grant-

Fallacies: ing this, an argument that has a term distributed in Illicit major; illicit minor

the conclusion but not in the premises has more in the

Examples: All horses are animals. conclusion than it does in the premises and is there-

Some dogs are not horses. fore invalid. Of course, it is always permissible to have Some dogs are not animals. more in a premise than appears in the conclusion,

Section 5.3 Rules and Fallacies 249

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). so it is perfectly all right for a term to be distributed in These arguments may be seen as invalid because a premise but not in the conclusion. each has true premises and a false conclusion. The

first draws an affirmative conclusion from a negative

Rule 3: Two Negative Premises Are Not premise, and the second draws a negative conclusion Allowed. from affirmative premises.

Fallacy: The logic behind Rule 4 may be seen as follows. If Exclusive premisesS , P , and M once again designate the minor, major,

Example: No fish are mammals. and middle terms, an affirmative conclusion always

Some dogs are not fish. states that the S class is contained either wholly or Some dogs are not mammals. partially in the P class. The only way that such a con-

clusion can follow is if the S class is contained either

This syllogism may be seen as invalid because it has wholly or partially in the M class, and the M class true premises and a false conclusion. The defect wholly in the P class. In other words, it follows only stems from the fact that it has two negative premises. when both premises are affirmative. But if, for exam-

On reflection, Rule 3 should be fairly obvious. Let ple, the S class is contained either wholly or partially

S, P, and M once again designate the minor, major, and in the M class, and the M class is separate either middle terms. Now, if the P class and the M class are wholly or partially from the P class, such a conclu-separate either wholly or partially, and the S class and sion will never follow. Thus, an affirmative conclu-the M class are separate either wholly or partially, noth- sion cannot be drawn from negative premises. ing is said about the relation between the S class and Conversely, a negative conclusion asserts that the the P class. These two classes may be either distinct or S class is separate either wholly or partially from identical in whole or in part. Venn diagrams may be the P class. But if both premises are affirmative, used effectively to illustrate the fact that no conclusion they assert class inclusion rather than separation. can be validly drawn from two negative premises. Thus, a negative conclusion cannot be drawn from Rule 4: A Negative Premise Requires a affirmative premises.

As a result of the interaction of these first four

Negative Conclusion, and�a�Negative rules, it turns out that no valid syllogism can have Conclusion Requires a Negative Premise. two particular premises. This result is convenient

Fallacy: to keep in mind, because it allows us to identify as Drawing an affirmative conclu-invalid any standard-form syllogism in which both sion from a negative premise premises start with �some.� Because it is logically or derivable from the first four rules, a separate rule to

Drawing a negative conclusion this effect is not given here. from affirmative premises

Examples: Rule 5: If Both Premises Are Universal, the All crows are birds.

Some wolves are not crows. Conclusion Cannot Be Particular.

Some wolves are birds. Fallacy: Existential fallacy

All triangles are three-angled Example: All mammals are animals. polygons. All tigers are mammals. All three-angled polygons are

Some tigers are animals.

three-sided polygons.

Some three-sided polygons are The example has two universal premises and a par-not triangles. ticular conclusion, so it violates Rule�5. It commits

the existential fallacy from the Boolean standpoint.

250Chapter5 Categorical Syllogisms

The reason the syllogism is invalid from the Bool- term is to draw a Venn diagram. The critical term ean standpoint is that the conclusion asserts that is the one that corresponds to the circle that is all tigers exist, whereas the premises make no such shaded except for one area. In the case of two such assertion. From the Boolean standpoint, universal circles, it is the one that corresponds to the Venn premises have no existential import. circle containing the circled X on which the conclu-

In applying Rule 5, keep in mind that the existen- sion depends. Another way of identifying the criti-

tial fallacy is a fallacy that occurs when a syllogism is cal term is through examination of the distributed invalid merely because the premises lack existential terms in the syllogism. The critical term is the one import. Thus, if a syllogism is invalid for some other that is superfluously distributed. In other words, it is reason (that is, if it commits some other fallacy), the term that, in the premises, is distributed in more it does not commit the existential fallacy. Hence, occurrences than is necessary for the syllogism to before deciding that a syllogism breaks Rule 5, make obey the first two rules. Here are three examples:

certain that no other rule is violated. If a syllogism All Md are P. No M d are P d. does break one of the other four rules, Rule 5 does All S d are M. All M d are S. not apply. Some S are P. Some S are not P d.

Aristotelian Standpoint All P d are M. All M d are S. Any categorical syllogism that breaks one of the first Some S are P. four rules is invalid from the Aristotelian standpoint. The distributed terms are tagged with a super-However, if a syllogism breaks only Rule 5, it is valid from the Aristotelian standpoint on condition that the script �d.� In the first syllogism, M must be distrib-uted to satisfy Rule 1, but critical term denotes at least one existing thing. (The S , in the second premise, need not be distributed to satisfy any rule. Thus, by critical term is the term listed in the farthest right-the hand column of the table of conditionally valid syllo-superfluous distribution rule, S is the term that gistic forms presented in Section 5.1.) In the example must denote actually existing things for the syllo-gism to be valid from the Aristotelian standpoint. given in connection with Rule 5, the critical term is In the second syllogism, �tigers,� and the syllogism breaks no other rules, so it P must be distributed in the first premise to satisfy Rule 2, and is valid from the Aristotelian standpoint. The conclu-M must be sion asserts that tigers exist, and from the Aristotelian distributed once to satisfy Rule 1; but M is distrib-uted twice. Thus, standpoint the premises imply their existence. On the M is the term that must denote existing things for the syllogism to be valid from other hand, consider the following example: the Aristotelian standpoint. In the third syllogism, All mammals are animals. M must be distributed to satisfy Rule 1, but P need All unicorns are mammals. not be distributed to satisfy any rule. Thus, in this Some unicorns are animals. syllogism, P is the critical term. In this example, the critical term is �unicorns.� You may recall that the existential fallacy from Since unicorns do not exist, the premises have no the Boolean standpoint first appeared in Section 4.3, existential import from the Aristotelian standpoint. where it arose in connection with the modern square Thus, the syllogism is invalid from the Aristotelian of opposition. Also, the existential fallacy from the standpoint, and it commits the existential fallacy Aristotelian standpoint first appeared in Section 4.5, from that standpoint. Of course, it also commits where it arose in connection with the traditional square the existential fallacy from the Boolean standpoint. of opposition. The two versions of the existential fal-

In addition to consulting the table of condition- lacy that appear in connection with Rule 5 stem from

ally valid forms, one way of identifying the critical the same mistake as it relates to categorical syllogisms.

Section 5.3 Rules and Fallacies 251

Finally, if you turn to the table of conditionally the conclusion, �All S are P,� we can determine by

valid forms in Section 5.1, you will see that all of simple reasoning or a Venn diagram that the syllo-the forms listed there break Rule 5. All of them have gism is valid. Note that only Rules 1, 2, and 4 were universal premises and a particular conclusion, and used in producing this first step in our proof, but the they break no other rule. Thus, all of them commit resulting syllogism obeys the unused rules as well. the existential fallacy from the Boolean standpoint. A similar process applies to the steps that follow. But if the Aristotelian standpoint is adopted and the Next, we consider a syllogism having an E propo-critical term refers to something that does not exist, sition for its conclusion. By Rules 1 and 2, all three the Aristotelian standpoint gives exactly the same terms are distributed in the premises, and by Rules results as the Boolean. (See Section 4.3.) Thus, under 3 and 4, one premise is negative and the other affir-these conditions, all of the syllogistic forms in the mative. Because three terms are distributed in the conditionally valid table commit the existential fal- premises and there are only two premises, one of the lacy from the Aristotelian standpoint as well. premises must distribute two terms. Accordingly,

this premise must be an E proposition. Furthermore,

Proving the Rules the other premise, which is affirmative and which

distributes the third term, must be an A proposition.

The foregoing discussion has shown that if a syllo- From this we conclude that there are four possible gism breaks any one of the five rules, it cannot be sets of premises: �All S are M � and �No M are P� (or valid from the Boolean standpoint. Thus, we have its converse), and �All P are M� and �No M are S� shown that each of the rules is a necessary condition (or its converse). Since converting an E proposition for validity. The question remains, however, whether has no effect on validity, we may ignore the converse a syllogism�s breaking none of the rules is a sufficient of these propositions. If we now combine the two condition for validity. In other words, does the fact given sets of premises with the conclusion, �No S are that a syllogism breaks none of the rules guarantee P,� simple reasoning or a pair of Venn diagrams will its validity? The answer to this question is �yes,� but establish the validity of the two resulting syllogisms. unfortunately there appears to be no quick method Next, consider a syllogism having an I proposi-for proving this fact. Therefore, if you are willing tion for its conclusion. By Rule 1, M is distributed to take your instructor�s word for it, you may stop in at least one premise, and by Rule 4, both premises reading this section now and proceed to the exer- are affirmative. Further, by Rule 5, both premises cises. The proof that follows is somewhat tedious, cannot be universal. Thus, at least one premise is an and it proceeds by considering four classes of syl- I proposition. However, since the other premise dis-logisms having A,�E, I, and O propositions for their tributes a term, that premise must be an A proposi-conclusions. tion. Accordingly, the four possible sets of premises

Let us first suppose we are given a valid syllo- are �All M are S� and �Some M are P� (or its con-

gism having an A proposition for its conclusion. verse), and �All M are P� and �Some M are S� (or its Once again, suppose that P, S, and M designate the converse). Again, since converting an I proposition major, minor, and middle terms, respectively. Then, has no effect on validity, we may ignore the converse by Rules 1 and 2, both M and S are distributed in of these propositions. Then if we combine the two the premises. Further, by Rule 4, both premises are given pairs of premises with the conclusion, �Some affirmative. Now, since I propositions distribute nei- S are P,� simple reasoning or a pair of Venn dia-ther term, and A propositions distribute only one grams will establish the validity of the two resulting term, both premises must be A propositions, and syllogisms. S must be distributed in one and M in the other. Last, we consider a syllogism having an O propo-Accordingly, the premises are �All S are M� and �All sition for its conclusion. By Rules 1 and 2, both M M are P.� If we now combine these premises with and P are distributed in the premises. Also, by Rules

252Chapter5 Categorical Syllogisms

3 and 4, one premise is negative and the other affir- the converted propositions, simple reasoning or a mative, and by Rule 5, both premises cannot be uni- Venn diagram will establish the validity of the single versal. However, both premises cannot be particu- resulting syllogism. lar (I and O), because then only one term would be This procedure proves that the five rules collec-distributed. Therefore, the premises are either A and tively provide a sufficient condition for the valid-O or E and I. In regard to the first of these alter- ity of any syllogism from the Boolean standpoint. natives, either M is the subject of the A statement Since eight distinct inferences or Venn diagrams and P is the predicate of the O, or P is the subject were needed to accomplish it, this shows that of the A statement and M is the predicate of the O. there are really only eight significantly distinct This gives the premises as �All M are S� and �Some syllogisms that are valid from the Boolean stand-M are not P,� and �All P are M � and �Some S are not point. The other seven are variations of these that M.� When these pairs of premises are combined with result from converting one of the premises. For the conclusion, �Some S are not P,� simple reasoning syllogisms having particular conclusions and uni-or a pair of Venn diagrams will establish the validity versal premises about existing things, an analo-of the two resulting syllogisms. Finally, considering gous procedure can be used to prove that the first the other alternative (E and I), the resulting four sets four rules collectively provide a sufficient con-of premises are �No M are P� (or its converse) and dition for the validity of any syllogism from the �Some M are S� (or its converse). Again ignoring Aristotelian standpoint.

Exercise 5.3

I. Reconstruct the following syllogistic forms and use the five rules for syllogisms to

determine if they are valid from the Boolean standpoint, conditionally valid from the Aristotelian standpoint, or invalid. For those that are conditionally valid, identify the condition that must be fulfilled. For those that are invalid from either the Boolean or Aristotelian standpoint, name the fallacy or fallacies committed. Check your answers by constructing a Venn diagram for each.

★1. AAA-3 11. AII-2

2. IAI-2 12. AIO-3

3. EIO-1 ★13. AEE-4

★4. AAI-2 14. EAE-4

5. IEO-1 15. EAO-3

6. EOO-4 ★16. EEE-1

★7. EAA-1 17. EAE-1

8. AII-3 18. OAI-3

9. AAI-4 ★19. AOO-2

★10. IAO-3 20. EAO-1

II. Use the five rules to determine whether the following standard-form syllogisms are valid

from the Boolean standpoint, valid from the Aristotelian standpoint, or invalid. For those

Section 5.3 Rules and Fallacies 253

that are invalid from either the Boolean or Aristotelian standpoint, name the fallacy or fallacies committed. Check your answer by constructing a Venn diagram for each.

★1. Some nebulas are clouds of gas.

Some clouds of gas are objects invisible to the naked eye.

Therefore, some objects invisible to the naked eye are nebulas.

2. No individuals sensitive to the difference between right and wrong are people

who measure talent and success in terms of wealth.

All corporate take-over experts are people who measure talent and success in terms of wealth.

Therefore, no corporate take-over experts are individuals sensitive to the difference between right and wrong.

3. No endangered species are creatures loved by the timber industry.

All spotted owls are endangered species.

Therefore, some spotted owls are not creatures loved by the timber industry.

★ 4. Some cases of affirmative action are not measures justified by past discrimination.

No cases of affirmative action are illegal practices.

Therefore, some illegal practices are not measures justified by past discrimination.

5. All transparent metals are good conductors of heat.

All transparent metals are good conductors of electricity.

Therefore, some good conductors of electricity are good conductors of heat.

6. All members of the National Rifle Association are people opposed to gun control.

All members of the National Rifle Association are law-abiding citizens. Therefore, all law-abiding citizens are people opposed to gun control.

★7. No searches based on probable cause are violations of Fourth Amendment rights.

Some warrantless searches are violations of Fourth Amendment rights. Therefore, some warrantless searches are not searches based on probable cause.

8. All war zones are places where abuse of discretion is rampant.

Some places where abuse of discretion is rampant are international borders. Therefore, some international borders are war zones.

9. All inside traders are people subject to prosecution.

Some executives with privileged information are not people subject to prosecution.

Therefore, some executives with privileged information are inside traders.

★10. All successful flirts are masters at eye contact.

All masters at eye contact are people genuinely interested in others.

Therefore, some people genuinely interested in others are successful flirts.

III. Answer �true� or �false� to the following statements:

1. If a standard-form categorical syllogism violates one of the first four rules, it

may still be valid.

2. If a valid standard-form syllogism has an E statement as its conclusion, then

both the major and minor terms must be distributed in the premises.

254Chapter5 Categorical Syllogisms

3. If a standard-form syllogism has two I statements as premises, then it

is invalid.

4. If a standard-form syllogism has an E and an O statement as premises, then no

conclusion follows validly.

5. If a standard-form syllogism has an I statement as its conclusion, then Rule 2

cannot be violated.

6. If a valid standard-form syllogism has an O statement as its conclusion, then its

premises can be an A and an I statement.

7. If a valid standard-form syllogism has an E statement as a premise, then its

conclusion can be an A statement.

8. If a standard-form syllogism breaks only Rule 5 and its three terms are��dogs,�

�cats,� and �animals,� then the syllogism is valid from the Boolean standpoint.

9. If a standard-form syllogism breaks only Rule 5 and its three terms are

�dogs,� �cats,� and �animals,� then the syllogism is valid from the Aristotelian standpoint.

10. If a standard-form syllogism breaks only Rule 5 and its three terms are �elves,�

�trolls,� and �gnomes,� then the syllogism is valid from the Aristotelian standpoint.

IV. Create five standard-form categorical syllogisms about flowers, each of which breaks

one of the five rules for syllogisms.

5.4 Reducing the Number of Terms

Categorical syllogisms, as they occur in ordinary All photographers are non-writers. spoken and written expression, are seldom phrased Some editors are writers. according to the precise norms of the standard-form Therefore, some non-photographers are not syllogism. Sometimes quantifiers, premises, or con- non-editors.

clusions are left unexpressed, chains of syllogisms This syllogism is clearly not in standard form are strung together into single arguments, and terms because it has six terms: �photographers,� �editors,� are mixed together with their negations in a single �writers,� �non-photographers,� �non-editors,� and argument. The final four sections of this chapter are �non-writers.� But because three of the terms are concerned with developing techniques for rework-complements of the other three, the number of terms ing such arguments in order to render them testable can be reduced to a total of three, each used twice by Venn diagrams or by the rules for syllogisms. in distinct propositions. To accomplish the reduc-In this section we consider arguments that con-tion, we can use the three operations of conversion, tain more than three terms but that can be modified obversion, and contraposition discussed in Chapter 4. to reduce the number of terms to three. Consider the But, of course, since the reworked syllogism must be following:

Section 5.4 Reducing the Number of Terms 255

Prominent 1940�2022Saul Kripke Logicians

A s a child, Saul Kripke demonstrated prodi-gious intellectual abilities. By the age of ten he had read all the plays of Shakespeare, and he discovered algebra, which he said he could have invented on his own. By fourteen he had mastered

geometry and calculus and became deeply involved

in philosophy. At seventeen he wrote a paper, pub-

lished in the prestigious Journal of Symbolic Logic,

that introduced a completeness theorem for modal

logic. Legend has it that when this paper reached the

attention of the Harvard mathematics department,

someone there invited him to apply for a teaching

position. He replied, �My mother said that I should Steve Pyke/Premium Archive /Getty Images finish high school and go to college first.� Prior to his

death, in 2022, Kripke was considered by many to be received honorary degrees from several universities, the world�s greatest living philosopher and logician. and in 2001 he received the Schock Prize (comparable

Saul Kripke was born the son of a rabbi in Bay to the Nobel Prize) in Logic and Philosophy.

Shore, New York, in 1940. He attended public grade Kripke is universally hailed for his work in modal school and high school in Omaha, Nebraska, and then logic where, in addition to proving its formal com-Harvard University where, during his sophomore year, pleteness, he created a semantics, now called Kripke he taught a graduate-level course at MIT. In 1962, semantics, in which a proposition is said to be neces-he graduated summa cum laude with a bachelor�s sarily true when it holds in all possible worlds and pos-degree in mathematics. After that, instead of going sibly true when it holds in some possible world. Also, to graduate school, he simply began teaching�first his book Naming and Necessity made groundbreaking at Harvard, then Rockefeller University, then Princeton contributions to the philosophy of language by intro-University, and finally CUNY Graduate Center. He has ducing a new theory of reference for proper names.

equivalent in meaning to the original one, we must Symbolized argument Reduced argument use these operations only on the kinds of statements All P are non-W. No P are W. for which they yield logically equivalent results. That Some E are W. Some E are W. is, we must use conversion only on E and I statements Some non-P are not Some E are not P. and contraposition only on A and O statements. non-E. Obversion yields logically equivalent results for all

four kinds of categorical statements. Because the first premise of the original argument

Let us rewrite our six-term argument using let- is an A statement and the conclusion an O state-

ters to represent the terms, and then obvert the first ment, and because the operations performed on premise and contrapose the conclusion in order to these statements yield logically equivalent results, eliminate the negated letters: the reduced argument is equivalent in meaning to

256Chapter5 Categorical Syllogisms

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). the original argument. The reduced argument is When tested with a Venn diagram or by means of in standard syllogistic form and may be evaluated the five rules, this argument will, of course, also be either with a Venn diagram or by the five rules found valid, and so the original argument is valid. for syllogisms. The application of these methods When using a Venn diagram, no unusual method is indicates that the reduced argument is valid. We needed; the diagram is simply lettered with the three conclude, therefore, that the original argument is terms � W,� �non-E,� and �non-P.� also valid. The most important point to remember in reducing

It is not necessary to eliminate the negated the number of terms is that conversion and contraposi-

terms in order to reduce the number of terms. It tion must never be used on statements for which they is equally effective to convert certain non-negated yield undetermined results. That is, conversion must terms into negated ones. Thus, instead of obvert- never be used on A and O statements, and contrapo-ing the first premise of the example argument sition must never be used on E and I statements. The and contraposing the conclusion, we could have operations that are allowed are summarized as follows: contraposed the first premise and converted and

then obverted the second premise. The operation Conversion: No A are B. No B are A. is performed as follows: Some A are B. Some B are A.

Symbolized argument Reduced argument Obversion: All A are B. No A are All P are non-W. All non-B.W are non-P. Some E are W. Some W are not non-E. No A are B. All A are non-B.

Some non-P are Some non-P are not Some A are B. Some A are not not non-E. non-E. non-B.

The reduced argument is once again equivalent to the Some A are Some A are original one, but now we must reverse the order of not B. non-B.

the premises to put the syllogism into standard form: Contraposition: All A are B. All non-B are

Some W are not non-E. non-A. All W are non-P. Some A are Some non-B Some non-P are not non-E. not B. are not non-A.

Exercise 5.4

I. Rewrite the following arguments using letters to represent the terms, reduce the number

of terms, and put the arguments into standard form. Then test the new forms with Venn diagrams or by means of the five rules for syllogisms to determine the validity or inva-lidity of the original arguments.

★1. Some intelligible statements are true statements, because all unintelligible

statements are meaningless statements and some false statements are meaning-ful statements.

2. Some people who do not regret their crimes are convicted murderers, so some

convicted murderers are people insusceptible of being reformed, since all people susceptible of being reformed are people who regret their crimes.

Section 5.4 Reducing the Number of Terms 257

3. All Peace Corps volunteers are people who have witnessed poverty and desola-

tion, and all people insensitive to human need are people who have failed to witness poverty and desolation. Thus, all Peace Corps volunteers are people sensitive to human need.

★4. Some unintentional killings are not punishable offenses, inasmuch as all cases

of self-defense are unpunishable offenses, and some intentional killings are cases of self-defense.

5. All aircraft that disintegrate in flight are unsafe planes. Therefore, no poorly

maintained aircraft are safe planes, because all well-maintained aircraft are air-craft that remain intact in flight.

6. No objects that sink in water are chunks of ice, and no objects that float in

water are things at least as dense as water. Accordingly, all chunks of ice are things less dense than water.

★7. Some proposed flights to Mars are inexpensive ventures, because all unmanned

space missions are inexpensive ventures, and some proposed flights to Mars are not manned space missions.

8. All schools driven by careerism are institutions that do not emphasize liberal

arts. It follows that some universities are not institutions that emphasize liberal arts, for some schools that are not driven by careerism are universities.

9. No cases of encephalitis are infections easily curable by drugs, since all diseases

that infect the brain are infections not easily curable by drugs, and all diseases that do not infect the brain are cases other than encephalitis.

★10. Some foreign emissaries are people without diplomatic immunity, so some people

invulnerable to arrest and prosecution are foreign emissaries, because no people with diplomatic immunity are people vulnerable to arrest and prosecution.

II. Create a categorical syllogism about people that contains two negated terms. Then reduce

the number of terms so the syllogism is in standard form. Finally, test the syllogism for validity using either the Venn-diagram method or the rules method.

5.5 Ordinary Language Arguments

Many arguments that are not standard-form cat- propositions that contain a total of three differ-egorical syllogisms as written can be translated into ent terms, each of which occurs twice in distinct standard-form syllogisms. In doing so we often propositions. Once translated, the argument can be use techniques developed in the last section of tested by means of a Venn diagram or the rules for Chapter 4�namely, inserting quantifiers, modify- syllogisms. ing subject and predicate terms, and introducing Since the task of translating arguments into copulas. The goal, of course, is to produce an argu- standard-form syllogisms involves not only con-ment consisting of three standard-form categorical verting the component statements into standard

258Chapter5 Categorical Syllogisms

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). form but adjusting these statements one to another This is the so-called �Barbara� syllogism and is, of

tain amount of practice may be required before it Boeing must be a manufacturer because it can be done with facility. In reducing the terms to so that their terms occur in matched pairs, a cer- course, valid. Here is another example: three matched pairs it is often helpful to identify hires riveters, and any company that hires

tions and to express this common factor through For this argument, the parameter �companies� sug-gests itself: the strategic use of parameters. For example, if some factor common to two or all three proposi- riveters is a manufacturer. all three statements are about people, the term All companies identical to Boeing are manu-�people� or �people identical to� might be used; facturers, because all companies identical to or if they are about times or places, the term Boeing are companies that hire riveters, and all �times� or �times identical to� or the term �places� companies that hire riveters are manufacturers.

or �places identical to� might be used. Here is an The first statement, of course, is the conclusion. example: When the syllogism is written in standard form, it

Whenever people put off marriage until they will be seen that it has, like the previous syllogism, are older, the divorce rate decreases. Today, the form AAA-1. people are putting off marriage until they are Here is another example:

The temporal adverbs �whenever� and �today� sug-older. Therefore, the divorce rate is decreas- If a piece of evidence is trustworthy, then ing today. it should be admissible in court. Polygraph tests are not trustworthy. Therefore, they gest that �times� should be used as the common should not be admissible in court.

factor. Following this suggestion, we have this: To translate this argument, using a single common

All times people put off marriage until factor is not necessary:

This is a standard-form categorical syllogism. Notice they are older are times the divorce rate All trustworthy pieces of evidence are pieces decreases. All present times are times of evidence that should be admissible in people put off marriage until they are older. court. No polygraph tests are trustworthy Therefore, all present times are times the pieces of evidence. Therefore, no polygraph divorce rate decreases. tests are pieces of evidence that should be admissible in court.

that each of the three terms is matched with an exact This syllogism commits the fallacy of illicit major duplicate in a different proposition. To obtain such and is therefore invalid. a match, it is sometimes necessary to alter the word-As was mentioned in Section 4.7, arguments con-ing of the original statement just slightly. Now if we taining an exceptive proposition must be handled in adopt the convention a special way. Let us consider one that contains an

M = times people put off marriage until they exceptive proposition as a premise:

are older

D = All of the jeans except the Levi�s are on sale. times the divorce rate decreases

P = Therefore, since the Calvin Klein jeans are present times

the syllogism may be symbolized as follows: The first premise is translated as two conjoined not Levi�s, they must be on sale.

All M are D. categorical propositions: �No Levi�s are jeans on All P are M. sale,� and �All jeans that are not Levi�s are jeans on All P are D. sale.� These give rise to two syllogisms:

Section 5.5 Ordinary Language Arguments 259

No Levi�s are jeans on sale. form, because it has four terms. If the second prem-No Calvin Klein jeans are Levi�s. ise is obverted, so that it reads �All Calvin Klein jeans Therefore, all Calvin Klein jeans are jeans on are jeans that are not Levi�s,� the syllogism becomes sale. an AAA-1 standard-form syllogism, which is valid. All jeans that are not Levi�s are jeans on sale. Each of these two syllogisms may be viewed as No Calvin Klein jeans are Levi�s. a pathway in which the conclusion of the original Therefore, all Calvin Klein jeans are jeans on argument might follow necessarily from the prem-sale. ises. Since it does follow via the second syllogism,

the original argument is valid. If both of the result-

The first syllogism, which is in standard form, ing syllogisms turned out to be invalid, the original

is invalid because it has two negative premises. argument would be invalid. The second one, on the other hand, is not in standard

Exercise 5.5

I. Translate the following arguments into standard-form categorical syllogisms, and then

use Venn diagrams or the rules for syllogisms to determine whether each is valid or invalid. See Section 4.7 for help with the translation.

★1. Physicists are the only scientists who theorize about dark matter, and Patricia

Burchat certainly does that. Therefore, Patricia Burchat must be a physicist.

2. Whenever suicide rates decline, we can infer that people�s lives are better

adjusted. Accordingly, since suicide rates have been declining in recent years, we can infer that people�s lives have been better adjusted in recent years.

3. Environmentalists purchase only fuel-efficient cars. Hence, Jeep Rubicons

must not be fuel efficient, since environmentalists do not purchase them.

★4. Whoever wrote the Declaration of Independence had a big impact on civili-

zation, and Thomas Jefferson certainly had that. Therefore, Thomas Jefferson wrote the Declaration of Independence.

5. There are public schools that teach secular humanism. Therefore, since

secular humanism is a religion, there are public schools that teach religion.

6. Any city that has excellent art museums is a tourist destination. Therefore,

Paris is a tourist destination, because it has excellent art museums.

★7. Miranda Lambert sings what she wants. Hence, since Miranda sings country

songs, it follows that she must want to sing country songs.

8. Not all interest expenses are tax deductible. Home-mortgage payments are

interest expenses. Thus, they are not tax deductible.

9. If a marriage is based on a meshing of neuroses, it allows little room for growth.

If a marriage allows little room for growth, it is bound to fail. Therefore, if a marriage is based on a meshing of neuroses, it is bound to fail.

★10. Endurance hikers cannot finish the trek unless they can tolerate hardship.

Whoever hikes Death Valley can tolerate hardship. Therefore, whoever hikes Death Valley can finish the trek.

260Chapter5 Categorical Syllogisms

11. Wherever icebergs are present, threats to shipping exist. Icebergs are not present

in the South Pacific. Hence, there are no threats to shipping in the South Pacific.

12. According to surveys, there are college students who think that Africa is in

North America. But anyone who thinks that has no knowledge of geography. It follows that there are college students who have no knowledge of geography.

★13. Diseases carried by recessive genes can be inherited by offspring of two carri-

ers. Thus, since cystic fibrosis is a disease carried by recessive genes, it can be inherited by offspring of two carriers.

14. All of the movies except the romantic comedies were exciting. Hence, the

action films were exciting, because none of them are romantic comedies.

15. Autistic children are occasionally helped by aversion therapy. But aversion

therapy is sometimes inhumane. Thus, autistic children are sometimes helped by inhumane therapy.

II. Create two arguments�one about desserts, the other about beverages�each of which

contains two or more of the following words: �whoever,� �whenever,� �only,� �the only,�

�not every,� �if,� �any,� �only if,� �a few,� �there is,� �none but.� Then, translate your argu-ments into standard-form syllogisms and test each for validity using a Venn diagram or the rules method.

5.6 Enthymemes

An enthymeme is an argument that is expressible sustaining his or her interest. On other occasions as a categorical syllogism but that is missing a prem- the arguer may want to slip an invalid or unsound ise or a conclusion. Examples: argument past an unwary listener or reader, and

The corporate income tax should be abol- this aim may be facilitated by leaving a premise or ished; it encourages waste and high prices. conclusion out of the picture.

Many enthymemes are easy to convert into syl-

Animals that are loved by someone should logisms. The reader or listener must first determine not be sold to a medical laboratory, and lost what is missing, whether it is a premise or conclu-pets are certainly loved by someone. sion, and then introduce the missing statement with

�Whatever encourages waste and high prices should The first enthymeme is missing the premise the aim of converting the enthymeme into a good argument. Attention to indicator words will often provide the clue as to the nature of the missing state-be abolished,� and the second is missing the con-ment, but a little practice can render this task virtu-clusion �Lost pets should not be sold to a medical ally automatic. The missing statement need not be laboratory.� expressed in categorical form; expressing it in the Enthymemes occur frequently in ordinary general context of the other statements is sufficient spoken and written English for several reasons. and is often the easier alternative. Once this is done, Sometimes it is simply boring to express every the entire argument may be translated into categori-statement in an argument. The listener or reader�s cal form and then tested with a Venn diagram or by intelligence is called into play when he or she is the rules for syllogisms. Example: required to supply a missing statement, thereby

Section 5.6 Enthymemes 261

Venus completes its orbit in less time than missing statement, attention to the rules for syllo-the earth, because Venus is closer to the sun. gisms may be helpful (if the resulting syllogism is Missing premise: Any planet closer to the sun to be valid). For example, if the missing statement completes its orbit in less time than the�earth. is a conclusion and one of the given premises is

Translating this argument into categorical form, we have Or if the missing statement is a premise and the negative, the missing conclusion must be negative.

All planets closer to the sun are planets that stated conclusion is universal, the missing premise complete their orbit in less time than the earth. must be universal. All planets identical to Venus are planets The enthymemes that we have considered thus closer to the sun. far have been fairly straightforward. The kinds of Therefore, all planets identical to Venus are enthymemes that occur in letters-to-the-editor col-planets that complete their orbit in less time umns of magazines and newspapers often require a than the earth. bit more creativity to convert into syllogisms. Con-

This syllogism is valid (and sound). sider the following:

Any enthymeme (such as the one about Venus) The United States� endless wars since 1945

that contains an indicator word is missing a prem- were authorized by the president, not, as the ise. This may be seen as follows. If an enthymeme Constitution requires, declared by Congress. contains a conclusion indicator, then the conclusion Therefore, all those wars have been unconsti-

is a premise. On the other hand, if the enthymeme (Ricardo Nicol) contains a premise indicator, then the conclusion follows it, which means that the missing statement tutional and illegal. precedes it, which means, again, that the missing In this argument, the conclusion is stated, so what statement is a premise. is missing is a premise. After introducing the miss-

If, however, an enthymeme contains no indica- ing premise, the argument can be expressed as an

beginning of this section), then the missing state- All wars not declared by Congress are uncon-ment could be either a premise or a conclusion. If tor words at all (such as the two enthymemes at the AAA-1 syllogism as follows: the two given statements are joined by a word such stitutional and illegal wars. as �and,� �but,� �moreover,� or some similar con- All wars fought by the United States since junction, the missing statement is usually a conclu- 1945 are wars not declared by Congress.

sion. If not, the first statement is usually the con- States since 1945 are unconstitutional and Therefore, all wars fought by the United clusion, and the missing statement is a premise. To illegal wars. test this latter alternative, it may help to mentally

insert the word �because� between the two state- The next example relates to college admission ments. If this insertion makes sense, the missing requirements:

statement is a premise. If the SAT and ACT were dropped as college

After the nature of the missing statement has admission requirements, there would still be

been determined, the next task is to write it out. a need for applicants to take a standardized To do so, one must first identify its terms. This can test and not gain admission solely on the be done by taking account of the terms that are basis of their grades. Grades can vary accord-

match up with each other. Once this pair of terms (Bill Seckler) is found, attention should be focused on the other given. Two of the terms in the given statements will ing to the standards of different schools. two terms. These are the ones that will be used to Here the author makes two points. One relates to form the missing statement. In constructing the the inadequacy of grades, the other to the need

262Chapter5 Categorical Syllogisms

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). for some standardized tests. Thus, we have two All admission tests needed for fairness are arguments: tests that must be taken.

No standards that can vary from school to school Some standardized tests are admission tests needed for fairness. are adequate criteria for admission to college. Therefore, some standardized tests are tests All grades are standards that can vary from that must be taken. school to school.

Therefore, no grades are adequate criteria The first is an EAE-1 syllogism; the second an AII-1. for admission to college. Both are valid.

Exercise 5.6

I. In the following enthymemes, determine whether the missing statement is a premise

or a conclusion. Then supply the missing statement, attempting whenever possible to convert the enthymeme into a valid argument. The missing statement need not be expressed as a standard-form categorical proposition.

★1. Some police chiefs undermine the evenhanded enforcement of the law, because

anyone who fixes parking tickets does that.

2. Any form of cheating deserves to be punished, and plagiarism is a form of cheating.

3. Zendaya is a talented actress. After all, she�s won a Primetime Emmy Award.

★4. A few fraternities have dangerous initiation rites, and those that do have no

legitimate role in campus life.

5. Only nonprofit organizations are exempt from paying taxes, so churches must

be exempt.

6. All of the operas except Mozart�s were well performed, and Carmen was not

written by Mozart.

★7. Not all text messages are welcome, but those from friends are.

8. Higher life-forms could not have evolved through merely random processes,

because no organized beings could have evolved that way.

9. None but great novels are timeless, and The Brothers Karamazov is a great novel.

★10. Attempts to reduce tuition have failed in recent years because administrative

salaries have exploded.

11. Wherever water exists, human life can be sustained, and water exists on the�moon.

12. If a symphony orchestra has effective fund-raisers, it will survive; and the

Cleveland symphony has survived for years.

★13. Mechanistic materialists do not believe in free will, because they think that

everything is governed by deterministic laws.

Section 5.6 Enthymemes 263

14. A contract to buy land is not enforceable unless it�s in writing, but our client�s

contract to buy land is in writing.

15. The only telescopes that are unaffected by the atmosphere are orbiting tele-

scopes, and the James Webb space telescope is in orbit.

II. Translate the enthymemes in Part I of this exercise into standard-form categorical syl-

logisms and test them for validity.

III. The following enthymemes were originally submitted in letters to the editor of vari-

ous magazines and newspapers. Convert them into valid standard-form syllogisms. In some cases two or more syllogisms may be required.

★1. If the Defense Department is so intent on fighting alcohol abuse, why does it

make alcohol so readily available and acceptable? Alcohol is tax free at post liquor stores, and enlisted men�s and officers� clubs make drinking almost a mandatory facet of military life.

(Diane Lynch)

2. All aid to Israel should be stopped at once. Why should the American taxpayer

be asked to send billions of dollars to Israel when every city in the United States is practically broke and millions of people are out of work?

(Bertha Grace)

3. When the state has a budget surplus of almost $100 billion, why would anyone

think that raising the income tax is smart?

(Paul Brown)

★4. To combat climate change, we need a national energy policy that places a fee

on carbon. This fee will help to price petroleum products in a manner that reflects the true cost that they are placing on society and will make renewable energy much more attractive.

(Jeff Stoddard)

5. The budget deficit will not be brought under control because to do so would

require our elected leaders in Washington to do the unthinkable�act coura-geously and responsibly.

(Bruce Crutcher)

6. The Constitution bans any law that is so vague that �men of common intelli-

gence must necessarily guess at its meaning.� Sexual harassment laws, however, are so vague that no one knows what they mean.

(Hans Bader)

★7. College students of today are the higher-income taxpayers of tomorrow. Con-

gress should consider financial aid as an investment in the financial future of our country.

(Carol A. Steimel)

8. Our genes and our environment control our destinies. The idea of conscious

choice is ridiculous. Yes, prisons should be designed to protect society, but they should not punish the poor slobs who were headed for jail from birth.

(Paul R. Andrews)

264Chapter5 Categorical Syllogisms

9. Encouraging toy-gun play gives children a clear message that the best way to

deal with frustration and conflict is with a gun. Is this the message that we want to be sending our kids?

(Patricia Owen)

★10. You want a great pet? Adopt one from a shelter. You want a purebred animal

and can�t find one at a shelter? Search online for rescue groups that are breed- specific. If you find an abandoned animal, consider making it yours. Any of these approaches is a win-win for people and animals, and they are less expen-sive than buying from breeders.

(Marilyn Shultz)

IV. Look through a magazine or newspaper or go online and identify five topics of current

interest. Construct an enthymeme involving each topic.

V. Translate the arguments in the following dialogue into standard-form categori-

cal syllogisms. The dialogue contains more than twenty arguments, and most are expressed in the form of enthymemes. The first translated syllogism may be found in the answer section.

Do Kids Make Parents Happy?

Tad and Lara, his fianc�e, are walking in the park. Tad pauses a moment to look at some children playing in the grass. Lara stops beside him.

�I know we�ve talked about this before,� she says, �but I wish you�d get over this

idea about us having a bunch of kids.�

�I�ll think about it,� Tad says. �But could you tell me once again about why you

don�t want any?�

�I think most couples believe that kids automatically bring happiness,� Lara says.

�But that�s simply not the case. I�ve read some recent studies about this. They show that childless couples experience higher levels of emotional well-being than couples with kids. Thus, childless couples are happier than couples with kids.�

�I�ve read those studies of yours,� Tad says, �but I don�t know what to make of

them. Consider this. Kids make you laugh, and laughter is essential to happiness. So kids do make you happy. Plus, in a way, having kids makes you immortal.�

�How do you figure that?�

�Okay, not immortal exactly, but I mean that they make you participate in a kind of

immortality. Your children grow up and have children, and those children have chil-dren, and on it goes. This feeling of contributing to the perpetuation of the human race makes you happy, don�t you think?�

�Frankly,� Lara replies, �that�s all a bit too removed from the �now� to suit me. May-

be at the beginning when a couple is expecting their first child, they tend to be happy. But once the baby arrives, the honeymoon ends. The newborn has to be fed every couple of hours, the parents aren�t getting any sleep, and those parents aren�t happy.�

�Well, very tired, yes, but unhappy?� says Tad. �Plus, it�s temporary and only lasts

for the first few months.�

�And after the first few months, you have toys and food all over the house and sticky

stuff on all the furniture. How happy is living in a constant mess going to make you?�

�Not very, I guess. But there�s more to life than a neat house. Having kids will

force us not to be selfish, and no one who�s selfish is truly happy. Also, kids promote

Section 5.6 Enthymemes 265

an atmosphere of growth. Watching kids learn, change, and grow makes parents happy. Don�t you think that�s true?�

�Sure,� Lara says, �in theory. In reality, growing means infants become toddlers

who scream, whine, and throw tantrums. Having to listen to all that racket makes no one happy. And then, life just gets more hectic. Kids constantly have to be driven to and from school, dancing lessons, Little League, the mall�all detracting from the happiness of the parents. Right?�

�Right,� Tad says, �but kids are fun. They�re constantly playing; and having playful

children around you brings happiness. Plus some kids grow up to accomplish extraor-dinary things. How could one not be happy at witnessing the success of one�s child?�

�Okay,� says Lara, �but kids take up a tremendous amount of time. Couples who

have kids often have no time for vacations�I know that wouldn�t make you happy. And some couples with kids don�t even have time for sex. How would you like that?�

�I wouldn�t,� Tad admits. �But maybe we could learn to manage our time better

than those other couples. Which leads me to something else: Having kids will force us to work together as a couple. That will deepen our relationship, and deeper re-lationships are happier relationships. And don�t forget unconditional love. Kids love you for who you are. They don�t care what you look like, whether you�re overweight, or what kind of car you drive. Being loved like that makes you happy.�

�Well, of course it does,� says Lara. �But what about money? Kids cost a fortune.

Food, clothing, medical bills, and a college education will leave us constantly broke. Can we be happy like that? And what about my career? I have goals, just like you. If I�m constantly pregnant or taking care of little kids, I�ll never achieve them. I can�t possibly be happy if that happens.�

�No,� Tad says. �We�d have to work that out. As we grow older, though, think about

what our lives will be like without any children. We�ll be lonely and cut off, with no one to call us and no grandchildren to visit us. Does that sound like happiness to you?�

�Ha!� Lara says. �You assume that the kids will eventually leave the house. You

forget that we live in the age of boomerang offspring. After they finish school, they come back home to live, and you never get rid of them. How does that grab you?�

�Well, we didn�t stay around long after college. I think when we decided to leave

the nest and get married, we made our parents truly happy.�

�Yeah,� Lara replies. �I think we did. So maybe we�ve proved your case. Some

kids, at least, end up making their parents happy.�

5.7 Sorites

A sorites is a chain of categorical syllogisms in All bloodhounds are dogs. which the intermediate conclusions have been left All dogs are mammals. out. The name is derived from the Greek word soros, No fish are mammals. meaning �heap,� and is pronounced �sōrīt�z,� with Therefore, no fish are bloodhounds.

the accent on the second syllable. The plural form is The first two premises validly imply the intermediate also �sorites.� Here is an example: conclusion �All bloodhounds are mammals.� If this

266Chapter5 Categorical Syllogisms

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). intermediate conclusion is then treated as a premise write it second. Continue in this way until all the

clusion follows validly. The sorites is thus composed All D are C. of two valid categorical syllogisms and is therefore and put together with the third premise, the final con- premises are listed: valid. The rule in evaluating a sorites is based on the No B are C. idea that a chain is only as strong as its weakest link. If All A are B. any of the component syllogisms in a sorites is invalid, Some E are A. the entire sorites is invalid. Some E are not D.

A Next, the intermediate conclusions are drawn. standard-form sorites is one in which each

of the component propositions is in standard form, Venn diagrams are useful in performing this step, each term occurs twice, the predicate of the conclu- and they serve simultaneously to check the valid-sion is in the first premise, and each suc- ity of each component syllogism: cessive premise has a term in common

with the preceding one.* The sorites in All D are C. (1) No B are D. the example is in standard form. Each No B are C . (2) No A are D. of the propositions is in standard form, All A are B . (3) Some E are not D. each term occurs twice; the predicate of Some E are A . the conclusion, �bloodhounds,� is in the Some E are not D . first premise; the other term in the first CBA premise, �dogs,� is in the second premise;

and so on.

We will now introduce two techniques X

for testing a sorites for validity. The first

technique involves three steps: (1) put BDADED the sorites into standard form, (2)�intro-

duce the intermediate conclusions, and

(3) test each component syllogism for validity. If The first intermediate conclusion, �No B are D,� each component is valid, the sorites is valid. Con- is drawn from the first two premises. The second, sider the following sorites form: �No A are D,� is drawn from the first intermediate

conclusion and the third premise. And the third

No B are C. conclusion, which is identical to the final conclu-

Some E are A. sion, is drawn from the second intermediate con-All A are B. clusion and the fourth premise. Since all conclu-All D are C. sions are drawn validly, the sorites is valid. On the Some E are not D. other hand, if at some step in the procedure no

To put the sorites form into standard form, the conclusion can be drawn, the sorites is invalid. premises must be rearranged. To do this find the The second technique for testing the validity of a premise that contains the predicate of the conclu- sorites is faster and simpler than the first one because sion and write it first. Then find the premise that it does not require that the intermediate conclusions contains the other term in the first premise and be derived. This second technique consists in apply-

ing five rules that closely resemble the rules for syl-

*Actually, there are two definitions of standard-form sorites: the Gocle- logisms. The rules are as follows:

Aristotelian version, the premises are arranged so that the nian and the Aristotelian. The one given here is the Goclenian. In the Rule 1: Each of the middle terms must be subject of the conclusion occurs in the first premise. distributed at least once.

Section 5.7 Sorites 267

Rule 2: If a term is distributed in the conclu- This sorites breaks Rule 1 because neither

sion, then it must be distributed in a occurrence of M in the first two premises is dis-premise. tributed. Thus, the sorites is invalid. Note that no

Rule 3: other rule is broken. Both of the K �s in lines 2 and Two negative premises are not allowed.

3 are distributed, one of the R�s in lines 3 and 4 is

Rule 4: A negative premise requires a nega- distributed, S is distributed in the conclusion and

tive conclusion, and a negative con- also in the first premise, there is a negative conclu-clusion requires a negative premise. sion and only one negative premise, and while the

Rule 5: If all the premises are universal, the conclusion is particular so is one of the premises.

conclusion cannot be particular. The logic behind the five rules is as follows. For

Rule 1, each of the middle terms in the sorites is also

Rule 1 refers to the �middle terms� in the sorites. These a middle term in one of the component syllogisms.

ises. The conclusion referred to in Rules 2, 4, and 5 is logisms has an undistributed middle, making the the final conclusion of the sorites. Also, as with categor-are the terms that occur in matched pairs in the prem- Thus, if Rule 1 is broken, one of the component syl-

entire sorites invalid.

ical syllogisms, if a sorites breaks only Rule 5, it is valid For Rule 2, when the sorites is in standard form, from the Aristotelian standpoint on condition that its the predicate of the conclusion must appear in each critical term refers to at least one existing thing. of the intermediate conclusions. Thus, if the predicate

The critical term can be identified in the same way of the conclusion is distributed, it must also be dis-

as it is for categorical syllogisms. There are three pos- tributed in each of the intermediate conclusions, and sible approaches. By entering the mood and figure also in the first premise. Otherwise, one of the com-of the last component syllogism of the sorites into ponent syllogisms would have either an illicit major the table of conditionally valid forms in Section 5.1, or an illicit minor, making the entire sorites invalid. the critical term is given in the farthest right-hand Analogously, if the subject of the conclusion is distrib-column. Applying the superfluous distribution rule, uted, it must also be distributed in the last premise. the critical term is the term in the premises that is Otherwise, the last component syllogism would have distributed in more occurrences than is necessary an illicit minor, making the entire sorites invalid. for the sorites to obey the first two rules. And in the For Rule 3, when a negative premise appears in Venn diagram for the final component syllogism the list of premises, the conclusion derived from that of the sorites, the critical term is the one that cor- premise must be negative, as must all subsequent con-responds to the circle that is all shaded except for clusions. Otherwise, one of the component syllogisms one area. In the case of two such circles, it is the one would break Rule 4 for syllogisms. If a second negative that corresponds to the Venn circle containing the premise should appear, the syllogism consisting of that circled X on which the conclusion depends. premise and the prior intermediate conclusion would

Before applying the rules, it helps to put the sorites commit the fallacy of exclusive premises, making the

into standard form; in either event the sorites must be entire sorites invalid. written so that the terms occur in matched pairs. The Similarly, for Rule 4, when a negative premise following sorites is in standard form, and the distrib- appears in the list of premises, all subsequent con-uted terms are marked with a superscript �d.� As is clusions must be negative. Otherwise, one of the apparent, there are three middle terms: M, K, and R. component syllogisms would break Rule 4 for syl-

All Sd are M. logisms. Conversely, if the conclusion of the sorites All Kd are M. is negative, either the last premise or the last inter-No Kd are Rd. mediate conclusion must be negative. Otherwise, Some F are R. the last component syllogism would break Rule 4 Some F are not Sd. for syllogisms. If the last intermediate conclusion is

268Chapter5 Categorical Syllogisms

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). negative, then either the prior premise or the prior One of the advantages of the rules method for testing intermediate conclusion must be negative. If we con- a sorites is that invalidity can often be detected through tinue this reasoning, we see that some prior premise immediate inspection. Once a sorites has been put into must be negative. standard form, any sorites having more than one nega-

For Rule 5, a particular conclusion has existential tive premise is invalid, and any sorites having a negative

import, while universal premises (from the Boolean premise and an affirmative conclusion (or vice versa) standpoint) do not. Thus, if all the premises are uni- is invalid. Also, as with syllogisms, any sorites having versal and the conclusion is particular, the sorites as more than one particular premise is invalid. This last a whole commits the existential fallacy. requirement is implied by the other rules.

Exercise 5.7

I. Rewrite the following sorites in standard form, reducing the number of terms when

necessary. Then supply the intermediate conclusions and test with Venn diagrams.

★1. No B are C. 6. All M are non-P.

Some D are C. Some M are S. All A are B. All K are P. Some D are not A. Some non-K are not non-S.

2. No C are D. ★7. All non-U are non-V.

All No U are non-W. A are B .

Some All V are Y. C are not B .

Some No X are W. D are not A.

All Y are non-X.

3. No S are M. All 8. All D are non-C. F

Some All non-B are non-A. are S.

All Some E are D. M are H.

Some All B are C. E are F.

H are not E.

★4. Some T are K. 9. All non-L are non-K. No Some non-A are not non-E. Some K are N. Some K are M. C are Q . All P are non-L. All T are C . No non-N are M. Some

Q are not N. No Q are non-P.

5. Some N are not Q. All B are C .

No C are D. ★10. All R are S. All E are A. No non-V are T. All A are B. No Q are non-R. Some E are not D. No non-Q are P.

All T are non-S. All V are non-P.

II. For the sorites in Part I above, rewrite each in standard form, reducing the number of

terms when necessary. Then use the five rules for sorites to test each for validity.

Section 5.7 Sorites 269

III. The following sorites are valid. Rewrite each sorites in standard form, using letters to

represent the terms and reducing the number of terms whenever necessary. Then use Venn diagrams or the rules method to prove each one valid.

★1. Whatever produces oxygen supports human life.

Rain forests produce oxygen.

Nothing that supports human life should be destroyed.

Rain forests should not be destroyed.

2. No restrictive trade policies fail to invite retaliation.

Trade wars threaten our standard of living.

Some Chinese trade policies are restrictive.

Policies that invite retaliation lead to a trade war.

Some Chinese trade policies threaten our standard of living.

3. Anything that poisons drinking water causes disease and death.

Chemicals percolating through the soil contaminate aquifers.

Dumped chemicals percolate through the soil.

Whatever contaminates aquifers poisons drinking water.

Dumped chemicals cause disease and death.

★4. Nothing that is brittle is ductile.

Superconductors are all ceramics.

Only ductile things can be pulled into wires.

Ceramics are brittle.

Superconductors cannot be pulled into wires.

5. Some college students purchase their term papers.

Any cheat is expelled from college.

No one will achieve his career goals who is expelled.

No one who purchases term papers is other than a cheat.

Some college students will not achieve their career goals.

6. Creation science does not favor the teaching of evolution.

Nothing should be taught that frustrates the understanding of life.

Whatever opposes the teaching of evolution impedes the learning of biology. Anything that enhances the understanding of life fosters the learning of biology. Creation science should not be taught.

★ 7. Any theory that attracts morons contaminates the public consciousness.

A few conspiracy theories are based on lies.

Theories that contaminate the public consciousness can harm innocent victims.

No conspiracy theory based on lies fails to attract morons.

There are conspiracy theories that can harm innocent victims.

8. Whatever retards population growth increases food availability.

Anything that prevents starvation enhances life.

Birth-control measures never accelerate population growth.

Anything that enhances life should be encouraged.

Whatever increases food availability prevents starvation.

Birth-control measures should not be discouraged.

270Chapter5 Categorical Syllogisms

9. A few countries allow ivory trading.

Whatever country resists elephant killing discourages poachers.

Any country that allows ivory trading encourages poachers.

No country that promotes the extinction of elephants should escape the condemnation of the civilized world.

Any country that supports elephant killing promotes the extinction of elephants. A few countries should be condemned by the civilized world.

★10. Anything that promotes skin cancer causes death.

Whatever preserves the ozone layer prevents the release of CFCs.

Nothing that resists skin cancer increases UV radiation.

Anything that destroys the ozone layer increases UV radiation.

There are packaging materials that release CFCs.

Nothing that causes death should be legal. Some packaging materials should be illegal.

IV. The following sorites are taken from Lewis Carroll�s Symbolic Logic. All are valid.

Rewrite each sorites in standard form, using letters to represent the terms and reducing the number of terms whenever necessary. Then use Venn diagrams or the rules method to prove each one valid.

★1. No ducks waltz.

No officers ever decline to waltz.

All my poultry are ducks.

My poultry are not officers.

2. No experienced person is incompetent.

Jenkins is always blundering.

No competent person is always blundering.

Jenkins is inexperienced.

3. No terriers wander among the signs of the zodiac.

Nothing that does not wander among the signs of the zodiac is a comet. Nothing but a terrier has a curly tail. No comet has a curly tail.

★4. All hummingbirds are richly colored.

No large birds live on honey.

Birds that do not live on honey are dull in color.

All hummingbirds are small.

5. All unripe fruit is unwholesome.

All these apples are wholesome.

No fruit grown in the shade is ripe.

These apples were grown in the sun.

6. All my sons are slim.

No child of mine is healthy who takes no exercise.

All gluttons who are children of mine are fat.

No daughter of mine takes any exercise.

All gluttons who are children of mine are unhealthy.

Section 5.7 Sorites 271

★7. The only books in this library that I do not recommend for reading are

unhealthy in tone.

The bound books are all well written.

All the romances are healthy in tone.

I do not recommend you to read any of the unbound books.

All the romances in this library are well written.

8. No interesting poems are unpopular among people of real taste.

No modern poetry is free from affectation.

All your poems are on the subject of soap bubbles.

No affected poetry is popular among people of real taste.

No ancient poem is on the subject of soap bubbles.

All your poems are uninteresting.

9. All writers who understand human nature are clever.

No one is a true poet unless he can stir the hearts of men.

Shakespeare wrote Hamlet.

No writer who does not understand human nature can stir the hearts of men.

None but a true poet could have written Hamlet. Shakespeare was clever.

★10. I trust every animal that belongs to me.

Dogs gnaw bones.

I admit no animals into my study unless they beg when told to do so.

All the animals in the yard are mine.

I admit every animal that I trust into my study.

The only animals that are really willing to beg when told to do so are dogs. All the animals in the yard gnaw bones.

V. Create a sorites having four premises about trees; then put it into standard form and test

it for validity using Venn diagrams or the rules method.

Summary

Categorical syllogism:

■ A deductive argument consisting of three categorical propositions

■ Containing three different terms

■ Each term occurs twice in distinct propositions

Terms in a categorical syllogism:

■ Major term: The term that occurs in the predicate of the conclusion

■ Minor term: The term that occurs in the subject of the conclusion

■ Middle term: The term that occurs twice in the premises

272Chapter5 Categorical Syllogisms

Premises of a categorical syllogism:

■ Major premise: The premise that contains the major term

■ Minor premise: The premise that contains the minor term

Standard-form categorical syllogism:

■ All three propositions are in standard form.

■ The two occurrences of each term are identical.

■ Each term is used in the same sense throughout the argument.

■ The major premise is listed first, minor premise second, conclusion last.

The validity of a categorical syllogism is determined by its mood and figure:

■ Mood: Consists of the letter names of the propositions in the syllogism

■ Figure: Is determined by how the occurrences of the middle term are arranged

Tables for categorical syllogisms:

■ Unconditionally valid: Lists all forms valid from the Boolean standpoint

■ Conditionally valid: Lists additional forms valid from the Aristotelian standpoint

■ Critical term: The term listed in the farthest right-hand column of the conditionally valid table

Venn diagrams: To test a syllogism:

■ Enter the information from the premises into the diagram.

■ Look at the completed diagram to see if it supports the conclusion.

■ If a syllogism having universal premises and a particular conclusion is not valid from the

Boolean standpoint, test it from the Aristotelian standpoint.

Rules for syllogisms: For a syllogism to be valid from the Boolean standpoint:

■ The middle term must be distributed in at least one premise;

■ A term distributed in the conclusion must also be distributed in a premise;

■ At least one premise must be affirmative;

■ A negative conclusion requires a negative premise, and vice versa;

■ A particular conclusion requires a particular premise.

If only the last rule is broken, a syllogism is valid from the Aristotelian standpoint on condition that the critical term refers to at least one existing thing.

Some syllogisms having more than three terms can be reduced to standard form:

■ Each additional term must be a complement of one of the others.

■ Conversion, obversion, and contraposition must be applied only when they yield logically

equivalent results.

Ordinary language syllogisms: The component propositions are not in standard form.

■ Can be converted into standard-form syllogisms by applying the techniques presented in

Section 4.7.

Enthymemes: Syllogisms that are missing a premise or conclusion:

■ Can be converted into standard-form syllogisms by supplying the missing statement.

Summary273

Sorites: A chain of syllogisms in which the intermediate conclusions are missing

Testing a sorites:

■ Using the five rules:

● Put the sorites into standard form. ● Apply the rules.

● If only the last rule is broken, a sorites is valid from the Aristotelian standpoint on

condition that the critical term refers to at least one existing thing.

■ Using Venn diagrams

● Put the sorites into standard form. ● Supply the missing intermediate conclusions. ● Test each component syllogism.

274Chapter5 Categorical Syllogisms

6Propositional Logic

6.1 Symbols and Translation

6.2 Truth Functions

6.3 Truth Tables for Propositions

6.4 Truth Tables for Arguments

6.5 Indirect Truth Tables

6.6 Argument Forms and Fallacies

6.1 Symbols and Translation

Learning propositional logic is empowering. Propositional logic introduces special symbols to simplify the expression of statements and argu-ments and expose their structure. Consider this argument: �If the Greenland ice sheets are melting, then climate change is a reality. Therefore, climate change is a reality since the Greenland ice sheets are melting.� Once translated into sym-bols, this argument will be immediately recognized as having a specifically named valid form, which means that it is valid.

Earlier chapters showed that the validity of a deductive argument is purely a function of its form. By knowing the form of an argument, we can often tell immediately whether it is valid or invalid. Unfortunately, however, ordinary lin-guistic usage often obscures the form of an argument. To dispel this obscurity, logic introduces various simplifying procedures. In Chapter 5, letters were used to represent the terms in a syllogism, and techniques were developed to reduce syllogisms to what is called standard form. In this chapter, form recognition is facilitated through the introduction of special symbols called operators, or connectives . When arguments are expressed in terms of these symbols, deter-mining validity often becomes a matter of mere visual inspection.

275

In the two previous chapters, the fundamen- It is not the case that T.

however, the fundamental elements are whole state- Either P or E. ments (or propositions). Statements are represented tal elements were terms. In propositional logic, D and C. by letters, and these letters are then combined by If N then F. means of the operators to form more-complex sym- B if and only if R.

in propositional logic, we must distinguish simple compound even though it contains only a single component ( A ). In general, negative statements statements from compound statements. A simple are interpreted as compound units consisting of an statement bolic representations. To understand the symbolic representation used In the first example, note that the statement is statement as a component. Here are some examples: affirmative statement and the phrase �it is not the is one that does not contain any other

case that.�

Fast foods tend to be unhealthy. The expressions �it is not the case that,� �and,� James Joyce wrote Ulysses. �or,� �if . . . then . . . ,� and �if and only if� are trans-Parakeets are colorful birds. lated by logical operators. The five logical operators*

are as follows:

The bluefin tuna is threatened

with extinction. OperatorNameLogical function Used to translate

may be selected to represent each dot conjunction and, also, moreover statement. Thus, Any convenient uppercase letter tilde negation not, it is not the case that ∼ ∼

to represent the first, F might be selected �� wedge disjunction or, unless J the second, P horseshoe implication if . . . then . . . , only if the third, and B the fourth. As will ⊃

be explained shortly, lowercase let- K triple bar equivalence if and only if ters are reserved for use as state-

ment variables.

A Saying that logical operators are used to �trans-compound statement is one that contains at

least one simple statement as a component. Here are late� these English expressions does not mean that some examples: the expressions and the operators are identical.

As in any translation (from English to French, for

It is not the case that the Taliban supports example), a certain distortion of meaning occurs. educating women. The meaning of such English expressions as �and,� Diana Krall sings jazz, and Christina Aguilera �or,� and �if and only if � is often vague and may sings pop. vary with context, whereas the meaning of the Either people get serious about conservation logical operators is clear, precise, and invariable. or energy prices will skyrocket. Thus, when we say that the logical operators may If nations spurn international law, then future be used to translate expressions in ordinary lan-wars are guaranteed. guage, we mean that the operators capture a cer-

tain aspect of their correlative English expressions.

The Broncos will win if and only if they run The precise character of this aspect is spelled out the ball.

Using uppercase letters to stand for the simple state- *Some textbooks use the ampersand, &, in place of the dot; the arrow, ments, these compound statements may be repre-S , in place of the horseshoe; and the double arrow, 4 , in place of the sented as follows: triple bar.

276Chapter6 Propositional Logic

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). in the next section of this chapter. The purpose G ∼ ∼ H is not a proper expression. But the tilde is the of the current section is to develop a familiarity only operator that can immediately follow another with the logical operators through practice in operator. Thus, it would be proper to write G ● ∼ ∼H. translation. In the Rolex examples, the tilde is used to negate a

When we use the operators to translate the previ- simple proposition, but it can also be used to negate

ous examples of compound statements, the results a compound proposition�for example ∼ ∼(G ● F ). are as follows: In this case, the tilde negates the entire expression

It is not the case that T. inside the parentheses. ∼ ∼ T

D and C. D ● C

Either P or E. P �� E These statements are all negations. The main If N then F. operator is a tilde. N ⊃ F

B if and only if R. B K R

The statement ∼ ∼T is called a negation. The statement D ● C is called a conjunctive statement (or a conjunction), and the statement P �� E is called a disjunctive statement (or a disjunction); in the At this point we should define what is called conjunctive statement, the components D and C are the main operator in a compound statement. The called conjuncts, and in the disjunctive statement main operator is the operator that has as its the components P and E are called disjuncts. The scope everything else in the statement. If there are statement N ⊃ F is called a conditional statement no parentheses in the statement, the main opera-(or a conditional), and it expresses the relation of tor will either be the only operator or, if there is material implication . Its components are called more than one, it will be the operator that is not a antecedent ( N ) and consequent (F ). Lastly, tilde. If there are parentheses, brackets, or braces B K R is called a biconditional statement (or in the statement, the main operator will be the a biconditional), and it expresses the relation of operator that lies outside all parentheses, brack-material equivalence. ets, and braces; if there is more than one such

Let us now use the logical operators to trans- operator, the main operator will be the one that

late additional English statements. The tilde is not a tilde. symbol is used to translate any negated simple For example, in the statement H ● ( J �� K ), the proposition: main operator is the dot, because�its scope extends

It is not the case that Rolex Rolex does not make computers. to everything else in the statement, whereas the ∼ ∼ R scope of the wedge extends only to the J and K . ∼ ∼ R In the statement ∼ ∼ ( K ● M ), the main operator is makes computers. the tilde because its scope extends to everything It is false that Rolex makes ∼ ∼R else. In the statement K ⊃ ∼ ∼(L ● M ), the main computers. operator is the horseshoe, because, once again, its

in front of the proposition it negates. All of the other the parentheses. operators are placed between two propositions. The dot symbol is used to translate such conjunc-Also, unlike the other operators, the tilde cannot As these examples show, the tilde is always placed Excluding the tilde, it is the only operator outside scope extends to everything else in the statement.

be used to connect two propositions. Thus, �moreover,� �although,� and �nevertheless�: tions as �and,� �also,� �but,� �however,� �yet,� �still,�

Section 6.1 Symbols and Translation 277

Tiffany sells jewelry, and must be introduced in the symbolic expression. If Gucci sells cologne. T ● G parentheses or brackets are not needed, �either� Tiffany sells jewelry, but does not affect the translation. A similar point Gucci sells cologne. T ● G applies to the word �both,� which is often used to Tiffany sells jewelry; however, introduce conjunctive statements. Here are some

Gucci sells cologne. disjunctive statements: T ● G

Tiffany and Ben Bridge sell jewelry. Alta allows snowboards or T ● B

to �Tiffany sells jewelry, and Ben Bridge sells jew- or Telluride does. A �� T elry.� To translate such a statement as a conjunction Alta allows snowboards unless of two simple statements, the original statement must Telluride does. Note that the last example is equivalent in meaning Either Alta allows snowboards Telluride does. A �� T be equivalent to a compound statement in English. A �� T For example, the statement �Mary and Louise are Unless Alta allows snowboards, friends� is not equivalent in meaning to �Mary is Telluride does. A �� T

a friend, and Louise is a friend,� so this statement From the English sense of these statements, it should be cannot be translated as M ● L . clear that A �� T is logically equivalent to T �� A. Also T

● ● G is logically equivalent to G T. Logically equivalent

These statements are all conjunctions. The main propositions necessarily have the same truth value. operator is a dot.

These statements are all disjunctions. The main

operator is a wedge.

The wedge symbol is used to translate �or�

and �unless.� A previous chapter explained that

�unless� is equivalent in meaning to �if not.� This The horseshoe symbol is used to translate �if . . . equivalence holds in propositional logic as well, then . . . ,� �only if, � and similar expressions that but in propositional logic it is usually simpler to indicate a conditional statement. The expressions equate �unless� with �or.� For example, the state- �in case,� �provided that,� �given that,� and �on con-ment �You won�t graduate unless you pass fresh- dition that� are usually translated in the same way as man English� is equivalent to �Either you pass �if.� By customary usage, the horseshoe symbol is freshman English or you won�t graduate� and also also used to translate �implies.� Although �implies� to �If you don�t pass freshman English, then you is used most properly to describe the relationship won�t graduate.� As the next section demonstrates, between the premises and conclusion of an argu-the wedge symbol has the meaning of �and/or�� ment, we may accept this translation as an alternate that is, �or� in the inclusive sense. Although �or� meaning for �implies.� According to this alternate and �unless� are sometimes used in an exclusive meaning, A implies B is translated A ⊃ B. sense, the wedge is usually used to translate them The function of �only if � is, in a sense, just the as well. reverse of �if.� For example, the statement �You will

The word �either,� which is often used to intro- catch a fish only if your hook is baited� does not

duce disjunctive statements, has primarily a punc- mean �If your hook is baited, then you will catch a tuational meaning. The placement of this word fish.� If it meant this, then everyone with a baited often tells us where parentheses and brackets hook would catch a fish. Rather, the statement

278Chapter6 Propositional Logic

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). means �If your hook is not baited, then you will not is a necessary condition for survival. Other things catch a fish,� which is logically equivalent to �If you besides having the flu might cause a person to catch a fish, then your hook was baited.� To avoid feel miserable, but that by itself is sufficient; other mistakes in translating �if � and �only if � remember things besides having air to breathe are required for this rule: The statement that follows �if � is always survival, but without air, survival is impossible. In the antecedent, and the statement that follows �only other words, air is necessary. if � is always the consequent. Thus, �C only if H � To translate statements involving sufficient and is translated C ⊃ H, whereas �C if H � is translated necessary conditions into symbolic form, place the H ⊃ C. Additional examples: statement that names the sufficient condition in

If Purdue raises tuition, then so the antecedent of the conditional and the statement

does Notre Dame. that names the necessary condition in the conse-P ⊃ N quent. The mnemonic device �SUN� may be con-Notre Dame raises tuition veniently used to keep this rule in mind. Turning if Purdue does. P ⊃ N the U sideways creates S ⊃ N , wherein S and N des-Purdue raises tuition only ignate sufficient and necessary conditions, respec-if Notre Dame does. P ⊃ N tively. Whatever is given as a sufficient condition Cornell cuts enrollment goes in the place of the S, and whatever is given as a provided that Brown does. B ⊃ C necessary condition goes in the place of the N:

Cornell cuts enrollment on Hilton�s opening a new hotel is a condition that Brown does. B ⊃ C sufficient condition for Marriott�s Brown�s cutting enrollment doing so. H ⊃ M

implies that Cornell does. B ⊃ C Hilton�s opening a new hotel is a

In translating conditional statements, it is essen- necessary condition for Marriott�s

tial not to confuse antecedent with consequent. Just doing so. M ⊃ H as 6 � 3 is not equal to 3 � 6, the statement A ⊃ B is The triple-bar symbol is used to translate the not logically equivalent to B ⊃ A. expressions �if and only if� and �is a sufficient and

necessary condition for�:

These statements are all conditionals JFK tightens security if (material implications). The main operator is a horseshoe. and only if O�Hare does. J K O

JFK�s tightening security is

a sufficient and necessary

condition for O�Hare�s

doing so. J K O

The horseshoe symbol is also used to translate Analysis of the first statement reveals that J K O is

statements phrased in terms of sufficient condi- logically equivalent to ( J ⊃ O) ● (O ⊃ J ). The state-tions and necessary conditions. Event A is said to ment �JFK tightens security only if O�Hare does� be a sufficient condition for event B whenever is translated J ⊃ O, and �JFK tightens security if the occurrence of A is all that is required for the O�Hare does� is translated O ⊃ J. Combining the occurrence of B. On the other hand, event A is said two English statements, we have (J ⊃ O) ● (O ⊃ J ), to be a necessary condition for event B when- which is just a longer way of writing J K O. A similar ever B cannot occur without the occurrence of A. analysis applies to the second statement. Because For example, having the flu is a sufficient condition the order of the two conjuncts can be reversed, for feeling miserable, whereas having air to breathe J K O is logically equivalent to O K J. However,

Section 6.1 Symbols and Translation 279

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). when translating such statements, we adopt the allergies or Zocor lowers convention that the letter representing the first cholesterol. P ● (A �� Z ) English statement is written to the left of the triple Prozac relieves depression bar, and the letter representing the second English or both Allegra combats statement is written to the right of the triple bar. allergies and Zocor lowers Thus, the examples given are translated J K O, and cholesterol. P �� (A ● Z )

not O K J. Prozac relieves depression

and Allegra or Zocor lowers

These statements are all biconditionals cholesterol. P ● (A �� Z )

(material equivalences).The main operator is a triple bar. If Merck changes its logo,

then if Pfizer increases sales,

then AbbVie will reorganize. M ⊃ (P ⊃ A)

If Merck�s changing its logo

implies that Pfizer increases

Whenever more than two letters appear in a sales, then AbbVie will

translated statement, we must use parentheses, reorganize. (M ⊃ P ) ⊃ A brackets, or braces to indicate the proper range of If Sanofi and Pfizer lower the operators. The statement A ● B �� C, for exam- prices or Novartis downsizes, ple, is ambiguous. When parentheses are intro- then Roche will expand duced, this statement becomes either (A ● B) �� C or production. [(S ● P) �� N] ⊃ RA ● (B �� C). These two statements are not logically equivalent. Thus, with statements such as these,

some clue must be found in the English statement Do not confuse these three statement forms: that indicates the correct placement of the parenthe- A if B ses in the symbolic statement. Such clues are usually A only if B given by commas and semicolons, by such words as A if and only if B �either� and �both,� and by the use of a single predi-

cate in conjunction with two or more subjects. The When a tilde appears in a symbolic expres-following examples illustrate the correct placement sion, by convention it is considered to affect only of parentheses and brackets: the unit that immediately follows it. For example, in the expression ∼ ∼ K �� M the tilde affects only Prozac relieves depression and the K; in the expression ∼ ∼ ( K �� M ) it affects Allegra combats allergies, or the entire expression inside the parentheses. In Zocor lowers cholesterol. ( P ● A ) �� Z English, the expression �It is not the case that K or Prozac relieves depression, M � is ambiguous, because the range of the negating and Allegra combats allergies words is indefinite. To eliminate this ambiguity, we or Zocor lowers cholesterol. P ● (A �� Z ) now adopt the convention that the negating words

Either Prozac relieves are considered to affect only the unit that follows depression and Allegra them. Thus, �It is not the case that K or M � is trans-combats allergies or Zocor lated ∼ ∼K �� M. lowers cholesterol. (P ● A) �� Z The statement �Not both S and T � is translated Prozac relieves depression ∼ ∼ (S ● T ). By an important rule called De Morgan�s and either Allegra combats rule, this statement is logically equivalent to ∼ ∼S �� ∼ ∼T.

280Chapter6 Propositional Logic

For example, the statement �Not both Steven and Some examples of symbolic arrangements that are Tyrone were fired� is equivalent in meaning to not well-formed formulas are �A ⊃ �� B,� �A ● B (�� Either Stephen was not fired or Tyrone was not C ),� and �∼ ∼�� B K ⊃ C.� fired.� Because the former statement is not equiv- A few informal cues for constructing well-alent in meaning to �Stephen was not fired and formed formulas follow. In these cues, the term Tyrone was not fired,� ∼ ∼(S ● T ) is not logically statements includes simple statements and com-equivalent to ∼ ∼S ● ∼ ∼T. Analogously, the statement pound statements. Remember, ~A is a compound �Not either S or T � is translated ∼ ∼(S �� T ), which by statement.

De Morgan�s rule is logically equivalent to ∼ ∼S ● ∼ ∼T. ● Statements cannot be combined without an oper-For example, �Not either Steven or Tyrone was ator occurring between them. fired� is equivalent in meaning to �Steven was not Not WFFs: fired and Tyrone was not fired.� Thus, ∼ ∼ ( S �� T ) is AB , A ( B ⊃ C ), ( A ⊃ B )( B �� C ) WFFs: not logically equivalent to ∼ ∼ S �� ∼ ∼ T. The following A ⊃ B , A �� ∼ ∼ B , A ● ( B K C )

examples illustrate these points: ● A tilde cannot immediately follow a statement,

Either Martina or Kathy Martina is not a winner, but Kathy is. but it can immediately precede any statement ∼ ∼ M ● K (except when it immediately follows a statement). Not both Martina and Kathy Not WFFs: are winners. ∼ ∼ (M ● K ) A ∼ ∼ , A ∼ ∼ B , ( A ● B ) ∼ ∼ WFFs: ∼ ∼ A , ∼ ∼ ( A ● B ), ∼∼ ∼∼ A

is not a winner. ∼ ∼M �� ∼ ∼K ● A tilde cannot immediately precede any other Both Martina and Kathy are operator.

not winners. ∼ ∼M ● ∼ ∼K Not WFFs: ∼ ∼● A, A∼ ∼�� B

Not either Martina or Kathy A dot, wedge, horseshoe, or triple bar must go is a winner. ● ∼ ∼ (M �� K ) immediately between statements. Neither Martina nor Kathy

is a winner. Not WFFs: ∼ ∼ (M �� K ) ●A, A ��, A ● ⊃ B

WFFs: A ● B, A ⊃ ∼ ∼B, A �� (B ● C )

● Parentheses, brackets, and braces must be inserted

Notice the function of �either� and �both�: to prevent ambiguity.

Not either A or B. Not WFFs: A ⊃ B �� C, A ● B K C Either not A or not B. WFFs: A ⊃ (B �� C), (A ● B) ⊃ C Not both A and B.

Both not A and not B.SummaryOperator

not, it is not the case that, it is false that

The symbolic expressions that we have used and, yet, but, however, moreover, nevertheless,

throughout this section to translate meaningful, still, also, although, both, additionally,

furthermore

unambiguous English statements are called well- or, unless v formed formulas (WFFs) . �WFFs� is usually if . . . then, only if, implies, given that, in case,

syntactically dition for, necessary condition for (Note: Do not correct arrangement of symbols. In confuse antecedent with consequent!) English, for example, the expression �there is a cat pronounced �woofs.� A well-formed formula is a provided that, on condition that, sufficient con-on the porch� is syntactically correct, but �Porch if and only if, is equivalent to, sufficient and on the is cat a there� is not syntactically correct. necessary condition for

Section 6.1 Symbols and Translation 281

Prominent 1646�1716Gottfried Wilhelm Leibniz Logicians

Gottfried Wilhelm Leibniz was a polymath who knew virtually everything that could be known at the time about nearly every area of intellectual endeavor. He also made important

contributions to many of them, including physics,

engineering, philos ophy, theology, history, law, politics, and philology. In mathematics Leibniz

invented differential and integral calculus

(independently of Newton) and the theory of

differential equations. He also discovered the

binary number system (used by all of today�s digital

computers), and he created the first calculating

machine that could add, subtract, multiply, and

divide. In metaphysics he created the famous theory

of monads which, among other things, explained Herzog Anton Ulrich-Museum/Wikimedia Commons the relation between the soul and the body.

Leibniz was born in Leipzig to prominent parents. what should have been Leibniz�s inheritance. When

His father (who died when Leibniz was six) was a this happened Leibniz figured that the best way of professor of moral philosophy at the city�s univer- satisfying his expensive tastes was to attach himself sity, and his mother was the daughter of a famous to the wealthy and the powerful, which he did with lawyer. As a child, Leibniz proved himself a prod- great success. He entered the employment of the igy. By age twelve, he was fluent in Latin and had a elector of Mainz, and by the age of twenty-four he passing knowledge of Greek, both of which he had held the rank of privy counselor of justice, one of learned by himself. By thirteen, he was deep into the highest positions in the government. His work the works of Aristotle and scholastic philosophy, as a diplomat allowed him the opportunity to travel and at fourteen he entered the University of Leipzig, widely and meet most of the prominent figures in where he studied philosophy, mathematics, and law. Europe. Later he worked for the Duke of Hanover, a After completing that program he began a doctoral position that also allowed much time for travel and program in law; however, when he applied for the independent study. degree at age twenty, he was refused because of Leibniz is sometimes called the father of symbolic his youth. Not to be outdone, Leibniz presented his logic for his work in developing the universal thesis to the University of Altdorf, and the profes- characteristic, a symbolic language in which any item sors there were so impressed that they immediately of information can be represented in a natural and awarded him the degree of Doctor of Laws and systematic way, together with the calculus ratiocinator, offered him a professorship. a deductive system for manipulating the symbols

Quite early in life Leibniz developed a taste for and deriving consequences. Given the dispassionate

the finer things in life, including expensive clothing; nature of this enterprise, Leibniz thought that it would long, flowing wigs; fancy carriages; and luxurious serve to resolve differences of opinion in religion, accommodations. However, following the death of theology, and philosophy. However, at age seventy, he his mother when he was eighteen, an uncle received died in Hanover before completing the project.

282Chapter6 Propositional Logic

Exercise 6.1

I. Translate the following statements into symbolic form using uppercase letters to rep-

resent affirmative English statements. Example: Suppose you are given the statement �If Facebook makes us narcissistic, then either Twitter or LinkedIn relieves our loneli-ness.� This would be translated F ⊃ (T �� L).

★1. Cartier does not make cheap watches.

2. Arizona has a national park but Nebraska does not.

3. Either Stanford or Tulane has an architecture school.

★4. Both Harvard and Baylor have medical schools.

5. If Chanel has a rosewood fragrance, then so does Lanvin.

6. Chanel has a rosewood fragrance if Lanvin does.

★7. Thomas Friedman writes incisive editorials if and only if Paul Krugman does.

8. Pen�lope Cruz wins best actress only if Kenneth Branagh wins best director.

9. Armani will launch a leather collection given that Gucci rejects skinny models.

★10. The Packers� winning most of their games implies that Aaron Rodgers is a great

quarterback.

11. Bill Gates does not support malaria research unless Larry Ellison does.

12. Mercedes will introduce an electric model only if Lexus and BMW do.

★ 13. Ariana Grande sings pop and either Kendrick Lamar does rap or Diana Krall

sings jazz.

14. Either Ariana Grande sings pop and Kendrick Lamar does rap or Diana Krall

sings jazz.

15. Not both Toyota and Porsche make motorcycles.

★16. Both Toyota and Porsche do not make motorcycles.

17. Either Apple or Samsung makes smartphones.

18. Not either Ferrari or Maserati makes economy cars.

★19. Neither Ferrari nor Maserati makes economy cars.

20. Either Ferrari or Maserati does not make economy cars.

21. If Tucker Carlson spins the news, then if Don Lemon fights back, then Joy Reid

tells it straight.

★22. If Tucker Carlson�s spinning the news implies that Don Lemon fights back,

then Joy Reid tells it straight.

23. Tommy Hilfiger celebrates casual if and only if neither Ralph Lauren nor

Calvin Klein offers street chic.

Section 6.1 Symbols and Translation 283

24. If Saks promotes gift cards, then either Macy�s or Bloomingdale�s puts on a

fashion show.

★25. Either Rado does not make a sapphire watch or if Movado makes one then so

does Pulsar.

26. If either Lockheed Martin or Raytheon floods the world with weapons of war, then

neither General Dynamics nor Boeing reduces the likelihood of armed conflict.

27. If Electronic Arts and Activision Blizzard sell violent video games, then either

Colt or Smith & Wesson can expect increased profits.

★28. Mercury is a planet given that both Pluto and Ceres are not.

29. Saturn has rings, and Neptune is windy or Jupiter is massive.

30. Saturn has rings and Neptune is windy, or Jupiter is massive.

★31. Tiffany and Ben Bridge will release an emerald collection unless Zales and Kay

do not.

32. Chevron continues to pollute provided that British Petroleum does, but Koch

Industries and ExxonMobil deny climate change.

33. Either Pablo Picasso or Salvador Dali ignored perspective, but it is not the case

that both of them did.

★34. South Korea supports diplomacy; but if North Korea starts a war, then either

China or Japan will be involved.

35. It is not the case that both Iran gives up its nuclear program and Sudan or

Pakistan combats terrorism.

36. It is not the case that either Hezbollah will renounce violence or al-Qaeda and

ISIS will be defeated.

★37. If Monsanto poisons the human race, then if Nestl� buries us in bottles, then

Smithfield and Tyson feed us antibiotics.

38. If Maria Cantwell promotes alternative energy, then if Patty Murray supports

wilderness areas, then Chris Murphy�s advocating gun control implies that Susan Collins does so, too.

39. It is not the case that either Rory McIlroy and Ashleigh Barty play professional

football or Patrick Mahomes and Aliyah Boston play professional baseball.

★40. It is not the case that both Mikaela Shiffrin or Kevin Durant plays professional

tennis and Antonio Brown or J.J. Watt plays professional basketball.

41. Israel�s abandoning its settlements is a sufficient condition for the Palestinians�

declaring an end to hostilities.

42. Israel�s abandoning its settlements is a necessary condition for the Palestinians�

declaring an end to hostilities.

284Chapter6 Propositional Logic

★43. Israel�s abandoning its settlements is a sufficient and necessary condition for

the Palestinians� declaring an end to hostilities.

44. The Taliban�s being defeated is a sufficient condition for Pakistan�s winning the

war on terror only if Afghanistan�s securing its borders is a necessary condition for the UN�s stopping the opium trade.

45. Erin Burnett and Brooke Baldwin report international news if and only if Anna

Cabrera and Norah O�Donnell cover political developments.

★46. It is not the case that both Walmart�s jeopardizing worker safety implies that

Target does and Costco�s raising wages implies that Kmart does.

47. Cameron Diaz promotes environmental causes if Ben Affleck supports civil

liberties, provided that Sean Penn supports aid to Ukraine.

48. JP Morgan�s getting a bailout implies that Goldman Sachs cheats its clients,

given that Bank of America�s foreclosing on mortgages implies that neither Morgan Stanley nor Citigroup curtails obscene bonuses.

★49. If Christina Aguilera�s singing soul and Justin Timberlake�s singing pop are suf-

ficient and necessary conditions for Kelly Clarkson�s singing rock, then neither Beyonc� nor Shakira will do rap.

50. Nassau�s advertising sizzling night clubs is a necessary condition for Puerto

Vallarta�s offering luxury hotels; moreover, Cancun�s having turquoise waters and Acapulco�s promising powder-white beaches is a sufficient condition for Jamaica�s offering reggae music.

II. Translate the following statements into symbolic form using uppercase letters to represent

affirmative English statements. Use the letters given in parentheses to form the translations.

★1. Unless we reduce the incidence of child abuse, future crime rates will increase.

(R, F)

2. If pharmaceutical makers conceal test results, they are subject to substantial

fines. (P, S)

3. African safaris are amazing, but they are also expensive. (A, E)

★4. Cigarette manufacturers are neither honest nor socially responsible. (H, S)

5. Psychologists and psychiatrists do not both prescribe antidepressant drugs. (O, I)

6. If health maintenance organizations cut costs, then either preventive medicine

is emphasized or the quality of care deteriorates. (H, P, Q)

★7. A necessary condition for a successful business venture is good planning. (S, G)

8. If heroin is legalized, then its use may increase but criminal activity will

decline. (L, U, C)

9. Ozone depletion in the atmosphere is a sufficient condition for increased

cancer rates. (O, I)

Section 6.1 Symbols and Translation 285

★10. If affirmative action programs are dropped, then if new programs are not

created, then minority applicants will suffer. (A, N, M)

11. If Internet use continues to grow, then more people will become cyber addicts

and normal human relations will deteriorate. (I, M, N)

12. Human life will not perish unless either we poison ourselves with pollution or

a large asteroid collides with the earth. (H, P, L)

★13. Cooling a group of atoms to absolute zero and keeping them bunched

together is a necessary and sufficient condition for producing a Bose�Einstein condensate. (C, K, P)

14. If motion pictures contain subliminal sex messages or if they challenge the tra-

ditional family, then conservative politicians call for censorship. (M, C, P)

15. Either clear-cutting in national forests is halted and old-growth trees are

allowed to stand, or salmon runs will be destroyed and bird habitats obliterated. (C, O, S, B)

★16. Three-strikes laws will be enforced and longer sentences imposed only if hun-

dreds of new prisons are built, and that will happen only if taxes are increased. (T, L, H, I)

17. The Ebola virus is deadly, but it will become a major threat to humanity if and

only if it becomes airborne and a vaccine is not developed. (E, M, A, V )

18. If evolutionary biology is correct, then higher life-forms arose by chance, and

if that is so, then it is not the case that there is any design in nature and divine providence is a myth. (E, H, D, P)

★19. If the climate continues to warm, then more sea ice will melt; and if more sea

ice melts and fishing is not controlled, then the number of seals will decline and polar bears will become extinct. (C, M, F, N, P)

20. If corporate welfare continues, then taxpayer interests will be ignored and

billions of tax dollars will go to giant corporations; and if the latter occurs, then there will not be anything left for lower-income families and the budget will not be balanced. (C, T, B, A, U)

III. Determine which of the following are not well-formed formulas.

1. (S ● ∼ ∼T ) �� (∼ ∼U ● W )

2. ∼ ∼(K �� L) ● (⊃ G �� H )

3. (E ∼ ∼F ) �� (W K X )

4. (B ⊃ ∼ ∼T ) K ∼ ∼(∼ ∼C ⊃ U )

5. (F K ∼ ∼Q) ● (A ⊃ E �� T )

6. ∼ ∼D �� ∼ ∼[(P ⊃ Q) ● (T ⊃ R)]

7. [(D ● �� Q) ⊃ (P �� E )] �� [A ⊃ (● H )]

8. M(N ⊃ Q) �� (∼ ∼C ● D)

286Chapter6 Propositional Logic

9. ∼ ∼(F �� ∼ ∼G ) ⊃ [(A K E ) ● ∼ ∼H ]

10. (R K S ● T ) ⊃ ∼ ∼(∼ ∼W ● ∼ ∼X )

IV. Construct compound propositions involving at least three simple propositions about

the following topics. Then, symbolize your propositions as per Parts I and II of this exercise.

1. Political figures

2. NFL teams

3. Corporations

4. Brands of automobiles

5. Music or film celebrities

6.2 Truth Functions

The truth value of a compound proposition expressed further the degree to which symbolized expressions in terms of one or more logical operators is said to match the meaning of expressions in ordinary language. be a function of the truth values of its components.

proposition is completely determined by the truth This means that the truth value of the compound Definitions of the Logical Operators values of its components. If the truth values of the The definitions of the logical operators are presented in components are known, then the truth value of terms of statement variables, which are lowercase the compound proposition can be calculated from letters (p, q, r, s) that can stand for any statement. For the definitions of the logical operators. Accordingly, example, the statement variable p could stand for the a truth function is any compound proposition statements A, A ⊃ B, B �� C, and so on. whose truth value is completely determined by the Statement variables are used to construct statement truth values of its components. forms. A statement form is an arrangement of state-

Many compound propositions in ordinary lan- ment variables and operators such that the uniform

guage are not truth functions. For example, the substitution of statements in place of the variables statement �Mary believes that Paul is dishonest� is results in a statement. For example, ∼ ∼p and p ⊃ q are compound because it contains the statement �Paul statement forms because substituting the statements A is dishonest� as a component. Yet the truth value of and B in place of p and q, respectively, results in the the compound statement is not determined by the statements ∼ ∼A and A ⊃ B. For a slightly more com-truth value of the component, because Mary�s beliefs plicated example, (p ● q) ⊃ (q �� r) is a statement form, about Paul are not compelled by any attribute that because substituting A, B, and C in the place of p, q, Paul may or may not possess. and r, respectively, results in the statement (A ● B) ⊃

The first part of this section presents the definitions (B �� C). These statements are substitution instances of

of the five logical operators, the second part shows how the respective statement forms. they are used to compute the truth values of more- A compound statement is said to have a certain form complicated propositions, and the third examines if it can be produced by substituting statements in place

Section 6.2 Truth Functions 287

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). of the letters in that form. Thus, ∼ ∼A, ∼ ∼(A �� B), and Ferrari and Maserati make ∼ ∼[ A ● (B �� C )] are negations because they can be sports cars. F ● M

the form ∼ ∼p. Google and Apple make sports cars. G ● A Now let us consider the definition of the tilde produced by substituting statements in place of p in Ferrari and Google make sports cars. F ● G operator (negation). This definition is given by a The first statement is true, because both conjuncts truth table, an arrangement of truth values that are true; but the second and third statements are shows in every possible case how the truth value of false, because at least one of their conjuncts is false. a compound proposition is determined by the truth Turning now to the definition of the wedge oper-

negation shows how any statement having the form Disjunctionpqp �� q of a negation ( values of its simple components. The truth table for ator (disjunction), the truth table is as follows:

of the statement that is negated ( ∼ ∼p) is determined by the truth value T T T p ): T F T

Negationp ∼ ∼p F T T

T F F F F

F T The truth table indicates that the disjunction is

The truth table shows that ∼p is false when p is true that otherwise it is false. The truth-functional ∼ true when at least one of the disjuncts is true and and that ∼ ∼ p is true when p is false. This is exactly interpretation of �or� is that of inclusive disjunction: what we would expect, because it perfectly matches Cases in which the disjunction is true include the ordinary English usage. Examples: case when both disjuncts are true. This inclusive sense of �or� corresponds to many instances of It is not the case that McDonald�s ordinary usage, as the following examples illustrate: makes hamburgers. ∼ ∼ M

It is not the case that Starbucks Either Steven King or Cate makes hamburgers. ∼ ∼S Blanchett is a novelist. S �� C

The first statement is false because Either Steven King or M is true, and the second is true because J.K. Rowling is a novelist. S �� JS is false.

Let us now consider the definition of the dot Either Greta Thunberg or

operator (conjunction). The truth table that follows Tiger Woods is a novelist. G �� T

shows how any statement having the form of a con- The first two statements are true, because in each junction ( p ● q ) is determined by the truth values of case at least one of the disjuncts is true. The third is its conjuncts ( p , q ): false, because both disjuncts are false.

Conjunctionpqp ● q The match between the truth-functional

T definition of disjunction and ordinary usage is not T T

F in ordinary language is that of exclusive disjunction. T F Examples: T perfect, however. Sometimes the sense of a statement F F F F F

This truth table shows that a conjunction is true The commuter train is on either track A or when its two conjuncts are true and is false in all track B. other cases. This definition reflects ordinary lan- You can have either soup or salad with this guage usage almost as perfectly as negation. Con- meal.

sider the following conjunctive statements: Tammy is either ten or eleven years old.

288Chapter6 Propositional Logic

The sense of these statements excludes the possibility of For an intuitive approach to this problem, imagine both alternatives being true. Thus, if these statements that your logic instructor made the following statement: were translated using the wedge, a portion of their �If you get an A on the final exam, then you will get ordinary meaning would be lost. If the exclusive aspect an A for the course.� Under what conditions would of these �either . . . or . . .� statements is essential, the you say that your instructor had lied to you? Clearly, symbolic equivalent of �but not both� can be attached if you got an A on the final exam but did not get an A to their translations. Thus the first statement could be for the course, you would say that she had lied. This translated (A �� B) ● ∼ ∼(A ● B). outcome corresponds to a true antecedent and a false

Let us now consider the horseshoe operator consequent. On the other hand, if you got an A on the

(material implication, or conditional). Its truth table final exam and also got an A for the course, you would is as follows: say that she had told the truth (true antecedent, true

consequent). But what if you failed to get an A on the

Conditional final exam? Two alternatives are then possible: Either (material implication) pq p ⊃ q you got an A for the course anyway (false antecedent,

T T T true consequent) or you did not get an A for the course

F though, would you say that your instructor had lied to T T you. Giving her the benefit of the doubt, you would say T (false antecedent, false consequent). In neither case, F F F F T that she had told the truth.

Lastly, let us consider the definition of the triple-

The truth table shows that a conditional statement is bar operator (material equivalence, or biconditional). false when the antecedent is true and the consequent Its truth table is as follows: false and is true in all other cases. This truth-

functional interpretation of conditional statements Biconditional conforms in part with the ordinary meaning of (material �if . . . then . . .� and in part it diverges. Consider the equivalence) pq p K q following examples: T T T

T F F

If Nicole Kidman is an actor,

then so is Halle Berry. F T F N ⊃ H

If Nicole Kidman is an actor, F F T then so is Elon Musk. N ⊃ E The truth table shows that the biconditional is true when If Elon Musk is an actor, its two components have the same truth value and that

then so is Jennifer Lawrence. E otherwise it is false. These results are required by the fact ⊃ J

then so is Anderson Cooper. p). If p and q are either both true or both false, then p ⊃ qE ⊃ A and If Elon Musk is an actor, that p K q is simply a shorter way of writing (p ⊃ q) ● (q ⊃

Ap is false and q is true, are false. Thus, according to the truth-functional then q ⊃ p is false, again making the conjunction false. interpretation, the first statement is true and the second Thus, p K q is true when p and q have the same truth false. This result conforms in large measure to our value and false when they have opposite truth values. In these statements But if p is true and q is false, then p ⊃ q is false, making N, H, and J are true and E and the conjunction false. Similarly, if q ⊃ p are both true, making their conjunction true.

the last two statements is true. Although this result may The truth-table definition of the triple-bar expectations. But the truth-functional interpretation of

not conflict with our expectations, it is not at all clear symbol conforms quite closely with ordinary usage, as the following examples illustrate: why these statements should be considered true.

Section 6.2 Truth Functions 289

Bill Maher is a show host ( A �� D ) ⊃ E

if and only if Jimmy Fallon is. B K J First, we write the truth values of the simple Bill Maher is a show host propositions immediately below the respective letters if and only if Emma Watson is. B K E and bring the operators and parentheses down:

if and only if Rihanna is. E K Emma Watson is a show host ( A �� D ) ⊃ ER

( T �� F ) ⊃ F

In these statements, B and J are true and E and R false. Thus, from the truth- functional standpoint, the first Next, we compute the truth value of the proposition is true and the second false. This is what we would in parentheses and write it beneath the operator to ordinarily expect. The third statement, however, turns which it pertains:

out to be true because both of its components are false. ( A �� D ) ⊃ E While this result may not be what we would expect, ( T it does not violate our expectations either. Other �� F ) ⊃ F

biconditional statements having false components are T ⊃ F more obviously true. Example:

Finally, we use the last-completed line to obtain the

The Equal Rights Amendment is law truth value of the conditional, which is the main if and only if thirty-eight states have operator in the proposition: approved it.

This statement asserts what is required to ratify any ( A �� D ) ⊃ E amendment to the U.S. Constitution, and so it is ( T �� F ) ⊃ F

operators conform reasonably well with ordinary In summary, the definitions of the five logical F clearly true. T ⊃ F

linguistic usage. However, as the last part of this The final answer is circled. This is the truth value of section shows, the match is less than perfect. Before the compound proposition given that A is true and considering this question, though, let us use the D and E are false. operator definitions to compute the truth values of The general strategy is to build the truth values of more-complicated statements. the larger components from the truth values of the

smaller ones. In general, the order to be followed in

Computing the Truth Value entering truth values is this:

To compute the truth value of a more-complicated propositions 2. Tildes immediately preceding individual letters of Longer Propositions 1. Individual letters representing simple expression, use this procedure: Enter the truth

values of the simple components directly beneath 3. Operators joining letters or negated letters

the letters. Then use these truth values to compute 4. Tildes immediately preceding parentheses

the truth values of the compound components. The 5. And so on truth value of a compound statement is written beneath the operator representing it. Let us suppose, Here are some additional examples. As before, let for example, that we are told in advance that the A, B, and C be true, and D, E, and F false. Note that the simple propositions A, B, and C are true and D, E, computed truth values are written beneath the operators and F are false. We may then compute the truth to which they pertain. The final answers, which are value of the following compound proposition: written beneath the main operators, are circled.

290Chapter6 Propositional Logic

1. ( B ● C ) ⊃ (E ⊃ A ) She got married and had a baby. M ● B

( T ● T ) ⊃ (F ⊃ T ) She had a baby and got married. B ● M

T ⊃ T

T The first statement implies that the marriage

2. occurred first, and the baby came later, while the ∼ ∼ (C �� ∼ ∼ A) ⊃ ∼ ∼ B second statement implies that the baby came first. ∼ ∼ ( T �� ∼ ∼ T ) ⊃ ∼ ∼ T This implied meaning is lost in the truth-functional ∼ ∼ ( T �� F ) ⊃ F interpretation, because M ● B is logically equivalent ∼ ∼ T ⊃ F to B ● M . F ⊃ F Another kind of mismatch between the truth-T functional meaning of conjunctive statements and 3. [ ∼ ∼ (D �� F ) ● (B �� ∼ ∼ A) ] ⊃ ∼ ∼ (F ⊃ ∼ ∼ C ) their ordinary language meaning occurs when the [ ∼ ∼ (F �� F) ● ( T �� ∼ ∼ T )] ⊃ ∼ ∼ (F ⊃ ∼ ∼ T ) order of the conjuncts implies a causal connection. [ ∼ ∼ (F �� F) ● ( T �� F ) ] ⊃ ∼ ∼ (F ⊃ F ) Example: [ ∼ ∼ F ● T ] ⊃ ∼ ∼ T

[T ● T ] ⊃ F He fell off a ladder and broke his arm. F ● B

T ⊃ F He broke his arm and fell off a ladder. B ● F

If preferred, the truth values of the compound compo- The first statement implies that the fall caused the F

nents may be entered directly beneath the operators, such thing. However, the truth-functional meanings broken arm, but the second statement implies no

without using the line-by-line approach illustrated in of the two statements are logically equivalent. these examples. The following examples illustrate this For yet another example, consider the following: second approach, which is used in the next section:

1. [(D K This car is ugly, but it�s economical ∼ ∼ A) ● ∼ ∼ (C ● ∼ ∼ B)] K

to drive. U ● E

F T F T T T T F F T F

This car is economical to drive, but

∼ ∼ [(A ⊃ ∼ ∼ D) �� (C K E )] it�s ugly. E ● U

F T T T F T T F F

The first statement subtly implies that we should buy

2. ∼ ∼ {[(C ● ∼ ∼ E ) ⊃ ∼ ∼ (A ● ∼ ∼ B)] ⊃ this car, while the second implies that we should not

F T T T F T T T F F T T buy it. But once again, the truth-functional interpre-

[ tations are logically equivalent. ∼ ∼ (B �� D) K ( ∼ ∼ C �� E )]} Another instance where the truth-functional F T T F T F T F F interpretation of �and� differs from the ordinary

Further Comwparison with Ordinary linguistic meaning is offered by slang statements like this one: Language You go for that gun, and you�ll regret it. The first part of this section showed that the defini-tions of the five logical operators conform reasonably The sense of this statement is not that you will in well with ordinary linguistic usage. This part further fact go for the gun but rather that, if you go for that examines the extent of this match in meaning. gun, then . . . Accordingly, if this statement were

In regard to the dot operator, which is used to interpreted as a truth-functional conjunction, its

translate �and� and �but,� the match is often good; meaning would be distorted. but it fails, at least in part, when the meaning of a In regard to the wedge operator, which is used to conjunctive statement depends on the order of the translate �or� and �unless,� we saw that the wedge is conjuncts. Consider the following statements: defined as inclusive disjunction, but we observed

Section 6.2 Truth Functions 291

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). that the English word �or� sometimes has the sense of In all three examples the statement remains true exclusive disjunction. This same observation applies even though the antecedent might be false. In the to �unless.� In the following statements, �unless� has first, even though the temperature does not rise an inclusive sense: above 32�F at any particular moment, the law that

You won�t win the lottery unless you buy a governs the melting point of snow holds true. In ticket. other words, the statement (which expresses this

law) is true regardless of the truth value of the ante-

It will not rain unless there are clouds in the sky. cedent. In the second, the mere fact that Figure A

The meaning of the first statement includes the case might not be a triangle does not affect the fact that of buying a ticket and not winning, and the mean- a triangle, by definition, has three sides. Thus, the ing of the second includes the case of there being statement (which expresses this fact) is true regard-clouds and no rain. In statements like these, where less of the truth value of the antecedent. The third �unless� has an inclusive sense, using the wedge statement expresses a logical relationship between symbol to translate �unless� results in no loss statements. This logical relationship remains of meaning. unchanged regardless of what the terms A, B, and

On the other hand, in the following statements C are taken to represent. Thus, if A, B, and C rep-

�unless� is used in the exclusive sense: resent �dogs,� �cats,� and �birds,� respectively, both

Pork is not properly cooked unless the meat antecedent and consequent turn out to be false, but is white. the conditional statement remains true.

As these examples illustrate, the definition of the

These logs will make a nice campfire unless horseshoe operator matches the meaning of some they are wet. conditional statements in ordinary language very

The first statement suggests that the meat cannot be well. However, in general, the match is far from per-fect. The source of the mismatch stems from the fact white and at the same time not be properly cooked, that the horseshoe operator designates the material and the second suggests that the logs cannot be wet conditional, or truth-functional conditional. The and at the same time be used to make a nice camp-material conditional is a kind of conditional state-fire. Thus, if these statements are translated using the ment whose truth value depends purely on the truth wedge operator, part of the meaning will be left out. or falsity of the antecedent and consequent and not If this additional part is essential, it can be included on any inferential connection between antecedent by adding the symbolic equivalent of �but not both� and consequent. Since many conditional statements to the translation. in ordinary language express such an inferential In connection with the horseshoe operator, we connection, when the horseshoe operator is used to saw that a question arose when the antecedent of a translate them, part of their meaning is left out. For conditional statement turned out to be false. Why, example, compare the following two statements: under this circumstance, should the conditional statement be said to be true? For an example of If Shakespeare wrote Hamlet, then the sun some conditional statements that conform to the rises in the east. truth-functional interpretation, consider the If ice is lighter than water, then ice floats in following: water.

If the temperature rises above 32�F, then the The first statement expresses no inferential connec-snow will begin to melt. tion between antecedent and consequent, so using If Figure A is a triangle, then Figure A has the horseshoe operator to translate it results in no three sides. loss of meaning. However, the second statement does If all A are B and all B are C, then all A are C. express such a connection. The fact that ice is lighter

292Chapter6 Propositional Logic

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). than water is the reason why it floats. Accordingly, that if I were he, then I would be rich. Similarly, from when the horseshoe operator is used to translate the our knowledge that all fish are cold blooded, we con-second statement, this special meaning is lost. clude that if dolphins were fish, then they would be

The fact that the material conditional ignores cold blooded. On the other hand, we reason that the

inferential connections between antecedent and second two are false from our knowledge that lead consequent allows for conflicts between the truth- is heavier than air and our knowledge that suicide functional interpretation of a conditional statement results in death. Because the truth value of subjunc-and the ordinary interpretation. Consider, for exam- tive conditionals is so closely tied to inferences like ple, the following: these and is so unrelated to the truth or falsity of the

If Tammy Duckworth advocates the use of components, subjunctive conditionals are generally cocaine, then she is a good senator. not considered to be truth functional at all, and the

horseshoe operator is not used to translate them.

If Chicago is in Michigan, then Chicago is But if they were interpreted truth-functionally, note very close to Miami. that all four of these statements would turn out true,

According to their ordinary language interpretation, because they have false antecedents. These observations about conditional state-both of these statements are false. Good senators ments apply equally to biconditionals. Just as the do not advocate the use of cocaine, and Michigan horseshoe operator expresses material implication, is far from Miami. Yet, when these statements are the triple-bar operator expresses material equiva-interpreted as material conditionals, both turn out to lence. As such, it ignores any inferential connection be true, because their antecedents are false. In cases between its component statements, and, as a result, like these, when the truth- functional interpretation conflicts can arise between the ordinary meaning of a conditional statement conflicts with the ordinary of a biconditional and its truth-functional meaning. language interpretation, using the horseshoe operator Here are two examples of biconditionals expressed to translate it may not be appropriate. in the indicative mood that illustrate such a�conflict: While inferential relations between antecedent

and consequent often play some role in conditionals Mount Everest is easy to climb if and only if it

we just considered), they play a dominant role in The Department of Defense building is a conditional statements expressed in the subjunctive expressed in the indicative mood (such as those is twenty-five miles high. mood. Consider, for example, the following: hexagon if and only if it has eight sides.

According to the ordinary interpretation, these

If I were Mark Zuckerberg, then I would be rich. statements are false. Mountains that are twenty-five If dolphins were fish, then they would be miles high are not easy to climb, and hexagons, by cold blooded. definition, have six sides, not eight. Yet, when these If the Washington Monument were made of statements are interpreted as expressing material lead, then it would be lighter than air. biconditionals, both are true, because in each case If President Kennedy had taken his own life, the component statements are false. In cases like then he would be alive today. these, when the ordinary meaning of a bicondi-

tional conflicts with the truth-functional meaning,

Subjunctive conditionals are often called using the triple-bar operator to translate it may not

counterfactual conditionals because their antecedents be appropriate. Furthermore, as with subjunctive are typically false. As a result, the only way of conditionals, subjunctive biconditionals are gen-determining their truth value in ordinary language erally not considered to be truth functional at all, is through some kind of inference. Thus, from our so the triple-bar operator is not used to translate knowledge that Mark Zuckerberg is rich, we reason them.

Section 6.2 Truth Functions 293

Exercise 6.2

I. Identify the main operator in the following propositions.

★1. ∼ ∼(A �� M ) ● ∼ ∼(C ⊃ E )

2. (G ● ∼ ∼P ) ⊃ ∼ ∼(H �� ∼ ∼W )

3. ∼ ∼[P ● (S K K )]

★4. ∼ ∼(K ● ∼ ∼O) K ∼ ∼(R �� ∼ ∼B)

5. (M ● B) �� ∼ ∼[E K ∼ ∼(C �� I )]

6. ∼ ∼[(P ● ∼ ∼R) ⊃ (∼ ∼E �� F )]

★7. ∼ ∼[(S �� L) ● M] ⊃ (C �� N )

8. [∼ ∼F �� (N ● U )] K ∼ ∼H

9. E ● [(F ⊃ A) K (∼ ∼G �� H )]

★10. ∼ ∼[(X �� T ) ● (N �� F )] �� (K ⊃ L)

II. Write the following compound statements in symbolic form, then use your knowledge

of the historical events referred to by the simple statements to determine the truth value of the compound statements.

★1. It is not the case that Hitler ran the Third Reich.

2. Nixon resigned the presidency and Lincoln wrote the Gettysburg Address.

3. France bombed Pearl Harbor, or Lindbergh crossed the Atlantic.

★4. Hitler ran the Third Reich and Nixon did not resign the presidency.

5. Edison invented the telephone, or Custer was killed by Native Americans.

6. Julius Caesar conquered America if Napoleon ruled France.

★7. Washington was assassinated only if Edison invented the telephone.

8. Lincoln wrote the Gettysburg Address if and only if France bombed Pearl

Harbor.

9. It is not the case that either Julius Caesar conquered America or Washington

was assassinated.

★10. If Hitler ran the Third Reich, then either Custer was killed by Native Americans

or Einstein discovered aspirin.

11. Either Lindbergh crossed the Atlantic and Edison invented the telephone or

both Nixon resigned the presidency and it is false that Edison invented the telephone.

12. Lincoln�s having written the Gettysburg Address is a sufficient condition for

Julius Caesar�s having conquered America if and only if Washington�s being assassinated is a necessary condition for Custer�s having been killed by Native Americans.

294Chapter6 Propositional Logic

★13. Both Hitler ran the Third Reich and Lindbergh crossed the Atlantic if neither

Einstein discovered aspirin nor France bombed Pearl Harbor.

14. It is not the case that Custer was killed by Native Americans unless both Nixon

resigned the presidency and Edison invented the telephone.

15. Custer was killed by Native Americans, and Lincoln wrote the Gettysburg Address

only if either Washington was assassinated or Julius Caesar conquered America.

III. Determine the truth values of the following symbolized statements. Let A, B, and C be

true and X, Y, and Z be false. Circle your answer.

★1. A ● X

2. B ● ∼ ∼Y

3. X �� ∼ ∼Y

★ 4. ∼ ∼C �� Z

5. B ⊃ ∼ ∼Z

6. Y ⊃ ∼ ∼A

★ 7. ∼ ∼X ⊃ Z

8. B K Y

9. ∼ ∼C K Z

★10. ∼ ∼(A ● ∼ ∼Z )

11. ∼ ∼B �� (Y ⊃ A)

12. A ⊃ ∼ ∼(Z �� ∼ ∼Y )

★13. (A ● Y ) �� (∼ ∼Z ● C )

14. ∼ ∼(X �� ∼ ∼B ) ● (∼ ∼Y �� A)

15. (Y ⊃ C ) ● ∼ ∼(B ⊃ ∼ ∼X )

★16. (C K ∼ ∼A) �� (Y K Z )

17. ∼ ∼(A ● ∼ ∼C ) ⊃ (∼ ∼X ⊃ B)

18. ∼ ∼[(B �� ∼ ∼C ) ● ∼ ∼(X �� ∼ ∼Z )]

★19. ∼ ∼[∼ ∼(X ⊃ C ) K ∼ ∼(B ⊃ Z )]

20. (X ⊃ Z ) ⊃ [(B K ∼ ∼X ) ● ∼ ∼(C �� ∼ ∼A)]

21. [(∼ ∼X �� Z ) ⊃ (∼ ∼C �� B)] ● [(∼ ∼X ● A) ⊃ (∼ ∼Y ● Z )]

★22. ∼ ∼[(A K X ) �� (Z K Y )] �� [(∼ ∼Y ⊃ B) ● (Z ⊃ C )]

23. [(B ● ∼ ∼C ) �� (X ● ∼ ∼Y )] ⊃ ∼ ∼[(Y ● ∼ ∼X ) �� (A ● ∼ ∼Z )]

24. ∼ ∼{ ∼ ∼[(C �� ∼ ∼B) ● (Z �� ∼ ∼A)] ● ∼ ∼[∼ ∼(B �� Y ) ● (∼ ∼X �� Z )]}

★25. (Z ⊃ C ) ⊃ {[(∼ ∼X ⊃ B) ⊃ (C ⊃ Y )] K [(Z ⊃ X ) ⊃ (∼ ∼Y ⊃ Z )]}

Section 6.2 Truth Functions 295

IV. When possible, determine the truth values of the following symbolized statements. Let

A and B be true, and Y and Z be false. P and Q have unknown truth values. If the truth value of the statement cannot be determined, write �undetermined.�

★1. A �� P

2. Q �� Z

3. Q ● Y

★4. Q ● A

5. P ⊃ B

6. Z ⊃ Q

★7. A ⊃ P

8. P K ∼ ∼P

9. (P ⊃ A) ⊃ Z

★10. (P ⊃ A) K (Q ⊃ B)

11. (Q ⊃ B) ⊃ (A ⊃ Y )

12. ∼ ∼(P ⊃ Y ) �� (Z ⊃ Q)

★13. ∼ ∼(Q ● Y ) K ∼ ∼(Q �� A)

14. [(Z ⊃ P ) ⊃ P] ⊃ P

15. [Q ⊃ (A �� P)] K [(Q ⊃ B) ⊃ Y ]

V. Create three simple propositions that you know to be true, and three simple propo-

sitions that you know to be false. Then select letters to represent these six simple propositions, and combine these letters in various ways to produce five compound propositions. Finally, use the rules for the logical operators to determine the truth value of the five compound propositions (as in Part III of this exercise).

6.3 Truth Tables for Propositions

As we saw in Section 6.2, a truth table is an arrange- each line represents one possible arrangement of truth ment of truth values that shows, in every possible values, the total number of lines is equal to the number case, how the truth value of a compound proposition of possible combinations of truth values for the simple is determined by the truth value of its simple com- propositions. If there is only one simple proposition, p, ponents.* To construct a truth table, the first step is the number of lines in the truth table is two, because to determine the number of lines (or rows). Because there are only two possible truth values for p: true and

false. If there are two simple propositions, p and q, there

*An alternate technique, called Truth Trees, is presented in MindTap. are four lines, because there are four combinations

296Chapter6 Propositional Logic

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). of truth values: p true and q true, p true and q false, p Next, we take the number we obtained for A, namely false and q true and p false and q false. The relationship two, and divide that in half, which gives us one. So, between the number of different simple propositions beneath B, we write one true, one false, one true, and and the number of lines in the truth table is expressed one false:

as follows: AB (A �� ∼ ∼ B) ⊃ BNumber of different Number of lines T T simple propositions in truth table T F

1 2 F T 2 4 F F

3 8 This arrangement gives us all possible combinations

4 16 A of truth values for and B: A true, B true; A true, B 5 32 false; A false, B true; and A false, B false. 6 64 To complete the truth table, we use the columns

The relationship between these two columns of of truth values beneath the letters on the left to com-pute the column of truth values for the given propo-numbers is expressed by the formula, sition. Beginning inside the parentheses, we com-nL = 2 pute the truth value of ∼ ∼ B on each line, which is the

table, and n the number of different simple propositions. AB ( A �� ∼ ∼ B ) ⊃ B Let us now construct a truth table for a compound T T where negation of the truth value of B: L designates the number of lines in the truth proposition. We may begin with a fairly simple one: F

T F T

( A F T F �� ∼ ∼ B ) ⊃ B

The number of different simple propositions is two. F F T Thus, the number of lines in the truth table is four. Next, we look at the column beneath A, on the left, We begin by writing the simple propositions (A, B) and the column beneath the tilde, to produce the to the left of the compound proposition, separated column beneath the wedge:

we write the four lines for the truth table: AB ( A �� ∼ ∼ B) ⊃ B T T by a vertical line. Then, beneath this arrangement, T F

AB (A �� ∼ ∼ B) ⊃ B T F T T __________________ F T F F __________________ F F T T

__________________ Lastly, we look at the column beneath the wedge, __________________ and the column beneath B on the left, to produce

The next step is to enter truth values for the two let- the column beneath the horseshoe. Since the horse-ters on the left. For shoe is the main operator, we place a box around the A , we divide the total number of lines (four) in half. This gives us two, so we write two column beneath it to indicate that it represents the T�s and two F�s beneath A: entire compound proposition:

AB (AAB (A �� ∼ ∼ B) ⊃ B �� ∼ ∼ B) ⊃ B

T T T T F T T T F T T F F F T F F T F F F T T F

Section 6.3 Truth Tables for Propositions 297

The truth table tells us that the truth value of the CDE (C ● ∼ ∼ D) ⊃ E compound proposition is true when B is true and T T T F F false when B is false, regardless of the truth value T T F F F of A. T F T T T

Next, let us construct a truth table for a propo- T F F T T

sition containing three simple components: F T T F F

F T F F F

(C ● ∼ ∼ D) ⊃ E F F T F T

Since there are three simple components, there are F F F F T eight lines in the truth table. We begin by writing the Lastly, we look at the column beneath the dot and the three simple components (C, D, E) on the left, and column beneath E on the left to produce the column we separate them from the compound proposition beneath the horseshoe. We outline the column with a vertical line: beneath the horseshoe to indicate that it represents

the entire compound proposition:

CDE (C ● ∼ ∼ D) ⊃ E

________________________ CDE (C ● ∼ ∼ D) ⊃ E ________________________ T T T F F T ________________________ T T F F F T ________________________ T F T T T T ________________________ T F F T T F ________________________ F T T F F T ________________________ F T F F F T ________________________ F F T F T T

F F F F T T

First, we enter truth values for the three letters on

we divide that number in half and enter four T�s and compound proposition is false when C is true, D is false, and the left. Since there are eight lines in the truth table, The truth table shows that the truth value of the four F�s beneath E is false, and otherwise it is true. C . Then we enter two T�s, two F�s, two T�s, and two F�s beneath An alternate method for constructing truth D ; and one T, one F, and so forth, beneath tables can be shorter than the conventional method E :

just presented. Let us return to the compound

CDE (C ● ∼ ∼ D) ⊃ E proposition considered at the beginning of this T T T section. For this alternate method, we dispense T T F with the two columns of truth values on the left-T F T hand side, and we begin by entering truth values T F F directly beneath the A and B in the compound F T T proposition: F T F

F F T ( A �� ∼ ∼ B) ⊃ B F F F T T

Then we look at the column beneath T F D on the left

to compute the truth value for ∼ ∼D, and we use that F F T F

column together with the column beneath C on the left to produce the column beneath the dot: Then we enter the truth values beneath the tilde:

298Chapter6 Propositional Logic

(A �� ∼ ∼ B) ⊃ B the truth values of its components. By inspecting T F T the column of truth values under the main operator, T T F we can determine how the compound proposition F F T should be classified: F T F

Next, we use the column of truth values beneath Amain operator Column under Statement classification

and beneath the tilde to produce the column beneath all true tautologous the wedge: (logically true)

(A all false self-contradictory �� ∼ ∼ B) ⊃ B (logically false) T T F T at least one true, contingent T T T F

F F F T at least one false F T T F

As the truth table we developed earlier indi-

Lastly, we use the column beneath the wedge and the cates, (C ● ∼ ∼D) ⊃ E is a contingent propo sition. The column beneath B inside the parentheses to produce column under the main operator contains at least the column beneath the horseshoe. We place a box one T and at least one F. In other words, the truth around this final column to indicate that it repre- value of the compound proposition is �contingent� sents the entire compound proposition: on the truth values of its components. Sometimes

it is true, sometimes false, depending on the truth

(A �� ∼ ∼ B) ⊃ B values of the components. T T F T T On the other hand, consider the following truth T T T F F tables: F F F T T

F T T F F [ ( G ⊃ H ) ● G ] ⊃ H

T T T T T T T

Again, the truth table shows that the truth value of T F F F T T F the compound proposition is true when B is true F T T F F T T and false when B is false, without regard to the truth F T F F F T F

produce truth tables later in this chapter. ( G �� H ) K ( ∼ ∼ G ● ∼ ∼ H ) T T T F F T F F T value of A. This alternate method will be used to

T T F F F T F T F

Classifying Statements F T T F T F F F T

Truth tables may be used to determine whether F F F F T F T T F the truth value of a compound statement depends The proposition on the top is tautologous (logically solely on its form or whether it also depends on true or a tautology) because the column under the the specific truth values of its components. A com- main operator is all true. The one on the bottom is pound statement is said to be a logically true or self- contradictory (logically false) because the main tautologous statement if it is true regardless of operator column is all false. In neither case is the the truth values of its components. It is said to be truth value of the compound proposition contingent a logically false or self-contradictory state- on the truth values of the components. The one on ment if it is false regardless of the truth values of the top is true regardless of the truth values of its its components. And it is said to be a contingent components�in other words, necessarily true. The statement if its truth value varies depending on one on the bottom is necessarily false.

Section 6.3 Truth Tables for Propositions 299

If a proposition is either logically true or logi- For example, the following two propositions are

cally false, its truth value depends merely on its logically equivalent. The main operator columns form and has nothing to do with its content. As a of their respective truth tables are identical. Note result, such statements do not make any genuine that for proper comparison the columns under K assertions about things in the world. For example, must be identical and the columns under L must

is not raining� provides no information about the K ⊃ L ∼ ∼ L ⊃ ∼ ∼ K weather. Similarly, the self-contradictory statement the tautologous statement �It is either raining or it be identical. �It is raining and it is not raining� provides no T T T F T T F T information about the weather. On the other hand, T F F T F F F T Logically the contingent statement �It is raining in the moun- F T T F T T T F equivalent

F T F T F T T F

tains� does provide information about the weather.

For any two propositions that are logically

Comparing Statements equivalent, the biconditional statement formed by

joining them with a triple bar is tautologous. Thus,

Truth tables may also be used to determine how (K ⊃ L) K (∼ ∼L ⊃ ∼ ∼K ) is tautologous. This is easy to two propositions are related to each other. Two see because the columns under the main operators propositions are said to be logically equivalent of K ⊃ L and ∼ ∼L ⊃ ∼ ∼K are identical.

each line under their main operators, and they are K ⊃ LK ● ∼ ∼ Lcontradictory statements if they have opposite statements if they have the same truth value on The next two propositions are contradictory: truth values on each line under their main operators. T T T T F F T If neither of these relations hold, the propositions T F F T T T F Contradictory are either consistent or inconsistent. Two (or more) F T T F F F T

F T F F F T F

turn out to be true, and they are first line of each truth table the column under inconsistent the main operator turns out true. This means that it statements if there is no line on which both (or is possible for both propositions to be true, which is all) of them turn out to be true. By comparing the is at least one line on which both (or all) of them The next two propositions are consistent. On the propositions are consistent statements if there main operator columns, one can determine which is the meaning of consistency: the case. However, because the first two relations are K �� LK ● L stronger than (and may overlap) the second two, the T T T T T T first two relations should be considered first. T T F T F F Consistent

Columns under F F F F F F main operators F T T F F T same truth value Finally, the next two propositions are inconsistent. logically Relation

on each line There is no line in the columns under the main equivalent

opposite truth value operators where the truth values are both true: contradictory

on each line K K LK ● ∼ ∼ L there is at least one T T T T F F T consistent

line on which the truth T F F T T T F Inconsistent

values are both true F F T F F F T

there is no line on F T F F F T F inconsistent

which the truth Any pair of propositions is either consistent

values are both true or inconsistent. Furthermore, some consistent

300Chapter6 Propositional Logic

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). propositions are also logically equivalent, and some such a position are consistent, then there is at least a inconsistent propositions are either contradictory or possibility that the position makes sense. This is so logically equivalent. Because of this partial overlap, because there will be at least one line in the group of pairs of propositions are usually first classified in truth tables where all of the person�s statements are terms of the stronger of these relations, which are true. On the other hand, if the statements are incon-logical equivalence and contradiction. If neither sistent, then there is no possibility that the position of these stronger relations applies, then the pair makes sense. In this case, there is no line in the truth of propositions is classified in terms of the weaker tables where all of the statements are true. Conjoining relations, consistency and inconsistency. the statements results in a self-contradiction.

Unlike logical equivalence and contradiction, The truth tables for consistency and logical

which usually relate exactly two propositions, consis- equivalence also illustrate the important difference tency and inconsistency often apply to larger groups between two propositions being factually true and of propositions. For consistency, the only require- their being logically equivalent. For example, the ment is that there be at least one line in the group of statements �Water boils at 100�C� and �The cur-truth tables where all of the propositions are true, and rent population of the United States is over 300 mil-for inconsistency the only requirement is that there lion� are both true in the present actual world. This be no such line. As a result of these requirements, real-world situation conforms to the one truth-table the statement consisting of the conjunction of a line on which both statements are true. As a result group of inconsistent propositions will always be self- of this line, the two statements are consistent. How-contradictory, whereas the statement consisting of the ever, they are not logically equivalent, because their conjunction of a group of consistent propositions will truth values are not necessarily the same. The truth never be self-contradictory. value of the second proposition might change in

Consistency and inconsistency are important the future, while that of the first would remain the

because, among other things, they can be used to same. An analogous distinction, incidentally, holds evaluate the overall rationality of a person�s stated between two statements having actually opposite position on something. If the statements expressing truth values and their being contradictory.

Exercise 6.3

I. Use truth tables to determine whether the following symbolized statements are tautolo-

gous, self-contradictory, or contingent.

★1. N ⊃ (N ⊃ N )

2. (G ⊃ G) ⊃ G

3. (S ⊃ R) ● (S ● ∼ ∼R)

★4. [(E ⊃ F ) ⊃ F ] ⊃ E

5. (∼ ∼K ⊃ H ) K ∼ ∼(H �� K )

6. (M ⊃ P ) �� (P ⊃ M )

★7. [(Z ⊃ X ) ● (X �� Z )] ⊃ X

Section 6.3 Truth Tables for Propositions 301

8. [(C ⊃ D) ● ∼ ∼C ] ⊃ ∼ ∼D

9. [X ⊃ (R ⊃ F )] K [(X ⊃ R) ⊃ F ]

★10. [G ⊃ (N ⊃ ∼ ∼G )] ● [(N K G) ● (N �� G)]

11. [(Q ⊃ P ) ● (∼ ∼Q ⊃ R)] ● ∼ ∼(P �� R)

12. [(H ⊃ N ) ● (T ⊃ N )] ⊃ [(H �� T ) ⊃ N ]

★13. [U ● (T �� S )] K [(∼ ∼T �� ∼ ∼U ) ● (∼ ∼S �� ∼ ∼U )]

14. {[(G ● N ) ⊃ H ] ● [(G ⊃ H ) ⊃ P ]} ⊃ (N ⊃ P )

15. [(F �� E ) ● (G �� H )] K [(G ● E ) �� (F ● H )]

II. Use truth tables to determine whether the following pairs of symbolized statements are

logically equivalent, contradictory, consistent, or inconsistent. First, determine whether the pairs of propositions are logically equivalent or contradictory; then, if these rela-tions do not apply, determine if they are consistent or inconsistent.

★1. ∼ ∼D �� B ∼ ∼(D ● ∼ ∼B )

2. F ● M ∼ ∼(F �� M )

3. ∼ ∼K ⊃ LK ⊃ ∼ ∼L

★4. R �� ∼ ∼SS ● ∼ ∼R

5. ∼ ∼A K X (X ● ∼ ∼A) �� (A ● ∼ ∼X )

6. H K ∼ ∼G (G ● H ) �� (∼ ∼G ● ∼ ∼H )

★7. (E ⊃ C ) ⊃ LE ⊃ (C ⊃ L)

8. N ● (A �� ∼ ∼E ) ∼ ∼A ● (E �� ∼ ∼N )

9. M ⊃ (K ⊃ P ) (K ● M ) ⊃ P

★10. W K (B ● T ) W ● (T ⊃ ∼ ∼B)

11. G ● (E �� P ) ∼ ∼(G ● E ) ● ∼ ∼(G ● P )

12. R ● (Q �� S ) (S �� R) ● (Q �� R)

★13. H ● (K �� J ) ( J ● H ) �� ( H ● K )

14. Z ● (C K P ) C K (Z ● ∼ ∼P )

15. Q ⊃ ∼ ∼(K �� F ) (K ● Q) �� (F ● Q)

III. Use truth tables to obtain the answers to the following exercises. The letters to be used

in constructing the truth tables are given in parentheses.

★1. Renowned economist Harold Carlson makes the following prediction:

�The balance of payments will decrease if and only if interest rates remain steady; however, it is not the case that either interest rates will not remain steady or the balance of payments will decrease.� What can we say about Carlson�s prediction? (B, I)

302Chapter6 Propositional Logic

2. A high school principal made this statement to the school board: �Either music

is not dropped from the curriculum or the students will become cultural philis-tines; furthermore, the students will not become cultural philistines if and only if music is dropped from the curriculum.� Assuming the principal is correct, what has she told us about music and the students? (Hint: Construct a truth table for the principal�s statement and examine the line on which the statement turns out true.) (M, C)

3. Christina and Alejandro are having a discussion about their plans for the eve-

ning. Christina: �If you don�t love me, then I�m certainly not going to have sex with you.� Alejandro: �Well, that means that if I do love you, then you will have sex with me, right?� Is Alejandro correct? (Hint: Construct a truth table for each statement and compare them.) (L, S)

★ 4. Two astronomers are discussing supernovas. Dr. Frank says, �Research has

established that if a supernova occurs within ten light-years of the earth, then life on Earth will be destroyed.� Dr. Harris says, �Research has also established that either a supernova will not occur within ten light-years of the earth or life on Earth will not be destroyed.� Is it possible that both astronomers are correct? If so, what can we determine about the occurrence of a supernova? (S, L)

5. Antonia Martinez, who is running for the state senate, makes this statement:

�Either a tax reduction is feasible only if both educational costs do not increase and the welfare program is abolished, or a tax reduction is feasible and either the welfare program will not be abolished or educational costs will increase.� What has Martinez told us about taxes, educational costs, and welfare? (T, E, W)

6. Automotive expert Frank Goodbody has this to say about Japanese imports:

�If Mitsubishi is the sportiest, then both Toyota is the most trouble-free and Nissan is not the lowest priced. If Nissan is the lowest priced, then both Toyota is not the most trouble-free and Mitsubishi is the sportiest.� Is it possible that Goodbody is correct in his assessment? If so, what may we conclude about Mitsubishi, Toyota, and Nissan? (M, T, N)

★7. Two stockbrokers are having a discussion. One claims that Netmark will intro-

duce a new product if and only if both Datapro cuts its workforce and Compu-cel expands production. The other claims that Datapro will cut its workforce, and Compucel will expand production if and only if Netmark introduces a new product. Is it possible that both stockbrokers are right? If so, what have they told us about these companies? (N, D, C)

8. Eric Carson sums up his beliefs about God as follows: �God exists if and only

if either life is meaningful or the soul is not immortal. God exists and the soul is immortal. If God exists, then life is not meaningful.� Is it possible that Eric�s beliefs make sense? (G, L, S)

9. Camila, Serenity, and Amanda witnessed a bank robbery. At trial, Camila testi-

fied that Lefty did not enter the bank, and if Howard pulled a gun, then Conrad collected the money. Serenity testified that if Howard did not pull a gun, then Lefty entered the bank. Amanda testified that if Conrad collected the money,

Section 6.3 Truth Tables for Propositions 303

then Howard pulled a gun. Is it possible that all three witnesses told the truth? If so, what can we conclude about Lefty, Howard, and Conrad? (L, H, C)

★10. Nicole Ramirez expresses her philosophy as follows: �If the mind is identical to

the brain, then personal freedom does not exist and humans are not responsible for their actions. If personal freedom does not exist, then the mind is identical to the brain. Either humans are responsible for their actions or the mind is not identical to the brain. If personal freedom exists, then humans are responsible for their actions.� Is it possible that Nicole�s philosophy makes sense? If so, what does it say about the mind, personal freedom, and responsibility? (M, P, H)

IV. Did you work Exercise 6.1, IV? If not, work it now. Then take the symbolized com-

pound statements you created for this exercise and construct a truth table for each. Finally, categorize each statement as tautologous, self-contradictory, or contingent.

6.4 Truth Tables for Arguments

Truth tables provide the standard technique for test- Juvenile killers are not as responsible for ing the validity of arguments in propositional logic. their crimes as adults are. To construct a truth table for an argument, follow Therefore, execution is not a justifiable these steps: punishment.

1. Symbolize the arguments using letters to repre- The first step is to symbolize the argument: sent the simple propositions. J ⊃ E2. Write out the symbolized argument, plac-∼ ∼ J ing a single slash between the premises and a ∼ ∼ E double slash between the last premise and the

conclusion. Now a truth table may be constructed. Since the

3. Draw a truth table for the symbolized argument symbolized argument contains two different letters,

as if it were a proposition broken into parts, out- the truth table has four lines. Make sure that iden-lining the columns representing the premises tical letters have identical columns beneath them. and conclusion. Here are the columns for the individual letters:

4. Look for a line in which all of the premises are J ⊃ E / ∼ ∼ J / / ∼ ∼ E true and the conclusion is false. If such a line T T T T exists, the argument is invalid; if not, it is valid.

T F T F

For example, let us test the following argument F T F T

for validity: F F F F

If juvenile killers are as responsible for their The truth table is now completed, and the col-crimes as adults are, then execution is a umns representing the premises and conclusion are justifiable punishment. outlined:

304Chapter6 Propositional Logic

J ⊃ E / ∼ ∼ J / / ∼ ∼ E Truth tables provide a convenient illustration of the T T T F T F T fact that any argument having inconsistent premises T F F F T T F is valid regardless of what its conclusion may be, and F T T T F F T any argument having a tautologous conclusion is valid F T F T F T F regardless of what its premises may be. Example:

Inspection of the third line reveals that both of the The sky is blue. premises are true and the conclusion is false. The The sky is not blue. argument is therefore invalid. Therefore, Paris is the capital of France.

Another example: S / ∼ ∼ S / / P

If insider trading occurs, then investors will not T F T T trust the securities markets. If investors do not T F T F trust the securities markets, then business in F T F T general will suffer. Therefore, if insider trading F T F F

occurs, then business in general will suffer. Since the premises of this argument are inconsistent,

The completed truth table is this: there is no line on which the premises are both true.

T F F T F T T T O Accordingly, there is no line on which the premises ⊃ ∼ ∼ T / ∼ ∼ T ⊃ B / / O ⊃ B are both true and the conclusion false, so the argu- T T T ment is valid. Of course, the argument is unsound, T F F T F T T F T F F because it has a false premise. Another example: T T T F T F T T T T T T T T F T F F F T F F Bern is the capital of Switzerland. Therefore, F T F T F T T T F T T it is either raining or it is not raining.

Inspection of the truth table reveals that there is no F T F T F T T F F T F B / / R �� ∼ ∼ R F T T F T F T T F T T T T T F T F T T F T F F F F T F T F T T F F T T F T line on which both premises are true and the conclu-F F T T F

sion is false. The argument is therefore valid. The conclusion of this argument is a tautology. Accord-The logic behind the method of truth tables is easy ingly, there is no line on which the premise is true and to understand. By definition, a valid argument is one in the conclusion false, and so the argument is valid. Inci-which it is not possible for the premises to be true and dentally, it is also sound, because the premise is true. the conclusion false. A truth table presents every pos-The conditional statement having the conjunc-sible combination of truth values that the components tion of an argument�s premises as its antecedent and of an argument may have. Therefore, if no line exists the conclusion as its consequent is called the argu-on which the premises are true and the conclusion ment�s corresponding conditional . For example, false, then it is not possible for the premises to be true the corresponding conditional of the second argu-and the conclusion false, in which case the argument is ment tested in this section is [( O ⊃ ∼ ∼ T ) ● ( ∼ ∼ T ⊃ valid. Conversely, if there is a line on which the prem-B )] ⊃ ( O ⊃ B ). For any valid argument (such as this ises are true and the conclusion false, then it is possible one), the corresponding conditional is a tautology. for the premises to be true and the conclusion false, and This is easy to see. In any valid argument, there is the argument is invalid. Accordingly, to test the validity no line on which the premises are all true and the of an argument, use this procedure: conclusion false. Thus, in the corresponding con-Look for a line that has all true premises and a ditional, there is no line on which the antecedent is false conclusion. If you find such a line, the ar-true and the consequent false, so the corresponding gument is invalid. If not, the argument is valid. conditional is true on every line.

Section 6.4 Truth Tables for Arguments 305

Prominent Ada Byron, Countess of Lovelace

Logicians1815�1852

A da Byron, born in 1815, was the only child of the English poet George Gordon (Lord Byron) and Annabella Milbanke. By the time Ada was born, Lady Byron had grown to detest her husband, and she did everything in her power to ensure that

Ada would grow up to be as much unlike him as pos-

sible. Lady Byron had a talent for mathematics, which

her husband did not share in the least, so she hired a

series of tutors to instruct young Ada in that discipline.

One of those tutors was the famous mathematician

and logician Augustus De Morgan, with whom Ada

became a close friend.

In 1835, Ada Byron married William King, and when

King was elevated to Earl of Lovelace three years later, UK Government Art Collection website/Wikimedia Commons Ada became Countess of Lovelace. The couple had

three children, but the duties of wife and mother did envisioned how it could be used not only to solve problems not interrupt Ada�s study of mathematics and logic. in mathematics, but to produce music and graphics as Two years prior to her marriage, Ada met Charles well. Because of this work , Ada Byron is usually credited Babbage, an early inventor of mechanical computers, with being the first computer programmer. and when she was first shown Babbage�s Difference Ada had suffered problems with her health ever Engine, she immediately grasped all the intricacies since childhood, and as she grew older, these prob-of its operation. A few years later, when Babbage pro - lems were aggravated by alcohol abuse. In 1852, posed a design for the Analytic Engine, Ada wrote a she died from cancer at the relatively young age of lengthy program for the new machine, and she thirty-six.

Exercise 6.4

I. Translate the following arguments into symbolic form. Then determine whether each

is valid or invalid by constructing a truth table for each. The letters to be used in con-structing the truth tables are given in parentheses.

★1. If national elections deteriorate into TV popularity contests, then smooth-

talking morons will get elected. Therefore, if national elections do not dete-riorate into TV popularity contests, then smooth-talking morons will not get elected. (N, S)

2. Brazil has a huge foreign debt. Therefore, either Brazil or Argentina has a huge

foreign debt. (B, A)

306Chapter6 Propositional Logic

3. If fossil-fuel combustion continues at its present rate, then the greenhouse

effect will worsen. If the greenhouse effect worsens, then world temperatures will rise. Therefore, if fossil-fuel combustion continues at its present rate, then world temperatures will rise. (F, G, W)

★ 4. If there are dried-up riverbeds on Mars, then water once flowed on the Martian

surface. There are dried-up riverbeds on Mars. Therefore, water once flowed on the Martian surface. (D, W)

5. If high school graduates are deficient in reading, they will not be able to com-

pete in the modern world. If high school graduates are deficient in writing, they will not be able to compete in the modern world. Therefore, if high school graduates are deficient in reading, then they are deficient in writing. (R, C, W)

6. The disparity between rich and poor is increasing. Therefore, political con-

trol over economic equality will be achieved only if restructuring the eco-nomic system along socialist lines implies that political control over economic equality will be achieved. (D, P, R)

★7. Einstein won the Nobel Prize either for explaining the photoelectric effect or

for the special theory of relativity. But he did win the Nobel Prize for explaining the photoelectric effect. Therefore, Einstein did not win the Nobel Prize for the special theory of relativity. (P, S)

8. If microchips are made from diamond wafers, then computers will gener-

ate less heat. Computers will not generate less heat and microchips will be made from diamond wafers. Therefore, synthetic diamonds will be used for jewelry. (M, C, S)

9. Either the USS Arizona or the USS Missouri was not sunk in the attack on Pearl

Harbor. Therefore, it is not the case that either the USS Arizona or the USS Missouri was sunk in the attack on Pearl Harbor. (A, M)

★10. If racial quotas are adopted for promoting employees, then qualified employ-

ees will be passed over; but if racial quotas are not adopted, then prior discrim-ination will go unaddressed. Either racial quotas will or will not be adopted for promoting employees. Therefore, either qualified employees will be passed over or prior discrimination will go unaddressed. (R, Q, P)

II. Determine whether the following symbolized arguments are valid or invalid by con-

structing a truth table for each.

★ 1. K ⊃ ∼ ∼K5. K K ∼ ∼L

∼ ∼K ∼ ∼(L ● ∼ ∼K)

2. K ⊃ L R ⊃ R

R6. Z

3. E ⊃ (Z ⊃ E) P K ∼ ∼ N

N �� P ★7. ∼ ∼(W ● ∼ ∼X)

★ ∼ ∼(X4. ∼ ∼ ( G ● M ) ● ∼ ∼W )

M �� ∼ ∼G X �� W ∼ ∼G

Section 6.4 Truth Tables for Arguments 307

8. C K D ★19. A ⊃ B

E �� ∼ ∼D (A ● B) ⊃ C E ⊃ CA ⊃ (C ⊃ D)

A ⊃ D

9. A K (B �� C )

∼ ∼C �� B 20. ∼ ∼A �� R

★ R ⊃ C10. J ⊃ ( K ⊃ L ) C ⊃ ∼ ∼N K A ⊃ B ∼ ∼(N ● ∼ ∼C )

( ⊃ ( J ⊃ L) A �� C J �� K ) ⊃ L

★11. 21. (G �� S) ⊃ S ∼ ∼ ( K K S ) S K ( G �� S ) S ⊃ ∼ ∼ ( R �� K ) ( G R ● ∼ ∼ S ) ⊃ ∼ ∼G �� ∼ ∼ S ( S ● ∼ ∼ G ) ⊃ ∼ ∼ S12. E ⊃ ( F ● G ) ∼ ∼ G �� ∼ ∼ S F ⊃ ( G ⊃ H ) G ● S E ⊃ H22. ★ ( Q ● E ) ⊃ ( F �� H) 13. A ⊃ ( N �� Q ) F ⊃ H ∼ ∼ ( N �� ∼ ∼ A ) H ⊃ E A ⊃ Q Q ● ∼ ∼ F

14. G ⊃ HE �� F

15. RT ∼ ∼K K H ( N �� K ) ⊃ T L ⊃ M ( K ● T ) ⊃ N M ∼ 23. (N ∼ H ● ∼ ∼T) K K �� G ⊃ R K G

⊃ NT �� ∼ ∼N

N ⊃ LN ⊃ K

L �� N24. C ⊃ (P �� E)

★ P ⊃ ∼ ∼C16. S ⊃ T

SE ⊃ (C ⊃ ∼ ∼ T● P) ∼ ∼ P ⊃ E T ⊃ S

S (C �� E) ⊃ P �� ∼ ∼ T

17. C �� ∼ ∼E W ⊃ X

X ⊃ W ★25. D K (R ⊃ J) X ⊃ YJ ⊃ ∼ ∼D Y ⊃ XD �� ∼ ∼R W K YR ⊃ (D ● J)

18. K K (L �� (J �� R) ⊃ D D M ) ∼ ∼ ( R �� J ) L ⊃ M M ⊃ K K �� L K ⊃ L

308Chapter6 Propositional Logic

III. The following dialogue contains eleven arguments. Translate each into symbolic form,

and then use truth tables to determine whether each is valid or invalid.

Android Rights

�I just came from Professor Shaw�s class on the philosophy of human nature,� Nick says to his friend Erin, as he meets her in the hallway. �We discussed the question of whether an android could be a person and whether they have rights.�

�Sounds like an interesting class,� Erin replies, �but an android could never be a

person.�

�Why is that?� Nick asks.

�It�s really quite simple,� she says. �If an android is a person, then it�s rational. But

no android is rational, so it�s not a person.�

�But wait,� Nick says. �Androids can solve problems, and they can also deliberate.

And if they can either deliberate or solve problems, then they�re rational. Right? So androids are rational, after all.�

�No they�re not,� Erin says. �If an android is rational, then it�s conscious, and if it�s

conscious, then it has reflective mental activity�it can reflect on its own act of think-ing. But no android can do that, so it�s not rational.�

�How do you know that no android has reflective mental activity?� he asks. �Because an android has reflective mental activity only if it has a soul,� Erin says. �And

it�s ridiculous to think that an android could have a soul. Hence it has no reflective mental activity.�

�But consider this,� Nick says. �Either a soul is a material entity or it�s a nonmaterial

entity. You would agree with that, wouldn�t you?�

�Of course,� Erin replies.

�Okay,� says Nick. �Now let me finish the argument. If a soul is a material entity,

then if an android is material, it could easily have a soul. But if a soul is a nonmaterial entity, then if God could infuse a soul into it, then it could have a soul. Now an android is material and God could infuse a soul into an android�after all, God can do anything. Thus, an android could have a soul.�

�Well, I know that Descartes considered humans to be machines with souls, but to

me it�s crazy to think that God would infuse a soul into a computer. He might as well infuse a soul into a pile of rocks. Anyway, let me try another approach,� Erin says. �If an android is a person, then it has free will. But if androids are programmed, then they have no free will. Androids are just computers made to appear like humans, and every computer is programmed. Hence, once again, an android is not a person. What do you think of that?�

�By your reasoning,� Nick replies, �even humans may not be free.�

�How is that?� Erin asks.

�Well,� he says, �whatever we do is caused by our biological makeup or by our

social conditioning. If it�s caused by our biological makeup, then it�s determined. If it�s caused by our social conditioning, then it�s determined, too. And if it�s determined, then it�s not free. Thus, whatever we do is not free.�

�Not so,� Erin objects. �Our actions may be influenced by our biological makeup

and our social conditioning, but they�re not caused by them. Not strictly. And if they�re not strictly caused by them, they�re not determined by them, and if they�re not deter-mined by them, then they�re free. Thus, our actions are free.�

Section 6.4 Truth Tables for Arguments 309

�Well, I don�t know what it means for our actions to be influenced by something yet

not be determined,� Nick replies. �If X is influenced by Y, then X is caused by Y, and if X is caused by Y, then X is determined by Y. Thus, if X is influenced by Y, then X is determined by Y.�

�I think you�re equivocating on the meaning of cause,� Erin replies. �But if you�re

unconvinced, how about this: If an android is a person, then it has feelings. And if it has feelings, then it can feel love or compassion. But no android loves anything. Just imagine two computers in love. The very thought is absurd. So is one android feeling compassion for another. Thus, an android cannot be a person.�

�Well, look at it this way,� Nick replies. �Feelings are either mental or they�re physi-

cal. If they�re mental, then they�re brain states, and if they�re brain states then androids could have them�because all brain states are just arrangements of atoms. If feelings are physical, then androids could have them�because, once again, all physical things are just arrangements of atoms. Thus, androids can have feelings.�

Erin laughs. �That�s the worst reasoning I�ve ever heard in my life. Sure, feelings

may be accompanied by physical states, but they�re not identical with them. Anyway, before I have to head off to class, tell me this: Do you really think that androids could be persons?�

�I think it�s possible,� Nick replies.

�So, would you say that an android has rights? Say the right to vote?�

�That depends,� he says.

�On what?� Erin asks.

�On whether it�s a Democrat or Republican,� he replies.

�Uh-huh.� She sighs. �Well, I�m off to class. Bye.�

IV. Construct five symbolized arguments having at least two premises each. Then deter-

mine whether each is valid or invalid by constructing a truth table.

6.5 Indirect Truth Tables

Indirect truth tables provide a shorter and faster Indirect truth tables can also be used to test a method for testing the validity of arguments series of statements for consistency. In Section 6.3 than do ordinary truth tables. This method is we showed how ordinary truth tables are used to especially applicable to arguments that contain test pairs of statements for consistency and we a large number of different simple propositions. noted that consistency was a relation that applied For example, an argument containing five differ- to any group of propositions. In this section, we ent simple propositions would require an ordi- use indirect truth tables to test groups of three, nary truth table having thirty-two lines. The four, five, and six propositions for consistency. indirect truth table for such an argument, on the Given the abbreviated nature of indirect truth other hand, would usually require only a single tables, this evaluation can usually be done much line and could be constructed in a fraction of the more quickly than it can with ordinary truth time required for the ordinary truth table. tables.

310Chapter6 Propositional Logic

Preliminary Skills A ⊃ B

Using indirect truth tables requires developing the skill F T F to work backward from the truth value of the main Similarly, suppose you are given a disjunctive state-operator of a compound statement to the truth values ment with these truth values:

of its simple components. Suppose, for example, that A �� B you are given a conjunctive statement that is true: F T

A ● B Then you know immediately that B is true, because

T if a disjunctive statement is true, at least one of the

one way, you know immediately that both A and BA �� B are true: Because a conjunctive statement can be true in only disjuncts must be true:

F T T

A ● B Computing the truth values for the simple com-T T T ponents of a compound proposition, as we have

Suppose, on the other hand, that you are given a just done, requires a thorough knowledge of the

conditional statement that is false: truth-table definitions of the five operators given A in Section 6.2. But after a little practice with ⊃ B examples such as these, this skill becomes almost

F automatic. Since a conditional statement can be false in only

one way, you know that A is true and B is false: Testing Arguments for Validity

A ⊃ B To construct an indirect truth table for an argu-T F F ment, we begin by assuming that the argument is

But suppose you are given a conditional statement invalid. That is, we assume that it is possible for the premises to be true and the conclusion false. Truth that is true: values corresponding to true premises and false

A ⊃ B conclusion are entered beneath the main operators

T for the premises and conclusion. Then, working

Since a conditional statement can be true in three backward, the truth values of the separate compo-nents are derived. If no contradiction is obtained ways, you cannot compute the truth values of A and in the process, this means that it is indeed possible B . It could be the case that A is true and B is true, or for the premises to be true and the conclusion false, A is false and B is true, or A is false and B is false. But as originally assumed, so the argument is therefore let us suppose, in relation to this example, that you invalid. If, however, the attempt to make the prem-have one more piece of information. Suppose you ises true and the conclusion false necessarily leads know that B is false: to a contradiction, it is not possible for the prem-

A ⊃ B ises to be true and the conclusion false, in which

T F case the argument is valid. Consider the following

Then you know immediately that symbolized argument: A is false. If A were true, then the truth value under the horseshoe ∼ ∼A ⊃ (B �� C ) would have to be false. But since this truth value is ∼ ∼B given as true, A must be false: C ⊃ A

Section 6.5 Indirect Truth Tables 311

We begin as before by writing the symbolized argu- A contradiction now appears in the truth values ment on a single line, placing a single slash between assigned to the first premise, since T ⊃ F is F. the premises and a double slash between the last The inconsistent truth values are circled. Because premise and the conclusion. Then we assign T to the every step was strictly necessitated by some prior premises and F to the conclusion: step, we have shown that it is impossible for the

∼ premises to be true and the conclusion false. The ∼ A ⊃ (B �� C ) / ∼ ∼ B / / C ⊃ A argument is therefore valid. T T F Sometimes a single row of truth values is not suf-

We can now derive the truth values of ficient to prove an argument valid. Example: B , C , and A,

as follows: ∼ ∼ A ⊃ B / B ⊃ A / A ⊃ ∼ ∼ B / / A ● ∼ ∼ B

∼ T T T F ∼ A ⊃ (B �� C ) / ∼ ∼ B / / C ⊃ A

T T F T F F Since a conditional statement can be true in any

These truth values are now transferred to the first one of three ways, and a conjunctive statement can

premise: be false in any one of three ways, merely assigning ∼ truth to the premises and falsity to the conclu-∼ A ⊃ ( B �� C ) / ∼ ∼ B / / C ⊃ A sion of this argument is not sufficient to obtain

T F T F T T T F T F F the truth values of any of the component state-

We thus have a perfectly consistent assignment ments. When faced with a situation such as this, of truth values, which makes the premises true the best way of proceeding is either to list all of and the conclusion false. The argument is there- the possible ways that one of the premises can be fore invalid. If an ordinary truth table were con- true or all of the possible ways that the conclusion structed for this argument, it would be seen that can be false, and to proceed from there. For this the argument fails on the line on which argument, let us select the conclusion. Since A is false, B the conclusion is a false conjunctive statement, is false, and C is true. This is the exact arrange-ment found in the indirect truth table we just the first conjunct can be true and the other finished. false, the first can be false and the other true,

Here is another example. As always, we begin by or they can both be false. Thus, we obtain the

assigning T to the premises and F to the conclusion: following:

A ⊃ (B �� C ) / B ⊃ D / A / / ∼ ∼ C ⊃ D ∼ ∼ A ⊃ B / B ⊃ A / A ⊃ ∼ ∼ B / / A ● ∼ ∼ B T T T F T T T T F F T

From the conclusion we can now derive the truth T T T T F F T F T T F F F T

values of C and D, which are then transferred to the first two premises: Extending the truth values of A and B to the prem-

A ises, we obtain the following result: ⊃ (B �� C ) / B ⊃ D / A / / ∼ ∼ C ⊃ D

T F T F T T F F F ∼ ∼ A ⊃ B / B ⊃ A / A ⊃ ∼ ∼ B / / A ● ∼ ∼ B

The truth value of T T TTF T T F F T B is now derived from the second TFTF T T F F T F premise and transferred, together with the truth T TTF T F F F T value of A , to the first premise:

A ⊃ (B �� C ) / B ⊃ D / A / / ∼ ∼ C ⊃ D Since each line necessarily leads to a contradiction, TTFF F F T F T T F F F the argument is valid. If a contradiction had been

312Chapter6 Propositional Logic

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). avoided on some line, the argument would, of are assigned haphazardly or by guessing, then course, be invalid, because it would be possible for nothing has been proved. The objective is not the premises to be true and the conclusion false. to produce a contradiction but to avoid one (if Note that in this argument it is not necessary possible). to fill out all the truth values on any one line to For example, in the following indirect truth table

contradiction is necessarily derived within the A ⊃ B / C ⊃ B / / A ⊃ C context of a single premise. be forced into a contradiction. On each line the a contradiction is apparent in the first premise:

If an indirect truth table requires more than TTF F T F T F F

one line, the method to be followed is this: Either Yet the argument is invalid. The contradiction select one of the premises and compute all of that appears is not required by the assignment of the ways it can be made true, or select the con- truth to the premises and falsity to the conclusion. clusion and compute all of the ways it can be The following indirect truth table, which is done made false. This selection should be dictated by correctly, proves the argument invalid:

conclusion can be made false in only two ways, A ⊃ B / C ⊃ B / / A ⊃ C T T T the requirement of simplicity. For example, if the while each of the premises can be made true in F T T T F F

three ways, then select the conclusion. On the The second point is that for valid arguments other hand, if one of the premises can be made the order in which the truth values are assigned true in only two ways while the conclusion can be may affect where the contradiction turns up. That made false in three ways, then select that premise. is, depending on the order of assignment, the If neither of these situations prevails, then select contradiction may appear in the first premise, the conclusion. second premise, third premise, and so on. But, of

Having made your selection, proceed to com- course, the order of assignment does not affect the

pute the truth values of each line, beginning with final determination of validity. the first. If no contradiction is derived on this line, The last point is that it is essential that iden-stop! The argument has been proved invalid. If a tical letters be assigned identical truth values. contradiction is derived on the first line, proceed For example, if the letter A appears three to the second line. If no contradiction is derived times in a certain symbolized argument and the on this line, then, again, the argument has been truth value T is assigned to it in one occurrence, proved invalid. If a contradiction is derived, then the same truth value must be assigned to it proceed to the third line, and so on. Remember, the in the other occurrences as well. After the truth objective is to produce a line having no contradic- table has been completed, each letter should tion. Once such a line is produced, the argument be rechecked to ensure that one and the same has been proved invalid, and no further work need truth value has been assigned to its various be done. If, on the other hand, each line necessarily occurrences. leads to a contradiction, the argument is valid.

Three final points need to be made about

indirect truth tables for arguments. First, if a Argument is Contradiction valid contradiction is obtained in the assignment of Assume argument invalid (true truth values, every step leading to it must be premises, false logically implied by some prior step. In other conclusion) Argument is No as assumed words, the contradiction must be unavoidable. contradiction (i.e., invalid) If a contradiction is obtained after truth values

Section 6.5 Indirect Truth Tables 313

Testing Statements for Consistency Statements are Contradiction inconsistent The method for testing a series of statements for Assume statements consistency is similar to the method for testing consistent (assume arguments. We begin by writing the statements on Statements are all of them are true) No as assumed a line, separating each with a single slash mark. contradiction (i.e., consistent) (Since we have no conclusion, we use no double slash marks.) Then we assume that the statements

are consistent. We assign a T to the main

operator of each, and we then compute the truth Here is another example. The statements are writ-values of the components. If this computation ten on a single line, and a T is assigned to each of the

statements are not as we assumed them to be. That A ⊃ (B ● C ) / C ⊃ ∼ ∼ A / B �� A / B ⊃ C is, they are inconsistent. But if no contradiction T leads necessarily to a contradiction, the main operators: is reached, the statements are consistent. Here is T T T

an example: Since all of the statements can be true in three ways,

A we select one of them (the fourth) and figure all of B �� the ways it can be true: B ⊃ (C �� A)

C ⊃ ∼ ∼ BA ⊃ (B ● C ) / C ⊃ ∼ ∼ A / B �� A / B ⊃ C ∼ ∼A T T T T T T

First, we write the statements on a single line sepa- F T T rated by a single slash mark; we then assign T to each F T F of the main operators: Filling out the first line leads to no contradiction, so

the statements are consistent:

A �� B / B ⊃ (C �� A) / C ⊃ ∼ ∼ B / ∼ ∼ A

T T T T A ⊃ (B ● C ) / C ⊃ ∼ ∼ A / B �� A / B ⊃ C

F T T T T T T T F T T F T T T

Next, we compute the truth values of the com- F T T ponents. First we compute the truth value of A. F T F

and compute the truth value of in the second statement is reached, we stop. The statements are consistent. C. Finally, the Only if all three lines had led to a contradiction truth values of C and B are carried to the third would these statements be inconsistent. statement: enter the truth value of B. Next, we avoid a contradiction. As soon as no contradiction B ment and compute the truth value of As with testing arguments, the objective is to Next, we enter this truth value in the first state-

A �� B / B ⊃ (C �� A) / C ⊃ ∼ ∼ B / ∼ ∼ A F T T T T T T F TTF T T F Rule for all multiline indirect truth tables

Contradiction is Go to next line(if there is one).

Since this computation leads necessarily to a contra- derived diction (third statement), the group of statements is No contradiction Stop. Argument is invalid/ inconsistent. is derived Statements are consistent.

314Chapter6 Propositional Logic

Prominent Augustus De Morgan 1806�1871Logicians

T he English logician and mathematician Augustus De Morgan is famous for the development of predicate quantifica-tion and the invention of relational algebra� an algebra fundamental to the work of Russell

and Whitehead in their Principia Mathematica.

He is known to all students of symbolic logic owing

to his formulation of what came to be known as De

Morgan�s rule of inference.

De Morgan was born in Madura, India, where his

father was employed by the East India Company. He

became blind in one eye not long after birth, and

after his family returned to England his fellow stu- Hulton Archive/Getty Images dents often taunted him and played cruel tricks on

him because of his disability. When he was ten, his thirty years, until, at age sixty, he became involved father died coming home from another trip to India. in another administrative dispute�this time over a This left the boy under the influence of his mother, a decision that De Morgan considered to be in viola-devout Anglican who wanted her son to become an tion of the university�s stated policy of religious neu-Anglican priest. trality. He again resigned in protest, bringing his aca-

De Morgan obtained a BA from Trinity College, demic career to an end. Though active in academic

Cambridge, and might have received a master�s politics, he curiously abstained from all involvement degree but for the fact that Cambridge then required in national politics. An acquaintance once remarked all candidates to take a theological test. Because of that �he never voted in an election, and never visited his commitment to the ideal of religious neutrality, the House of�Commons.� De Morgan refused to take this test. Perhaps in rebel- De Morgan was proud of his son, George, who lion against his mother�s influence, he developed a became a distinguished mathematician in his own lifelong habit of avoiding churches, claiming that he right, and he was disconsolate at his son�s untimely could not bear hearing sermons. Following his refusal death, which occurred when De Morgan was sixty-to take the exam, he continued his studies at Uni- two. The death of a daughter, during that same versity College London, a new institution founded on period, compounded his grief, and De Morgan died the principle of religious neutrality. At age twenty- three years later, possibly owing to grief and stress. two, he became the first professor of mathematics De Morgan was known for his sense of humor and there. his interest in odd facts. He produced an almanac of

Three years into his professorship, De Morgan full moons spanning a 4,000-year period. A lunar crater

became involved in a disagreement with the admin- is named after him. He liked to point out that he was istration regarding its treatment of a colleague. De x years old in the year x2 (43 in 1849). He also enjoyed Morgan led a faculty protest, and in the end he composing bits of verse in imitation of famous poets� resigned his position on the faculty. Five years later, for example, �Great fleas have little fleas upon their he returned to his former position after his replace- backs to bite �em, and little fleas have lesser fleas, and ment accidentally drowned. He remained there for so ad infinitum� (after Jonathan Swift).

Section 6.5 Indirect Truth Tables 315

Exercise 6.5

I. When possible, compute the truth values of the simple components in the following

compound propositions. If no truth value can be computed, write a question mark (?) under the letter or letters with unknown truth value.

★1. K �� D14. G ● B

F T F

2. Q ⊃ N15. S K Q

T T F

3. B K D ★16. G �� K

T F F

★4. N ⊃ G17. N ⊃ ∼ ∼P

T T T T

5. S ⊃ B18. ∼ ∼A K D

T F F T

6. K ● B ★19. ∼ ∼L ⊃ M

T T F

★7. C �� A20. E ⊃ ∼ ∼M

T F T F

8. S K E21. ∼ ∼N K R

T F F

9. M ⊃ R ★22. ∼ ∼(H �� B)

T T T

★10. H �� J23. Q ⊃ (R ● S )

T T T F

11. E K P24. K ⊃ ∼ ∼(S K M )

T F T F F

12. H �� S ★25. A �� ∼ ∼(C ● ∼ ∼R)

T F T

★13. C ⊃ P

II. Use indirect truth tables to determine whether the following arguments are valid or

invalid.

★1. B ⊃ C3. P ⊃ (Q ● R)

∼ ∼C R ⊃ S ∼ ∼BP ⊃ S

2. ∼ ∼E �� F ★4. ∼ ∼(I K J )

∼ ∼E ∼ ∼(I ⊃ J ) ∼ ∼F

316Chapter6 Propositional Logic

5. W ⊃ (X ⊃ Y ) 11. (A �� B) ⊃ (C ● D)

X ⊃ (Y ⊃ Z ) (∼ ∼A �� ∼ ∼B) ⊃ E W ⊃ (X ⊃ Z ) (∼ ∼C �� ∼ ∼D) ⊃ E

6. A ⊃ (B �� C ) 12. F ⊃ G

C ⊃ (D ● E ) ∼ ∼H �� I ∼ ∼B (G �� I ) ⊃ J A ⊃ ∼ ∼E ∼ ∼J

★ ∼ ∼(F �� H ) 7. G ⊃ H

H ⊃ I ★13. (A �� B) ⊃ (C ● D) ∼ ∼J ⊃ G (X �� ∼ ∼Y ) ⊃ (∼ ∼C ● ∼ ∼W ) ∼ ∼I (X �� Z ) ⊃ (A ● E ) J ∼ ∼X

8. J ⊃ (∼ ∼L ⊃ ∼ ∼K ) 14. ∼ ∼G ⊃ (∼ ∼H ● ∼ ∼I )

K ⊃ (∼ ∼L ⊃ M ) J ⊃ H (L �� M ) ⊃ N K ⊃ (L ● M ) J ⊃ NK �� J

9. L P ● ● G ( Q �� R )

( P ● R) ⊃ ∼ ∼(S �� T ) 15. N �� ∼ ∼O (∼ ∼S �� ∼ ∼T ) ⊃ ∼ ∼(P ● Q) P �� O

★ (N �� Q) ⊃ (R ● S) 10. ( M �� N ) ⊃ O S S K TP ⊃ Q O ⊃ (R ⊃ T ) ⊃ ( N �� P ) O ⊃ ( T ⊃ U ) M ⊃ ( ∼ ∼ Q ⊃ N ) U (Q ⊃ M) ⊃ ∼ ∼ P N

K O

III. Use indirect truth tables to determine whether the following groups of statements are

consistent or inconsistent.

★ 1. K K (R �� M ) 5. P �� ∼ ∼S

K ● ∼ ∼RS �� ∼ ∼T

2. F K (A ● X �� ∼ ∼J ∼ ∼ P ) J M ⊃ ∼ ∼KT �� ∼ ∼X

A �� ∼ ∼P ⊃ ( P ● S )

S ⊃ ∼ ∼F6. (Q �� K ) ⊃ C

3. C ⊃ (F ● ∼ ∼L) ( G �� ∼ ∼ Q ) ⊃ ( F �� B ) Q A ● ∼ ∼F (C ● F ) ⊃ (N �� L)

∼ ∼ ● ∼ ∼N ( F �� Q )

B ⊃ N ★7. S ⊃ (R K A)

★ R K (W �� T ) 4. ( N �� C ) K E S (F �� N ) ⊃ QA ⊃ (W ● ∼ ∼R)

N ● U ⊃ ∼ ∼ ( C �� H )

HU ⊃ T ⊃ E

C ⊃ H

Section 6.5 Indirect Truth Tables 317

8. ★10. A ( E �� Z �� H ) ⊃ ( K ● D ) ⊃ A (T D ⊃ ( M ● ● F ) B ) Z ⊃ ( M B ⊃ ∼ ∼ E ● Q) Q ⊃ ∼ ∼ F ∼ ∼ ( H �� K ) T ⊃ ∼ ∼ M D ⊃ B M ⊃ A9. G ⊃ P P ⊃ ( A ● ∼ ∼ G ) ( R �� T ) ⊃ G Y ⊃ R B ⊃ T Y �� B

IV. Create a group of four compound statements, similar to Exercise 6.5, III, 2, such that: (1)

the compound statements as a whole contain at least four simple statements, (2) one of the compound statements is a conjunction, (3) at least two compound statements have three or more simple components, and (4) each compound statement shares a simple statement in common with another compound statement in the group. Use an indirect truth table to test the group of compound statements for consistency. If the compound statements are consistent, construct a similar group of compound statements that turns out inconsistent, and vice versa.

6.6 Argument Forms and Fallacies

Many of the arguments that occur in propositional are statement variables. Analogously, an argument logic have forms that bear specific names and can be form is an arrangement of statement variables immediately recognized as either valid or invalid. and operators such that the uniform substitution The first two parts of this section present some of of statements in place of the variables results in an the more common ones and explain how they are argument. The resulting argument is said to be a recognized. The third part discusses ways of refuting substitution instance of its related argument two of these forms, constructive and destructive form. For example, the following arrangement of dilemmas. The fourth part presents a word of cau- statement variables and an operator is an argument tion relating to invalid forms. Finally, the fifth part form:

applies to real-life arguments some of the principles p ● q developed in the first two parts. p

If the statements A (Alicia is a chemist.) and B (Bianca

Six Valid Argument Forms is a biologist.) are uniformly substituted in place of

In Section 6.2, we defined a statement form as an the p and q of the argument form, we obtain the fol-arrangement of statement variables and operators lowing substitution instance, which is an argument: such that the uniform substitution of statements Alicia is a chemist and Bianca in place of the variables results in a statement. An A ● B is a biologist. example of a statement form is p ● q, where p and qA Alicia is a chemist.

318Chapter6 Propositional Logic

We can think of the relationship between an argument following reasoning process: The conclusion states and its form as analogous to the relationship between that we do not have p, because if we did have p, then an animal and its skeleton. Just as a skeleton can carry a (by the first premise) we would have q, and we do lot of flesh, so can an argument form, so to speak. The not have q (by the second premise). Any argument following argument has the form we just considered. that has the form of modus tollens is a valid argu-

[G ment. Example: ⊃ ( J �� K )] ● [(H �� N ) ⊃ L]

[G If Japan cares about endangered species, ⊃ ( J �� K )]

then it has stopped killing whales. C ⊃ S

A valid argument form is an argument form that sat- Japan has not stopped killing whales. ∼ ∼S isfies the truth-table test. Therefore, Japan does not care about

called modus ponens (�asserting mode�). It con- The next valid argument form we consider is sists of a conditional premise, a second premise that The first valid argument form we consider is endangered species. ∼ ∼C

asserts the antecedent of the conditional premise, pure hypothetical syllogism . It consists of two premises and one conclusion, all of which are hypo-and a conclusion that asserts the consequent: thetical (conditional) statements, and is defined as

follows:

Modus ponens (MP):

p ⊃ qPure hypothetical syllogism (HS):p

qp ⊃ q

Any argument having the form of q ⊃ rmodus ponens is a p ⊃ r valid argument. Example:

If three million children die yearly from Any argument that has the form of a pure hypo-starvation, then something is wrong thetical syllogism (that is, any argument that can

with food distribution. be produced by uniformly substituting statements T ⊃ S three million children die yearly from in place of the variables in the form) is a valid argu-

starvation. ment. Example: T Therefore, something is wrong with If world population continues to grow, food distribution. then cities will become hopelessly S

Closely associated with overcrowded. W ⊃ Cmodus ponens is

argument form consisting of one conditional prem- (�denying mode�). Modus tollens is a valid overcrowded, then pollution will tollens If cities become hopelessly modus ise, a second premise that denies the consequent become intolerable. C ⊃ P of the conditional premise, and a conclusion that Therefore, if world population denies the antecedent. It is defined as follows: continues to grow, then pollution

will become intolerable. W ⊃ P

Modus tollens (MT): The validity of a pure hypothetical syllogism

Although a little harder to understand than p is grounded in the fact that the premises link ⊃ q together like a chain. In the population argu-∼ ∼ q ment, the consequent of the first premise is ∼ ∼ p identical to the antecedent of the second. modus If the premises fail to link together in this way, the ponens , modus tollens can be understood by the argument may be invalid. Example:

Section 6.6 Argument Forms and Fallacies 319

If Jamie Foxx is a man, then Since both Harvard and Amherst are in Massa-Jamie Foxx is a human being. M ⊃ H chusetts, the premises are true and the conclusion If Jamie Foxx is a woman, then is false. Thus, the argument is invalid. Because the Jamie Foxx is a human being. W ⊃ H wedge symbol designates inclusive disjunction, the Therefore, if Jamie Foxx is a disjunctive premise includes the possibility of both man, then Jamie Foxx is a woman. M ⊃ W disjuncts being true. Thus, for the argument to be

The premises of this argument are true, and the con- valid, the other premise must eliminate one of the clusion is false. Thus, the argument is invalid. disjuncts.

The next valid form we consider is called A constructive dilemma is a valid argument

disjunctive syllogism. It consists of a disjunctive form that consists of a conjunctive premise made up statement and another statement that negates the of two conditional statements, a disjunctive premise left-hand disjunct. that asserts the antecedents in the conjunctive prem-

ise (like modus ponens), and a disjunctive conclu-

Disjunctive syllogism (DS): sion that asserts the consequents of the conjunctive

p premise. It is defined as follows: �� q

∼ ∼p

qConstructive dilemma (CD):

The validity of this form can be easily checked by a ( p ⊃ q ) ● (r ⊃ s) p �� r truth table. Now, given that validity is purely a func- q �� s tion of the form of an argument, any argument pro-

duced by uniformly substituting statements in place Any argument that has the form of a constructive of the variables in this argument form is a valid argu- dilemma is a valid argument. Example: ment. Such an argument is said to have the form of a disjunctive syllogism. The following argument was If we choose nuclear power, produced in this way and is therefore valid: then we increase the risk of a

nuclear accident; but if we

Either Harvard or Princeton is in choose conventional power, New Jersey. H �� P then we add to the Harvard is not in New Jersey. ∼ ∼H greenhouse effect. (N ⊃ I ) ● (C ⊃ A) Therefore, Princeton is in New Jersey. P We must choose either

The validity of a disjunctive syllogism arises from nuclear power or

the fact that one of the premises presents two alterna- conventional power. N �� C tives and the other premise eliminates one of those Therefore, we either increase alternatives, leaving the other as the conclusion. This the risk of a nuclear accident so-called �method of elimination� is essential to the or add to the greenhouse

should present two alternatives and the other premise The destructive dilemma is also a valid argu-should affirm one of those alternatives, the argument is ment form. It is similar to the constructive dilemma validity of a disjunctive syllogism. If one premise effect. I �� A invalid (unless the conclusion is a tautology). Example: in that it includes a conjunctive premise made

Either Harvard or Amherst is in up of two conditional statements and a disjunc-Massachusetts. H �� A tive premise. However, the disjunctive premise Harvard is in Massachusetts. H denies the consequents of the conditionals (like Therefore, Amherst is not in modus tollens ), and the conclusion denies the Massachusetts. ∼ ∼A antecedents:

320Chapter6 Propositional Logic

Destructive dilemma (DD): Given that this argument has true premises and a

∼ Denying the antecedent consists of a ∼ q �� ∼ ∼ s conditional premise, a second premise that ( p ⊃ q) ● (r ⊃ s) false conclusion, it is clearly invalid. ∼ ∼p �� ∼ ∼r denies the antecedent of the conditional, and a

Any argument that has the form of a destructive conclusion that denies the consequent:

dilemma is a valid argument. Example:

If we are to reverse the Denying the antecedent (DA):

greenhouse effect, then we p ⊃ q must choose nuclear power; ∼ ∼p but if we are to lower the ∼ ∼q

risk of a nuclear accident, Any argument that has the form of denying the ante-conventional power. (R ⊃ N ) ● cedent is an invalid argument. Example: then we must choose

(L ⊃ C )

We will either not choose If Napoleon was killed in a plane nuclear power or not crash, then Napoleon is dead. K ⊃ D choose conventional power. ∼ ∼N �� ∼ ∼C Napoleon was not killed in a Therefore, we will either not plane crash. ∼ ∼K reverse the greenhouse Therefore, Napoleon is not dead. ∼ ∼D

effect or not lower the Again, this argument has true premises and a false risk of a nuclear accident. ∼ ∼R �� ∼ ∼L conclusion, so it is clearly invalid.

Two Invalid Argument Forms

The invalid forms we will consider are closely associ- Refuting Constructive and

sists of one conditional premise, a second premise . Affirming the consequent con- forms in propositional logic, we may return for a closer look at two of them, constructive and that asserts the consequent of the conditional, and a destructive dilemmas. Arguments having these conclusion that asserts the antecedent: called affirming the consequent and denying Now that we are familiar with several argument the antecedent ated with modus ponens and modus tollens. These are Destructive Dilemmas

forms occur frequently in public debate, where

Affirming the consequent (AC): an arguer may use them to trap an opponent.

Any argument that has the form of affirming the p Since both forms are valid, the only direct mode ⊃ q of defense available to the opponent is to prove q the dilemma unsound. This can be done by prov-p ing at least one of the premises false. If the con-junctive premise ( otherwise called the �horns of consequent is an invalid argument.* The following the dilemma�) is proven false, the opponent is argument has this form and is therefore invalid: said to have �grasped the dilemma by the horns.�

If Napoleon was killed in a plane This, of course, may be done by proving either crash, then Napoleon is dead. K ⊃ D one of the conditional statements false. If, on the Napoleon is dead. D other hand, the disjunctive premise is proven Therefore, Napoleon was false, the opponent is said to have �escaped killed in a plane crash. K between the horns of the dilemma.� The latter

strategy often involves finding a third alter-

*See �Note on Invalid Forms� later in this section. native that excludes the two that are given in

Section 6.6 Argument Forms and Fallacies 321

Grasping by the horns: The strategy to be followed in refuting a dilemma Prove the conjunctive is therefore this: Examine the disjunctive premise. premise false by proving e.g.: p q r s If this premise is a tautology, attempt to grasp the either conjunct false T F F F dilemma by the horns by attacking one or the other Escaping between the horns: of the conditional statements in the conjunctive Prove the disjunctive e.g.: p v r premise. If the disjunctive premise is not a tautology, premise false F F F then either escape between the horns by, perhaps,

finding a third alternative, or grasp the dilemma by

the horns�whichever is easier.

the disjunctive premise. If such a third alternative A third, indirect strategy for refuting a dilemma can be found, both of the given disjuncts will be involves constructing a counter-dilemma. This is proved false. Consider the following constructive typically done by changing either the antecedents dilemma: or the consequents of the conjunctive premise

If taxes increase, the economy will suffer, while leaving the disjunctive premise as it is, and if taxes decrease, needed government so as to obtain a different conclusion. If the services will be curtailed. Since taxes must dilemma in question is a constructive dilemma, either increase or decrease, it follows that the consequents of the conjunctive premise are the economy will suffer or that needed changed. Here are possible counterdilemmas for government services will be curtailed. the two dilemmas just presented:

It is easy to escape between the horns of this If taxes increase, needed government dilemma by arguing that taxes could be kept as they services will be extended, and if taxes are, in which case they would neither increase nor decrease, the economy will improve. decrease. Since taxes must either increase or

Some dilemmas, however, do not allow for the decrease, it follows that needed

possibility of escaping between the horns. Consider government services will be extended the following constructive dilemma: or the economy will improve.

If we encourage competition, we will If we encourage competition, we will make have no peace, and if we do not encourage progress, and if we do not encourage competition, we will make no progress. competition, we will have peace. Since we Since we must either encourage competition must either encourage competition or not or not encourage it, we will either have no encourage it, we will either make progress peace or make no progress. or have peace.

Since the disjunctive premise of this dilemma is

a tautology, it cannot be proven false. This leaves Constructing a counterdilemma falls short of a the strategy of grasping the dilemma by the horns, refutation of a given dilemma because it merely which may be done by proving either of the condi- shows that a different approach can be taken to a tional statements in the conjunctive premise false. certain problem. It does not cast any doubt on the One debater might want to attack the first condi- soundness of the original dilemma. Yet the strategy tional and argue that competition and peace can is often effective because it testifies to the cleverness coexist, while another might want to attack the of the debater who can accomplish it successfully. second and argue that progress can be achieved In the heat of debate, the attending audience is through some means other than encouraging often persuaded that the original argument has competition. been thoroughly demolished.

322Chapter6 Propositional Logic

Throughout this text we have seen that any substi- argument on the left has the form of affirming the consequent. The argument on the right does not Note on Invalid Forms valid form. According to this convention only the

tution instance of a valid argument form is a valid have this form because it is a substitution instance argument. For example, consider modus ponens: of the following valid form:

p ⊃ q ( p �� q) ⊃ ( p ● q ) pp ● q q p �� q

Literally any two statements uniformly substituted The validity of this form results from the fact that in the place of p and q will result in a valid argument. the conclusion follows from the second premise Thus, the following symbolized arguments both alone, without any involvement of the first have the form of modus ponens, and are accordingly premise. This fact may be easily checked with a valid: truth table. Here is another invalid form: S ⊃ T (K �� B ) ⊃ (N ● R )

SK �� B p ⊃ qTN ● Rr ⊃ q

In the first argument, p ⊃ rS and T are uniformly substituted in the place of p and q; in the second Both of the following symbolized arguments are argument, K �� B and N ● R are uniformly substituted substitution instances of this form, but only the one in the place of p and q. on the left is invalid:

However, this result does not extend to invalid

argument forms. Consider, for example, affirming K ⊃ L ∼C ⊃ A the consequent: R ⊃ L (C ⊃ E ) ⊃ A

K ⊃ R ∼C ⊃ (C ⊃ E )

p ⊃ q

q The argument on the right is valid because its con-p clusion is a tautology. Accordingly, only the argu-

ment on the left will be said to have the invalid form

in the place of p and q results in an invalid argu- The point of this discussion is that when we ment, and sometimes it does not. Both of the follow-Sometimes the uniform substitution of statements in question.

ing symbolized arguments are substitution instances through mere inspection, we have to exert caution attempt to determine the validity of arguments of affirming the consequent, but the one on the left with invalid forms. The mere fact that an argument is invalid while the one on the right is valid: is a substitution instance of an invalid form does

G ⊃ N (F �� D) ⊃ (F ● D) not guarantee that it is invalid. Before judging it

N F ● invalid we must make sure that it is not valid for D

G some other reason, such as its conclusion being a F �� D

tautology. However, as concerns the exercises at

To deal with this problem, we adopt a convention the end of this section, all of the arguments that about when an argument will be said to have an are substitution instances of invalid forms are invalid form. We will say that an argument has an invalid. In other words, none of them is like either invalid form if it is a substitution instance of that of the right-hand examples considered in these form and it is not a substitution instance of any paragraphs.

Section 6.6 Argument Forms and Fallacies 323

Summary and Application If K and R are substituted respectively in place of p

valid: bolized argument in question. However, not every attempt at argument recogni-Any argument having one of the following forms is and q in the modus ponens form, we obtain the sym-

p ⊃ qmodus ponens tion is as simple as this. For more-complicated cases p (MP) it helps to keep two points in mind: q

The order of the premises never affects the argu-

p ⊃ q pure hypothetical ment�s form. q ⊃ r syllogism (HS)

p ⊃ Negated letters can be substituted in place of r

the p, q, r, and s of an argument form just as can

( p ⊃ q) ● (r ⊃ s) constructive non-negated letters.

q In regard to the first point, consider these symbolized �� p �� r dilemma

s (CD)

arguments:

p ⊃ qmodus tollens

∼ ∼q (MT) N ∼ ∼ S ∼ ∼pN ⊃ BS �� F

p �� q disjunctive

∼ ∼p syllogism The argument on the left is modus ponens, and the q (DS) one on the right is a disjunctive syllogism. To see this

( p more clearly, simply switch the order of the premises. ⊃ q) ● (r ⊃ s) destructive

∼ ∼q In regard to the second point (involving negated �� ∼ ∼ s dilemma (DD)

∼ ∼p letters), consider these examples: �� ∼ ∼ r

Any argument having either of the following forms ∼ ∼G ⊃ ∼ ∼H ∼ ∼K ⊃ ∼ ∼M ∼ ∼ G ∼∼ ∼∼ M is invalid: ∼ ∼ H ∼∼ ∼∼ K

p ⊃ q affirming the The argument on the left is modus ponens, and the q consequent (AC) one on the right is modus tollens. To produce the p argument on the left, substitute ∼ ∼G in the place of pp ⊃ q denying the in the modus ponens form, and ∼ ∼H in the place of q. ∼ ∼p antecedent (DA) For the argument on the right, substitute ∼ ∼K in the ∼ ∼q place of p in the modus tollens form, and ∼ ∼M in the

ment forms, use the following procedure. First sym- argument recognition arises from the fact that many bolize the argument, using uppercase letters for the arguments can be translated in alternate ways. Con-simple propositions. Then see whether the symbol-sider, for example, this argument: In identifying arguments as having these argu- Another problem that complicates the task of place of q. ized argument fits the pattern of one of these forms.

For example, the following symbolized argument Either the witness lied or Bob is guilty. has the form of modus ponens, and is therefore valid: The witness told the truth.

Therefore, Bob is guilty.

K ⊃ R

K If we select L to represent �The witness lied,� then the R argument can be translated into symbols as follows:

324Chapter6 Propositional Logic

L �� B of constructive dilemma because the letters in the ∼ ∼L second premise are in the wrong order. We can B reverse the order of these letters by applying the

This symbolized argument is clearly an instance of commutativity rule: disjunctive syllogism. E �� M ( R ⊃ L ) ● (T ⊃ K )

On the other hand, if we select T to represent �The ∼ ∼ER �� T

witness told the truth,� then we have this translation: ML �� K

∼ ∼T �� B After these changes, the argument on the left is now T clearly an instance of disjunctive syllogism, and B the one on the right is an instance of constructive

Technically this is not an instance of disjunctive syl- dilemma. Here are some additional examples. In some logism, because the second premise, T, is not pre-cases, the symbolized argument must be rewritten ceded by a tilde. To avoid this kind of difficulty in using double negation or commutativity before it connection with alternative translations, we intro-fits the pattern of the argument form indicated. duce two rules. They should be obvious, but if there is any doubt about them they can be proved using ∼ ∼A ⊃ ∼ ∼B truth tables. The rules are as follows: ∼ ∼B ⊃ C HS�valid

p is logically equivalent ∼ ∼A ⊃ C

to ∼∼ ∼∼p. (Double Negation) A ⊃ ∼ ∼Bp �� q is logically equivalent B ⊃ ∼ ∼C invalid to q �� p. (Commutativity) A ⊃ ∼ ∼C

According to the first rule, double tildes may be ∼ ∼A ⊃ ∼ ∼B either inserted or deleted prior to any statement, and B MT�valid according to the second rule the order of the com- A ponents in a disjunctive statement may be reversed.

Applying the double-negation rule to the second ∼ ∼A ⊃ B premise of the symbolized argument above, we have A DA�invalid

∼ ∼B

∼ ∼T �� B

∼∼ ∼∼T ∼ ∼A �� ∼ ∼BBA DS�valid

After this change, the argument is now an instance ∼ ∼B

of disjunctive syllogism. ∼ ∼A �� B

For examples of how the commutativity rule is ∼ ∼A invalid

applied, consider these symbolized arguments: B

M �� E (R ⊃ L) ● (T ⊃ K ) (A ⊃ ∼ ∼B ) ● (∼ ∼C ⊃ D) ∼ ∼E T �� R A �� ∼ ∼C CD�valid ML �� K ∼ ∼B �� D

Technically the argument on the left is not an

instance of disjunctive syllogism because the (∼ ∼A ⊃ B ) ● (C ⊃ ∼ ∼D) letters in the first premise are in the wrong order, B �� ∼ ∼D invalid and the argument on the right is not an instance A �� ∼ ∼C

Section 6.6 Argument Forms and Fallacies 325

A they were necessary to the security of a free �� ∼ ∼ BB DS�valid state. A The killings were not necessary to the

A ⊃ ∼ ∼B Therefore, the killings were not protected by security of a free state. ∼ ∼B AC�invalid the Second Amendment.

A Here is another example: A

A In a time when an entire nation believes ⊃ B MP�valid B in Murphy�s law (that if anything can go

wrong, it surely will) and has witnessed

A �� C serious accidents in the highly regulated, (A ⊃ B ) ● (C ⊃ D) CD�valid supposedly fail-safe nuclear industry, it�s B �� D fascinating that people can persist in the

Let us now see how the argument forms fantasy that an error will not occur in the

presented in this section can be used to interpret (Burk Gossom, area of nuclear weaponry. the structure of some real-life arguments. Newsweek) The following letter to the editor relates to a Although this argument allows for more than shooting at Saugus High School in Santa Clarita, one analysis, clearly the arguer presents two California. The author argues that the shooting main reasons why we can expect an accident in was not protected by the Second Amendment to the area of nuclear weaponry: �Murphy�s law� the Constitution: (which everyone believes to be true) dictates it,

and accidents have occurred in the area of nuclear

Can someone please tell me to which �well- power (which is presumed fail-safe). Thus, at regulated militia� the Saugus High School the very least, we can extract two modus ponens

shooter belonged? And why were the killing arguments from this selection: of two classmates and the wounding of three others �necessary to the security of a free If everyone believes Murphy�s law, then we state�? can expect accidents in nuclear weaponry.

(Brian Lipson) Everyone believes Murphy�s law.

Therefore, we can expect accidents in

missing certain parts. When the missing parts are If accidents have occurred in nuclear power, inserted, it can be interpreted as two cases of This argument is enthymematic; in other words, it is nuclear weaponry.

tollensmodus then we can expect accidents in nuclear : weaponry.

If the Saugus High School shooter was Accidents have occurred in nuclear power. protected by the Second Amendment, Therefore, we can expect accidents in then he was part of a well-regulated nuclear weaponry.

militia. Many arguments that we encounter in ordinary The shooter was not part of a well-regulated life can be interpreted as instances of valid argument militia. forms. After being so interpreted, however, not all Therefore, the shooter was not protected by will turn out sound. The invalid forms (denying the the Second Amendment. antecedent and affirming the consequent) should If the Saugus High School killings were be reserved for the relatively few arguments that protected by the Second Amendment, then are clearly invalid as originally expressed.

326Chapter6 Propositional Logic

Exercise 6.6

I. Interpret the following symbolized arguments in light of the eight argument forms pre-

sented in this section. In some cases a symbolized argument must be rewritten using commutativity or double negation before it becomes an instance of one of these forms. Those not having a named form are invalid.

★1. N ⊃ C11. (K �� B) ● (N �� Q)

∼ ∼C K �� N ∼ ∼NB �� Q

2. S ⊃ F12. X

F ⊃ ∼ ∼LX ⊃ ∼ ∼E S ⊃ ∼ ∼L ∼ ∼E

3. A �� ∼ ∼Z ★13. P �� ∼ ∼S

∼ ∼Z S AP

★4. (S ⊃ ∼ ∼P ) ● (∼ ∼S ⊃ D) 14. B ● T S �� ∼ ∼ S T ∼ ∼ P �� D ∼ ∼ B

5. ∼ ∼N15. ∼ ∼R �� ∼ ∼Q ∼ ∼ N ⊃ T (G ⊃ Q) ● (H ⊃ R) T ∼ ∼ G �� ∼ ∼ H

6. M �� ∼ ∼B ★16. ∼ ∼G ⊃ H ∼ ∼ M H ∼ ∼ B ∼ ∼ G

★ ★7. (E ⊃ N ) ● (∼ ∼L ⊃ ∼ ∼K ) 17. K ⊃ ∼ ∼C ∼ ∼ N �� K C ∼ ∼ E �� L ∼ ∼ K8. W ⊃ ∼ ∼ M18. ( I ⊃ M ) ● (∼ ∼O ⊃ A) ∼ ∼ M ∼ ∼ O W �� I M �� A9. ∼ ∼ B ⊃ ∼ ∼ L ★ 19. X ⊃ ∼ ∼ G ⊃ ∼ ∼ F B W ⊃ ∼ ∼ G ⊃ ∼ ∼ F L W ⊃ X10. F ⊃ O20. ∼ ∼ L ⊃ ∼ ∼ U F ∼ ∼ L O ∼ ∼ U

II. Translate the following arguments into symbolic notation and then interpret them

in light of the eight argument forms presented in this section. In some cases, a sym-bolized argument must be rewritten using commutativity or double negation

Section 6.6 Argument Forms and Fallacies 327

before it becomes an instance of one of these forms. Those not having a named form are invalid. The letters to be used in translating the statements are given in parentheses.

★1. A Boeing 757 crashed into the Pentagon on September 11, 2001, only if two

giant engines were found outside the building. It is not the case that two giant engines were found outside the building. Therefore, a Boeing 757 did not crash into the Pentagon on 9/11. (B, G)

2. If Michelangelo painted the ceiling of the Sistine Chapel, then he was familiar

with stories from the Old Testament. Michelangelo was familiar with stories from the Old Testament. Therefore, Michelangelo painted the ceiling of the Sistine Chapel. (M, F)

3. If you enter the teaching profession, you will have no money for vacations; and

if you do not enter the teaching profession, you will have no time for vacations. Since you must either enter or not enter the teaching profession, it follows that either you will have no money or no time for vacations. (E, M, T)

★4. Either the wealthiest people are the happiest, or it is not the case that money

can buy everything. The wealthiest people are not the happiest. Therefore, money cannot buy everything. (W, M)

5. Either drivers are forbidden to send text messages, or the highways will not

become safer. Drivers are forbidden to send text messages. Therefore, the high-ways will become safer. (D, H)

6. If the sun is a variable star, then its energy will drop drastically at some point in

the future. If the sun�s energy drops drastically at some point in the future, then the earth will become a giant ice ball. Therefore, if the sun is a variable star, then the earth will become a giant ice ball. (S, E, G)

★7. Nano-thermite was present in the debris from the World Trade Center. But

if that is so, then the buildings were brought down by controlled demolition. Therefore, the buildings were brought down by controlled demolition. (N, B)

8. If TV viewing provides genuine relaxation, then TV enhances the quality of

life. But TV viewing does not provide genuine relaxation. Therefore, TV does not enhance the quality of life. (G, E)

9. If high school clinics are to stem the tide of teenage pregnancy, then they must

dispense birth-control devices; but if they want to discourage wanton sex, then they must not dispense these devices. Since high school clinics must either dispense or not dispense birth-control devices, either they will not stem the tide of teenage pregnancy, or they will not discourage wanton sex. (S, D, I)

★10. If limits are imposed on medical-malpractice suits, then patients will not be

adequately compensated for their injuries; but if the cost of malpractice insur-ance continues to rise, then physicians will be forced out of business. Limits will not be imposed, and the cost of malpractice insurance will not continue to rise. Therefore, patients will be adequately compensated and physicians will not be forced out of business. (L, A, C, F)

328Chapter6 Propositional Logic

11. If Prohibition succeeded in the 1920s, then efforts to control cocaine and

heroin will succeed today. But Prohibition did not succeed in the 1920s. There-fore, efforts to control cocaine and heroin will not succeed today. (P, E)

12. If life is always better than death, then people do not take their own lives.

People do take their own lives. Therefore, life is not always better than death. (L, T)

★13. If we want to arrest criminals, then police must engage in high-speed chases;

but if we want to protect motorists, then police must not engage in high-speed chases. Since police must either engage or not engage in high-speed chases, either we will not arrest criminals or not protect motorists. (A, E, P)

14. Either industrial pollutants will be more stringently controlled, or air quality

will continue to fall. Industrial pollutants will be more stringently controlled. Therefore, air quality will not continue to fall. (I, A)

15. Insurance companies contribute millions of dollars to political campaigns. But

if that is so, then meaningful insurance reform is impossible. Therefore, mean-ingful insurance reform is impossible. (I, M)

★16. If congressional districts are gerrymandered, then minority votes will make no

difference. Minority votes will make no difference. Therefore, congressional districts are gerrymandered. (C, M)

17. Either the dinosaurs were not cold blooded or they were not the ancestors of

modern birds. The dinosaurs were the ancestors of modern birds. Therefore, the dinosaurs were not cold blooded. (C, A)

18. If coal burning continues, then heavy metals will be released into the

atmosphere. If heavy metals are not released into the atmosphere, then nervous-system damage will decrease. Therefore, if coal burning does not con-tinue, then nervous-system damage will decrease. (C, H, N)

★ 19. If sea levels rise twenty feet worldwide, then coastal cities from New York

to Sydney will be inundated. If the ice sheets on Antarctica slip into the sea, then sea levels will rise twenty feet worldwide. Therefore, if the ice sheets on Antarctica slip into the sea, then coastal cities from New York to Sydney will be inundated. (S, C, I)

20. If tax credits are given for private education, then the government will be

supporting religion; but if tax credits are not given for private education, then some parents will end up paying double tuition. Either tax credits will or will not be given for private education. Therefore, either the gov-ernment will be supporting religion, or some parents will end up paying double tuition. (T, G, P)

III. Identify the following dilemmas as either constructive or destructive. Then suggest a

refutation for each by escaping between the horns, grasping by the horns, or construct-ing a counterdilemma.

★1. If Melinda spends the night studying, she will miss the party; but if she does

not spend the night studying, she will fail the test tomorrow. Melinda must

Section 6.6 Argument Forms and Fallacies 329

either spend the night studying or not studying. Therefore, she will either miss the party or fail the test. (S, M, F)

2. If we build our home in the valley, it will be struck by floods; and if we build

it on the hilltop, it will be hit by lightning. Since we must either build it in the valley or on the hilltop, our home will either be struck by floods or hit by light-ning. (V, F, H, L)

3. If psychotherapists respect their clients� right to confidentiality, then they will

not report child abusers to the authorities; but if they have any concern for the welfare of children, then they will report them. Psychotherapists must either report or not report child abusers to the authorities. Therefore, psychothera-pists either have no respect for their clients� right to confidentiality or no con-cern for the welfare of children. (C, R, W)

★4. If corporations are to remain competitive, then they must not spend money to

neutralize their toxic waste; but if the environment is to be preserved, then cor-porations must spend money to neutralize their toxic waste. Corporations either will or will not spend money to neutralize their toxic waste. Therefore, either they will not remain competitive, or the environment will be destroyed. (C, S, E)

5. If physicians pull the plug on terminally ill patients, then they risk being

charged with murder; but if they do not pull the plug, they prolong their patients� pain and suffering. Since physicians with terminally ill patients must do one or the other, either they risk being charged with murder or they prolong their patients� pain and suffering. (P, M, S)

6. If the Mitchells get a divorce, they will live separately in poverty; but if they

stay married, they will live together in misery. Since they must either get a divorce or stay married, they will either live separately in poverty or together in misery. (D, S, M, T)

★7. If college students want courses that are interesting and rewarding, then they

must major in liberal arts; but if they want a job when they graduate, then they must major in business. College students will either not major in liberal arts, or they will not major in business. Therefore, either they will not take courses that are interesting and rewarding, or they will not have a job when they graduate. (C, L, J, B)

8. If merchants arrest suspected shoplifters, then they risk false imprisonment;

but if they do not arrest them, they risk loss of merchandise. Merchants must either arrest or not arrest suspected shoplifters. Therefore, they will either risk false imprisonment or loss of merchandise. (A, F, L)

9. If school kids want to avoid being bullied, then they must report the bully to

authorities; but if they want to avoid revenge by the bully, then they must not report the bully. School kids must either report or not report a bully to authori-ties. Therefore, either the bullying will continue, or the bully will exact revenge. (S, R, A)

★10. If we prosecute suspected terrorists, then we risk retaliation by other terror-

ists; but if we release them, then we encourage terrorism. Since we must either

330Chapter6 Propositional Logic

prosecute or release suspected terrorists, we either risk retaliation by other ter-rorists or we encourage terrorism. (P, R, T, E)

IV. The following dialogue contains at least fifteen arguments. Translate each into sym-

bolic notation, and then interpret them in light of the eight argument forms pre-sented in this section.

A Little Help from a Friend

�I can only talk for a minute,� Liz says to her friend, Amy. �I have this bio midterm tomorrow, and I�m afraid I�m going to fail it.�

�Okay,� Amy replies. �I�ll cut out after a minute or two. But why are you so afraid?�

�Because I really haven�t studied at all. I figure if I don�t pull an all-nighter, I�ll fail the

test. But I can�t fail the test, so I must pull the all-nighter.�

�I don�t envy you,� says Amy, digging in her purse. �But I have this little tablet of

Adderall that might get you through. As I see it, either you take the tab or you�ll fall asleep by midnight. But you can�t fall asleep, so you must take the tab.�

Liz stares at the little orange pill. �Adderall is for attention deficit disorder. You don�t

have that, do you?�

�No,� says Amy. �I got the tab from my boyfriend, Zach, who talked the health clinic

out of a whole bottle by faking ADD.�

�Wow,� says Liz. �Do you think it�s safe for me to take it?�

�Of course,� replies Amy. �Zach takes it all the time when he needs an extra

spurt of energy, and he�s had no adverse reactions. I�ve even tried it once or twice. If it�s safe for Zach, then it�s safe for me, and if it�s safe for me, then it�s safe for you. The conclusion is obvious.�

�And do you really think the Adderall will help me pass the test?� asks Liz. �Absolutely,� says Amy. �If you take it, you�ll be totally focused for the test, and you

must be focused. Hence, you take it, girl.�

�I don�t know,� Liz says as she takes a closer look at the little pill. �If I take it, then

I�ll feel like an athlete who takes performance-enhancing drugs. I�ll feel like I�m put-ting myself at an unfair advantage over the other students. I don�t want to do that, so maybe I shouldn�t take it.�

She thinks for a moment and sighs. �Either I take the Adderall or I don�t. If I take it,

I�ll feel like I cheated, but if I don�t take it, I�ll fail the test. Thus, I�ll feel like I cheated, or I�ll fail the test. Either way, I lose.�

Amy smiles. �There�s another way of looking at it. Either you take the Adderall or

you don�t. If you take it, you�ll pass the test, and if you don�t take it, you�ll have a clear conscience. Thus, you�ll either pass the test or you�ll have a clear conscience. Either way, you win.�

�Very clever,� says Liz, �but that really doesn�t solve my problem.�

�Okay,� says Amy. �But maybe your problem isn�t as bad as you think. Consider

this: Older people take drugs all the time to help their memory and their sex lives. If it�s okay for them to take those drugs�and it is�then it�s okay for you to take the Ad-derall. You shouldn�t sweat it.�

�That�s true,� says Liz, �but those older people suffer from a medical condition. If I

had ADD, I�d be justified in taking the Adderall, but I don�t. So I�m not justified.�

�Let�s look at it another way,� says Amy. �You could get through the night with lots

of coffee and Red Bull. If it�s okay to drink coffee and Red Bull�and it is�then it�s okay to take the Adderall. The conclusion is clear.�

Section 6.6 Argument Forms and Fallacies 331

�Not quite,� says Liz. �Coffee and Red Bull aren�t really comparable to Adderall�at

least not if it�s as good as you say it is. Suppose that I�m faced with this option: Either I drink lots of coffee and Red Bull, or I take the Adderall. I would say no to the coffee and Red Bull because it�s less effective, and it leaves me frazzled. Thus, I would take the Adderall. See, the two are not the same.�

�Maybe not,� says Amy, �but think about it this way. We take advantage of new

technology every day without giving it a second thought. We use smartphones in-stead of landlines because they�re more convenient. We use lightbulbs instead of candles because we see better with them. And we use Adderall instead of coffee because it makes us sharper. If it�s ethically okay to use smartphones, then it�s okay to use Adderall; and it�s certainly okay to use smartphones�just as it�s okay to use lightbulbs. Hence, it�s okay to use Adderall.�

�The problem with that line of reasoning,� Liz observes, �is that using lightbulbs

and smartphones doesn�t put anyone at a competitive advantage. Everyone uses them, so we�re all on an equal footing. But not every student uses Adderall to pass a test. If everyone used it, I would have no problem with it. But not everyone uses it, so I do have a problem.�

�I can see your point,� Amy says. �At fifteen or twenty bucks a pop on the under-

ground market, not every student can afford Adderall. If it were cheap, then everyone would use it. But it�s not cheap, so many students don�t.�

Amy smiles again at her friend. �Messy stuff,� she says. �So, what do you think you�ll

do?�

�I don�t know.� Liz groans and puts her face in her hands. �But leave that tab on my

desk. I�ll see how it goes. . . . I�ll give myself until midnight.�

V. The following selections were taken from letters-to-the-editor newspaper columns.

Each contains one or more arguments, but the exact form of the argument may be hidden or ambiguous. Use the argument forms presented in this section to structure the selections as specifically named arguments.

★1. There is a simple way to put a big dent in the national human-organ shortage:

Allocate organs first to the people who have agreed to donate their own. Giving organs first to registered donors would persuade more people to register, and that would make the allocation system fairer. People who aren�t willing to share the gift of life should go to the end of the waiting list.

(David J. Undis)

2. Okay, I�ve tried it for a week again this year, but I still don�t like daylight saving

time. My grass is brown enough already�it doesn�t need another hour of daylight each day. Let�s turn the clocks back to the way God intended� standard time.

( Jim Orr)

3. The religious right, in its impassioned fervor to correct our alleged moral

wrongs and protect the rights of our unborn �children,� may one day realize its ultimate goal of a constitutional amendment banning abortion. And what will the punishment be for those caught performing or receiving an abortion? The death penalty, of course.

(David Fisher)

332Chapter6 Propositional Logic

★4. Most educators believe math instructors ought to emphasize group problem

solving. If group problem solving is so important (and I think it is), why do we place such emphasis on individual testing? The national math test is a�mistake.

(Frederick C. Thayer)

5. If voluntary school prayer for our children is going to make them more moral,

then just think what mandatory church attendance on Sunday could do for the rest of us.

(Roderick M. Boyes)

6. A country that replaces the diseased hearts of old white men but refuses to

feed schoolchildren, pay women adequately, educate adolescents, or care for the elderly�that country is doomed. We are acting as if there is no tomorrow. Where is our shame?

(Robert Birch)

★ 7. We cannot afford to close the library at Central Juvenile Hall. These young

people, in particular, need to have access to ideas, dreams, and alternative ways of living. It can make the difference for many students who might become interested in reading for the first time in their lives while in Juvenile Hall.

(Natalie S. Field)

8. If the death penalty deters one person from becoming a murderer, it is effec-

tive. There are also some other important reasons for having the death penalty. First, the families and friends of innocent victims have the right to see effective retribution. Second, terminating the life of a killer is more economical than keeping him in jail at the taxpayer�s expense. Third, everyone will have greater respect for the judicial system when justice is carried out.

(Doug Kroker)

9. Regarding the bill to require parental consent for a minor�s abortion, I would

like to point out that the pious platitudes about parental authority quickly fall by the wayside when the minor wants to keep the baby and the parents say, �Don�t be silly! You have an abortion and finish your education.� If the parents can veto a minor�s abortion, shouldn�t they also be able to require one? Better the choice, either pro or con, be left to the girl/woman herself.

( Jane Roberts)

★10. More than a million adult videos are rented each week. Nor, as the propagan-

dists would have you believe, does viewing such material lead to violent sex crimes. If it did, there would be over one million such crimes per week.

(Lybrand P. Smith)

VI. Create the following arguments:

1. A modus ponens about social media

2. A modus tollens relating to sports

3. A disjunctive syllogism concerning political parties

4. A pure hypothetical syllogism dealing with the Middle East

Section 6.6 Argument Forms and Fallacies 333

5. A constructive dilemma about health care

6. A destructive dilemma concerning the minimum wage

7. An affirming the consequent relating to North Korea

8. A denying the antecedent about privacy rights

Summary

Propositional Logic:

■ The fundamental units are whole statements (propositions).

■ Simple statements are represented by uppercase letters (A, B, C, etc.).

■ These are combined via logical operators to form compound statements.

■ The logical operators:

● Tilde (∼ ∼) forms negations (�not,� �it is not the case that�).

● Dot ( ) forms conjunctions (�and,� �also,� �moreover,� etc.).

● Wedge (��) forms disjunctions (�or,� �unless�). ● Horseshoe ( ⊃) forms conditionals (�if . . . then,� �only if,� etc.).

● Triple bar (K) forms biconditionals (�if and only if,� etc.).

Truth Table:

■ An arrangement of truth values that shows in every possible case how the truth value of a

compound statement is determined by the truth values of its components.

■ Used to define the meaning of the five logical operators:

● ∼ ∼p is true only when p is false.

● ● pq is true only when both p and q are true.

● p �� q is false only when both p and q are false.

● p ⊃ q is false only when p is true and q is false.

● pq is true only when p and q have the same truth value. K

■ Used to classify individual compound statements:

● Tautologous: Truth values under main operator are all true.

● Self-contradictory: Truth values under main operator are all false.

● Contingent: Under main operator : at least one true, at least one false.

■ Used to compare one compound statement with another:

● Logically equivalent: Truth values under main operators are the same on each line.

● Contradictory: Truth values under main operators are opposite on each line.

● Consistent: There is at least one line under main operators where all of the truth values are true.

● Inconsistent: There is no line under main operators where all of the truth values are true.

334Chapter6 Propositional Logic

■ Used to test arguments for validity:

● Invalid: There is a line on which all the premises are true and the conclusion false.

● Valid: There is no such line.

Indirect Truth Table: A usually shorter truth table constructed by first assigning truth values to the main operators and then working backward to the simple components.

■ Used to test arguments for validity:

● Begin by assuming the premises true and the conclusion false.

● Valid: Assumption necessarily leads to a contradiction.

● Invalid: Assumption does not necessarily lead to a contradiction.

■ Used to test a series of statements for consistency:

● Begin by assuming all of the statements true.

● Inconsistent: Assumption necessarily leads to a contradiction.

● Consistent: Assumption does not necessarily lead to a contradiction.

Argument Forms and Fallacies:

■ Valid forms:

● Modus ponens: p ⊃ q / p // q

● Modus tollens: p ⊃ q / ∼ ∼q // ∼ ∼p

● Pure hypothetical syllogism: p ⊃ q / q ⊃ r // p ⊃ r

● Disjunctive syllogism: p �� q / ∼ ∼p // q

● ● Constructive dilemma: ( p ⊃ q ) (r ⊃ s) / p �� r // q �� s

● ● Destructive dilemma ( p ⊃ q ) (r ⊃ s) / ∼ ∼q �� ∼ ∼s // ∼ ∼p �� ∼ ∼r

■ Invalid forms (fallacies):

● Affirming the consequent: p ⊃ q / q // p

● Denying the antecedent: p ⊃ q / ∼ ∼p // ∼ ∼q

■ Logical equivalencies:

● p is logically equivalent to ∼∼ ∼∼p.

● p �� q is logically equivalent to q �� p.

Summary335

7Natural Deduction in�Propositional Logic

7.1 Rules of Implication I

7.2 Rules of Implication II

7.3 Rules of Replacement I

7.4 Rules of Replacement II

7.5 Conditional Proof

7.6 Indirect Proof

7.7 Proving Logical Truths

7.1 Rules of Implication I

Learning about natural deduction is empowering. Using natural deduction to derive the conclusion of an argument is like presenting an argu-ment to a jury in a murder trial. Suppose the evidence includes the location and posture of the victim�s body, the angle the bullet entered the body, and the loca-tion of the defendant at the moment the victim was shot. The defense attorney will put these facts together, step by step, to prove that the defendant could not have shot the victim.

Natural deduction is a method for deriving the conclusion of valid argu-ments expressed in the symbolism of propositional logic. The method con-sists of using rules of inference (valid argument forms) to derive either a con-clusion directly, or a series of intermediate conclusions that links the premises of an argument with the stated conclusion. Natural deduction gets its name from the fact that it resembles the ordinary step-by-step reasoning process people use in daily life. It also resembles the method used in geometry to derive theo-rems relating to lines and figures; but whereas each step in a geometrical proof depends on some mathematical principle, each step in a logical proof depends on a rule of inference.

336

Natural deduction is similar in some respects for short) says that given two conditional statements

to the truth-table method for testing arguments. on lines by themselves such that the consequent of Both methods can be used to prove a valid argu- one is identical with the antecedent of the other, we ment valid; but natural deduction is largely useless can assert on a line by itself a conditional statement for invalid arguments, so we still need truth tables. whose antecedent is the antecedent of the first con-However, the method of natural deduction is more ditional and whose consequent is the consequent of illuminating than truth tables are. Natural deduction the second conditional. Note in the rule that the two shows how a conclusion �comes out� of the prem- premises hook together like links of a chain. ises, whereas truth tables show nothing of the sort. Disjunctive syllogism says that given a dis-Also, while truth tables are relatively automatic and junctive statement and the negation of the left-hand mechanical, natural deduction requires insight and disjunct on lines by themselves, we can assert the creativity. As a result, most students find that natural right-hand disjunct on a line by itself. Because of the deduction is challenging and fun, while truth tables way this rule is written, only the right-hand disjunct can be tedious, especially for long arguments. can be asserted as the conclusion. However, once

The first eight rules of inference are called rules we are supplied with the commutativity rule (see

of implication because they consist of simple, valid Section�7.3), we will be able to switch the order of argument forms whose premises imply their conclu- the disjuncts in the first line, and this will allow us sions. The first four rules of implication should be to assert what was originally the left-hand disjunct familiar from Section 6.6. They are listed together as the conclusion. Disjunctive syllogism is otherwise with an illustration of their use as follows: called the method of elimination. The short premise

1. Modus ponens (MP) eliminates one of the alternatives in the disjunctive p premise, leaving the other as the conclusion. ⊃ q If Su Lin is a panda, then Su Lin is cute. These four rules will be sufficient to derive the

p Su Lin is a panda. conclusion of many simple arguments in propo-

q Su Lin is cute. sitional logic. Further, once we are supplied with 2. Modus tollens (MT ) all eighteen rules together with conditional proof,

p ⊃ q If Koko is a koala, then Koko is cuddly. the resulting system will be sufficient to derive

∼ ∼q Koko is not cuddly. the conclusion of any valid argument in proposi-∼ ∼p Koko is not a koala. tional logic. Conversely, since each rule is a valid

3. Pure hypothetical syllogism (HS) argument form unto itself, any conclusion derived

p ⊃ q If Leo is a lion, then Leo roars. from their correct use results in a valid argument. q ⊃ r If Leo roars, then Leo is fierce. Applying the rules of inference rests on the abil-p ⊃ r If Leo is a lion, then Leo is fierce. ity to visualize more- or less-complex arrangements

4. Disjunctive syllogism (DS) of simple propositions as substitution instances of

p the rules. For a fairly simple substitution instance of �� q Scooter is either a mouse or a rat. modus ponens , consider the following: ∼ ∼ p Scooter is not a mouse. q Scooter is a rat. 1. ∼ ∼A ⊃ Bp ⊃ q

ment and its antecedent on lines by themselves, we can Modus ponens 2. ∼ ∼Ap says that given a conditional state-3. Bq assert its consequent on a line by itself. Modus tollens When ∼ ∼A and B are mentally substituted, respec-says that given a conditional statement and the negation tively, in place of the p and q of the modus ponens of its consequent on lines by themselves, we can assert rule, then you should be able to see that the argu-the negation of its antecedent on a line by itself. Pure ment on the left is an instance of the rule. The fact hypothetical syllogism (�hypothetical syllogism� that A is preceded by a tilde is irrelevant.

Section 7.1 Rules of Implication I 337

Here is a more complex example: The first step is to symbolize the argument, number-

1. (A ing the premises and writing the intended conclu-● B) ⊃ (C �� D) p ⊃ q sion to the right of the last premise, separated by a 2. A ● Bp slash mark: 3. C �� Dq

In this case, if you mentally substitute 1. A ⊃ ∼ ∼BA ● B and C �� D , respectively, in place of p and q in the rule, you can 2. C ⊃ B see that the argument on the left is an instance of 3. A / ∼ ∼C

modus ponens. This example illustrates the fact that The next step is to derive the conclusion through any pair of compound statements can be uniformly a series of inferences. But before writing the first substituted in place of p and q to produce a substitu-line, it is important to keep two points in mind. tion instance of the rule. First, always begin by trying to �find� the conclu-Finally, the order of the premises never makes a sion in the premises, and second, always look at the difference: main operator of the conclusion for a clue about

1. Ap how to derive it. In this particular proof, the first 2. A ⊃ (B ⊃ C ) p ⊃ q point is most helpful. The conclusion to be derived 3. B ⊃ Cq is ∼ ∼C, and we see that C�appears in the antecedent

In this case, if you mentally substitute A and BCtollens ⊃ of line 2. We could derive ∼ ∼C from line 2 by modus if we had ∼ ∼ B , so now we look for ∼ ∼ B . Turn-in place of p and q , you can see, once again, that the ing our attention to line 1, we see that we� could argument on the left is an instance of modus ponens . derive ∼ ∼ B by modus ponens if we had A , and we The fact that the order of the premises is reversed do have A on line 3. Thus, we have now thought makes no difference. through the entire proof, and we can begin to write B it out. First, we derive ∼ ∼ by modus ponens from lines 1 and 3: These arguments are all instances of

modus ponens (MP):

1. A ⊃ ∼ ∼B 2. C ⊃ B 3. A / ∼ ∼C 4. ∼ ∼B 1, 3, MP

struct a proof. Such a proof consists of a sequence of Now let us use the rules of inference to con- The justification for line 4 is written to the right, directly beneath the slash mark. If you have trou-ble understanding how line 4 was derived, imag-propositions, each of which is either a premise or is ine substituting A and ∼ ∼ B in place of p and q in derived from preceding propositions by application the modus ponens rule. Then you can see that lines of a rule of inference and the last of which is the con-1, 3, and 4 constitute a substitution instance of clusion of the original argument. Let us begin with that rule. the following example: The final step is to derive ∼ ∼ C from lines 2 and 4 modus tollens If the Astros switch leagues, then the Braves will by :

not win the pennant. 1. A ⊃ ∼ ∼B If the Cubs retain their manager, then the Braves 2. C ⊃ B will win the pennant. 3. A / ∼ ∼C

The Astros will switch leagues. Therefore, the 5. 4. ∼ ∼B 1, 3, MP ∼ ∼C 2, 4, MT

Cubs will not retain their manager.

338Chapter7 Natural Deduction in�Propositional Logic

The proof is now complete. The justification for line 5 Here is another example.

is written directly beneath the justification for line 4. 1. F ⊃ G

These arguments are all instances of 2. F �� Hmodus tollens (MT): 3. ∼ ∼G

( D F) KG ( M v N) TK ( M v N ) [(H v K) ( L v N)] T ⊃ The intended conclusion is F I. When we attempt ( v 4. H ⊃ (G ⊃ I ) / F ⊃ ID v F) G [(H v K) (L v N )] to find it in the premises, we see no such statement.

The next example is already translated into symbols: mentioned earlier, which is to look at the main oper-This tells us that we should turn to the second point 1. A ⊃ B ator of the conclusion for a clue about how to derive 2. ∼ ∼A ⊃ (C �� D) it. In this case the main operator is a horseshoe, so 3. ∼ ∼B we ask: What rule gives us a horseshoe�that is, a 4. ∼ ∼C / D conditional statement? The answer is: pure hypo-

Once again, to derive the conclusion, always begin thetical syllogism. by trying to �find� it in the premises. The intended Having tentatively settled on pure hypotheti-conclusion is cal syllogism for deriving the conclusion, we now D , and after inspecting the premises we find look for two conditional statements, which, when D in line 2. If we had the consequent of that line, combined, will yield F ⊃ I. In line 1 we find F ⊃ G C �� D , on a line by itself, we could derive D by disjunctive syllogism if we had and in line 4 we find G ⊃ I, but before we can ∼ ∼ C . And we do have apply pure hypothetical syllogism, we must obtain ∼ ∼ C on line 4. Also, we could derive C �� D by modus ponensG ⊃ if we had ∼ ∼ A , so now we look for ∼ ∼ I on a line by itself. Examining line 4, we see A . Turning our attention to line 1, we see that we could that G ⊃ I could be derived by modus ponens, if

derive we had H on a line by itself, and examining line 2, ∼ ∼ A by modus tollens if we had ∼ ∼ B , and we do have we see that H could be derived by disjunctive syl-∼ ∼ B on line 3. Thus, we have now thought

through the entire proof, and we can write it out: logism if we had ∼ ∼F on a line by itself. Turning to 1. A line 1, we see that ∼ ∼ F could be derived by modus ⊃ Btollens

2. if we had ∼ ∼G on a line by itself, and we do ∼ ∼ A ⊃ (C �� D) have ∼ ∼ G on line 3. Thus, we have now thought 3. ∼ ∼ B through the entire proof, and we can write it out: 4. ∼ ∼ C / D

5. ∼ ∼A 1, 3, MT 1. F ⊃ G 6. C �� D 2, 5, MP 2. F �� H 7. D 4, 6, DS 3. ∼ ∼G

directly beneath the slash mark preceding the intended As usual, the justification for each line is written 4. H ⊃ (G ⊃ I ) / F ⊃ I 5. ∼ ∼ F 1, 3, MT 6. H 2, 5, DS conclusion. If you have trouble understanding line 6, 7. G ⊃ I 4, 6, MP imagine substituting ∼ ∼ A and C �� D in place of p and q 8. F ⊃ I 1, 7, HS in the modus ponens rule. Then you can see that lines 2, In addition to showing how useful it is to begin 5, and 6 constitute a substitution instance of that rule. a proof by looking at the main operator of the

These arguments are all instances of pure hypothetical conclusion, this proof teaches another important syllogism (HS): lesson about every proof. Every line in a proof can

be used multiple times. In line 5, we used line 1 in

conjunction with the modus tollens rule, and in line 8 we used that same line again in conjunction with

Section 7.1 Rules of Implication I 339

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). pure hypothetical syllogism. Any line can be used In applying the four rules of inference introduced as many times as we need it. in this section, we have noted that various expres-

sions first had to be obtained on lines by themselves.

These arguments are all instances of disjunctive If this procedure is not followed, the resulting proof syllogism (DS): will likely be invalid. For an example of an invalid

application of modus ponens, consider the following:

1. A ⊃ (B ⊃ C ) 2. B

The next example is more complex: 3. C 1, 2, MP (invalid)

1. ∼ ∼(A ● B) �� [∼ ∼(E ● F ) ⊃ (C ⊃ D)] This inference is invalid because B ⊃ C must first be 2. ∼ ∼ ∼ ∼(A ● B) obtained on a line by itself. In deriving the conclusion 3. ∼ ∼(E ● F ) of an argument we always assume the premises are true.

sion in the premises, we see no such statement. But B ⊃ C is true. Thus, B ⊃ C cannot be treated as a premise. We do not know if it is true or false. then we notice that the main operator of the conclu-Here are some additional examples of invalid sion is a horseshoe, and we find C ⊃ D on line 1 and inferences: Again, when we attempt to find the intended conclu- entail that B ⊃ C is true. What line�1 says is that if A is true, then 4. D ⊃ G / C ⊃ G But if we assume line 1 of this proof is true, this does not D ⊃ G on line 4. We could derive the conclusion by pure hypothetical syllogism if we could obtain C ⊃ D 1. A �� B on a line by itself. Examining line 1, we see that we 2. A could derive C ⊃ D by modus ponens if we could 3. B 1, 2, DS (invalid�line 2 must obtain both ∼ ∼(E ● F ) ⊃ (C ⊃ D) and ∼ ∼(E ● F ) on negate A in line 1, not assert it )

lines by themselves, and we see that ∼ ∼(E ● F ) appears 1. A ⊃ B on line 3. Also, examining line 1, we see that we could 2. B derive ∼ ∼( E ● F )�⊃ (C ⊃ D) by disjunctive syllogism if 3. A 1, 2, MP (invalid�line 2 must assert we had ∼ ∼ ∼ ∼(A ● B) on a line by itself, and we do have the antecedent of line 1, not the it on line 2. Thus, we can now write out the proof: consequent )

1. ∼ ∼(A ● B) �� [∼ ∼(E ● F ) ⊃ (C ⊃ D)] 1. A ⊃ B 2. ∼ ∼ ∼ ∼(A ● B) 2. A ⊃ C 3. ∼ ∼(E ● F ) 3. B ⊃ C 1, 2, HS (invalid�the conse-4. D ⊃ G / C ⊃ G quent of one conditional must 5. ∼ ∼(E ● F ) ⊃ (C ⊃ D) 1, 2, DS be identical with the anteced-6. C ⊃ D 3, 5, MP ent of the other) 7. C ⊃ G 4, 6, HS

1. A ⊃ B

If you have trouble seeing how lines 5 and 6 are 2. ∼ ∼A

derived, for line 5 imagine substituting 3. ∼ ∼B 1, 2, MT (invalid�line 2 must ∼ ∼ ( A ● B ) and ∼ negate the consequent of line ∼ ( E ● F ) ⊃ ( C ⊃ D ), in place of the p and q of the disjunctive syllogism rule. Then you can see that 1, not the antecedent )

lines 1, 2, and 5 constitute a substitution instance of 1. (A ⊃ B) ⊃ C that rule. For line 6, imagine substituting ∼ ∼( E ● F ) 2. ∼ ∼B and (C ⊃ D) in place of p and q in the modus ponens 3. ∼ ∼A 1, 2, MT (invalid�A ⊃ B must rule. Then you can see that lines 5, 3, and 6 consti- first be obtained on a line by tute a substitution instance of that rule. itself )

340Chapter7 Natural Deduction in�Propositional Logic

1. A ⊃ (B ⊃ C ) Strategy 3: If the conclusion contains a negated 2. C ⊃ D letter and that letter appears in the 3. B ⊃ ⊃ D 1, 2, HS (invalid�B ⊃ C must antecedent of a conditional statement

first be obtained on a line by in the premises, consider obtaining itself ) the negated letter via modus tollens:

1. (A 1. C ⊃ B �� B) ⊃ C

2. 2. A ⊃ B ∼ A

3. B 1, 2, DS (invalid�A 3. ∼ ∼B / ∼ ∼A � � B must

first be obtained on a line by 4. ∼ ∼A 2, 3, MT

itself ) Strategy 4: If the conclusion is a conditional

applying the first four rules of inference. We conclude this section with some strategies for statement, consider obtaining it via pure hypothetical syllogism:

Strategy 1: 1. B ⊃ C Always begin by attempting to �find� 2. C ⊃ A the conclusion in the premises. If 3. A ⊃ B / A ⊃ C the conclusion is not present in its 4. A ⊃ C 1, 3, HS entirety in the premises, look at the Strategy 5: If the conclusion contains a letter main operator of the conclusion. This that appears in a disjunctive state-will provide a clue as to how the con-ment in the premises, consider clusion should be derived. obtaining that letter via disjunctive Strategy 2: If the conclusion contains a letter syllogism: that appears in the consequent of a 1. A ⊃ B con ditional statement in the prem-2. A ises, consider obtaining that letter via �� C 3. ∼ ∼ A / Cmodus ponens: 4. C 2, 3, DS 1. A ⊃ B Of course, these strategies apply to deriving any 2. C �� A line prior to the conclusion, just as they apply to 3. A / B deriving the conclusion. 4. B 1, 3, MP

Exercise 7.1

I. For each of the following lists of premises, derive the conclusion and supply the justifi-

cation for it. There is only one possible answer for each problem.

★ (1) 1. G ⊃ F(3) 1. R ⊃ D

2. ∼ ∼F2. E ⊃ R3. _______ ____ 3. _______ ____

(2) 1. S ★(4) 1. B �� C

2. S ⊃ M2. ∼ ∼B3. _______ ____ 3. _______ ____

SectionSection 7.1 7.1 Rules of Implication I Rules of Implication I 341341

(5) 1. N3. ∼ ∼ ∼ ∼F

2. N �� F4. F �� ∼ ∼L3. N ⊃ K5. _______ ____

4. _______ ____ (14) 1. N ⊃ ∼ ∼E

(6) 1. ∼ ∼J �� P2. ∼ ∼ ∼ ∼S

2. ∼ ∼J3. ∼ ∼E �� ∼ ∼S3. S ⊃ J4. ∼ ∼S �� N4. _______ ____ 5. _______ ____

★(7) 1. H ⊃ D(15) 1. ∼ ∼R ⊃ ∼ ∼T

2. F ⊃ T2. ∼ ∼T �� B3. F ⊃ H3. C ⊃ ∼ ∼R4. _______ ____ 4. ∼ ∼C

(8)5. _______ ____ 1. S ⊃ W

2. ∼ ∼S ★(16) 1. ∼ ∼K3. S �� N2. ∼ ∼K ⊃ ∼ ∼P

(9)4. G ⊃ Q1. F ⊃ ∼ ∼ A5. 4. _______ ____ 3. ∼ ∼K �� G

2. _______ ____ N ⊃ A

3. ∼ ∼F(17) 1. F �� (A ⊃ C ) 4. ∼ ∼A2. A �� (C ⊃ F )

★ 4. ∼ ∼F(10) 1. H ⊃ A5. 5. _______ ____ 3. A

2. _______ ____ A

3. A �� M(18) 1. (R ⊃ M) ⊃ D4. G ⊃ H2. M ⊃ C

(11)4. ∼ ∼M1. W �� B5. 5. _______ ____ 3. D ⊃ (M �� E )

2. _______ ____ W

3. B ⊃ T ★(19) 1. (S �� C ) ⊃ L4. W ⊃ A2. ∼ ∼S

(12)4. S ⊃ (K ⊃ L) 1. K ⊃ ∼ ∼ R5. 5. _______ ____ 3. ∼ ∼L

2. _______ ____ ∼ ∼ R

3. R �� S(20) 1. (A �� W ) ⊃ (N ⊃ Q) 4. R ⊃ T2. Q ⊃ G

★ 4. (Q ⊃ G) ⊃ (A �� N ) (13) 1. ∼ ∼ C ⊃ ∼ ∼ F5. 5. _______ ____ 3. ∼ ∼A

2. _______ ____ L ⊃ F

342342ChapterChapter77 Natural Deduction in�Propositional Logic Natural Deduction in�Propositional Logic

II. The following symbolized arguments are missing a premise. Write the premise

needed to derive the conclusion (last line ), and supply the justification for the con-clusion. Try to construct the simplest premise needed to derive the conclusion.

★(1) 1. B �� K(11) 1. T ⊃ N

2. ______ 2. G ⊃ T

★ 1. C ⊃ H ★(13) 1. ∼ ∼S ⊃ ∼ ∼B 2. ______ 2. R �� ∼ ∼ B 3. R ⊃ H ____ 3. ∼ ∼ B ⊃ ∼ ∼ S(5) 1. F ⊃ N 4. ______ 2. N ⊃ T 5. ∼ ∼ ∼ ∼ B ____ 3. (4) 2. K ⊃ T 3. T �� ∼ ∼G ______ 4. ______ 3. ∼ ∼ K ____ 5. G ____ ______ (14) 1. ∼ ∼ R ⊃ D 4. ∼ ∼ F ____ 2. (3) 2. F ⊃ T ____ ______ G 3. S ____ (12) 1. ⊃ C 2. M �� G1. (2) 4. ______ 1. N ⊃ S 5. 3. K ____ 3. H �� T ∼ ∼J ⊃ ∼ ∼R

(6) 1. W �� T 3. N �� ∼ ∼R

2. A ⊃ W 4. ______

★ 4. A ⊃ T ____ (15) 1. ∼ ∼S �� ∼ ∼P M 3. ______ 5. ∼ ∼F ⊃ ∼ ∼R ____

(7) 1. ⊃ B 2. ∼ ∼K ⊃ P

2. Q ⊃ M 3. ∼ ∼P ⊃ F 3. ______ 4. ______ 4. M ____ 5. ∼ ∼P ____

(8) 1. C �� L ★(16) 1. J ⊃ E

2. L ⊃ T 2. B �� ∼ ∼J 3. ______ 3. ∼ ∼Z ⊃ J 4. L ____ 4. ______

(9) 5. J ____ 1. E ⊃ N

2. T �� ∼ ∼E(17) 1. (H ⊃ C ) ⊃ A 3. S ⊃ E 2. N ⊃ (F ⊃ K ) 4. ______ 3. (E ● R) ⊃ K 5. E ____ 4. ______

★ (10) 5. H ⊃ K ____ 1. H ⊃ A

2. S ⊃ H(18) 1. (S ⊃ M) ⊃ G 3. ∼ ∼M �� H 2. S ⊃ (M ● G ) 4. ______ 3. G ⊃ (R ⊃ ∼ ∼S ) 5. ∼ ∼H ____ 4. ______

5. ∼ ∼S ____

SectionSection 7.1 7.1 Rules of Implication I Rules of Implication I 343343

★(19) 1. (W �� ∼ ∼F ) ⊃ H(20) 1. (H ● A) �� T

2. (H �� G ) ⊃ ∼ ∼F 2. ∼ ∼S ⊃ (P ⊃ T ) 3. T ⊃ (F ⊃ G ) 3. (N �� T ) ⊃ P 4. ______ 4. ______ 5. ∼ ∼F ____ 5. T ____

III. Use the first four rules of inference to derive the conclusions of the following sym-

bolized arguments.

★(1) 1. ∼ ∼C ⊃ (A ⊃ C ) (11) 1. G ⊃ [∼ ∼O ⊃ (G ⊃ D)]

(2) 3. ∼ ∼O / D1. F �� ( D ⊃ T ) ∼ 2. ∼ ∼C / ∼ ∼A 2. O �� G

2. ∼ ∼F(12) 1. ∼M �� (B �� ∼ ∼T )

★ 2. ∼ ∼W / ∼ ∼T ∼ ∼ ( K ● B ) ⊃ 3. ∼ ∼ E / ∼ ∼ L ★ (13) 1. R ( G �� ∼ ∼ A ) 2. ( G �� ∼ ∼ A ) ⊃ ∼ ∼S(4) (3) 3. ∼ ∼ ∼ ∼M1. ( K ● B ) �� ( L ⊃ E ) 4. 3. 2. B ⊃ W D / T(5) 2. 1. P ⊃ (G ⊃ T ) 3. G ⊃ S Q ⊃ ( T ⊃ E ) 4. R/ ∼ ∼A 3. P 4. Q / G ⊃ E(14) 1. ( L K N ) ⊃ C 2. ( L K N ) �� ( P ⊃ ∼ ∼E ) 1. ∼ ∼ W ⊃ [ ∼ ∼ W ⊃ ( X ⊃ W )] 3. ∼ ∼ E ⊃ C 2. ∼ ∼ W / ∼ ∼ X 4. ∼ ∼ C / ∼ ∼ P(6) 1. J ⊃ ( K ⊃ L ) (15) 1. ∼ ∼ J ⊃ [ ∼ ∼ A ⊃ ( D ⊃A)] 2. L �� J 2. J �� ∼ ∼ A 3.

★ ★ ∼ ∼L / ∼ ∼K ∼ 3. ∼J / ∼ ∼D(7) 1. ∼ ∼ S ⊃ D ★ (16) 1. ( B ⊃ ∼ ∼ M ) ⊃ (T ⊃ ∼ ∼S ) 2. ∼ ∼ S �� ( ∼ ∼ D ⊃ K ) 2. B ⊃ K 3. ∼ ∼ D / K 3. K ⊃ ∼ ∼ M(8) 1. A ⊃ ( E ⊃ ∼ ∼ F ) 4. ∼ ∼ S ⊃ N / T ⊃ N 2. H �� ( ∼ ∼ F ⊃ M ) (17) 1. H �� ( Q �� F ) 3. A 2. R 4. ∼ ∼ H / E ⊃ M �� ( Q ⊃ R ) 3. R �� ∼ ∼ H(9) 1. ∼ ∼ G ⊃ ( G �� ∼ ∼ A) 4. ∼ ∼ R / F 2. ∼ ∼ A ⊃ ( C ⊃ A ) 3. ∼ ∼ G / ∼ ∼ (18) 1. ∼ ∼ A ⊃ ( B ⊃ ∼ ∼ C ) C 2. ∼ ∼ D ⊃ ( ∼ ∼ C ⊃ A ) (10) 1. N ⊃ ( J ⊃ P ) 3. D �� ∼ ∼ A 2. ( J ⊃ P ) ⊃ ( N ⊃ J ) 4. ∼ ∼ D / ∼ ∼ B 3. N / P

344344ChapterChapter77 Natural Deduction in�Propositional Logic Natural Deduction in�Propositional Logic

★ (19) 1. ∼ ∼G ⊃ [G �� (S ⊃ G )] (23) 1. (R ⊃ F ) ⊃ [(R ⊃ ∼ ∼G) ⊃ (S ⊃

2. (S �� L) ⊃ ∼ ∼GQ)]

(20) 3. ∼ ∼G ⊃ F1. H ⊃ [ ∼ ∼ E ⊃ ( C ⊃ ∼ ∼ D )] 4. 3. S �� L / L 2. (Q ⊃ F ) ⊃ (R ⊃ Q)

2. Q ⊃ ∼ ∼G / S ⊃ F ∼ ∼ D ⊃ E

3. E �� H(24) 1. ∼ ∼A ⊃ [A �� (T ⊃ R)]

(21) 3. (T �� D) ⊃ ∼ ∼R1. ∼ ∼ B ⊃ [( A ⊃ K ) ⊃ ( B �� ∼ ∼ K )] 4. 4. ∼ ∼E / ∼ ∼C 2. ∼ ∼R ⊃ [R �� (A ⊃ R)]

2. T �� D / D ∼ ∼ J ⊃ K

3. A ⊃ ∼ ∼J ★(25) 1. ∼ ∼N ⊃ [(B ⊃ D) ⊃ (N �� ∼ ∼E )]

★ 3. B ⊃ D(22) 1. ( C ⊃ M ) ⊃ ( N ⊃ P ) 4. 4. ∼ ∼B/ ∼ ∼A 2. (B ⊃ E ) ⊃ ∼ ∼N

2. D ⊃ E / ∼ ∼D ( C ⊃ N ) ⊃ ( N ⊃ M ) 3. (C ⊃ P ) ⊃ ∼ ∼M

4. C ⊃ N/ ∼ ∼C

IV. Translate the following arguments into symbolic form and use the first four rules

of inference to derive the conclusion of each. The letters to be used for the sim-ple statements are given in parentheses after each exercise. Use these letters in the order in which they are listed.

★1. If OxyContin is addictive, then either the pharmaceutical industry has stopped

making it or thousands of patients are becoming addicted to it. The pharmaceuti-cal industry has not stopped making OxyContin. Also, OxyContin is addictive. Therefore, thousands of patients are becoming addicted to it. (O, P, T)

2. If people who are poor live as long as people who are rich, then longevity is

unaffected by wealth. Either the less affluent have stopped smoking or it is not the case that longevity is unaffected by wealth. It is not the case that the less affluent have stopped smoking. Therefore, people who are poor do not live as long as people who are rich. (P, L, S)

3. If wind energy is available only if grid operators purchase it, then batteries

will store extra energy only if consumers will realize savings. If wind energy is available, then batteries will store extra energy. Batteries will store extra energy only if grid operators purchase wind energy. Therefore, if wind energy is avail-able, then consumers will realize savings. (W, G, B, C)

★4. If teachers are allowed to conduct random drug searches on students only

if teachers are acting in loco parentis, then if teachers are acting in loco parentis, then students have no Fourth Amendment protections. Either stu-dents have no Fourth Amendment protections or if teachers are allowed to conduct random drug searches on students, then teachers are acting in loco parentis. It is not the case that students have no Fourth Amendment protec-tions. Therefore, teachers are not allowed to conduct random drug searches on students. (R, L,�F )

5. Either funding for nuclear fusion will be cut or if sufficiently high temper-

atures are achieved in the laboratory, nuclear fusion will become a reality.

SectionSection 7.1 7.1 Rules of Implication I Rules of Implication I 345345

Either the supply of hydrogen fuel is limited, or if nuclear fusion becomes a reality, the world�s energy problems will be solved. Funding for nuclear fusion will not be cut. Furthermore, the supply of hydrogen fuel is not limited. Therefore, if sufficiently high temperatures are achieved in the laboratory, the world�s energy problems will be solved. (C, H, R, S, E )

6. Either the continents are not subject to drift or if Antarctica was always located

in the polar region, then it contains no fossils of plants from a temperate climate. If the continents are not subject to drift, then Antarctica contains no fossils of plants from a temperate climate. But it is not the case that Antarctica contains no fossils of plants from a temperate climate. Therefore, Antarctica was not always located in the polar region. (D, L, F )

★7. If terrorists take more hostages, then terrorist demands will be met if and only if

the media give full coverage to terrorist acts. Either the media will voluntarily limit the flow of information or if the media will recognize they are being exploited by terrorists, they will voluntarily limit the flow of information. Either the media will recognize they are being exploited by terrorists or ter-rorists will take more hostages. The media will not voluntarily limit the flow of information. Therefore, terrorist demands will be met if and only if the media give full coverage to terrorist acts. (H, D, A, V, R)

8. Either we take recycling seriously or we will be buried in garbage. If we incin-

erate our garbage only if our health is jeopardized, then we do not take recy-cling seriously. If our landfills are becoming exhausted, then if we incinerate our garbage, then toxic ash will be produced. If toxic ash is produced, then our health is jeopardized. Our landfills are becoming exhausted. Therefore, we will be buried in garbage. (R, B, I, H, L, T )

9. If fossil fuel companies promote climate disinformation only if millions of

people die, then either industry executives are innocent or they are guilty of racketeering. If fossil fuel companies promote climate disinformation, then the climate crisis will worsen. If the climate crisis worsens, then millions of people will die. Industry executives are not innocent. Therefore, industry executives are guilty of racketeering. (F, M, I, G, W)

★10. If the death penalty is not cruel and unusual punishment, then either it is cruel

and unusual punishment or if society is justified in using it, then it will deter other criminals. If the death penalty is cruel and unusual punishment, then it is both cruel and unusual and its use degrades society as a whole. It is not the case that both the death penalty is cruel and unusual and its use degrades society as a whole. Furthermore, the death penalty will not deter other criminals. There-fore, society is not justified in using the death penalty. (C, J, D, U )

V. Create four symbolized arguments consisting of at least three premises each,

and whose proof requires at least two of the rules introduced in this section. The arguments should look like those in Part III of this section. Then derive the conclusion of each.

346346ChapterChapter77 Natural Deduction in�Propositional Logic Natural Deduction in�Propositional Logic

7.2 Rules of Implication II

Four additional rules of implication are listed presented in Section 7.4, any argument that is a sub-here. Constructive dilemma should be familiar stitution instance of the destructive dilemma form from Chapter 6. The other three are new. They are can be easily converted into a substitution instance of listed together with an illustration of their use as constructive dilemma. Destructive dilemma, there-follows.p fore, is not needed as a rule of inference.

5. Constructive dilemma (CD) These arguments are both instances of

(pconstructive dilemma (CD): ⊃ q) ● (r ⊃ s) If Oscar is a dog, then you�ll

have fleas, and if Oscar is a

cat, then you�ll have fur balls.

p �� r Oscar is either a dog or a cat.

q �� s You�ll have either fleas or fur balls. Simplification states that if two propositions

6. Simplification (Simp) are given as true on a single line, then each of them

p is true separately. According to the strict interpre-● q Eliza has long legs and runs fast. tation of the simplification rule, only the left-hand p Eliza has long legs. conjunct may be stated in the conclusion. Once the

7. Conjunction (Conj ) commutativity rule for conjunction has been pre-

p sented, however (see Section 7.3), we will be justi-Roxy has big eyes. fied in replacing a statement such as H ● K with q Roxy has a tail. K ● H . Once we do this, the K will appear on the p ● q Roxy has big eyes and a tail. left, and the appropriate conclusion is K .

8. Addition (Add)

p These arguments are all instances of Theo has spots.

simplification (Simp):

p �� q Theo has either spots or stripes.

Like the previous four rules, these are fairly easy to

understand, but if there is any doubt about them their Conjunction states that two propositions�for validity may be proven by means of a truth table. example, H and K�asserted separately on differ-

as involving two modus ponens steps. The first premise propositions may be conjoined in whatever order we states that if we have Constructive dilemma can be understood ent lines may be conjoined on a single line. The two

p then we have q, and if we have choose (either H ● K or K ● H ) without appeal to the

r then we have s. But since, by the second premise, commutativity rule for conjunction. we do have either p or r, it follows by modus ponens that we have either q or s. Constructive dilemma is the These arguments are all instances of conjunction (Conj): only form of dilemma that will be included as a rule of

inference. By the rule of transposition, which will be

pSome textbooks include a rule called absorption by which the state-Addition states that whenever a proposition ment form p ⊃ (q ● p) is deduced from p ⊃ q. This rule is necessary is asserted on a line by itself it may be joined dis-conditional proof, to be introduced shortly. junctively with any proposition we choose. In other only if conditional proof is not presented. This volume opts in favor of

Section 7.2 Rules of Implication II 347

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). words, if G is asserted to be true by itself, it follows Seeing that L �� T does not appear as such in that G �� H is true. This may appear somewhat puz- the premises, we look at the main operator of zling at first, but once one realizes that G �� H is a the conclusion, which is a wedge. This makes much weaker statement than G by itself, the puzzle- us think of constructive dilemma and addition, ment should disappear. The new proposition must, each of which would give us a statement contain-of course, always be joined disjunctively (not con- ing a wedge. Finding L and T as the consequents junctively) to the given proposition. If G is stated on of two distinct conditional statements causes us a line by itself, we are not justified in writing G ● H as to think that the conclusion can be derived via a consequence of addition. constructive dilemma. If a constructive dilemma

can be set up, it will need a disjunctive state-

ment as its second premise, and such a state-

These arguments are all instances of addition (Add): ment appears on line 4. Furthermore, each of the

components of this statement, K and M, appears as the antecedent of a conditional statement,

exactly as they both should for a dilemma. The

trated. Consider the following argument: The use of these four rules may now be illus- only statement that is missing now is M ⊃ T. Inspecting line 2 we see that we can obtain M ⊃ N via simplification, and putting this together with

1. A ⊃ B line 3 gives us M ⊃ T via hypothetical syllogism. 2. (B �� C ) ⊃ (D ● E ) The completed proof may now be written out: 3. A / D

As usual, we begin by looking for the conclusion in 1. K ⊃ L the premises. 2. (M ⊃ N ) ● SD appears in the consequent of the second premise, which we can derive via simplifica- 3. N ⊃ T tion if we first obtain 4. K �� M / L �� TB �� C . This expression as such

does not appear in the premises, but from lines�1 6. M 5. M ⊃ N 2, Simp and 3 we see that we can derive B by itself via modus 7. (K ⊃ T 3, 5, HS ponens. Having obtained B, we can derive B �� C via 8. L �� ⊃ L) ● (M ⊃ T ) 1, 6, Conj T 4, 7, CD

addition. The proof has now been thought through

and can be written out as follows: Another example:

1. A 1. ∼ ∼M ● N ⊃ B

2. (B �� C ) ⊃ (D ● 2. P ⊃ ME )

3. A / D 3. Q ● R 4. B 1, 3, MP 4. (∼ ∼P ● Q ) ⊃ S / S �� T

6. D ● �� When we look for 5. B �� C 4, Add ES T in the premises we find S in 2, 5, MP 7. D 6, Simp the consequent of line 4 but no T at all. This signals

Another example: of an argument contains a letter not found in the an important principle: Whenever the conclusion

1. K ⊃ L premises, addition must be used to introduce the 2. (M ⊃ N ) ● S missing letter. Addition is the only rule of infer-3. N ⊃ T ence that can introduce new letters. To introduce 4. K �� M / L �� TT by addition, however, we must first obtain S on a

348Chapter7 Natural Deduction in�Propositional Logic

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). line by itself. S can be derived from line 4 via modus 1. P �� (S ● T ) ponens if we first obtain ∼ ∼P ● Q. This, in turn, can 2. S 1, Simp (invalid�S ● T must be derived via conjunction, but first ∼ ∼P and Q first be obtained on a line must be obtained individually on separate lines. Q by itself )

can be derived from line 3 via simplification and 1. K ∼ ∼P from line 2 via modus tollens, but the latter 2. K ● L 1, Add (invalid�the conclu-step requires that we first obtain ∼ ∼M on a line sion must be a disjunctive

simplification, the proof is now complete. It may 1. M �� N be written out as follows: by itself. Since this can be derived from line 1 via statement)

2. M 1, Simp (invalid�simplifica-

1. ∼ ∼M ● N tion is possible only with a 2. P ⊃ M conjunctive premise; line 1 3. Q ● R is a disjunction)

5. 4. ( ∼ ∼P ● Q ) ⊃ S / S �� T 1. G ⊃ H ∼ ∼ M 1, Simp 2. G ⊃ (H �� J ) 1, Add (improper�J must 6. ∼ ∼ P 2, 5, MT be added to the whole line, 7. Q 3, Simp not just to the consequent 8. ∼ ∼ P ● Q 6, 7, Conj of line 1) 9. S 4, 8, MP

10. S 1. L ⊃ M �� T 9, Add

2. L ⊃ N

Addition is used together with disjunctive syllo- 3. M ● N 1, 2, Conj (invalid�M and

gism to derive the conclusion of arguments having N must first be obtained on inconsistent premises. As we saw in Chapter 6, such lines by themselves)

arguments are always valid. The procedure is illus- 1. ∼ ∼(P ● Q) trated as follows: 2. ∼ ∼P 1, Simp (invalid�parenthe-

ses must be removed first )

1. S

2. ∼ ∼S / T 1. ∼ ∼(P �� Q) 3. S �� T 1, Add 2. ∼ ∼P 4. T 2, 3, DS 3. Q 1, 2, DS (invalid�parentheses

must be removed first )

With arguments of this sort, the conclusion is

always introduced via addition and then separated The use of addition in the G ⊃ H example is via disjunctive syllogism. Since addition can be called �improper� because the letter that is added is used to introduce any letter or arrangement of let- not added to the whole line. It turns out, however, ters we choose, it should be clear from this example that even though the addition rule is not correctly that inconsistent premises validly entail any con- applied here, the inference is still valid. Hence, this clusion whatever. inference is not called �invalid,� as the others are.

To complete this presentation of the eight rules As for the last two examples, a rule will be pre-

of implication, let us consider some of the typical sented in the next section (De Morgan�s rule) that ways in which they are misapplied. Examples are will allow us to remove parentheses preceded by as follows: negation signs. But even after the parentheses have

Section 7.2 Rules of Implication II 349

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). been removed from these examples, the inferences Strategy 8: If the conclusion is a disjunctive state-remain invalid. ment, consider obtaining it via con-

a few strategies for applying the last four rules of 1. (A ⊃ B) ● (C ⊃ D) implication: Like the previous section, this one ends with structive dilemma or addition:

2. B ⊃ C

Strategy 6: 3. AC / B If the conclusion contains a letter that �� D

appears in a conjunctive statement 4. B �� D 1, 3, CD

in the premises, consider obtaining 1. A �� C that letter via simplification: 2. B

1. A 3. C ⊃ D / B �� D ⊃ B 4. B �� D 2, Add 2. C ● B

3. C ⊃ A / CStrategy 9: If the conclusion contains a letter not 4. C 2, Simp found in the premises, addition must

Strategy 7: be used to introduce that letter. If the conclusion is a conjunctive statement, consider obtaining it via Strategy 10: Conjunction can be used to set up conjunction by first obtaining the indi-constructive dilemma: vidual conjuncts:

1. A ⊃ B

1. A ⊃ C 2. C ⊃ D 2. B 3. A �� C / B �� D 3. ∼ ∼C / B ● ∼ ∼C 4. (A ⊃ B ) ● (C ⊃ D) 1, 2, Conj 4. B ● ∼ ∼C 2, 3, Conj 5. B �� D 3, 4, CD

Exercise 7.2

I. For each of the following lists of premises, derive the indicated conclusion and com-

plete the justification. In problems 4 and 8 you can add any statement you choose.

★(1) 1. S �� H ★(4) 1. H

3. R ⊃ G(5) 1. R ● (N �� K ) 4. 2. B ● E 2. ____ ____ , Add

_____ ____ , Simp 2. (G ● T ) �� S

(2) 1. (N ⊃ T ) ● (F ⊃ Q) 3. (Q ● C ) ⊃ ( J ● L)

3. N �� F(6) 1. ∼ ∼R �� P 4. 2. (N ⊃ R) �� (F ⊃ M ) 4. _____________ ____ , Simp

_______________ ____ , CD 2. (P ⊃ ∼ ∼D) ● (∼ ∼R ⊃ S )

(3) 1. D 3. (∼ ∼R ⊃ A) ● (P ⊃ ∼ ∼N )

2. W 4. _____________ ____ , CD 3. ____ ____ , Conj

350350ChapterChapter77 Natural Deduction in�Propositional Logic Natural Deduction in�Propositional Logic

★(7) 1. (Q �� K ) ● ∼ ∼B(9) 1. ∼ ∼B ⊃ 2. ( M ● R ) D 2. F ��N 3. ( W ● S ) �� ( G ● F ) 3. ____ ____ , Conj 4. _____________ ____ , Simp ★ (10) 1. S �� ∼ ∼ C(8) 1. E ● G 2. ( S ⊃ ∼ ∼ L ) ● ( ∼ ∼ C ⊃ M ) 2. ______ ____ , Add 3. ( ∼ ∼ N ⊃ S ) ● ( F ⊃ ∼ ∼ C ) 4. ______________ ____ , CD

II. In the following symbolized arguments, derive the line needed to obtain the

conclusion (last line), and supply the justification for both lines.

★ (6) 1. N(1) 1. G ⊃ N

2. 2. N ⊃ F G ● K

3. 3. (N ⊃ A) ______ ____● (F ⊃ C )

(2) 1. 5. A �� C ____ ∼ ∼ A ★ 4. 4. ______________ ____ G �� T ____

2. A �� E(7) 1. J 3. 2. ∼ ∼L ______ ____

(3) 1. 4. _____ ____ B ⊃ N 5. 4. 3. F ⊃ L ∼ ∼ A ● E ____ 2. ∼ ∼F B ● J ____ �� K(8) 1. ( E ⊃ B ) 3. K ⊃ R ● ( Q ⊃ N ) 2. K ⊃ E 4. _____ ____ 3.

★ 4. _______________ ____ (4) 1. T 5. 5. B ⊃ K N �� R ____

2. E ⊃ K ____ T ⊃ G

3. (T �� U ) ⊃ H(9) 1. G �� N 4. ___________ ____ 2. ∼ ∼G

(5) 1. 4. ___________ ____ S ⊃ E 5. 5. H ____ 3. ∼ ∼G ⊃ (H ● R)

2. H ____ E �� ( S ● P )

3. ∼ ∼E ★(10) 1. M 4. ________ ____ 2. (M ● E ) ⊃ D 5. S ____ 3. E

4. __________ ____ 5. D ____

III. Use the first eight rules of inference to derive the conclusions of the following

symbolized arguments:

★ (3) 1. E ⊃ (A(1) 1. ∼ ∼ M ⊃ Q ● C )

2. 2. A ⊃ (F R ⊃ ∼ ∼ T ● E ) 3. 3. E / F ∼ ∼ M �� R / Q �� ∼ ∼ T

(2) ★(4) 1. (H �� ∼ ∼B) ⊃ R1.N ⊃ ( D ● W ) �� 2. ( H ∼ ∼ M ) ⊃ P 2. D ⊃ KR 3. N / N ● 3. H / K ● P

SectionSection 7.2 7.2 Rules of Implication II Rules of Implication II 351351

★ ★(5) 1.G ⊃ (S ● T ) ★(19) 1. (U ● ∼∼ ∼∼P ) ⊃ Q �� 2. ( S T ) ⊃ J 2. ∼ ∼ O ⊃ U 3. G / J 3. ∼ ∼ P ⊃ O 4. ∼ ∼ O(6) 1. ( L �� T ) ⊃ ( B ● ● T / Q G ) 2. L ● ( K K R ) / L ● B(20) 1. ( M �� N ) ⊃ ( F ⊃ G ) 2. D ⊃ ∼ ∼ C(7) 1. ( ∼ ∼ F �� X ) ⊃ ( P �� T ) 3. ∼ ∼ C ⊃ B2. F ⊃ P 4. M 3. ∼ ∼ P / T ● H 5. D �� F / B�� G(8) 1. ( N ⊃ B ) ● ( O ⊃ C ) 2. Q ⊃ ( N �� O ) (21) 1. ( F ● M ) ⊃ ( S �� T ) 2. ( ∼ ∼ S �� A ) ⊃ F 3. Q / B �� C 3. ( ∼ ∼ S �� B ) ⊃ M(9) 1. ( U �� W ) ⊃ ( T ⊃ R ) 4. ∼ ∼ S ● G / T 2. U ● H ★ (22) 1. ( ∼ ∼ K ● ∼ ∼ N ) ⊃ 3. ∼ ∼ R ● ∼ ∼ J / U ● ∼ ∼ T [( ∼ ∼ P ⊃ K ) ● ( ∼ ∼ R ⊃ G )] (10) 1. ( D �� E ) ⊃ ( G ● H ) 2. K ⊃ N2. G ⊃ ∼ ∼ D 3. ∼ ∼ N ● B 3. D ● F / M 4. ∼ ∼ P �� ∼ ∼ R / G(11) 1. ( B �� F ) ⊃ ( A ⊃ G ) (23) 1. ( ∼ ∼ A �� D ) ⊃ ( B ⊃ F ) 2. ( B �� E ) ⊃ ( G ⊃ K ) 2. ( B �� C ) ⊃ ( A ⊃ E ) 3. B ● ∼ ∼ H / A ⊃ K 3. A �� B(12) 1. ( P ⊃ R ) ⊃ ( M ⊃ P ) 4. ∼ ∼ A / E �� F 2. ( P �� M ) ⊃ ( P ⊃ R ) (24) 1. ( J ⊃ K ) ● ( ∼ ∼ O ⊃ ∼ ∼ P ) 3. P �� M / R �� P 2. ( L ⊃ J ) ● ( ∼ ∼ M ⊃ ∼ ∼ O ) (13) ★ 1. ( C ⊃ N ) ● E 3. ∼ ∼ K ⊃ ( L �� ∼ ∼ M ) 2. D �� ( N ⊃ D ) 4. ∼ ∼ K ● G / ∼ ∼ P 3. ∼ ∼ D / ∼ ∼ C �� P ★ (25) 1. ( ∼ ∼ M ● ∼ ∼ N ) ⊃ [( ∼ ∼ M �� H ) ⊃ (K ● L )] (14) 1. F ⊃ ( ∼ ∼ T ● A ) 2. ∼ ∼ M ● ( C ⊃ D ) 2. ( ∼ ∼ T �� G ) ⊃ ( H ⊃ T ) 3. ∼ ∼ N ● ( F K G ) / K ● ∼ ∼ N 3. F ● O / ∼ ∼ H ● ∼ ∼ T(26) 1. ( P �� S ) ⊃ ( E ⊃ F ) 2. ( P �� T ) ⊃ ( G ⊃ H ) (15) 1. ( ∼ ∼ S �� B ) ⊃ ( S �� K ) 2. ( K �� ∼ ∼ D ) ⊃ ( H ⊃ S ) 3. ( P �� U ) ⊃ ( E �� G ) 3. ∼ ∼ S ● W / ∼ ∼ H 4. P / F �� H

★ (27) 1. (S ⊃ Q) ● (Q ⊃ ∼ ∼S) (16) 1. ( C �� ∼ ∼ G ) ⊃ ( ∼ ∼ P ● L )

2. 2. S ( ∼ ∼ P ● C ) ⊃ ( C ⊃ D ) �� Q 3. 3. ∼ ∼Q / P C ● ● ∼ ∼ R R / D �� R

(17) 1. ★(28) 1. (D ⊃ B) ● (C ⊃ D ) [ A �� ( K ● J )] ⊃ ( ∼ ∼ E ● ∼ ∼ F ) 2. 2. M ⊃ [ A ● ( B ⊃ D ) ● ( E ⊃ C ) ( P �� R )] 3. B �� E / D �� B 3. M ● U / ∼ ∼ E ● A(29) 1. ( R ⊃ H ) ● ( S ⊃ I ) (18) 1. ∼ ∼ H ⊃ ( ∼ ∼ T ⊃ R ) 2. ( ∼ ∼ H ● ∼ ∼ L ) ⊃ ( R �� S ) 2. H �� ( E ⊃ F ) 3. ∼ ∼ H 3. ∼ ∼ T ● ( K ⊃ T ) �� E 4. H �� ∼ ∼ L / I �� M 4. ∼ ∼ H ● D / R �� F

352352ChapterChapter77 Natural Deduction in�Propositional Logic Natural Deduction in�Propositional Logic

(30) 1. 4. Q ( W ● X ) ⊃ ( Q �� R ) �� S ∼ 5. ∼ Q ● H / R 2. ( S �� F ) ⊃ ( Q �� W ) 3. ( S �� G ) ⊃ ( ∼ ∼ Q ⊃ X )

IV. Translate the following arguments into symbolic form and use the first eight rules

of inference to derive the conclusion of each. Use the letters in the order in which they are listed.

★ 1. If topaz is harder than quartz, then it will scratch quartz and also feldspar.

Topaz is harder than quartz and it is also harder than calcite. Therefore, either topaz will scratch quartz or it will scratch corundum. (T, Q, F, C, O)

2. If clear-cutting continues in primary forests and the Endangered Species Act

is not repealed, then either the Endangered Species Act will be repealed or thousands of animal species will become extinct. Clear-cutting continues in primary forests. The Endangered Species Act will not be repealed. Therefore, thousands of animal species will become extinct. ( C, E, T )

3. If either executive salaries are out of control or exorbitant bonuses are

paid, then either shareholders will be cheated or ordinary workers will be paid less. Executive salaries are out of control and the rich are get-ting richer. If shareholders are cheated, then future investors will stay away; also, if ordinary workers are paid less, then consumer spending will decline. If either future investors stay away or consumer spending declines, then the economy will suffer. Therefore, the economy will suffer. (S, B, C, P, R, F, D, E )

★ 4. Either animals are mere mechanisms or they feel pain. If either animals feel

pain or they have souls, then they have a right not to be subjected to needless pain and humans have a duty not to inflict needless pain on them. It is not the case that animals are mere mechanisms. Therefore, animals have a right not to be subjected to needless pain. ( M, P, S, R, D)

5. If half the nation suffers from depression, then if either the insurance compa-

nies have their way or the psychiatrists have their way, then everyone will be taking antidepressant drugs. If either half the nation suffers from depression or sufferers want a real cure, then it is not the case that everyone will be taking antidepressant drugs. Half the nation suffers from depression. Therefore, it is not the case that either the insurance companies or the psychiatrists will have their way. (H, I, P, E, W )

6. If either parents get involved in their children�s education or the school year

is lengthened, then if the children learn phonics, their reading will improve and if they are introduced to abstract concepts earlier, their math will im-prove. If either parents get involved in their children�s education or nebu-lous subjects are dropped from the curriculum, then either the children will learn phonics or they will be introduced to abstract concepts earlier. Parents will get involved in their children�s education, and writing lessons

SectionSection 7.2 7.2 Rules of Implication II Rules of Implication II 353353

will be integrated with other subjects. Therefore, either the children�s read-ing or their math will improve. (P, S, L, R, I, M, N, W )

★7. If either manufacturers will not concentrate on producing a superior product

or they will not market their product abroad, then if they will not concentrate on producing a superior product, then the trade deficit will worsen. Either manufacturers will concentrate on producing a superior product or the trade deficit will not worsen. Manufacturers will not concentrate on producing a superior product. Therefore, today�s business managers lack imagination. (C, M, T, B )

8. If either medical fees or malpractice awards escape restrictions, then healthcare

costs will soar and millions of low-income people will go uninsured. If the lawyers get�their way, then malpractice awards will escape restrictions. If the doctors get their way, then medical fees will escape restrictions. Either the doctors or the lawyers will get their way, and insurance companies will resist reform. Therefore, healthcare costs will soar. (F, A, H, P, L, D, I )

9. If we are less than certain the human fetus is a person, then we must give it

the benefit of the doubt. If we are certain the human fetus is a person, then we must accord it the right to live. If either we must give the fetus the benefit of the doubt or accord it the right to live, then we are not less than certain the fetus is human and it is not merely a part of the mother�s body. Either we are less than certain the human fetus is a person or we are certain about it. If we are certain the human fetus is a person, then abortion is immoral. Therefore, abortion is immoral. (L, G, C, A, M, I )

★ 10. If the assassination of terrorist leaders violates civilized values and also is not

effective in the long run, then if it prevents terrorist atrocities, then it is effec-tive in the long run. If the assassination of terrorist leaders violates civilized values, then it is not effective in the long run. The assassination of terrorist leaders violates civilized values and is also illegal. If the assassination of ter-rorist leaders is not effective in the long run, then either it prevents terrorist atrocities or it justifies acts of revenge by terrorists. Therefore, the assassination of terrorist leaders justifies acts of revenge by terrorists and also is not effective in the long run. (V, E, P, I, J )

V. Create four symbolized arguments consisting of at least three premises each, and

whose proof requires at least two of the rules introduced in this section. The argu-ments should look like those in Part III of this section. Then derive the conclusion of each.

354354ChapterChapter77 Natural Deduction in�Propositional Logic Natural Deduction in�Propositional Logic

7.3 Rules of Replacement I

Unlike the rules of implication, which are basic The rule may be summarized as follows: When argument forms, the ten rules of replacement moving a tilde inside or outside a set of parentheses, a are expressed in terms of pairs of logically equiva- dot switches with a wedge and vice versa. lent statement forms, either of which can replace the Commutativity asserts that the truth value of a other in a proof sequence. To express these rules, conjunction or disjunction is unaffected by the order a new symbol, called a double colon (::), is used in which the components are listed. In other words, to designate logical equivalence. This symbol is a the component statements may be commuted, or metalogical symbol in that it makes an assertion not switched for one another, without affecting the truth about things but about symbolized statements: It value. The validity of this rule should be immedi-asserts that the expressions on either side of it have ately apparent. You may recall from arithmetic that the same truth value regardless of the truth values of the commutativity rule also applies to addition and their components. Underlying the use of the rules of multiplication and asserts, for example, that 3 + 5 replacement is an axiom of replacement, which equals 5 + 3, and that 2 � 3 equals 3 � 2. However, it asserts that within the context of a proof, logically does not apply to division; 2 � 4 does not equal 4 � 2. equivalent expressions may replace each other. The A similar lesson applies in logic: The commutativity first five rules of replacement are as follows: rule applies only to conjunction and disjunction; it

does not apply to implication.

9. De Morgan�s rule (DM): Associativity states that the truth value of

∼ ∼( p ● q ) :: (∼ ∼p �� ∼ ∼q) a conjunctive or disjunctive statement is unaf-∼ ∼ ( p �� q ) :: ( ∼ ∼ p ● ∼ ∼ q) fected by the placement of parentheses when

10. Commutativity (Com): the same operator is used throughout. In other ( p �� q ) :: (q �� p) words, the way in which the component proposi- ( p ● q ) :: (q ● p) tions are grouped, or associated with one another, 11. Associativity (Assoc): can be changed without affecting the truth value.

[ The validity of this rule is quite easy to see, but if p �� (q �� r)] :: [(p �� q) �� r] there is any doubt about it, it may be readily checked [ p ● (q ● r)] :: [(p ● q) ● r] by means of a truth table. You may recall that the 12. Distribution (Dist): associativity rule also applies to addition and multi- [ p ● (q �� r)] :: [(p ● q) �� (p ● r)] plication and asserts, for example, that 3 + (5 + 7) [ p �� (q ● r)] :: [(p �� q) ● (p �� r)] equals (3 + 5) + 7, and that 2 � (3 � 4) equals 13. Double negation (DN ): (2 � 3) � 4. But it does not apply to division: p:: ∼∼ ∼∼p (8 � 4) � 2 does not equal 8 � (4 � 2). Analo-

gously, in logic, the associativity rule applies only

De Morgan�s rule (named after the nineteenth- to conjunctive and disjunctive statements; it does

century logician Augustus De Morgan) was discussed not apply to conditional statements. Also note, in Section 6.1 in connection with translation. There it when applying this rule, that the order of the let-was pointed out that �Not both p and q� is logically ters remains unchanged; only the placement of the equivalent to �Not p or not q,� and that �Not either parentheses changes. p or q� is logically equivalent to �Not p and not q.� Distribution, like De Morgan�s rule, applies only When applying De Morgan�s rule, one should keep in to conjunctive and disjunctive statements. When mind that it holds only for conjunctive and disjunctive a proposition is conjoined to a disjunctive state-statements (not for conditionals or biconditionals). ment in parentheses or disjoined to a conjunctive

Section 7.3 Rules of Replacement I 355

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). statement in parentheses, the rule allows us to put application of De Morgan�s rule, even though the rule

nents inside the parentheses, and also to go in the 1. S ⊃ ∼ ∼(T ● U ) reverse direction. In the first form of the rule, a that proposition together with each of the compo- is applied only to the consequent of line 1: statement is distributed through a disjunction, and 2. S ⊃ (∼ ∼T �� ∼ ∼U ) 1, DM (valid ) in the second form, through a conjunction. While Another way of viewing this distinction is that the the rule may not be immediately obvious, it is easy rules of implication are �one-way� rules, whereas the to remember: The operator that is at first outside the rules of replacement are �two-way� rules. The rules parentheses goes inside, and the operator that is at of implication allow us to proceed only from the first inside the parentheses goes outside. Note also premise lines of a rule to the conclusion line, but the how distribution differs from commutativity and rules of replacement allow us to replace either side of associativity. The latter two rules apply only when an equivalence expression with the other side. the same operator (either a dot or a wedge) is used Application of the first five rules of replacement throughout a statement. Distribution applies when a may now be illustrated. Consider the following dot and a wedge appear together in a statement. argument:

little explanation. The rule states simply that pairs Double negation is fairly obvious and needs 1. A ⊃ ∼ ∼(B ● C ) 2. A ● C / ∼ ∼B of tildes immediately adjacent to one another may be either deleted or introduced without affecting the Examining the premises, we find B in the conse-truth value of the statement. quent of line 1. This leads us to suspect that the con-

There is an important difference between the clusion can be derived via modus ponens. If this is

rules of implication, treated in the first two sec- correct, the tilde would then have to be taken inside tions of this chapter, and the rules of replacement. the parentheses via De Morgan�s rule and the result-The rules of implication derive their name from the ing ∼ ∼C eliminated by disjunctive syllogism. The fol-fact that each is a simple argument form in which lowing completed proof indicates that this strategy

in natural deduction, certain lines in a proof must 1. A ⊃ ∼ ∼(B ● C ) be interpreted as substitution instances of the argu-the premises imply the conclusion. To be applicable yields the anticipated result: ment form in question. Stated another way, the rules 2. A ● C / ∼ ∼B of implication are applicable only to 3. A 2, Simp whole lines in

a proof. For example, step 3 in the following proof 5. 4. ∼ ∼(B ● C ) 1, 3, MP

∼ ∼B �� ∼ ∼C 4, DM

is not a legitimate application of modus ponens, 6. C ● A 2, Com because the first premise in the modus ponens rule is 7. C 6, Simp applied to only a part of line 1. 8. ∼ ∼ ∼ ∼C 7, DN

1. A 9. ∼ ∼C �� ∼ ∼B 5, Com ⊃ (B ⊃ C )

2. B 10. ∼ ∼B 8, 9, DS

3. C 1, 2, MP ( invalid ) The rationale for line 6 is to get C on the left side The rules of replacement, on the other hand, are so that it can be separated via simplification. Simi-

not rules of implication but rules of logical equiva- larly, the rationale for line 9 is to get ∼ ∼C on the left lence. Since, by the axiom of replacement, logically side so that it can be eliminated via disjunctive syl-equivalent statement forms can always replace one logism. Line 8 is required because, strictly speaking, another in a proof sequence, the rules of replacement the negation of ∼ ∼C is ∼ ∼ ∼ ∼C�not simply C. Thus, C can be applied either to a whole line or to any part of a must be replaced with ∼ ∼ ∼ ∼C to set up the disjunc-line. Step 2 in the following proof is a quite legitimate tive syllogism. If your instructor permits it, you can

356Chapter7 Natural Deduction in�Propositional Logic

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). combine commutativity and double negation with Application of the associativity rule is illustrated other inferences on a single line, as the following in the next proof: shortened proof illustrates. However, we will avoid

this practice throughout the remainder of the book. 1. M �� (N �� O)

2. ∼ ∼O / M �� N

1. A ⊃ ∼ ∼(B ● C ) 3. (M �� N ) �� O 1, Assoc 2. A ● C / ∼ ∼B 4. O �� (M �� N ) 3, Com 3. A 2, Simp 5. M �� N 2, 4, DS 4. ∼ ∼(B ● C ) 1, 3, MP

5. ∼ ∼B �� ∼ ∼C 4, DM Before O can be eliminated via disjunctive syl-6. C 2, Com, Simp logism from line 1, it must be moved over to the 7. ∼ ∼B 5, 6, Com, DN, DS left side. Associativity and commutativity together

Another example: In some arguments the attempt to �find� the con-accomplish this objective. 1. D ● (E �� F ) clusion in the premises is not immediately success-2. ∼ ∼D �� ∼ ∼F / D ● E ful. When confronted with such an argument, one

should often begin by �deconstructing� the conclu-

The conclusion requires that we get D and E together. sion using the rules of replacement. In other words, Inspection of the first premise suggests distribution as one should first apply the rules of replacement to the the first step in achieving this. The completed proof is as conclusion to see how it is put together. After this is follows: done, how the premises entail the conclusion may

1. D ● (E �� F ) be evident. This procedure is justified by the fact

2. ∼ ∼D �� ∼ ∼F / D ● that the rules of replacement are two-way rules. As E

3. (D ● E ) �� (D ● a result, after the conclusion is deconstructed, it can F ) 1, Dist

4. (D ● F ) �� (D ● be derived by using the same rules in reverse order. E ) 3, Com

5. ∼ ∼(D ● Here is an example of such an argument: F ) 2, DM

6. D ● E 4, 5, DS

1. K ⊃ (F �� B )

Some proofs require that we use distribution in 2. G ● K / (F ● G ) �� (B ● G )

the reverse manner. Consider this argument:

If immediate inspection does not reveal how

1. (G ● H ) �� (G ● J ) the conclusion should be derived, we may begin by

The conclusion can be obtained from line 2 via The form of the conclusion suggests the distribution rule, but first we must use commutativity to move 2. (G �� K ) ⊃ L / L applying the rules of replacement to the conclusion.

itself. Since the G�s to the left-hand side. The deconstruction modus ponens if we first obtain G �� K on a line by all, it must be introduced by addition. Doing this proceeds as follows: K does not occur in the first premise at

requires in turn that we obtain G on a line by itself. (F ● G ) �� (B ● G ) Distribution applied to line 1 provides the solution: (G ● F ) �� (B ● G ) Com

1. (G (G ● F ) �� (G ● B ) Com ● H ) �� (G ● J ) G ● (F �� B ) Dist 2. (G �� K ) ⊃ L / L

3. G ● (H �� J ) 1, Dist Now we see that if we can obtain G on a line by itself, 4. G 3, Simp and F �� B on a line by itself, we can combine them on 5. G �� K 4, Add a single line via the conjunction rule. We can then 6. L 2, 5, MP

Section 7.3 Rules of Replacement I 357

Prominent 1908�2000Willard Van Orman Quine Logicians

Prior to his death in the year 2000, Willard Van Orman Quine was widely considered to be, as Stuart Hampshire put it, �the most distinguished and influential of living philosophers.� At that time, over

2,000 scholarly articles had been written about his work.

Quine was born in Akron, Ohio, in 1908 to a father

who founded a heavy equipment company and a

mother who taught elementary school. He earned

his bachelor�s degree in mathematics from Oberlin

College, where he graduated summa cum laude in

1930. He then entered Harvard University, where he

switched to philosophy so he could study under Alfred AP Images/Julia Malakia North Whitehead. He earned his PhD in a record

two years. Except for four years during World War II,

when he served in the Navy decoding messages from As a boy, Quine had a fascination with collect-German submarines, Quine remained affiliated with ing stamps and drawing maps, which, as an adult, Harvard for the remainder of his life. he translated into a zest for world travel. He visited

Quine wrote twenty-two books, the first five of which 118 countries, became fluent in six different lan-

dealt with mathematical logic. One of the goals of the guages, delivered lectures all over the world, and earlier books was to show how the foundations of math- was awarded the first Schock Prize (Stockholm, ematics could be laid out in less than a fourth of the 1993) and the Kyoto Prize (Tokyo, 1996). He was space taken by Whitehead and Russell�s Principia Math- married twice; raised two children from each mar-ematica. One of his most famous publications was �Two riage; loved Dixieland jazz; and played the banjo, Dogmas of Empiricism,� which shook the pillars of ana- mandolin, and piano. He was singularly unpreten-lytic philosophy by undermining the sacrosanct distinc- tious, had an unfailing curiosity about a vast range tion between analytic and synthetic statements. As a res- of topics, and delighted in teaching freshman logic ult of this work, even the truths of logic and mathematics as well as advanced courses in philosophy. He died became subject to the dictates of empirical experience. in Boston at the age of ninety-two.

derive the conclusion via distribution and commu- 5. K 4, Simp tativity. Inspection of the premises reveals that G can 6. F �� B 1, 5, MP be derived from line 2 of the premises by simplifica- 7. G ● (F �� B) 3, 6, Conj tion, and F 8. (G �� B can be derived from line 1 by modus ● F ) �� (G ● B) 7, Dist ponens 9. (F . The completed proof is as follows:● G ) �� (G ● B) 8, Com

10. (F ● G ) �� (B ● G ) 9, Com

1. K ⊃ (F �� B)

2. G ● K / (F ● G ) �� (B ● Here are some strategies for applying the first five G )

3. G 2, Simp rules of replacement. Most of them show how these

4. K ● G 2, Com rules may be used together with other rules.

358Chapter7 Natural Deduction in�Propositional Logic

Strategy 11: Conjunction can be used to set up 3. A �� (B ● C ) 1, Dist

De Morgan�s rule: 4. B ● C 2, 3, DS

1. ∼ ∼A 1. A ● (B �� C ) 2. ∼ ∼B 2. ∼ ∼(A ● B) 3. ∼ ∼A ● ∼ ∼B 1, 2, Conj 3. (A ● B) �� (A ● C ) 1, Dist 4. ∼ ∼(A �� B) 3, DM 4. A ● C 2, 3, DS

Strategy 12:Strategy 15: Constructive dilemma can be used Distribution can be used in two ways

to set up De Morgan�s rule: to set up simplification:

1. (A ⊃ ∼ ∼B) ● (C ⊃ ∼ ∼D) 1. A �� (B ● C ) 2. A 2. (A �� C�� B) ● (A �� C ) 1, Dist 3. ∼ ∼B 3. A �� ∼ ∼ D 1, 2, CD�� B 2, Simp 4. ∼ ∼(B ● D) 3, DM

1. (A ● B) �� (A ● C )

Strategy 13: 2. A Addition can be used to set up De ● (B �� C ) 1, Dist

Morgan�s rule: 3. A 2, Simp

1. Strategy 16: ∼ ∼ A If inspection of the premises does 2. ∼ ∼A �� ∼ ∼B 1, Add not reveal how the conclusion 3. ∼ ∼(A ● B) 2, DM should be derived, consider using

the rules of replacement to decon-

Strategy 14: Distribution can be used in two ways struct the conclusion. (See the final

to set up disjunctive syllogism: example in this section.)

1. (A �� B) ● (A �� C )

2. ∼ ∼A

Exercise 7.3

I. For each of the following lists of premises, derive the indicated conclusion and

complete the justification. For double negation, avoid the occurrence of triple tildes. Exercise 6 has two possible answers.

★(1) 1. ∼ ∼(E ⊃ H )

2. ∼ ∼(N �� G )

3. ∼ ∼A �� D

4. _______ ____ , DM

(2) 1. G ⊃ (N ⊃ K )

2. R �� (D ⊃ F )

3. S ● (T �� U )

4. __________ ____ , Dist

(3) 1. M �� (G �� T )

2. P ● (S ⊃ N )

3. D ● (R �� K )

4. __________ ____ , Assoc

SectionSection 7.3 7.2 Rules of Replacement I Rules of Implication II 359359

★(4) 1. B ⊃ W

2. G K F

3. S ● A

4. ______ ____ , Com

(5) 1. ∼ ∼ ∼ ∼R �� T

2. ∼ ∼N �� ∼ ∼B

3. ∼ ∼A ⊃ ∼ ∼H

4. ________ ____ , DN

(6) 1. (F �� N ) ⊃ (K ● D)

2. (H ● Z ) �� (H ● W )

3. (P ⊃ H ) �� (P ⊃ N )

4. _______________ ____ , Dist

★(7) 1. ∼ ∼(G ● ∼ ∼Q)

2. ∼ ∼(K K ∼ ∼B)

3. ∼ ∼T ⊃ ∼ ∼F

4. _________ ____ , DM

(8) 1. G ⊃ (∼ ∼L ⊃ T )

2. L K (∼ ∼R ⊃ ∼ ∼C )

3. J ⊃ (S �� ∼ ∼N )

4. ___________ ____ , Com

(9) 1. S ⊃ (M ⊃ D)

2. (K ● G ) �� B

3. (E ● H ) ● Q

4. __________ ____ , Assoc

★(10) 1. ∼ ∼R �� ∼ ∼P

2. ∼ ∼F ⊃ ∼ ∼W

3. G ● ∼ ∼A

4. ________ ____ , DM

(11) 1. ∼ ∼B �� E

2. ∼ ∼E ● ∼ ∼A

3. ∼ ∼C ⊃ ∼ ∼R

4. ________ ____ , DN

(12) 1. ∼ ∼G ● (S ⊃ A)

2. ∼ ∼S ⊃ (B ● K )

3. ∼ ∼Q �� (T ● R )

4. __________ ____ , Dist

★(13) 1. F ⊃ (∼ ∼S �� M )

2. H ⊃ (∼ ∼L ● ∼ ∼D)

3. N ⊃ (∼ ∼G ⊃ ∼ ∼C )

4. __________ ____ , DM

(14) 1. F ⊃ (P ⊃ ∼ ∼E )

2. C �� (S ● ∼ ∼B)

360360ChapterChapter77 Natural Deduction in�Propositional Logic Natural Deduction in�Propositional Logic

3. M ● (R ● ∼ ∼T )

4. ___________ ____ , Assoc

(15) 1. (D �� ∼ ∼K ) ● (D �� ∼ ∼W )

2. (S �� ∼ ∼Z ) �� (P �� ∼ ∼T ) 3. (Q ⊃ ∼ ∼N ) ● (Q ⊃ ∼ ∼F ) 4. _________________ ____ , Dist

II. In the following symbolized arguments, derive the line needed to obtain the

conclusion (last line ), and supply the justification for both lines.

★ (9) 1. ∼ ∼L(1) 1. K �� M �� C

2. 2. L ∼ ∼ C

3. 3. ______ ____ _____ ____

4. 4. M ____ K ____

(2) 1. ★(10) 1. D G ⊃ ( R �� (N �� N ) ● H ) 2. __________ ____ 2. ∼ ∼ R ● ∼ ∼ N 3. D 3. __________ ____ �� N ____ 4. ∼ ∼ G ____ (11) 1. ( K �� E ) ● ( K �� G ) 2. ∼ ∼ K(3) 1. H ● T 3. ______________ ____ 2. _____ ____ 4. E 3. T ____ ● G ____

★ ( (12) 1. N ⊃ T ) (4) 1. ( L ● ● (F ⊃ Q) S ) ● F 2. F 2. ________ ____ �� N 3. ______________ ____ 3. L ____ 4. T �� Q ____ (5) 1. ∼ ∼ B �� K ★ (13) 1. ( M 2. �� G ) �� T ________ ____ 2. ∼ ∼ M 3. ∼ ∼ ( B ● ∼ ∼ K ) ____ 3. __________ ____ (6) 1. C ⊃ ∼ ∼ A 4. G �� T ____ 2. A(14) 1. ( ∼ ∼ A ⊃ T ) 3. ______ ____ ● ( ∼ ∼ S ⊃ K ) 2. ∼ ∼ ( A 4. ∼ ∼ C ____ ● S ) 3. ______________ ____ ★ (7) 1. ( D ● M ) �� ( D ● N ) 4. T �� K ____ 2. ______________ ____ (15) 1. ∼ ∼ R 3. D ____ 2. _______ ____ (8) 1. ( U �� T ) ⊃ R 3. ∼ ∼ ( R ● T ) ____ 2. T �� U 3. __________ ____ 4. R ____

III. Use the first thirteen rules of inference to derive the conclusions of the following

symbolized arguments:

★ �� (3) 1. J (K ● L) (1) 1. ( ∼ ∼ M ⊃ P ) ● ( ∼ ∼ N ⊃ Q ) 2. 2. ∼ ∼ ( M ● ∼ ∼ K / J N ) / P �� Q

(2) 1. ★(4) 1. ∼ ∼(N ∼ ∼ S / ∼ ∼ ( F ● ● T ) S ) ∼ 2. T / ∼N

SectionSection 7.3 7.2 Rules of Replacement I Rules of Implication II 361361

★(5) 1. H ⊃ ∼ ∼A 3. (G �� C ) ⊃∼ ∼H / ∼ ∼(K �� H ) ∼ 2. A / ∼ ( H �� ∼ ∼ A ) ★ (22) 1. S �� ( I ● ∼ ∼ J ) (6) 1. R ⊃ ∼ ∼ B 2. S ⊃ ∼ ∼ R 2. D �� R 3. ∼ ∼ J ⊃ ∼ ∼ Q / ∼ ∼ ( R ● Q ) 3. B / D(23) 1. ( J �� F ) �� M(7) 1. T ⊃ ( B �� E ) 2. ( J �� M ) ⊃ ∼ ∼ P 2. ∼ ∼ E ● T / B 3. ∼ ∼ F / ∼ ∼ ( F �� P )

(8) 1. (O �� M ) ⊃ S(24) 1. (K ● P ) �� (K ● Q ) ∼ 2. ∼ S / ∼ ∼ M 2. P ⊃ ∼ ∼ K / Q �� T

(9) 1. Q �� (L �� C ) ★ (25) 1. E �� ∼ ∼(D �� C ) ∼ 2. ∼ C / L �� Q 2. ( E �� ∼ ∼ D ) ⊃ C / E

★(10) 1. (K ● H ) �� (K ● L) (26) 1. A ● ( F ● L) ∼ 2. ∼ L / H 2. A ⊃ ( U ��W ) 3. F ⊃ ( U �� X ) / U �� (W(11) 1. ∼ ∼ ( ∼ ∼ E ● ∼ ∼ N ) ⊃ T ● X ) 2. G ⊃ ( N �� E ) / G ⊃ T(27) 1. ( T ● R ) ⊃ P 2. ( ∼ ∼ P ● R ) (12) 1. H ● ● G ( C ● T ) 3. ( ∼ ∼ T �� N ) ⊃ H / H 2. ∼ ∼ ( ∼ ∼ F ● T ) / F

★(13) 1. (E ● ★ (28) 1. P �� ( I ● L ) I ) �� ( M ● U ) ⊃ 2. ( P �� I ) ∼ ∼( L �� C ) 2. ∼ ∼ E / ∼ ∼ ( E �� ∼ ∼ M ) 3. ( P ● ∼ ∼ C ) ⊃ ( E ● F ) / F �� D(14) 1. ∼ ∼ ( J �� K ) (29) 1. B 2. B ⊃ K �� ( S ● N ) 2. B ⊃ ∼ ∼ S 3. S ⊃ B / ∼ ∼ S ● ∼ ∼ J 3. S ⊃ ∼ ∼ N / B �� W(15) 1. ( G ● H ) �� ( M ● G ) (30) 1. ( ∼ ∼ M �� E ) ⊃ ( S ⊃ U ) 2. G ⊃ ( T ● A ) / A 2. ( ∼ ∼ Q �� E ) ⊃ ( U ⊃ H ) ★ (16) 1. ( Q ● N ) �� ( N ● T ) 3. ∼ ∼ ( M �� Q ) / S ⊃ H 2. ( Q �� C ) ⊃ ∼ ∼ N / T ★ (31) 1. ( ∼ ∼ R �� D ) ⊃ ∼ ∼ ( F ● G ) (17) 1. ∼ ∼ ( U �� R ) 2. ( F ● R ) ⊃ S 2. ( ∼ ∼ R �� N ) ⊃ ( P ● H ) 3. F ● ∼ ∼ S / ∼ ∼ ( S �� G ) 3. Q ⊃ ∼ ∼ H / ∼ ∼ Q(32) 1. ∼ ∼ Q ⊃ ( C ● B ) (18) 1. ∼ ∼ ( F ● A ) 2. ∼ ∼ T ⊃ ( B ● H ) 2. ∼ ∼ ( L �� ∼ ∼ A ) 3. ∼ ∼ ( Q ● T ) / B 3. D ⊃ ( F �� L ) / ∼ ∼ D(33) 1. ∼ ∼ ( A ● G ) 2. ∼ ∼ ( A ● E ) ★ (19) 1. [( I �� M ) �� G ] ⊃ ∼ ∼ G 3. G �� E / ∼ ∼ ( A ● F ) 2. M �� G / M ★ (34) 1. ( M ● N ) �� ( O ● P ) (20) 1. E ⊃ ∼ ∼ B 2. ( N �� O ) ⊃ ∼ ∼ P / N 2. U ⊃ ∼ ∼ C 3. ∼ ∼ ( ∼ ∼ E ● ∼ ∼ U ) / ∼ ∼ ( B ● C ) (35) 1. ( T ● K ) �� ( C ● E ) 2. K ⊃ ∼ ∼ E(21) 1. ∼ ∼ ( K �� F ) 3. E ⊃ ∼ ∼ C / T ● K 2. ∼ ∼ F ⊃ ( K �� C )

362362ChapterChapter77 Natural Deduction in�Propositional Logic Natural Deduction in�Propositional Logic

IV. Translate the following arguments into symbolic form and then use the first

thirteen rules of inference to derive the conclusion of each. Use the translation letters in the order in which they are listed.

★1. Either healthcare costs are skyrocketing and they are attributable to greedy

doctors, or healthcare costs are skyrocketing and they are attributable to greedy hospitals. If healthcare costs are skyrocketing, then both the government should intercede and health care may have to be rationed. Therefore, healthcare costs are skyrocketing and health care may have to be rationed. (S, D, H, I, R)

2. Either the ancient Etruscans were experienced city planners and they invented

the art of writing or they were highly skilled engineers and they invented the art of writing. If the ancient Etruscans were bloodthirsty numskulls (as scholars once thought), they did not invent the art of writing. Therefore, the ancient Etruscans were not bloodthirsty numskulls (as scholars once thought). (C, I, H, B)

3. It is not the case that either the earth�s molten core is stationary or that it con-

tains no iron. If it is not the case that both the earth�s molten core is stationary and has a regular topography, then either the earth�s core contains no iron or the direction of the earth�s magnetic field is subject to change. Therefore, the direction of the earth�s magnetic field is subject to change. (S, C, R, D)

★ 4. Either mosquito genes can be cloned or mosquitoes will become resistant to all

insecticides and the incidence of encephalitis will increase. If either mosquito genes can be cloned or the incidence of encephalitis increases, then mosqui-toes will not become resistant to all insecticides. Therefore, either mosquito genes can be cloned or mosquitoes will multiply out of control. (G, R, E, M )

5. Protein engineering will prove to be as successful as genetic engineering, and new

enzymes will be developed for producing food and breaking down industrial wastes. If protein engineering proves to be as successful as genetic engineering and new en-zymes are developed for breaking down industrial wastes, then it is not the case that new enzymes will be developed for producing food but not medicines. Therefore, protein engineering will prove to be as successful as genetic engineering and new enzymes will be developed for producing medicines. (E, P, B, M )

6. If workers have a fundamental right to a job, then unemployment will be virtu-

ally nonexistent but job redundancy will become a problem. If workers have no fundamental right to a job, then production efficiency will be maximized but job security will be jeopardized. Workers either have or do not have a funda-mental right to a job. Therefore, either unemployment will be virtually non-existent or production efficiency will be maximized. (F, U, R, P, S)

★7. If China is to reduce its huge trade surplus, then it must either convince its

citizens to spend more or it must move its manufacturing facilities to other countries. It is not the case that China will either increase its imports or con-vince its citizens to spend more. Furthermore, it is not the case that China will either allow foreign construction companies to compete on an equal footing or move its manufacturing facilities to other countries. Therefore, China will not reduce its huge trade surplus. (R, C, M, I, A)

SectionSection 7.3 7.2 Rules of Replacement I Rules of Implication II 363363

8. If women are by nature either passive or uncompetitive, then it is not the case that

there are lawyers who are women. If men are by nature either insensitive or with-out the ability to nurture, then it is not the case that there are kindergarten teachers who are men. There are lawyers who are women and kindergarten teachers who are men. Therefore, it is not the case that either women by nature are uncompeti-tive or men by nature are without the ability to nurture. (P, U, L, I, W, K )

9. It is not the case that either the sun�s interior rotates faster than its surface or Ein-

stein�s general theory of relativity is wrong. If the sun�s interior does not rotate faster than its surface and eccentricities in the orbit of Mercury can be explained by solar gravitation, then Einstein�s general theory of relativity is wrong. Therefore, eccen-tricities in the orbit of Mercury cannot be explained by solar gravitation. (S, E, M )

★ 10. Either school dropout programs are not as effective as they could be, or they provide

basic thinking skills and psychological counseling to their students. Either school dropout programs are not as effective as they could be, or they adequately prepare their students for getting a job and working effectively with others. Either school dropout programs do not provide psychological counseling to their students or they do not provide adequate preparation for working effectively with others. Therefore, school dropout programs are not as effective as they could be. (E, B, P, G, W )

V. The following dialogue contains eight arguments. Translate each into symbolic form

and then use the first thirteen rules of inference to derive the conclusion of each.

New Cradle

Chloe, with youngster in tow, is strolling across campus when she bumps into her friend Dylan. �Hi,� she says. �You must be coming from class.�

�Yes, I am,� he says. �Social psychology. And who is the little one?�

�This is my niece Ashley. She was three last month. Ashley, say �Hi� to Dylan. We�re

heading over to the campus day-care center. Do you want to join us?�

�Sure,� he says. �At least as far as the cafeteria.�

�You know, kids are really a kick in the pants,� she says. �That�s why my heart goes

out to a same-sex couple I know.�

�And why is that?� Dylan asks.

�They�re trying to adopt a baby, but the only adoption agency in this small town

refuses to help them because the agency is religiously affiliated. I�m talking about the New Cradle agency. Those people at New Cradle think same-sex marriage is sinful.�

�Well, in light of this religious freedom legislation that�s been coming down the pike

lately, the people at New Cradle may be within their rights.�

�How is that?� Chloe asks.

�It goes like this,� he says. �If religious freedom is a right, then the people at New

Cradle have a right to practice it as they choose. And if they have a right to practice it as they choose, then if they think same-sex marriage violates their religion, then they can refuse to help same-sex couples adopt a baby. Now religious freedom is a right and those people think same-sex marriage violates their religion. Hence they can refuse to help same-sex couples adopt a baby.�

�But wait just a minute,� Chloe says. �New Cradle is engaged in a commercial activ-

ity, just like a restaurant, and some of its babies come from out of state. So I would offer you this argument. If New Cradle receives government support or is engaged in

364364ChapterChapter77 Natural Deduction in�Propositional Logic Natural Deduction in�Propositional Logic

interstate commerce, then if it discriminates against same-sex couples, then it violates those couple�s civil rights. Now if New Cradle refuses to help same-sex couples adopt, then it discriminates. New Cradle is engaged in interstate commerce and it refuses to help same-sex couples adopt. Thus, New Cradle violates those couples� civil rights.�

�I can see that New Cradle faces a dilemma,� Dylan replies. �If it helps same-sex

couples adopt a baby, then it will violate its own principles; but if it does not help them, then it will violate constitutional principles. And if New Cradle abandons its own principles, then it will close up shop and lay off its employees. But New Cradle will never lay off its employees, and it will either help same-sex couples adopt a baby or not help them. Therefore, New Cradle will violate constitutional principles.�

�Another point we must not forget,� Chloe adds, �is that same-sex couples now have

the right to marry. And, if that is the case, they have a right to raise a family. And if they have a right to raise a family and they have good parenting skills, then if New Cradle can refuse to help them adopt a baby, then religion is misappropriated and the public good is subverted. Same-sex couples do have good parenting skills, and the public good will not be subverted. Therefore, New Cradle cannot refuse to help them adopt a baby.�

�I think you�re right,� says Dylan, �that equality plays a big role in this issue. Our country is

based on the principle of equality. If straight couples have a right to the services of an adoption agency, and obviously they do, then either same-sex couples have such a right or they are not equal to straight couples. And if same-sex couples have a right to the services of an adop-tion agency, then New Cradle must change its policy and other religiously affiliated agencies must do the same. Now if our country is based on the principle of equality, then same-sex couples are equal to straight couples. Thus, as you say, New Cradle must change its policy.�

�Yet,� Dylan continues, �up till now we�ve said nothing about the welfare of the baby

who will be adopted. If New Cradle helps same-sex couples adopt a baby, then it�s important that the baby grow up well adjusted. But if the baby is to be well adjusted, then it must have both a male role model and a female role model. If the parents are both men, then the baby will not have a female role model. If they are both women, then it will not have a male role model. If the couple is a same-sex couple, then the parents will be either both men or both women, and the couple is a same-sex couple. Therefore, New Cradle must not help same-sex couples adopt a baby.�

�I think your reasoning is a bit shortsighted,� Chloe replies. �Let�s suppose a same-

sex couple adopts a baby. If a same-sex couple adopts a baby, then the parents will be either both men or they will be both women. This much I grant you. But if they are both men, then surely they will have close female friends. And if that is so, then there will be both male and female role models. If the parents are both women, then surely they will have close male friends. And if that is so, then there will be both male and female role models. If there are both male and female role models, then the baby will be well adjusted. And if the baby will be well adjusted, then New Cradle should help with the adoption. Thus, the conclusion follows, that New Cradle should help with the adoption.�

�Interesting point,� Dylan says. �But I wonder if those surrogate role models would

be as effective as male and female parents. Anyway, here�s where I get off. See you again soon, I hope.�

�Yeah, me too,� says Chloe.

�Bye-bye, Ashley. It was nice meeting you.�

�G�bye,� she says, waving her hand.

VI. Create four symbolized arguments consisting of at least three premises each, and whose

proof requires at least two of the rules introduced in this section. The arguments should look like those in Part III of this section. Then derive the conclusion of each.

SectionSection 7.3 7.2 Rules of Replacement I Rules of Implication II 365365

7.4 Rules of Replacement II

The remaining five rules of replacement are as follows: Exportation is also fairly easy to understand. It

14. Transposition (Trans): asserts that the statement �If we have both p and q,

then we have r� is logically equivalent to �If we have

( p ⊃ q) :: (∼ ∼q ⊃ ∼ ∼p) p, then if we have q, then we have r.� As an illustra-15. Material implication (Impl): tion of this rule, the statement �If Bob and Sue told

( the truth, then Jim is guilty� is logically equivalent to p ⊃ q) :: ( ∼ ∼ p �� q) �If Bob told the truth, then if Sue told the truth, then 16. Material equivalence (Equiv): Jim is guilty.�

( Tautologyp , the last rule introduced in this sec-K q) :: [( p ⊃ q) ● (q ⊃ p)]

( tion, is obvious. Its effect is to eliminate redundancy p K q) :: [( p ● q) �� ( ∼ ∼ p ● ∼ ∼ q)]

in disjunctions and conjunctions.

17. Exportation (Exp): The following proofs illustrate the use of these [( p ● q) ⊃ r] :: [ p ⊃ (q ⊃ r)] five rules.

1. ∼ ∼A / A ⊃ B

p In this argument, the conclusion contains a letter not :: 18. Tautology ( Taut ):

( p �� p) found in the premise. Obviously, addi tion must be

p:: ( p ● p) used to introduce the B. The material implication rule

Transposition completes the proof: asserts that the antecedent and

consequent of a conditional statement may switch 1. ∼ ∼A / A ⊃ B places if and only if tildes are inserted before both 2. ∼ ∼A �� B 1, Add or tildes are removed from both. The rule is fairly 3. A ⊃ B 2, Impl

easy to understand and is easily proved by a truth Here is another example: table. 1. F ⊃ GMaterial implication is less obvious than 2. F �� G / G transposition, but it can be illustrated by substi-To derive the conclusion of this argument, some tuting actual statements in place of the letters. For example, the statement � If you bother me, then method must be found to link the two premises together and eliminate the I�ll punch you in the nose� ( B ⊃ P ) is logically F . Hypothetical syl-logism provides the solution, but first the second equivalent to �Either you stop bothering me or I�ll punch you in the nose� ( ∼ ∼ B �� P premise must be converted into a conditional. Here ). The rule states that a horseshoe may be replaced by a wedge if the is the proof: left-hand component is negated, and the reverse 1. F ⊃ G replacement is allowed if a tilde is deleted from the 2. F �� G / G left-hand component. 3. ∼ ∼ ∼ ∼F �� G 2, DN

Material equivalence has two formulations. 4. ∼ ∼F ⊃ G 3, Impl

The first is the same as the definition of material 5. ∼ ∼F ⊃ ∼ ∼ ∼ ∼G 4, DN equivalence given in Section 6.1. The second for- 6. ∼ ∼G ⊃ F 5, Trans mulation is easy to remember through recalling the 7. ∼ ∼G ⊃ G 1, 6, HS two ways in which p K q may be true. Either p and 8. ∼ ∼ ∼ ∼G �� G 7, Impl q 9. G �� G 8, DN are both true or p and q are both false. This, of course, is the meaning of [( 10. G 9, Taut p ● q ) �� ( ∼ ∼ p ● ∼ ∼ q )].

366Chapter7 Natural Deduction in�Propositional Logic

Another example: 4. S ⊃ (T ⊃ P) / P K R 1. J 5. Q ⊃ R 3, Trans ⊃ (K ⊃ L) / K ⊃ ( J ⊃ L)

The conclusion can be obtained by simply rearrang- 6. P ⊃ R 1, 5, HS

tion provides the simplest method: 8. R ⊃ P 2, 7, HS 9. (P ⊃ R) ● (R ⊃ P) 6, 8, Conj 1. J ing the components of the single premise. Exporta- 7. (S ● T ) ⊃ P 4, Exp

2. ( J ⊃ (K ⊃ L ) / K ⊃ ( J ⊃ L) 10. P K R 9, Equiv ● K ) ⊃ L 1, Exp

3. (K ● J ) ⊃ L 2, Com As we saw in Section 7.3, if it is not readily appar- 4. K ⊃ ( J ⊃ L) 3, Exp ent how the conclusion should be derived, we can Another example: use the rules of replacement to deconstruct the con-

1. M clusion. This will usually provide insight on how ⊃ N

2. M ⊃ O / M ⊃ (N ● O) best to proceed. Again, this technique is justified

As with the F and G example, some method must be As a result, they can be applied in reverse order in because the rules of replacement are two-way rules.

found to link the two premises together. In this case, the completed proof. Here is an example: however, hypothetical syllogism will not work. The solution lies in setting up a distribution step: 1. ∼ ∼ S ⊃ K

1. M 2. S ⊃ (R �� M ) / ∼ ∼R ⊃ (∼ ∼M ⊃ K ) ⊃ N

3. In deconstructing the conclusion, the form of the ∼ 2. M ⊃ O / M ⊃ (N ● O) 4. conclusion suggests exportation, and the result ∼ ∼M �� N 1, Impl

5. ( of this step suggests De Morgan�s rule. For further ∼ ∼ M �� ∼M �� O 2, Impl

N ) ● (∼ ∼M �� O) 3, 4, Conj

6. ∼ ∼M �� (N ● O) 5, Dist insight, we apply transposition to the latter step. 7. M ⊃ (N ● O) 6, Impl Each step follows from the one preceding it:

Another example: ∼ ∼R ⊃ (∼ ∼M ⊃ K )

4. S 3. ⊃ (S ● T ) ∼ ∼K ⊃ ∼ ∼ ∼ ∼(R �� M) Trans R ∼ ∼ ⊃ ∼ ∼ Q ∼ ∼ K ⊃ (R �� M) DN ⊃ (T ⊃ P ) / P K R Now, examining the premises in light of the decon- 1. P ⊃ Q ∼ ∼(R �� M) ⊃ K DM 2. R (∼ ∼R ● ∼ ∼M) ⊃ K Exp

The conclusion is a biconditional, and there are only struction suggests that we begin by setting up a two ways that a biconditional can be obtained from hypothetical syllogism. This will give us the last such premises�namely, via the two formulations of step in the deconstruction. We can then obtain the the material equivalence rule. The fact that the prem-conclusion by repeating the deconstruction steps in ises are all conditional statements suggests the first reverse order. The completed proof is as follows: formulation of this rule. Accordingly, we must try to obtain P ⊃ R and R ⊃ P. Again, the fact that the 1. ∼ ∼S ⊃ K premises are themselves conditionals suggests hypo- 2. S ⊃ (R �� M) / ∼ ∼R ⊃ (∼ ∼M ⊃ K ) thetical syllogism to accomplish this. Premises 1 and 3. ∼ ∼K ⊃ ∼ ∼ ∼ ∼S 1, Trans 3 can be used to set up one hypothetical syllogism; 4. ∼ ∼K ⊃ S 3, DN premises 2 and 4 provide the other. Here is the proof: 5. ∼ ∼K ⊃ (R �� M) 2, 4, HS

6. ∼ ∼(R �� M) ⊃ ∼ ∼ ∼ ∼K 5, Trans

1. P ⊃ Q 7. ∼ ∼(R �� M) ⊃ K 6, DN 2. R ⊃ (S ● T ) 8. (∼ ∼R ● ∼ ∼M) ⊃ K 7, DM 3. ∼ ∼R ⊃ ∼ ∼Q 9. ∼ ∼R ⊃ (∼ ∼M ⊃ K ) 8, Exp

Section 7.4 Rules of Replacement II 367

Here is another example: Strategy 17: Material implication can be used to

In deconstructing the conclusion, the form of the 1. K set up hypothetical syllogism: ⊃ M 1. ∼ ∼ A �� B 2. L ⊃ M / (K �� L) ⊃ M 2. ∼ ∼ B �� C 3. A ⊃ B 1, Impl premises suggests that we use some procedure that 4. B ⊃ C 2, Impl will combine M separately with K and L . This, in 5. A ⊃ C 3, 4, HS turn, suggests distribution; but before we can use Strategy 18: Exportation can be used to set up distribution, we must eliminate the horseshoe via modus ponens: material implication. The deconstruction is as 1. (A ● B) ⊃ C follows: 2. A

(K �� L) ⊃ M 3. A ⊃ (B ⊃ C ) 1, Exp ∼ ∼(K �� L) �� M Impl 4. B ⊃ C 2, 3, MP

(∼ ∼K ● ∼ ∼L) �� M DM Strategy 19: Exportation can be used to set up M �� ( ∼ ∼ K ● ∼ ∼ L) Com modus tollens: (M �� ∼ ∼ K ) ● ( M �� ∼ ∼ L ) Dist 1. A ⊃ (B ⊃ C ) ( ∼ ∼ K �� M ) ● ( M �� ∼ ∼ L ) Com 2. ∼ ∼ C ( ∼ ∼ K �� M ) ● ( ∼ ∼ L �� M ) Com 3. (A ● B) ⊃ C 1, Exp (K ⊃ M ) ● ( ∼ ∼ L �� M ) Impl 4. ∼ ∼ (A (K ⊃ M ) ● ● B) 2, 3, MT ( L ⊃ M ) Impl

Now, examining the premises in light of the last line Strategy 20: Addition can be used to set up mate-rial implication: of the deconstruction suggests that we begin by join-ing the premises together via the conjunction rule. 1. A The conclusion can then be obtained by reversing 2. A �� ∼ ∼B 1, Add the steps of the deconstruction: 3. ∼ ∼B �� A 2, Com

4. B ⊃ A 3, Impl

2. L 1. K ⊃ MStrategy 21: Transposition can be used to set up ⊃ M / (K �� L) ⊃

3. (K hypothetical syllogism: ⊃ M

M ) ● (L ⊃ M ) 1, 2, Conj

4. (∼ ∼K �� M ) ● (L ⊃ M ) 3, Impl 1. A ⊃ B 5. (∼ ∼K �� M ) ● (∼ ∼L �� M ) 4, Impl 2. ∼ ∼C ⊃ ∼ ∼B 6. (M �� ∼ ∼K ) ● (∼ ∼L �� M ) 5, Com 3. B ⊃ C 2, Trans 7. (M �� ∼ ∼K ) ● (M �� ∼ ∼L ) 6, Com 4. A ⊃ C 1, 3, HS

8. M �� (∼ ∼K ● ∼ ∼L) 7, Dist Strategy 22: Transposition can be used to set up 9. (∼ ∼K ● ∼ ∼L) �� M 8, Com constructive dilemma: 10. ∼ ∼(K �� L) �� M 9, DM

11. (K �� L) ⊃ M 10, Impl 1. (A ⊃ B) ● (C ⊃ D)

Note that whenever we use this strategy of working 2. ∼ ∼B �� ∼ ∼D 3. ( ∼ ∼ B ⊃ ∼ ∼ A) ● (C ⊃ D) 1, Trans backward from the conclusion, the rules of replace-4. ( ∼ ∼ B ⊃ ∼ ∼ A) ● ( ∼ ∼ D ⊃ ∼ ∼ C ) 3, Trans ment are the only rules we may use. We may not use 5. ∼ ∼ A �� ∼ ∼ C 2, 4, CD the rules of implication, because these rules are one-Strategy 23: Constructive dilemma can be used way rules. This section ends with some strategies that show to set up tautology: how the last five rules of replacement can be used 1. (A ⊃ C ) ● (B ⊃ C ) together with various other rules. 2. A �� B

368Chapter7 Natural Deduction in�Propositional Logic

3. C �� C 1, 2, CD Strategy 25: Material implication can be used to 4. C 3, Taut set up distribution:

Strategy 24: Material implication can be used to 1. A ⊃ (B ● C )

set up tautology: 2. ∼ ∼A �� (B ● C ) 1, Impl

1. A 3. ( ∼ ∼A �� B) (∼ ∼A �� C ) 2, Dist ⊃ ∼ ∼ A ● 2. ∼ ∼A �� ∼ ∼A 1, Impl

3. ∼ ∼A 2, Taut

Exercise 7.4

I. For each of the following lists of premises, derive the indicated conclusion and

complete the justification. For tautology, derive a conclusion that is simpler than the premise.

★(1) 1. H �� F

2. N �� ∼ ∼S

3. ∼ ∼G �� Q

4. ______ ____ , Impl

(2) 1. R ⊃ (S ⊃ N )

2. T ⊃ (U �� M )

3. K ● (L ⊃ W )

4. __________ ____ , Exp

(3) 1. G K R

2. H ⊃ P

3. ∼ ∼F �� T

4. ______ ____ , Trans

★(4) 1. (B ⊃ N ) ● (N ⊃ B)

2. (R �� F ) ● (F �� R)

3. (K ⊃ C ) �� (C ⊃ K ) 4. _______________ ____ , Equiv

(5) 1. E �� ∼ ∼E

2. A �� A

3. G ● ∼ ∼G

4. ______ ____ , Taut

(6) 1. S �� ∼ ∼M

2. ∼ ∼N ● ∼ ∼T

3. ∼ ∼L ⊃ Q

4. _______ ____ , Trans

★(7) 1. ∼ ∼C ⊃ ∼ ∼F

2. D �� ∼ ∼P

3. ∼ ∼R ● Q

4. ________ ____ , Impl

Section 7.4 Rules of Replacement II 369

(8) 1. E ⊃ (R ● Q)

2. (G ● N ) ⊃ Z

3. (S ⊃ M ) ⊃ P

4. ___________ ____ , Exp

(9) 1. (D ● H ) �� (∼ ∼D ● ∼ ∼H )

2. (F ⊃ J ) ● (∼ ∼F ⊃ ∼ ∼J )

3. (N �� T ) ● (∼ ∼N �� ∼ ∼T )

4. ________________ ____ , Equiv

★(10) 1. L ⊃ (A ⊃ A)

2. K ⊃ (R �� ∼ ∼R )

3. S ⊃ (G ● G )

4. __________ ____ , Taut

(11) 1. K ● (S �� B )

2. ∼ ∼F ⊃ ∼ ∼J

3. ∼ ∼E �� ∼ ∼M

4. _________ ____ , Trans

(12) 1. H ⊃ (K ● J )

2. (N �� E ) ⊃ B

3. C ⊃ (H ⊃ A)

4. __________ ____ , Exp

★(13) 1. (A ⊃ ∼ ∼C ) ● (C ⊃ ∼ ∼A)

2. (W ⊃ ∼ ∼T ) ● (∼ ∼T ⊃ W )

3. (M ⊃ ∼ ∼E ) ● (∼ ∼M ⊃ E )

4. _________________ ____ , Equiv

(14) 1. (∼ ∼K �� M) K S

2. T �� (F ● G )

3. R K (N ● ∼ ∼H )

4. __________ ____ , Impl

(15) 1. (S �� S) ⊃ D

2. K ⊃ (T ● ∼ ∼T )

3. (Q ⊃ Q) ⊃ M

4. __________ ____ , Taut

II. In the following symbolized arguments, derive the line needed to obtain the

conclusion (last line), and supply the justification for both lines.

★(1) 1. ∼ ∼J �� M(3) 1. C ⊃ A

2. M ⊃ B 2. A ⊃ C 3. ______ ____ 3. _____ ____ 4. J ⊃ B ____ 4. C K A ____

(2) 1. ( J ● F ) ⊃ N ★(4) 1. (G ⊃ K ) ● (T ⊃ K )

2. J 2. G �� T 3. ______ ____ 3. _______________ ____ 4. F ⊃ N ____ 4. K ____

370Chapter7 Natural Deduction in�Propositional Logic

(5) 1. (G ⊃ B) ● (∼ ∼C ⊃ ∼ ∼H ) ★(10) 1. H ⊃ ∼ ∼H

2. G �� H 2. _______ ____ 3. _________________ ____ 3. ∼ ∼H ____

4. B �� C ____ (11) 1. ∼ ∼S

(6) 1. J ⊃ (M ⊃ Q) 2. ______ ____

3. __________ ____ (12) 1. M ⊃ (M ⊃ D) 4. 2. J ● M 3. S ⊃ K ____

Q ____ 2. ___________ ____

★(7) 1. H ⊃ (∼ ∼C �� R) 3. M ⊃ D ____ 2. ___________ ____ ★(13) 1. (N ⊃ A) ● (∼ ∼N ⊃ ∼ ∼A) 3.

(H ● C ) ⊃ R ____ 2. _________________ ____

(9) 1. 4. __________ ____ 2. _____ ____ T ⊃ N ____ 3. E K R ____ K ⊃ ( A ⊃ F ) (15) 1. Q ⊃ ( ∼ ∼ W ⊃ ∼ ∼G ) 2. ∼ ∼ F 2. _____________ ____ 3. (8) 1. ∼ ∼G ⊃ ∼ ∼T 3. N K A ____ 2. G ⊃ N(14) 1. E ● R 3.

4. __________ ____ 3. (Q ● G) ⊃ W ____ ∼ ∼ ( K ● A ) ____

III. Use the eighteen rules of inference to derive the conclusions of the following sym-

bolized arguments.

★(1) 1. (S ● K ) ⊃ R

2. K / S ⊃ R

(2) 1. T ⊃ (F �� F )

2. ∼ ∼(F ● F ) / ∼ ∼T

(3) 1. G ⊃ E

2. H ⊃ ∼ ∼E / G ⊃ ∼ ∼H

★(4) 1. S K Q

2. ∼ ∼S / ∼ ∼Q

(5) 1. ∼ ∼N �� P

2. (N ⊃ P) ⊃ T / T

(6) 1. F ⊃ B

2. B ⊃ (B ⊃ J ) / F ⊃ J

★(7) 1. (B ⊃ M) ● (D ⊃ M )

2. B �� D / M

(8) 1. Q ⊃ (F ⊃ A)

2. R ⊃ (A ⊃ F )

3. Q ● R / F K A

(9) 1. T ⊃ (∼ ∼T �� G )

2. ∼ ∼G / ∼ ∼ T

★ (10) 1. (B ⊃ G ) ● (F ⊃ N )

2. ∼ ∼(G ● N ) / ∼ ∼(B ● F )

Section 7.4 Rules of Replacement II 371

(11) 1. ( J ● R) ⊃ H

2. (R ⊃ H ) ⊃ M

3. ∼ ∼(P �� ∼ ∼J ) / M ● ∼ ∼P

(12) 1. T / S ⊃ T

★(13) 1. K ⊃ (B ⊃ ∼ ∼M )

2. D ⊃ (K ● M ) / D ⊃ ∼ ∼B

(14) 1. (O ⊃ C ) ● (∼ ∼S ⊃ ∼ ∼D)

2. (E ⊃ D) ● (∼ ∼E ⊃ ∼ ∼C ) / O ⊃ S

(15) 1. ∼ ∼(U ● W ) ⊃ X

2. U ⊃ ∼ ∼U / ∼ ∼(U �� ∼ ∼X )

★(16) 1. T ⊃ R

2. T ⊃ ∼ ∼R / ∼ ∼ T

(17) 1. S �� ∼ ∼N

2. ∼ ∼S �� Q / N ⊃ Q

(18) 1. M ⊃ (U ⊃ H )

2. (H �� ∼ ∼U ) ⊃ F / M ⊃ F

★(19) 1. ∼ ∼R �� P

2. R �� ∼ ∼P / R K P

(20) 1. ∼ ∼H ⊃ B

2. ∼ ∼H ⊃ D

3. ∼ ∼(B ● D) / H

(21) 1. J ⊃ (G ⊃ L) / G ⊃ ( J ⊃ L)

★(22) 1. S ⊃ (L ● M )

2. M ⊃ (L ⊃ R) / S ⊃ R

(23) 1. F ⊃ (A ● K )

2. G ⊃ (∼ ∼A ● ∼ ∼K )

3. F �� G / A K K

(24) 1. (I ⊃ E ) ⊃ C

2. C ⊃ ∼ ∼C / I

★(25) 1. T ⊃ G

2. S ⊃ G / (T �� S) ⊃ G

(26) 1. H ⊃ U / H ⊃ ( U �� T )

(27) 1. Q ⊃ (W ● D) / Q ⊃ W

★(28) 1. P ⊃ (∼ ∼E ⊃ B)

2. ∼ ∼(B �� E ) / ∼ ∼P

(29) 1. (G ⊃ J ) ⊃ (H ⊃ Q)

2. J ● ∼ ∼Q / ∼ ∼ H

(30) 1. I �� (N ● F )

2. I ⊃ F / F

372Chapter7 Natural Deduction in�Propositional Logic

★ (31) 1. K K R

2. K ⊃ (R ⊃ P )

3. ∼ ∼P / ∼ ∼R

(32) 1. C ⊃ (∼ ∼L ⊃ Q)

2. L ⊃ ∼ ∼C

3. ∼ ∼Q / ∼ ∼C

(33) 1. (E ⊃ A) ● (F ⊃ A)

2. E �� G

3. F �� ∼ ∼G / A

★ (34) 1. (F ● H ) ⊃ N

2. F �� S

3. H / N �� S

(35) 1. T ⊃ (H ● J )

2. (H �� N ) ⊃ T / T K H

(36) 1. T ⊃ ∼ ∼(A ⊃ N )

2. T �� N / T K ∼ ∼N

★ (37) 1. (D ⊃ E ) ⊃ (E ⊃ D)

2. (D K E ) ⊃ ∼ ∼(G ● ∼ ∼H ) 3. E ● G / G ● H

(38) 1. (O ⊃ R) ⊃ S

2. (P ⊃ R) ⊃ ∼ ∼S / ∼ ∼R

(39) 1. (L �� P) ⊃ U

2. (M ⊃ U ) ⊃ I

3. P / I

★ (40) 1. A K W

2. ∼ ∼A �� ∼ ∼W

3. R ⊃ A / ∼ ∼(W �� R)

(41) 1. (S �� T ) ⊃ (S ⊃ ∼ ∼T )

2. (S ⊃ ∼ ∼T ) ⊃ (T ⊃ K ) 3. S �� T / S �� K

(42) 1. G K M

2. G �� M

3. G ⊃ (M ⊃ T ) / T

★ (43) 1. O ⊃ (Q ● N )

2. (N �� E ) ⊃ S / O ⊃ S

(44) 1. H K I

2. H ⊃ (I ⊃ F )

3. ∼ ∼(H �� I ) ⊃ F / F

★ (45) 1. P ⊃ A

2. Q ⊃ B / ( P �� Q ) ⊃ (A �� B)

Section 7.4 Rules of Replacement II 373

IV. Translate the following arguments into symbolic form and then use the eighteen

rules of inference to derive the conclusion of each. Use the translation letters in the order in which they are listed.

★1. If feral pigs invade a territory, then they destroy local flora and fauna. It is not

the case that both feral pigs destroy local flora and fauna and hunters should not be allowed to kill them. Therefore, if feral pigs invade a territory, then hunters should be allowed to kill them. (F, D, H)

2. If there is a direct correlation between what a nation spends for health care

and the health of its citizens, then America has the lowest incidence of disease and the lowest mortality rates of any nation on Earth. But America does not have the lowest mortality rates of any nation on Earth. Therefore, there is not a direct correlation between what a nation spends for health care and the health of its citizens. (C, D, M )

3. It is not the case that strict controls exist on either the manufacture or the sale

of handguns. Therefore, if strict controls exist on the sale of handguns, then the use of handguns in the commission of crimes has decreased. (M, S, U )

★4. If birth-control devices are made available in high school clinics, then the

incidence of teenage pregnancy will decrease. Therefore, if both birth-control information and birth-control devices are made available in high school clinics, then the incidence of teenage pregnancy will decrease. (D, P, I )

5. If Congress enacts a law that either establishes a religion or prohibits the

free exercise of religion, then that law is unconstitutional. Therefore, if Congress enacts a law that establishes a religion, then that law is unconstitu-tional. (E, P, U )

6. If selecting a Supreme Court justice creates intense political upheaval and ex-

treme efforts are expended to block judicial candidates, then Supreme Court rulings are contaminated with political bias. Selecting a Supreme Court justice creates intense political upheaval. Therefore, if extreme efforts are expended to block judicial candidates, then Supreme Court rulings are contaminated with political bias. (S, E, C)

★7. If grade-school children are assigned daily homework, then their achieve-

ment level will increase dramatically. But if grade-school children are assigned daily homework, then their love for learning may be dampened. Therefore, if grade-school children are assigned daily homework, then their achievement level will increase dramatically but their love for learning may be dampened. (G, A, L)

8. If two black holes collide, then Einstein�s general theory is proven true and

gravitational waves are propagated throughout the universe. Either two black holes collide, or space-time is distorted and Einstein�s general theory is proven true. Therefore, Einstein�s general theory is proven true. ( T, E, G, S )

9. If parents are told that their unborn child has Tay�Sachs disease, then if they go

ahead with the birth, then they are responsible for their child�s pain and suffering.

374374ChapterChapter77 Natural Deduction in�Propositional Logic Natural Deduction in�Propositional Logic

Therefore, if parents are not responsible for their child�s pain and suffering, then if they go ahead with the birth, then they were not told that their unborn child had Tay�Sachs disease. (T, G, R)

★10. Vitamin E is an antioxidant and a useless food supplement if and only if it

does not reduce heart disease. It is not the case either that vitamin E does not reduce heart disease or is not an antioxidant. Therefore, vitamin E is not a useless food supplement. (A, U, R)

V. The following dialogue contains ten arguments. Translate each into symbolic form

and then use the eighteen rules of inference to derive the conclusion of each.

Is This the End?

Brian and Lucia are at the memorial service of a mutual friend who had died suddenly the week before. �I�m still shocked to think that Karl is gone,� Lucia says.

�I know you were quite close to him,� Brian says. �But do you think in some sense

Karl could still be with us? I mean do you think there could be such a thing as postmor-tem persistence of consciousness�life after death, as most people say?�

�I wish there were,� Lucia replies, �and that�s what makes death so tragic. As I see

it, the mind is totally dependent on the brain, and if that�s so, when the brain dies, the mind dies. If the mind dies, then consciousness dies, too. Thus, if the brain dies, then consciousness dies�which means there�s no life after death.�

�But what makes you think that the mind is totally dependent on the brain?�

Brian asks.

�Our day-to-day experience provides lots of evidence,� Lucia replies. �If you drink

alcohol, your mind is affected. If you smoke marijuana, your mind is affected. If your mind is affected by these things, then you have firsthand experience that the mind is dependent on the brain. Thus, if you either smoke marijuana or drink alcohol, then you have firsthand experience that the mind is dependent on the brain.�

�So the mind is affected by the brain. Anyone with ordinary sensation knows that,�

Brian retorts. �If your eye receives a visual stimulus, then that stimulus is sent to the brain and your mind is affected. If your ear receives an auditory stimulus, then your mind is affected. Thus, if either your eye or your ear receives a stimulus, then your mind is affected. But that doesn�t prove that the mind is necessarily dependent on the brain. And there are lots of reasons for saying that it isn�t.�

�What reasons are those?� Lucia asks.

�Well, we learned about Plato in Introduction to Philosophy,� Brian replies. �And

Plato held that the mind can conceive ideal objects such as perfect justice and perfect triangularity. Now, if either of these concepts came through the senses, then perfect ideals exist in nature. But no perfect ideals exist in nature. And if the concept of triangu-larity did not come through the senses, then the mind produced it independently of the brain. But if that is the case or the concept of triangularity is innate, then the mind is not necessarily dependent on the brain. I�ll leave the conclusion up to you.�

�Very interesting,� Lucia replies, �but I question whether the mind is really capable

of conceiving ideal objects such as perfect justice and perfect triangularity. For me, these things are just words. But there are other reasons for thinking that the mind is

Section 7.4 Rules of Replacement II 375

necessarily dependent on the brain. Consider the visual cortex. The visual cortex is part of the brain. If the visual cortex isn�t stimulated, there is no visual sensation. But if visual sensation occurs only if the visual cortex is stimulated, and if the visual cortex is part of the brain, then visual sensation is dependent on the brain. And if that is true and visual sensation is a function of the mind, then the mind is necessarily dependent on the brain. Therefore, if visual sensation is a function of the mind, then the mind is necessarily dependent on the brain.�

�Furthermore,� Lucia continues, �there are many cases where strokes have caused

loss of memory, and also loss of speech. But if remembering is a mental function, then if the mind is not necessarily dependent on the brain, then strokes do not cause loss of memory. Therefore, if remembering is a mental function, then the mind is necessarily dependent on the brain.�

�It may indeed be the case,� Brian replies, �that memory�or at least certain kinds

of memory�are dependent on the brain. And the same may be true of sensation. But that doesn�t prove that consciousness as such is brain dependent. It seems to me that consciousness as such is a nonmaterial process, and that it can occur only in a non-material�entity, such as a soul. And if those two claims are true and the soul is immortal, then consciousness survives the death of the body. Thus, if the soul is immortal, then consciousness survives the death of the body.�

�If memory goes with the brain,� Lucia replies, �then I wonder if the consciousness

you speak of is in any way your consciousness. But setting that aside, are there any reasons for thinking that the soul is immortal?�

�I think there are,� Brian replies. �If the soul is nonmaterial, then it has no parts,

and if it has no parts, then it can�t come �a-part��in other words it can�t disinte-grate. And if it can�t disintegrate, then if nothing can destroy it, then it is immortal. But the soul can be destroyed only if God destroys it, and God does not destroy souls. Therefore, if the soul is nonmaterial, then it is immortal. I think Leibniz invented that argument.�

�Fine,� Lucia says. �But what makes you think that you have a nonmaterial soul in

the first place?�

�Well,� Brian replies, �according to Descartes, I am essentially either a mind or a

body. But if I can doubt that I have a body, then I am not essentially a body. And I can doubt that I have a body. For example I can imagine that I am in The Matrix, and that all of my sensations are illusions. If I am essentially a mind, then if the essence of mind is to be nonextended, then I am a nonextended substance. But the essence of mind, being different from the essence of body, is to be nonextended. And if I am a nonex-tended substance, then I am (or have) a nonmaterial soul. Therefore, I am (or have) a nonmaterial soul.�

�Your argument is so abstruse that I don�t find it very persuasive,� says Lucia, as she

scratches her head. �I think the evidence is overwhelming that humans are the product of biological evolution, and if that is true and humans have souls, then there is a point in the course of evolution where humans either received or developed a soul. But there is no evidence that humans ever received a soul. Also, there is no evidence that humans ever developed a soul. Therefore, humans do not have souls.�

�Wow, that sounds pretty far out,� Brian replies. �Well, it looks like the service is

ready to start, so we�ll have to hang this up. But maybe we can continue it at a later date.�

�Maybe we can,� Lucia replies.

376Chapter7 Natural Deduction in�Propositional Logic

VI. Create four symbolized arguments consisting of at least three premises each,

and whose proof requires at least two of the rules introduced in this section. The arguments should look like those in Part III of this section. Then derive the conclusion of each.

7.5 Conditional Proof

Conditional proof is a method for deriving a con- follow from the second premise via modus ponens. In ditional statement (either the conclusion or some other words, if we assume that we have A, we can get intermediate line) that offers the usual advantage of E. But this is exactly what the conclusion says. Thus, being both shorter and simpler to use than the direct we have just proved that the conclusion follows from method. Moreover, some arguments have conclu- the premises. sions that cannot be derived by the direct method, The method of conditional proof consists of so some form of conditional proof must be used on incorporating this simple thought process into the them. Conditional proof may thus be seen as com- body of a proof sequence. A conditional proof for pleting the rules of inference. The method consists this argument requires only eight lines and is sub-of assuming the antecedent of the required condi- stantially simpler than a direct proof:

tional statement on one line, deriving the conse- 1. A ⊃ (B ● C ) quent on a subsequent line, and then �discharging� 2. (B �� D) ⊃ E / A ⊃ E this sequence of lines in a conditional statement that 3. A ACP exactly replicates the one that was to be obtained. 4. B ● C 1, 3, MP

Any argument whose conclusion is a conditional 5. B 4, Simp

statement is an immediate candidate for conditional 6. B �� D 5, Add proof. Consider the following example: 7. E 2, 6, MP

1. A ⊃ (B ● 8. A ⊃ E 3�7, CP C )

2. (B �� D) ⊃ E / A ⊃ E Lines 3 through 7 are indented to indicate their

Using the direct method to derive the conclusion of hypothetical character: They all depend on the this argument would require a proof having at least assumption introduced in line 3 via ACP (assump-twelve lines, and the precise strategy to be followed tion for conditional proof ). These lines, which con-in constructing it might not be immediately obvious. stitute the conditional proof sequence, tell us that if Nevertheless, we need only give cursory inspection to we assume A (line 3), we can derive E (line 7). In the argument to see that the conclusion does indeed line 8 the conditional sequence is discharged in the follow from the premises. The conclusion states that conditional statement A ⊃ E, which simply reiter-

if we have ates the result of the conditional sequence. Since A , we then have E . Let us suppose, for a moment, that we do have line 8 is not hypothetical, it is written adjacent to A . We could then derive B ● the original margin, under lines 1 and 2. A vertical C from the first premise via modus ponens . Sim-plifying this expression we could derive line is added to the conditional sequence to B , and from this we could get emphasize the indentation. B �� D via addition. E would then

Section 7.5 Conditional Proof 377

Prominent 1848�1925Gottlob Frege Logicians

T he German mathematician, logician, and phi- matters worse, in losopher Gottlob Frege (pronounced fray-ga) 1902, while Vol-was born in Wismar, a small town in northern ume 2 was in proof, Germany on the Baltic Sea. His parents taught at a Fre ge received a private girls� school, which his father had helped to letter from Bertrand found. Frege attended the local gymnasium, where he Russell that left him studied mathematics, and then the University of Jena, �thunderstruck.� He where he studied mathematics, philosophy, and chem- was later to remark istry. After two years he transferred to the University of that it had destroyed G�ttingen, earning a doctor�s degree in mathematics his entire life�s work. at age twenty-four. He then returned to the University Basic Law 5 of the of Jena, where he taught until retiring in 1917. While Grundgesetze pro-there he married Margaret Liesburg, who bore him at vides for the creation Pictorial Press Ltd/Alamy Stock Photo least two children. The children died young, but years of classes of things later the couple adopted a son, Alfred. merely by describing the properties of their mem-

Frege spent his entire life analyzing the concept of bers. So Russell invited Frege to create the class of

number, developing theories of logic and language, all classes that are not members of themselves, and and attempting to reduce arithmetic to logic. In 1879, he then asked whether this very class is a member of he published the Begriffsschrift (�Concept-Script�), itself. If it is a member of itself, then it is one of those a work written in the tradition of Leibniz that devel- classes that are not members of themselves; but if is ops a purely formal symbolic language to express not a member of itself, then, again, it is one of those any proposition in any area of human discourse. Five classes, and it is a member of itself. The derivation of years later he published Die Grundlagen der Arithme- this contradiction (which has come to be called Rus-tik (�The Foundations of Arithmetic�), a less technical sell�s paradox) meant that the axioms of the Grund-work containing few symbols that outlined his goal of gesetze were fatally inconsistent. Frege attempted a reducing arithmetic to logic. Then, nine years later he last- minute modification of his system, but the change published Volume 1 of Grundgesetze der Arithme- proved unworkable. tik (�Basic Laws of Arithmetic�), which attempted to Despite this setback, Frege is universally rec-accomplish the first phase of this reduction. ognized today as one of the most important logi-

None of these works were well received, for cians and philosophers of all time. Single-handedly

several reasons: They were ahead of their time, he developed quantification theory and predicate the symbolic notation of the technical works struck logic, and his analysis of the concept of number led readers as bizarre, and they were written in German, to a general theory of meaning that introduced the whereas most of the new work in logic was being important distinction between Sinn (�sense�) and done by English speakers. In fact, the last of these Bedeutung (�reference� ). Also, his work on concept works was so badly reviewed that Frege was forced clarification initiated the current movement known to publish Volume 2 at his own expense. To make as analytic philosophy.

378Chapter7 Natural Deduction in�Propositional Logic

The first step in constructing a conditional 7. G ⊃ H 4 � 6, CP

proof is to decide what should be assumed on 8. J ACP the first line of the conditional sequence. While 9. K ● L 2, 8, MP any statement whatsoever can be assumed on 10. K 9, Simp this line, only the right statement will lead to the 11. J ⊃ K 8�10, CP desired result. The clue is always provided by 12. (G ⊃ H ) ● ( J ⊃ K ) 7, 11, Conj the conditional statement to be obtained in the 13. H �� K 3, 12, CD

to be obtained is ( K ● L) ⊃ M, then K ● and the second J ⊃ K. These two lines are then L should conjoined and used together with line 3 to set up a be assumed on the first line. This line is always constructive dilemma, from which the conclusion indented and tagged with the designation �ACP. � is derived. must be assumed. For example, if the statement The first conditional proof sequence gives us G ⊃ H, end. The antecedent of this statement is what

the second step is to derive the consequent This proof sequence provides a convenient Once the initial assumption has been made,

the conditional sequence. To do this, we simply erning conditional proof. The rule states that after a conditional proof sequence has been dis-of the desired conditional statement at the end of opportunity to introduce an important rule gov-

ous line in the proof (including the assumed line), charged, no line in the sequence may be used as apply the ordinary rules of inference to any previ-

The third and final step is to discharge the condi- If, for example, line 5 in the proof just given were used as a justification for line 9 or line 12, this writing the result directly below the assumed line. a justification for a subsequent line in the proof.

antecedent of this conditional statement is what- rule would be violated, and the corresponding tional sequence in a conditional statement. The

sequence, and the consequent is whatever appears sequence is discharged, it is sealed off from the remaining part of the proof. The logic behind ever appears on the first line of the conditional inference would be invalid. Once the conditional

on the last line. For example, if A �� B is on the first line and this rule is easy to understand. The lines in a con-C ● D is on the last, the sequence is

ing line is always written adjacent to the original depend on the assumption stated in the first line. Because no mere assumption can provide any discharged by (A �� B) ⊃ (C ● ditional sequence are hypothetical in that they D ). This discharg-

indicating the first through the last lines of the line that depends on such an assumption. When a conditional sequence is discharged, the assump-sequence. tion on which it rests is expressed as the anteced-Conditional proof can also be used to derive (conditional proof ) together with the numerals genuine support for anything, neither can any margin and is tagged with the designation �CP�

a line other than the conclusion of an argument. ent of a conditional statement. This conditional statement can be used to support subsequent lines The following proof, which illustrates this fact, because it makes no claim that its antecedent is incorporates two conditional sequences one true. The conditional statement merely asserts after the other within the scope of a single direct that if its antecedent is true, then its consequent proof: is true, and this, of course, is what has been estab-

1. G ⊃ (H ● I ) lished by the conditional sequence from which it 2. J ⊃ (K ● L) is obtained. 3. G �� J / H �� K Just as a conditional sequence can be used

4. G ACP within the scope of a direct proof to derive a

5. H ● I 1, 4, MP desired statement, one conditional sequence can

6. H 5, Simp be used within the scope of another to derive a

Section 7.5 Conditional Proof 379

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). desired statement. The following proof provides No line in the sequence 5�8 could be used to sup-an example: port any line subsequent to line 9, and no line in the

sequence 3�10 could be used to support any line sub-

1. L ⊃ [M ⊃ (N �� O)] sequent to line 11. Lines 3 or 4 could, of course, be 2. M ⊃ ∼ ∼N / L ⊃ (∼ ∼M �� O) used to support any line in the sequence 5�8.

3. L ACP One final reminder regarding conditional proof 4. M ⊃ (N �� O) 1, 3, MP is that every conditional proof must be discharged. It 5. M ACP is absolutely improper to end a proof on an indented 6. N �� O 4, 5, MP line. If this rule is ignored, any conclusion one 7. ∼ ∼ N 2, 5, MP chooses can be derived from any set of premises. 8. O 6, 7, DS The following invalid proof illustrates this mistake: 9. M ⊃ O 5�8, CP

11. L 10. ∼ ∼M �� O 9, Impl 1. P / Q ⊃ R ⊃ ( ∼ ∼ M �� O) 3�10, CP 2. ∼ ∼ Q ACP

The rule introduced in connection with the previous 3. ∼ ∼Q �� R 2, Add example applies unchanged to examples of this sort. 4. Q ⊃ R 2, Impl

Exercise 7.5

I. Use conditional proof and the eighteen rules of inference to derive the conclusions

of the following symbolized arguments. Having done so, attempt to derive the con-clusions without using conditional proof.

★(1) 1. N ⊃ O

2. N ⊃ P / N ⊃ ( O ● P)

(2) 1. F ⊃ E

2. (F ● E ) ⊃ R / F ⊃ R

(3) 1. G ⊃ T

2. (T �� S) ⊃ K / G ⊃ K

★(4) 1. (G �� H ) ⊃ (S ● T )

2. (T �� U ) ⊃ (C ● D) / G ⊃ C

(5) 1. A ⊃ ∼ ∼(A �� E ) / A ⊃ F

(6) 1. J ⊃ (K ⊃ L)

2. J ⊃ (M ⊃ L)

3. ∼ ∼L / J ⊃ ∼ ∼ ( K �� M )

★(7) 1. M �� (N ● O) / ∼ ∼ N ⊃ M

(8) 1. P ⊃ (Q �� R)

2. (P ⊃ R) ⊃ (S ● T )

3. Q ⊃ R / T

(9) 1. H ⊃ (I ⊃ N )

2. (H ⊃ ∼ ∼I ) ⊃ (M �� N )

3. ∼ ∼N / M

380Chapter7 Natural Deduction in�Propositional Logic

★ (10) 1. C ⊃ (A ● D)

2. B ⊃ (A ● E ) / (C �� B) ⊃ A

(11) 1. M ⊃ (K ⊃ L)

2. (L �� N ) ⊃ J / M ⊃ ( K ⊃ J )

(12) 1. F ⊃ (G ● H ) / (A ⊃ F ) ⊃ (A ⊃ H )

★ (13) 1. R ⊃ B

2. R ⊃ (B ⊃ F )

3. B ⊃ (F ⊃ H ) / R ⊃ H

(14) 1. (F ● G) K H

2. F ⊃ G / F K H

(15) 1. C ⊃ (D �� ∼ ∼E )

2. E ⊃ (D ⊃ F ) / C ⊃ ( E ⊃ F )

★ (16) 1. Q ⊃ (R ⊃ S)

2. Q ⊃ (T ⊃ ∼ ∼U )

3. U ⊃ (R �� T ) / Q ⊃ ( U ⊃ S)

(17) 1. N ⊃ (O ● P )

2. Q ⊃ (R ● S ) / (P ⊃ Q) ⊃ (N ⊃ S)

(18) 1. E ⊃ (F ⊃ G )

2. H ⊃ (G ⊃ I )

3. (F ⊃ I ) ⊃ ( J �� ∼ ∼H ) / (E ● H) ⊃ J

★ (19) 1. P ⊃ [(L �� M) ⊃ (N ● O)]

2. (O �� T ) ⊃ W / P ⊃ ( M ⊃ W )

(20) 1. A ⊃ [B ⊃ (C ● ∼ ∼D)]

2. (B �� E ) ⊃ (D �� E ) / (A ● B) ⊃ (C ● E )

II. Translate the following arguments into symbolic form, using the letters in the order

in which they are listed. Then use conditional proof and the eighteen rules of infer-ence to derive the conclusion of each. Having done so, attempt to derive the conclu-sion without using conditional proof.

★ 1. If high-tech products are exported to Argentina, then domestic industries will

benefit. If the Argentinians can effectively utilize high-tech products, then their standard of living will improve. Therefore, if high-tech products are exported to Argentina and the Argentinians can effectively utilize them, then their stan-dard of living will improve and domestic industries will benefit. (H, D, U, S )

2. If the police take you into custody, then if they inform you that you have the

right to remain silent, then whatever you say will be used against you. If the police inform you that you have the right to remain silent, then if whatever you say will be used against you, then you should not say anything. Therefore, if the police take you into custody, then if they inform you that you have the right to remain silent, then you should not say anything. (P, I, W, S )

3. A doctor must disconnect a dying patient from a respirator if and only if the

fact that patients are self-determining implies that the doctor must follow the

Section 7.5 Conditional Proof 381

patient�s orders. If a dying patient refuses treatment, then the doctor must disconnect the patient from a respirator and the patient will die peacefully. Patients are self-determining. Therefore, if a dying patient refuses treatment, then the doctor must follow the patient�s orders. (D, S, F, R, P )

★4. If jails are overcrowded, then dangerous suspects will be released on their own

recognizance. If jails are overcrowded and dangerous suspects are released on their own recognizance, then crime will increase. If no new jails are built and crime increases, then innocent victims will pay the price of increased crime. Therefore, if jails are overcrowded, then if no new jails are built, then innocent victims will pay the price of increased crime. (J, D, C, N, I )

5. If astronauts attempt interplanetary space travel, then heavy shielding will be

required to protect them from solar radiation. If massive amounts of either fuel or water are carried, then the spacecraft must be very large. Therefore, if heavy shielding is required to protect the astronauts from solar radiation only if mas-sive amounts of fuel are carried, then if astronauts attempt interplanetary space travel, then the spacecraft must be very large. (A, H, F, W, L )

III. Create two symbolized arguments consisting of at least two premises each, and

which invite conditional proof to derive the conclusion. The arguments should look like those in Part I of this section. Then derive the conclusion of each.

7.6 Indirect Proof

Indirect proof is a technique similar to conditional 6. C 5, Simp proof that can be used on any argument to derive 7. ∼ ∼D 2, 6, MP either the conclusion or some intermediate line 8. D ● C 5, Com leading to the conclusion. It consists of assuming the 9. D 8, Simp negation of the statement to be obtained, using this 10. D ● ∼ ∼D 7, 9, Conj assumption to derive a contradiction, and then con- 11. ∼ ∼A 3�10, IP cluding that the original assumption is false. This The indirect proof sequence (lines 3�10) begins by last step, of course, establishes the truth of the state- assuming the negation of the conclusion. Since the ment to be obtained. The following proof sequence conclusion is a negated statement, it shortens the proof uses indirect proof to derive the conclusion: to assume A instead of�∼ ∼ ∼ ∼A. This assumption, which

1. (A is tagged �AIP� (assumption for indirect proof ), leads B) �� ⊃ (C ● D) to a contradiction in line 10. Since any assumption that 2. C ⊃ ∼ ∼ D / ∼ ∼ A leads to a contradiction is false, the indirect sequence

3. A AIP is discharged (line 11) by asserting the negation of the

4. A �� B 3, Add assumption made in line 3. This line is then tagged with

5. C ● D 1, 4, MP the designation �IP� (indirect proof ) together with the

382Chapter7 Natural Deduction in�Propositional Logic

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). numerals indicating the scope of the indirect sequence Just as a conditional sequence may be constructed from which it is obtained. within the scope of another conditional sequence, so

Indirect proof can also be used to derive an inter- a conditional sequence may be constructed within

mediate line leading to the conclusion. Example: the scope of an indirect sequence, and, conversely,

1. E ⊃ [(F �� G ) ⊃ (H ● an indirect sequence may be constructed within the J )]

2. E ● scope of either a conditional sequence or another ∼ ∼ ( J �� K ) / ∼ ∼ (F �� K )

3. E indirect sequence. The next example illustrates the 2, Simp

4. (F �� G ) ⊃ (H ● use of an indirect sequence within the scope of a con- J ) 1, 3, MP

5. ∼ ∼( J �� K ) ● E 2, Com ditional sequence:

6. ∼ ∼( J �� K ) 5, Simp 1. L ⊃ [∼ ∼M ⊃ (N ● O)] 7. ∼ ∼J ● ∼ ∼K 6, DM 2. ∼ ∼N ● P / L ⊃ (M ● P )

8. F AIP 3. L ACP

9. F �� G 8, Add 4. ∼ ∼M ⊃ (N ● O) 1, 3, MP

10. H ● J 4, 9, MP 5. ∼ ∼M AIP

11. J ● H 10, Com 6. N ● O 4, 5, MP

12. J 11, Simp 7. N 6, Simp

13. ∼ ∼J 7, Simp 8. ∼ ∼N 2, Simp

14. J ● ∼ ∼J 12, 13, Conj 9. N ● ∼ ∼N 7, 8, Conj

15. ∼ ∼F 8�14, IP 10. ∼ ∼ ∼ ∼M 5�9, IP 16. ∼ ∼K ● ∼ ∼J 7, Com 11. M 10, DN 17. ∼ ∼K 16, Simp 12. P ● ∼ ∼N 2, Com 18. ∼ ∼F ● ∼ ∼K 15, 17, Conj 13. P 12, Simp 19. ∼ ∼(F �� K ) 18, DM 14. M ● P 11, 13, Conj

The indirect proof sequence begins with the assump- 15. L ⊃ (M ● P ) 3�14, CP

tion of F (line 8), leads to a contradiction (line 14), The indirect sequence (lines 5�9) is discharged (line and is discharged (line 15) by asserting the negation 10) by asserting the negation of the assumption made of the assumption. One should consider indirect proof in line 5. The conditional sequence (lines 3�14) is dis-whenever a line in a proof appears difficult to obtain. charged (line 15) in the conditional statement that has

As with conditional proof, when an indi- the first line of the sequence as its antecedent and the

rect proof sequence is discharged, no line in the last line as its consequent. sequence may be used as a justification for a sub- Indirect proof provides a convenient way for prov-sequent line in the proof. In reference to the last ing the validity of an argument having a tautology proof, this means that none of the lines 8�14 could for its conclusion. In fact, the only way in which the be used as a justification for any of the lines 16�19. conclusion of many such arguments can be derived is Occasionally, this rule requires certain priorities through either conditional or indirect proof. in the derivation of lines. For example, for the For the following argument, indirect proof is the purpose of deriving the contradiction, lines 6 and easier of the two:

because line 7 is needed as a justification for line ∼ ∼(T �� ∼ ∼T ) AIP 3. ∼ ∼ T ● ∼ ∼ ∼ ∼ T 2, DM 16, which lies outside the indirect sequence. If 4. ∼ ∼ ∼ ∼ (T �� ∼ ∼ T ) 2�3, IP lines 6 and 7 had been included within the indirect 5. T sequence. But this would not have been advisable, 1. S / T �� ∼ ∼T 2. 7 could have been included as part of the indirect sequence, they would have had to be repeated after �� ∼ ∼T 4, DN the sequence had been discharged to allow ∼ ∼K to Here is another example of an argument having be derived on a line outside the sequence. a tautology as its conclusion. In this case, since the

Section 7.6 Indirect Proof 383

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). conclusion is a conditional statement, conditional syllogism yielding the negation of the assumption as

proof is the easier alternative: the last line of the conditional sequence. Then, dis-1. S charge the sequence and use tautology to derive the / T ⊃ T negation of the assumption outside the sequence.

2. T ACP Indirect proof can be viewed as a variety of con-

4. T 3, Taut 3. T �� T 2, Add ditional proof in that it amounts to a modification

5. T of the way in which the indented sequence is dis-⊃ T 2 � 4, CP

charged, resulting in an overall shortening of the

The similarity of indirect proof to conditional proof for many arguments. The indirect proof for

proof may be illustrated by returning to the first the argument just given is repeated as follows, with example presented in this section. In the proof that the requisite changes noted in the margin:

used to derive the conclusion: 1. (A �� B) ⊃ (C ● D) 2. C follows, conditional proof�not indirect proof�is ⊃ ∼ ∼D / ∼ ∼A

1. (A �� B) ⊃ (C ● D) 3. A AIP 2. C ⊃ ∼ ∼D / ∼ ∼A 4. A �� B 3, Add

3. A ACP 5. C ● D 1, 4, MP

4. A �� B 3, Add 6. C 5, Simp

5. C ● D 1, 4, MP 7. ∼ ∼D 2, 6, MP

6. C 5, Simp 8. D ● C 5, Com

7. ∼ ∼D 2, 6, MP 9. D 8, Simp

8. D ● C 5, Com 10. D ● ∼ ∼D 7, 9, Conj ______ changed }

9. D 8, Simp 11. ∼ ∼A 3�10, IP

10. D �� ∼ ∼A 9, Add The reminder at the end of the previous section 11. ∼ ∼

12. A ⊃ A 7, 10, DS regarding conditional proof pertains to indirect proof ∼ ∼

13. A 3�11, CP as well: It is essential that every indirect proof be dis-∼ ∼ A �� ∼ ∼ A 12, Impl

14. charged. No proof can be ended on an indented line. ∼ ∼ A 13, Taut

If this rule is ignored, indirect proof, like conditional

This example illustrates how a conditional proof can proof, can produce any conclusion whatsoever. The be used to derive the conclusion of any argument, following invalid proof illustrates such a mistake:

whether or not the conclusion is a conditional state- 1. P / Q ment. Simply begin by assuming the negation of the 2. Q AIP conclusion, derive contradictory statements on sepa- 3. Q �� Q 2, Add rate lines, and use these lines to set up a disjunctive 4. Q 3, Taut

Exercise 7.6

I. Use either indirect proof or conditional proof (or both) and the eighteen rules

of inference to derive the conclusions of the following symbolized arguments. Having done so, attempt to derive the conclusions without using indirect proof or conditional proof.

★(1) 1. (S �� T ) ⊃ ∼ ∼S / ∼ ∼S

(2) 1. (K ⊃ K ) ⊃ R

2. ( R �� M ) ⊃ N / N

384Chapter7 Natural Deduction in�Propositional Logic

(3) 1. (C ● D) ⊃ E

2. (D ● E ) ⊃ F / (C ● D) ⊃ F

★(4) 1. H ⊃ (L ⊃ K )

2. L ⊃ (K ⊃ ∼ ∼L) / ∼ ∼H �� ∼ ∼L

(5) 1. S ⊃ (T �� ∼ ∼U )

2. U ⊃ (∼ ∼T �� R )

3. (S ● U ) ⊃ ∼ ∼R / ∼ ∼S �� ∼ ∼U

(6) 1. ∼ ∼A ⊃ (B ● C )

2. D ⊃ ∼ ∼C / D⊃ A

★(7) 1. (E �� F ) ⊃ (C ● D)

2. (D �� G) ⊃ H

3. E �� G / H

(8) 1. ∼ ∼M ⊃ (N ● O)

2. N ⊃ P

3. O ⊃ ∼ ∼P / M

(9) 1. (R �� S) ⊃ T

2. (P �� Q) ⊃ T

3. R �� P / T

★(10) 1. K / S ⊃ (T ⊃ S)

(11) 1. (A �� B) ⊃ C

2. (∼ ∼A �� D) ⊃ E / C �� E

(12) 1. (K �� L) ⊃ (M ● N )

2. (N �� O) ⊃ (P ● ∼ ∼K ) / ∼ ∼K

★(13) 1. [C ⊃ (D ⊃ C )] ⊃ E / E

(14) 1. F / (G ⊃ H) �� (∼ ∼G ⊃ J )

(15) 1. B ⊃ (K ● M )

2. (B ● M ) ⊃ (P K ∼ ∼P ) / ∼ ∼B

★(16) 1. (N �� O) ⊃ (C ● D)

2. (D �� K ) ⊃ (P �� ∼ ∼C ) 3. (P �� G ) ⊃ ∼ ∼(N ● D) / ∼ ∼N

(17) 1. (R ● S) K (G ● H )

2. R ⊃ S

3. H ⊃ G / R K H

(18) 1. K ⊃ [(M �� N ) ⊃ (P ● Q)]

2. L ⊃ [(Q �� R) ⊃ (S ● ∼ ∼N )] / (K ● L) ⊃ ∼ ∼N

★(19) 1. A ⊃ [(N �� ∼ ∼N ) ⊃ (S �� T )]

2. T ⊃ ∼ ∼(F �� ∼ ∼F ) / A ⊃ S

Section 7.6 Indirect Proof 385

(20) 1. F ⊃ [(C ⊃ C ) ⊃ G]

2. G ⊃ {[H ⊃ (E ⊃ H )] ⊃ (K ● ∼ ∼K )} / ∼ ∼F

II. Translate the following arguments into symbolic form, using the letters in the order

in which they are listed. Then use indirect proof and the eighteen rules of inference to derive the conclusion of each. Having done so, attempt to derive the conclusion without using indirect proof.

★1. If government deficits continue at their present rate and a recession sets in,

then interest on the national debt will become unbearable and the government will default on its loans. If a recession sets in, then the government will not default on its loans. Therefore, either government deficits will not continue at their present rate or a recession will not set in. (C, R, I, D)

2. If either the sea turtle population continues to decrease or rescue efforts are

commenced to save the sea turtle from extinction, then nesting sanctuaries will be created and the indiscriminate slaughter of these animals will be halted. If either nesting sanctuaries are created or poachers are arrested, then if the indiscriminate slaughter of these animals is halted, then the sea turtle popula-tion will not continue to decrease. Therefore, the sea turtle population will not continue to decrease. (C, R, N, I, P)

3. If asbestos workers sue their employers, then if punitive damages are awarded,

then their employers will declare bankruptcy. If asbestos workers sue their em-ployers, then punitive damages will be awarded. If asbestos workers contract asbestosis, then either they will sue their employers or their employers will de-clare bankruptcy. Therefore, either asbestos workers will not contract asbesto-sis or their employers will declare bankruptcy. (S, P, B, C )

★4. If astronauts spend long periods in zero gravity only if calcium is resorbed

in their bodies, then astronauts on a Mars voyage will arrive with brittle bones. If astronauts attempt a voyage to Mars only if they spend long periods in zero gravity, then astronauts on a Mars voyage will arrive with brittle bones. Therefore, astronauts on a Mars voyage will arrive with brittle bones. (Z,�C,�B,�V )

5. Either deposits should be required on beer and soft-drink containers, or these

containers will be discarded along highways and the countryside will look like a dump. If these containers will be discarded either in parks or along highways, then deposits should be required on soft-drink containers. Therefore, deposits should be required on soft-drink containers. (B, S, H, C, P)

III. Create two symbolized arguments consisting of at least two premises each, and

which invite indirect proof to derive the conclusion. The arguments should look like those in Part I of this section. Then derive the conclusion of each.

386Chapter7 Natural Deduction in�Propositional Logic

7.7 Proving Logical Truths

Both conditional and indirect proof can be used 4. ∼ ∼ ∼ ∼P ● ∼ ∼(∼ ∼Q �� P ) 3, DM to establish the truth of a logical truth (tautol- 5. P ● ∼ ∼(∼ ∼Q �� P ) 4, DN ogy). Tautological statements can be treated as if 6. P ● (∼ ∼ ∼ ∼Q ● ∼ ∼P ) 5, DM they were the conclusions of arguments having 7. P ● (∼ ∼P ● ∼ ∼ ∼ ∼Q) 6, Com no premises. Such a procedure is suggested by the 8. (P ● ∼ ∼P ) ● ∼ ∼ ∼ ∼Q 7, Assoc fact that any argument having a tautology for its 9. P ● ∼ ∼P 8, Simp conclusion is valid regardless of what its premises 10. ∼ ∼ ∼ ∼[P ⊃ (Q ⊃ P )] 1�9, IP are. As we saw in the previous section, the proof 11. P ⊃ (Q ⊃ P ) 10, DN

for such an argument does not use the premises More-complex conditional statements are at all but derives the conclusion as the exclusive proved by merely extending the technique used consequence of either a conditional or an indirect in the first proof. In the following proof, notice sequence. Using this strategy for logical truths, we how each conditional sequence begins by assert-write the statement to be proved as if it were the ing the antecedent of the conditional statement to conclusion of an argument, and we indent the first be derived: line in the proof and tag it as being the beginning of either a conditional or an indirect sequence. In / [P ⊃ (Q ⊃ R)] ⊃ [(P ⊃ Q) ⊃ (P ⊃ R)] the end, this sequence is appropriately discharged 1. P ⊃ (Q ⊃ R) ACP to yield the desired statement form. 2. P ⊃ Q ACP

Tautologies expressed in the form of condi- 3. P ACP

tional statements are most easily proved via a 4. Q ⊃ R 1, 3, MP conditional sequence. The following example uses 5. Q 2, 3, MP two such sequences, one within the scope of the 6. R 4, 5, MP

other: 7. P ⊃ R 3�6, CP 1. P / P 8. (P ⊃ Q) ⊃ (P ⊃ R) 2�7, CP ⊃ (Q ⊃ P ) 9. [P ⊃ (Q ⊃ R)] ⊃ [(P ⊃ Q) ⊃ (P ⊃ R)] 1�8, CP ACP

2. Q ACP Tautologies expressed as equivalences are usually

5. Q 3. P �� P 1, Add proved using two conditional sequences, one after 4. P 3, Taut the other. Example: ⊃ P 2 � 4, CP

6. P ⊃ (Q ⊃ P ) 1�5, CP / P K [P ● (Q ⊃ P)]

Notice that line 6 restores the proof to the original 1. P ACP

margin�the first line is indented because it intro- 2. P �� ∼ ∼Q 1, Add duces the conditional sequence. 3. ∼ ∼Q �� P 2, Com

Here is a proof of the same statement using an 4. Q ⊃ P 3, Impl

indirect proof. The indirect sequence begins, as usual, 5. P ● (Q ⊃ P) 1, 4, Conj

with the negation of the statement to be proved: 7. P 6. P ⊃ [P ● (Q ⊃ P)] 1�5, CP

● (Q ⊃ P) ACP

/ P ⊃ (Q ⊃ P) 8. P 7, Simp

1. ∼ ∼[P ⊃ (Q ⊃ P )] AIP 9. [P ● (Q ⊃ P)] ⊃ P 7�8, CP

2. ∼ ∼[∼ ∼P �� (Q ⊃ P )] 1, Impl 10. {line 6} ● {line 9} 6, 9, Conj

3. ∼ ∼[∼ ∼P �� (∼ ∼Q �� P )] 2, Impl 11. P K [P ● (Q ⊃ P)] 10, Equiv

Section 7.7 Proving Logical Truths 387

Exercise 7.7

I. Use conditional proof or indirect proof and the eighteen rules of inference to estab-

lish the truth of the following tautologies.

★1.P ⊃ [(P ⊃ Q) ⊃ Q]

2. (∼ ∼P ⊃ Q) �� (P ⊃ R)

3.P K [P �� (Q ● P)]

★4. (P ⊃ Q) ⊃ [(P ● R) ⊃ (Q ● R)]

5. (P �� ∼ ∼Q) ⊃ [(∼ ∼P �� R) ⊃ (Q ⊃ R)]

6.P K [P ● (Q �� ∼ ∼Q)]

★7. (P ⊃ Q) �� (∼ ∼Q ⊃ P)

8. (P ⊃ Q) K [P ⊃ (P ● Q)]

9. [(P ⊃ Q) ● (P ⊃ R)] ⊃ [P ⊃ (Q ● R)]

★10. [∼ ∼(P ● ∼ ∼Q) ● ∼ ∼Q] ⊃ ∼ ∼P

11. (P ⊃ Q) �� (Q ⊃ P)

12. [P ⊃ (Q ⊃ R)] K [Q ⊃ (P ⊃ R)]

★13. (P ⊃ Q) ⊃ [(P ⊃ ∼ ∼Q) ⊃ ∼ ∼P]

14. [(P ⊃ Q) ⊃ R] ⊃ [(R ⊃ ∼ ∼R) ⊃ P]

15. (∼ ∼P �� Q) ⊃ [(P �� ∼ ∼Q) ⊃ (P K Q)]

★16. ∼ ∼[(P ⊃ ∼ ∼P) ● (∼ ∼P ⊃ P)]

17.P ⊃ [(Q ● ∼ ∼Q) ⊃ R]

18. [(P ● Q) �� R] ⊃ [(∼ ∼R �� Q) ⊃ (P ⊃ Q)]

★19.P K [P �� (Q ● ∼ ∼Q)]

20.P ⊃ [Q K (P ⊃ Q)]

II. Create a symbolized tautology (logical truth). Then use conditional proof or indi-

rect proof to prove it.

Summary

Natural Deduction in Propositional Logic:

■ A step-by-step method for proving the validity of propositional type arguments.

■ Shows exactly how the conclusion �comes out� of the premises.

388Chapter7 Natural Deduction in�Propositional Logic

■ Consists in applying eighteen rules of inference to the premises and deriving the conclusion as

the last line in a sequence of lines.

■ Success in using this method requires much practice.

Rules of Inference:

■ Rules of implication (�one-way � rules):

● The premise(s) can be used to derive the conclusion.

● The conclusion cannot be used to derive the premise(s).

■ Rules of replacement (� two-way � rules):

● Expressed as logical equivalencies.

■ Either side of the equivalence can replace the other.

● Can be used to �deconstruct� the conclusion for insight into how to derive it.

Conditional Proof:

■ A method for deriving a conditional statement.

■ Assume the antecedent of the desired conditional on an indented line.

■ Derive the consequent of the desired conditional statement.

■ Discharge the indented sequence in a conditional statement having the first line of the

sequence as the antecedent and the last line as the consequent.

■ This method can greatly simplify many proofs.

Indirect Proof:

■ A method for deriving any kind of statement.

■ Assume the negation of the desired statement (often this is the conclusion ) on an indented

line.

■ Derive a contradiction.

■ Any assumption that necessarily leads to a contradiction is false.

■ Discharge the indented sequence in a statement consisting of the negation of the first line of

the sequence.

Proving Logical Truths (Tautologies):

■ Use conditional proof to derive conditionals and biconditionals.

● Assume the antecedent of the conditional statement on an indented line.

● Derive the consequent.

● Discharge the indented sequence in the usual way.

● Biconditionals require two indented sequences.

■ Use indirect proof to derive any logical truth:

● Assume the negation of the logical truth on an indented line.

● Derive a contradiction.

● Discharge the indented sequence in the usual way.

Summary389

8Predicate Logic

8.1 Symbols and Translation

8.2 Using the Rules of Inference

8.3 Quantifier Negation Rule

8.4 Conditional and Indirect Proof

8.5 Proving Invalidity

8.6 Relational Predicates and Overlapping Quantifiers

8.7 Identity

8.1 Symbols and Translation

Learning about predicate logic is empowering. Suppose, while attending a Shakespeare festival, a friend poses this argument: �Romeo loves Juliet. Romeo is a Montague and Juliet is a Capulet. Therefore, there is a Montague who loves a Capulet.� This argument is clearly valid, but it cannot be converted into a standard form categorical syllogism, and it cannot be expressed as an argument in propositional logic. To prove its validity, we need predicate logic.

Techniques were developed in earlier chapters for evaluating two basically dif-ferent kinds of arguments. The chapter on categorical syllogisms dealt with arguments such as the following:

All episodes of amnesia are neurological disorders.

No cases of leukemia are neurological disorders.

Therefore, no episodes of amnesia are cases of leukemia.

In such arguments the fundamental components are terms, and the validity of the argument depends on the arrangement of the terms within the premises and conclusion.

390

The chapter on propositional logic, on the other The blank space immediately following the predicate

hand, dealt with arguments such as this: letter is not part of the predicate; rather, it indicates

If chronic stress is reduced, then relaxation the place for some lowercase letter that will repre-increases and health improves. sent the subject of the statement. Depending on If health improves, then people live longer. what lowercase letter is used, and on the additional Therefore, if chronic stress is reduced, then symbolism involved, symbolic predicates may be people live longer. used to translate three distinct kinds of statements:

In such arguments, the fundamental compo- singular statements, universal statements, and par-ticular statements. nents are not terms but statements . The validity A singular statement , you may recall from Sec-of these arguments depends not on the arrange-tion 4.7, is a statement that makes an assertion about a ment of the terms within the statements but on specifically named person, place, thing, or time. Trans-the arrangement of the statements themselves as lating a singular statement involves writing a lowercase simple units. letter corresponding to the subject of the statement Not all arguments, however, can be assigned to to the immediate right of the uppercase letter corre-one or the other of these two groups. There is a third sponding to the predicate. The letters that are allocated type that is a kind of hybrid, sharing features with to serve as abbreviated names of individuals are the both categorical syllogisms and propositional argu-first twenty-three letters of the alphabet ( a , b , c , . . . u , ments. Consider, for example, the following: v , w ). These letters are called individual constants . Kate Middleton is rich and beautiful. Here are some examples of translated statements: If a woman is either rich or famous, she

is happy.

Therefore, Kate Middleton is happy. StatementSymbolic translation

The validity of this argument depends on both the Socrates is mortal. Ms arrangement of the terms and the arrangement of Tokyo is populous. Pt the statements. Accordingly, neither syllogistic logic The Sun-Times is a Ns nor propositional logic alone is sufficient to estab- newspaper. lish its validity. What is needed is a third kind of King Lear is not a fairy ∼ ∼Fk

gistic logic and propositional logic. This third kind Berlioz was not a ∼ ∼Gbpredicate logic is called logic that combines the distinctive features of syllo- tale.

. German.

The fundamental component in predicate logic

is the predicate, symbolized by uppercase letters (A, B, C, . . . X, Y, Z ), called predicate symbols. Here are some examples of bare predicates: Compound arrangements of singular statements

may be translated by using the familiar connectives

English predicateSymbolic predicate of propositional logic. Here are some examples: ___ is a rabbit R __

___ is gigantic Symbolic G __

Statementtranslation

___ is helpless If Paris is beautiful, then Bp ⊃ TaH __ ___ is a doctor D __

Andre told the truth.

Section 8.1 Symbols and Translation 391

Symbolic alphabet (x, y, z). These letters are called individual

Irene is either a doctor or a in the universe, and they have individual constants Di �� L i as substitution instances. Statementvariablestranslation . They can stand for any item at random lawyer. The horseshoe operator and the universal quanti-

Senator Wilkins will Ew ⊃ Cw fier are combined to translate universal statements be elected only if he as follows: campaigns.

General Motors will prosper (Cn � � Ds) ⊃ PgStatement Symbolic Verbal if either Nissan is crippled by formtranslationmeaning a strike or Subaru declares All S are P. (x)(Sx ⊃ Px) For any x, bankruptcy. if x is an S, Indianapolis gets rain if R i K (Sc ● Sm) then x is and only if Chicago and a P. Milwaukee get snow. No S are P. (x)(Sx ⊃ ∼ ∼ Px) For any x,

Recall from Chapter 4 that a if x is an S, universal state-

ment then x is is a statement that makes an assertion about every member of its subject class. Such statements not a P.

are either affirmative or negative, depending on Because an individual variable can stand for any whether the statement affirms or denies that the item at random in the universe, the expression (x) members of the subject class are members of the (Sx ⊃ Px) means �If anything is an S, then it is a predicate class. The key to translating universal P.� Also (x)(Sx ⊃ ∼ ∼Px) means �If anything is an S, statements is provided by the Boolean interpretation then it is not a P.� The fact that these expressions of these statements (see Section 4.3): are equivalent to the Boolean interpretation of uni-

versal statements may be seen by recalling how the

Boolean Boolean interpretation is represented by Venn dia-

Statement forminterpretation grams (see Section 4.3). The Venn diagrams corre-All S are P. If anything is an S, then sponding to the two universal statement forms are

it is a P. as follows:

No S are P. If anything is an S, then

it is not a P. All S are P. No S are P.

According to the Boolean interpretation, univer-

sal statements are translated as conditionals. We

have a symbol (the horseshoe) to translate con-

ditional statements, so we may use it to translate SPSP universal statements. What is still needed, how-

ever, is a symbol to indicate that universal state- Where shading designates emptiness, the diagram ments make an assertion about every member of the on the left asserts that if anything is in the S circle, S class. This symbol is called the universal it is also in the P circle, and the one on the right quantifier. It is formed by placing a lowercase asserts that if anything is in the S circle, it is not in letter in parentheses, (x), and is translated as �for the P circle. This is exactly what is asserted by the any x.� The letters that are allocated for forming the symbolic expressions just given. These symbolic universal quantifier are the last three letters of the expressions may therefore be taken as being exactly

392Chapter8 Predicate Logic

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). synonymous with the Boolean interpretation of uni- quantifier governs the entire statement function in versal statements. parentheses�namely, Ax ⊃ Bx. But in the expres-

A possible source of confusion at this point con- sion (x)Ax ⊃ Bx, the universal quantifier governs

cerns the fact that both S and P in the symbolic only the statement function Ax. The same conven-expressions are predicates, whereas in the original tion is adopted for the existential quantifier, which statement forms S is the subject and P is the predicate. will be introduced presently. Any problem in this regard vanishes, however, once Recall from Chapter 4 that a particular statement one understands what happens when universal state- is a statement that makes an assertion about one or more ments are converted into conditionals. When so con- unnamed members of the subject class. As with univer-verted, S becomes the predicate of the antecedent and sal statements, particular statements are either affirma-P becomes the predicate of the consequent. In other tive or negative, depending on whether the statement words, in the conditional �If anything is an S, then it affirms or denies that members of the subject class are is a P,� both S and P are predicates. Thus, using predi- members of the predicate class. Also, as with universal cate symbolism to translate universal statements leads statements, the key to translating particular statements is to no difficulties. When translating these statements, provided by the Boolean interpretation: the point to remember is simply this: The subject of

the original statement is represented by a capital letter Boolean in the antecedent, and the predicate by a capital letter Statement forminterpretation

in the consequent. Here are some examples: Some S are P. At least one thing is an

S and it is also a P.

Symbolic Some S are not P. At least one thing is an

StatementtranslationS and it is not a P. All skyscrapers are tall. (x)(Sx ⊃ Tx) No frogs are birds. (x)(Fx ⊃ ∼ ∼Bx) In other words, particular statements are translated

All ambassadors are as conjunctions. Since we are already familiar with (x)(Ax ⊃ Sx) statesmen. the symbol for conjunction (the dot), the only addi-

No diamonds are rubies. tional symbol that we need in order to translate these (x)(Dx ⊃ ∼ ∼ Rx) statements is a symbol for existence. This is provided

∼ ∼ Bx, and so on are called statement functions. A thus: (∃x). This symbol is translated �there exists an statement function is the expression that remains x such that.� The existential quantifier is combined when a quantifier is removed from a statement. It with the dot operator to translate particular state-In these examples, the expressions Sx ⊃ Tx, Fx ⊃ variable to the right of a backward E in parentheses, by the existential quantifier, formed by placing a

is a mere pattern for a statement. It makes no defi- ments as follows: nite assertion about anything in the universe, has no

truth value, and cannot be translated as a statement. Statement Symbolic The variables that occur in statement functions are formTranslationverbal meaning called free variables because they are not bound Some S are ( ∃ x)(Sx ● Px) There exists an x by any quantifier. In contrast, the variables that P. such that x is an occur in statements are called bound variables . S and x is a P. In using quantifiers to translate statements, we ( Some S are ∃ x)(Sx ● ∼ ∼ adopt a convention similar to the one adopted for Px) There exists an the tilde operator. That is, the quantifier governs not P. x such that x is only the expression immediately following it. For an S and x is not example, in the statement ( a P. x )( Ax ⊃ Bx ) the universal

Section 8.1 Symbols and Translation 393

As in the symbolic expression of universal state- Many of the principles developed in syllogistic

ments, the letter x is an individual variable, which logic (see Section 4.7) may be carried over into can stand for any item in the universe. Accordingly, predicate logic. Specifically, it should be under-the expression (∃x)(Sx ● Px) means �Something stood that statements beginning with the words exists that is both an S and a P,� and (∃x)(Sx ● ∼ ∼Px) only and none but are exclusive propositions. means �Something exists that is an S and not a P.� When these statements are translated, the term To see the equivalence of these expressions with the occurring first in the original statement becomes Boolean (and Aristotelian) interpretation of particu- the consequent in the symbolic expression, and lar statements, it is again useful to recall how these the term occurring second becomes the anteced-statements are represented by Venn diagrams: ent. One of the few differences in this respect

between predicate logic and syllogistic logic con-

Some S are P. Some S are not P. cerns singular statements. In syllogistic logic,

singular statements are translated as universals,

while in predicate logic, as we have seen, they are

X X translated in a unique way. Here are some exam-

ples of a variety of statements:

SPSP

Where the X designates at least one existing item, the Symbolic Statementtranslation diagram on the left asserts that something exists that ∃ is both an S and a P , and the one on the right asserts There are happy marriages. ( x)(Mx ● Hx) that something exists that is an S and not a P. In Every pediatrician loses (x)(Px ⊃ L x) other words, these diagrams assert exactly the same sleep.

symbolic expressions, therefore, exactly express the Unicorns do not exist. ∼ ∼(∃x)Ux or (x)∼ ∼Ux Boolean (and Aristotelian) interpretation of particu-thing as the symbolic expressions just given. These Animals exist. (∃x)A x lar statements. Here are some examples: Anything is conceivable. (x)Cx

Sea lions are mammals. (x)(Sx ⊃ Mx)

Symbolic Sea lions live in these (∃x)(Sx ● L x)

Statementtranslation caves.

Some men are paupers. (∃x)(Mx ● Px) Egomaniacs are not (x)(E x ⊃ ∼ ∼Px)

Some diseases are not pleasant companions. ( ∃ x)(Dx ● ∼ ∼ Cx)

contagious. A few egomaniacs did not (∃x)(E x ● ∼ ∼A x)

Some jobs are boring. arrive on time. ( ∃ x)(Jx ● Bx)

Some vehicles are not Only close friends were (x)(Ix ⊃ Cx) ( ∃ x)(Vx ● ∼ ∼ Mx)

motorcycles. invited to the wedding.

None but citizens are (x)(E x ⊃ Cx)

The general rule to follow in translating state- eligible to vote.

to understand the meaning of the statement to be Girl Scout sells cookies. or translated. If the statement makes an assertion about ments in predicate logic is always to make an effort It is not the case that every ∼ ∼(x)(Gx ⊃ Sx) every member of its subject class, a universal quanti- ( ∃x)(Gx ● ∼ ∼Sx) fier should be used to translate it; but if it makes an Not a single psychologist ∼ ∼(∃x)(Px ● A x) assertion about only one or more members of this attended the convention. or class, an existential quantifier should be used. (x)(Px ⊃ ∼ ∼A x)

394Chapter8 Predicate Logic

As these examples illustrate, the general proce- statement reads �snakes and lizards.� If the transla-

dure in translating statements in predicate logic is to tion were rendered as (x)[Tx ⊃ (Sx ● Lx)] it would render universal statements as conditionals preceded mean that anything that thrives in the desert is both by a universal quantifier, and particular statements a snake and a lizard (at the same time). And this is as conjunctions preceded by an existential quantifier. surely not what is meant. For the same reason, the However, as the third and fifth examples indicate, second example is translated in terms of the disjunc-there are exceptions to this procedure. A statement tion Ox �� Lx even though the English reads �oranges that makes an assertion about literally everything in and lemons.� If the statement were translated (x) the universe is translated in terms of a single predicate [(Ox ● Lx) ⊃ Cx], it would mean that anything that preceded by a universal quantifier, and a statement is simultaneously an orange and a lemon (and there that asserts that some class of things simply exists is are none of these) is a citrus fruit. The same principle translated in terms of a single predicate preceded by is used in translating the sixth example, which, inci-an existential quantifier. The last two examples illus- dentally, reads �If anything is a cat or a dog, then if it trate that a particular statement is equivalent to a is frightened or harassed, it bites.� The third example negated universal, and vice versa. The first of these is employs the conjunction Rx ● Ax to translate ripe equivalent to �Some Girl Scouts do not sell cookies� apples. This, of course, is correct, because such a and the second to �No psychologists attended the thing is both ripe and an apple at the same time. The convention.� Actually, any quantified statement can fifth example illustrates the fact that �unless� may be be translated using either a universal or an existential translated as either �if not� or �or.� quantifier, provided that one of them is negated. The The operators of propositional logic can be used equivalence of these two forms of expression will be to form compound arrangements of universal and analyzed further in Section 8.3. particular statements, just as they can be used to

More-complex statements may be translated by form compound arrangements of singular state-

following the basic rules just presented. Examples: ments. Here are some examples:

StatementSymbolic translationSymbolic

Only snakes and lizards Statementtranslation (x)[Tx ⊃ (Sx �� L x)] thrive in the desert. If Elizabeth is a historian, He ⊃ (∃x)(Wx ● Hx)

Oranges and lemons then some women are (x)[(Ox �� L x) ⊃ Cx] historians. are citrus fruits. If some cellists are music (∃x)(Cx ● Mx) ⊃ Ripe apples are (x)[(Rx ● A x) ⊃ (C x ● Dx)] directors, then ( ∃ x)(Ox ● Px) crunchy and delicious. some orchestras are Azaleas bloom if (x)[Ax ⊃ (Bx K Fx)] properly led. and only if they are Either everything is alive (x)Ax �� ∼ ∼ Cb fertilized. or Bergson�s theory ⊃ Peaches are edible (x)[Px ( ∼ ∼ Rx ⊃ Ex)] is not correct. unless they are rotten. or All novels are interesting (x)(Nx ⊃ Ix) K

(x)[Px ⊃ (E x �� R x)] if and only if some (∃x)[(Nx ● Sx) ● ∼ ∼R x]

Cats and dogs bite if Steinbeck novels are not (x){(Cx �� Dx) ⊃

they are frightened romances. [(Fx �� Hx) ⊃ Bx]}

or harassed. Green avocados are (x)[(Gx ● A x) ⊃ ∼ ∼Px] ��

never purchased unless (x)[(R x ● A x) ⊃ E x]

Notice that the first example is translated in terms all the ripe ones are of the disjunction expensive. Sx �� Lx even though the English

Section 8.1 Symbols and Translation 395

We have seen that the general procedure is to that everything in the universe is both a cat and an

translate universal statements as conditionals pre- animal, which is clearly false. Thus, as these exam-ceded by a universal quantifier, and to translate ples illustrate, it is imperative that the two quanti-particular statements as conjunctions preceded by fiers not be confused with each other. an existential quantifier. Let us see what happens One final observation needs to be made. It was to these translations when they are preceded by the mentioned earlier that the letters x,� y, and z are wrong quantifier. Consider the false statement �No reserved for use as variables for translating univer-cats are animals.� This is correctly translated (x)(Cx sal and particular statements. In accord with this ⊃ ∼ ∼Ax). If, however, it were translated (∃x)(Cx ⊃ convention, the other twenty-three lowercase let-∼ ∼Ax), the symbolic statement would turn out to be ters (a, b, c, .�.�. u, v, w) may be used as names for true. This may be seen as follows. (∃x)(Cx ⊃ ∼ ∼Ax) translating singular statements. Thus, for example, is equivalent via material implication to (∃x)(∼ ∼Cx �Albert is a scientist� is translated Sa. But a question �� ∼ ∼Ax), which in turn is equivalent via De Mor- naturally arises with statements such as �Xerxes gan�s rule to (∃ x)∼ ∼(Cx ● Ax). The latter statement, was a king.� Should this statement be translated Kx? however, merely asserts that something exists that is The answer is no. Some other letter, for example not both a cat and an animal�for example, a dog� the second letter in the name, should be selected which is true. Again, consider the true statement instead of x. Maintaining this alphabetical conven-�Some cats are animals.� This is correctly trans- tion will help us avoid mistakes in the next section lated (∃x)(Cx ● Ax). If, however, it were translated when we use natural deduction to derive the con-(x)(Cx ● Ax), the symbolic statement would assert clusions of arguments.

Exercise 8.1

I. Translate the following statements into symbolic form. Avoid negation signs preceding

quantifiers. The predicate letters are given in parentheses.

★1. Elaine is a chemist. (C )

2. Nieshe is not a sales clerk. (S)

3. Neither Wordsworth nor Shelley was Irish. (I )

★4. Rachel is either a journalist or a newscaster. ( J, N )

5. Intel designs a faster chip only if Micron does. (D)

6. Belgium and France subsidize the arts only if Austria or Germany expand

museum holdings. (S, E )

★7. All maples are trees. (M, T )

8. Some grapes are sour. (G, S )

9. No novels are biographies. (N, B)

★10. Some holidays are not relaxing. (H, R)

11. If Gabriel is correct, then the Taj Mahal is made of marble. (C, M )

12. Gabriel is not correct only if the Taj Mahal is made of granite. (C, G )

★13. Tigers exist. (T )

396Chapter8 Predicate Logic

14. Anything that leads to violence is wrong. (L, W )

15. There are pornographic art works. (A, P )

★16. Not every smile is genuine. (S, G )

17. Every penguin loves ice. (P, L)

18. There is trouble in River City. (T, R)

★19. Whoever is a socialite is vain. (S, V )

20. Any caring mother is vigilant and nurturing. (C, M, V, N )

21. Terrorists are neither rational nor empathic. (T, R, E )

★22. Nobody consumed by jealousy is happy. (C, H )

23. Everything is imaginable. (I )

24. Ghosts do not exist. (G )

★25. A thoroughbred is a horse. (T, H )

26. A thoroughbred won the race. (T, W )

27. Not all mushrooms are edible. (M, E )

★28. Not any horse chestnuts are edible. (H, E )

29. A few guests arrived late. (G, A)

30. None but gentlemen prefer blondes. (G, P )

★31. A few cities are neither safe nor beautiful. (C, S, B)

32. There are no circular triangles. (C, T )

33. Snakes are harmless unless they have fangs. (S, H, F )

★34. Some dogs bite if and only if they are teased. (D, B, T )

35. An airliner is safe if and only if it is properly maintained. (A, S, P )

36. Some companies go bankrupt if sales decline. (C, B, S)

★37. Some children act up only if they are tired. (C, A, T )

38. The only musicians that are available are trombonists. (M, A, T )

39. Only talented musicians perform in the symphony. (T, M, P)

★ 40. Any well-made car runs smoothly. (W, C, R)

41. Not every foreign car runs smoothly. (F, C, R)

42. A good violin is rare and expensive. (G, V, R, E )

★ 43. Violins and cellos are stringed instruments. (V, C, S, I )

44. A room with a view is available. (R, V, A)

45. A room with a view is expensive. (R, V, E )

★ 46. Some French restaurants are exclusive. (F, R, E )

Section 8.1 Symbols and Translation 397

47. Some French caf�s are not recommended. (F, C, R)

48. Hurricanes and earthquakes are violent and destructive. (H, E, V, D)

★49. Tyrell is guilty if and only if all the witnesses committed perjury. (G, W, C )

50. If any witnesses told the truth, then either Parsons or Harris is guilty. (W, T, G )

51. If all mysteries are interesting, then Rebecca is interesting. (M, I )

★52. If there are any interesting mysteries, then Rebecca is interesting. (M, I )

53. Skaters and dancers are energetic individuals. (S, D, E, I )

54. Swiss watches are not expensive unless they are made of gold. (S, W, E, M )

★55. If all the buildings in Manhattan are skyscrapers, then the Chrysler building

is a skyscraper. (B, M, S)

56. Experienced mechanics are well paid only if all the inexperienced ones are lazy.

(E, M, W, L)

57. Balcony seats are never chosen unless all the orchestra seats are taken. (B, S, C, O, T )

★58. Some employees will get raises if and only if some managers are overly gener-

ous. (E, R, M, O)

59. The physicists and astronomers at the symposium are listed in the program

if they either chair a meeting or read a paper. ( P, A, S, L, C, R)

60. If the scientists and technicians are conscientious and exacting, then some

of the mission directors will be either pleased or delighted. (S, T, C, E, M, P, D)

II. Create five statements relating to social media. Then translate them into the symbolism

of predicate logic.

8.2 Using the Rules of Inference

The chief reason for using truth-functional opera- are required to remove quantifiers at the begin-tors (the dot, wedge, horseshoe, and so on) in ning of a proof sequence and to introduce them, translating statements into the symbolism of when needed, at the end of the sequence. These predicate logic is to allow for the application of four rules are called universal instantiation, uni-the eighteen rules of inference to derive the con- versal generalization, existential instantiation, and clusion of arguments via natural deduction. Since, existential generalization. The first two are used however, the first eight of these rules are appli- to remove and introduce universal quantifiers, cable only to whole lines in an argument, as long respectively, and the second two to remove and as the quantifier is attached to a line these rules introduce existential quantifiers. of inference cannot be applied�at least not to the Let us first consider universal instantiation. As kind of arguments we are about to consider. To an illustration of the need for this rule, consider the provide for their application, four additional rules following argument:

398Chapter8 Predicate Logic

All economists are social scientists. instantial letter. For an example of an operation Paul Krugman is an economist. that violates the rule expressed in this definition, Therefore, Paul Krugman is a social scientist. consider line 3 of the foregoing proof. If this line

This argument, which is clearly valid, is symbolized were instantiated as Ep ⊃ Sx, it would not be cor-as follows: rect because the x in Sx was not replaced with the

instantial letter p.

1. (x)(E x ⊃ Sx) Let us now consider universal generalization. 2. Ep / Sp The need for this rule may be illustrated through ref-

As the argument now stands, none of the first eight erence to the following argument:

rules of inference can be applied; as a result, there All psychiatrists are doctors. is no way in which the two premises can be com-All doctors are college graduates. bined to derive the conclusion. However, if the first Therefore, all psychiatrists are college premise could be used to derive a line that reads graduates. Ep ⊃ Sp , this statement could be combined with the This valid argument is symbolized as follows: second premise to yield the conclusion via modus

Paul Krugman is himself an item in the universe, ponens. Universal instantiation serves exactly this 1. (x)(Px ⊃ Dx) purpose. 2. (x)(Dx ⊃ Cx) / (x)(Px ⊃ Cx) The first premise states that for any item x in the universe, if that item is an E , then it is an S . But since Once universal instantiation is applied to the two premises, we will have lines that can be used to set up a hypothetical syllogism. But then we will have the first premise implies that if Paul Krugman is an E to reintroduce a universal quantifier to derive the , then Paul Krugman is an S . A line stating exactly this can be derived by universal instantiation (UI). conclusion as written. This final step is obtained by universal generalization (UG). The justification for In other words, universal instantiation provides such a step lies in the fact that both premises are uni-us with an instance of the universal statement versal statements. The first states that if anything is a ( x )( Ex ⊃ Sx ). In the completed proof, which fol-lows, the p in line 3 is called the instantial letter : P , then it is a D , and the second states that if anything is a D , then it is a C . We may therefore conclude that 1. (x)(Ex ⊃ Sx) if anything is a P , then it is a C . But because of the 2. Ep / Sp complete generality of this reasoning process, there 3. Ep ⊃ Sp 1, UI is a special way in which we must perform the uni-4. Sp 2, 3, MP versal instantiation step. Instead of selecting a spe-cifically named At this point the question might arise as to why instance, as we did in the previous modus ponens is applicable to lines� 2 and 3. In example, we must select a variable that can range Chapter 7 we applied modus ponens to lines of the over every instance in the universe. The variables at form p ⊃ q, but are we justified in applying it to a our disposal, you may recall from the previous sec-line that reads Ep ⊃ Sp? The answer is yes, because tion, are w, x, y, and z. Let us select y. The completed Ep and Sp are simply alternate ways of symbolizing proof is as follows:

simple statements. As so understood, these sym- 1. (x)(Px ⊃ Dx) bols do not differ in any material way from the p 2. (x)(Dx ⊃ Cx) / (x)(Px ⊃ Cx) and q of propositional logic. 3. Py ⊃ Dy 1, UI We may now give a general definition of instan-4. Dy ⊃ Cy 2, UI tiation. Instantiation is an operation that con-5. Py ⊃ Cy 3, 4, HS sists in deleting a quantifier and replacing every 6. (x)(Px ⊃ Cx) 5, UG variable bound by that quantifier with the same

Section 8.2 Using the Rules of Inference 399

As noted earlier, the expressions in lines 3, 4, and 5 If we instantiate the first line with respect to a, we are called statement functions. As such, they are mere can obtain Sa via modus ponens. But if it is true that patterns for statements; they have no truth value and Andrea Bocelli is a tenor, then it certainly follows cannot be translated as statements. Yet if we take cer- that there is at least one singer (namely, Andrea tain liberties, we might characterize line 5 as saying Bocelli). This last step is accomplished by existential

item at random in the universe. Line 6 can then be 1. (x)(Tx ⊃ Sx) seen as reexpressing this sense of line 5. �If it is a P, then it is a C, where �it� designates any generalization (EG). The proof is as follows:

As the two previous examples illustrate, we have 2. Ta / (∃x)Sx

two ways of performing universal instantiation. On 3. Ta ⊃ Sa 1, UI the one hand, we may instantiate with respect to a 4. Sa 2, 3, MP

5. (∃x)Sx 4, EG

constant, such as a or b, and on the other, with respect to a There are no restrictions on existential generaliza-variable , such as x or y . The exact way in which this operation is to be performed depends on the kind tion, and the operation can be performed when the of result intended. If we want some part of a universal instantial letter is either a constant (as in the Bocelli statement to match a singular statement on another example) or a variable. As an instance of the latter, line, as in the first example, we instantiate with consider the following sequence:

respect to a constant. But if, at the end of the proof, we 1. (x)(Px ⊃ Qx) want to perform universal generalization over some 2. (x)Px / (∃x)Qx part of the statement we are instantiating, then we 3. Py ⊃ Qy 1, UI must instantiate by using a variable. This latter point 4. Py 2, UI leads to an important restriction governing univer- 5. Qy 3, 4, MP sal generalization�namely, that we cannot perform 6. (∃x)Qx 5, EG

Consider the following Line 5 states in effect that everything in the universe this operation when the instantial letter is a constant. erroneous proof sequence:

is a Q. From this, the much weaker conclusion fol-

1. Ta lows (line 6) that something is a Q. If you should 2. (x)Tx 1, UG (invalid) wonder how an existential conclusion can be drawn

If from universal premises, the answer is that predicate Ta means �Albert is a thief, � then on the basis of logic assumes that at least one thing exists in the uni-this information, we have concluded (line 2) that verse. Hence, line 2, which asserts that everything in everything in the universe is a thief. Clearly, such an the universe is a P , entails that at least one thing is a inference is invalid. This illustrates the fact that uni-P . Without this assumption, universal instantiation versal generalization can be performed only when in line 4 would not be possible. the instantial letter (in this case a ) is a variable. We may now construct a definition of general-Let us now consider existential generalization . ization that covers both varieties. Generalization in The need for this operation is illustrated through the the inclusive sense is an operation that consists in following argument: (1) introducing a quantifier immediately prior to a

All tenors are singers. statement, a statement function, or another quanti-Andrea Bocelli is a tenor. fier, and (2) replacing one or more occurrences of Therefore, there is at least one singer. a certain instantial letter in the statement or state-

This argument is symbolized as follows: ment function with the same variable that appears

in the quantifier. For universal generalization, all

1. (x)(Tx ⊃ Sx) occurrences of the instantial letter must be replaced 2. Ta / (∃x)Sx with the variable in the quantifier, and for existential

400Chapter8 Predicate Logic

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). generalization, at least one of the instantial letters 1. (x)(Ax ⊃ Cx) must be replaced with the variable in the quantifier. 2. (∃x)(Ax ● Gx) / (∃x)(Gx ● Cx) Thus, both of the following cases of existential gen- 3. Ad ● Gd 2, EI eralization are valid (although the one on the left is 4. Ad ⊃ Cd 1, UI by far the more common version): 5. Ad 3, Simp

1. Fa ● Ga 1. Fa ● Ga 7. Gd 6. Cd 4, 5, MP ● Ad 3, Com

2. (∃x)(Fx ● Gx ) 1, EG 2. (∃x)(Fx ● Ga ) 1, EG 8. Gd 7, Simp

On the other hand, only one of the following 9. Gd ● Cd 6, 8, Conj

cases of universal generalization is valid: 10. (∃x)(Gx ● Cx) 9, EG

1. Fx Examination of this proof reveals an immediate ⊃ Gx 1. Fx ⊃ Gx restriction that must be placed on existential instan-2. (y)(Fy ⊃ Gy ) 1, UG 2. (y)(Fy ⊃ Gx ) 1, UG (invalid) tiation. The name that we have assigned to the par-ticular something in line 2 that is both an A and a The inference on the right is invalid because the x in G is a hypothetical name. It would be a mistake to Gx was not replaced with the variable in the quanti-conclude that this something really has that name. fier (that is, y ). Accordingly, we must introduce a restriction that Of course, it may happen that the instantial letter prevents us from ending the proof with some line is the same as the variable that appears in the quan-that includes the letter d . If, for example, the proof tifier. Thus, the operation � Gx, therefore ( x ) Gx � were ended at line 9, we would be concluding that counts as a generalization. Cases of generalization the something that is a G and a C really does have where a quantifier is introduced prior to another the name d . This, of course, would not be legitimate, quantifier will be presented in Section 8.6. because d is an arbitrary name introduced into the The need for existential instantiation can be proof for mere convenience. To prevent such a mis-illustrated through the following argument: take, we require that the name selected for existential All attorneys are college graduates. instantiation not appear to the right of the slanted Some attorneys are golfers. line adjacent to the last premise that indicates the Therefore, some golfers are college conclusion to be derived. Since the last line in the graduates. proof must be identical to this line, such a restriction prevents us from ending the proof with a line that The symbolic formulation is as follows: contains the existential name. 1. (x)(Ax ⊃ Cx) Further examination of this proof indicates 2. ( ∃ x)(Ax ● Gx) / ( ∃ x)(Gx ● Cx) another important restriction on existential instan-If both quantifiers can be removed, the conclusion tiation. Notice that the line involving existential can be derived via simplification, modus ponens, instantiation is listed before the line involving uni-and conjunction. The universal quantifier can be versal instantiation. There is a reason for this. If the removed by universal instantiation, but to remove order were reversed, the existential instantiation step the existential quantifier we need existential instan- would rest on the illicit assumption that the some-tiation. Line 2 states that there is something that is thing that is both an A and a G has the same name both an A and a G. Existential instantiation con- as the name used in the earlier universal instan-sists in giving this something a name, for example, tiation step. In other words, it would involve the �David.� We will call this name an �existential name� assumption that the something that is both an A and because it is obtained through existential instantia- a G is the very same something named in the line tion. The completed proof is as follows: Ad ⊃ Cd. Of course, no such assumption is legitimate.

Section 8.2 Using the Rules of Inference 401

To keep this mistake from happening, we introduce applied only to whole lines in a proof. The following the restriction that the name introduced by existen- sequence illustrates a violation of this restriction:

tial instantiation be a new name not occurring ear- 1. (x)Px ⊃ (x)Qx lier in the proof sequence. The following defective 2. Py ⊃ Qy 1, UI (invalid) proof illustrates what can happen if this restriction is violated: In line 2, universal instantiation is applied to both

To see that this proof is indeed defective, let 10. ( 1. ( the antecedent and consequent of the first line. ∃ x)(Fx ● Ax) To derive line 2 validly, the first line would have 2. ( ∃ x)(Fx ● Ox) / ( ∃ x)(Ax ● Ox) to read ( x )( Px ⊃ Qx ). With this final restriction in 3. Fb ● Ab 1, EI mind, the four new rules of inference, developed 4. Fb ● Ob 2, EI (invalid) on the previous pages, may now be summarized. In 5. Ab ● Fb 3, Com the formulation that follows, the symbols ℱ x and 6. Ab 5, Simp ℱ y represent any statement function�that is, any 7. Ob ● Fb 4, Com symbolic arrangement containing individual vari-8. Ob 7, Simp ables, such as 9. Ab ● Ob 6, 8, Conj Ax ⊃ Bx , Cy ⊃ ( Dy �� Ey ), or Gz ● Hz . The symbol ∃ x)(Ax ● Ox) 9, EG ℱ a represents any statement ; that is, any symbolic arrangement containing individual F stand constants (or names), such as Ac ⊃ Bc , Cm ⊃ ( Dm for fruits, A for apples, and O for oranges. The argu-�� Em ), or Gw ● Hw. And the symbol ℱ is a predicate ment that results is: variable that represents any predicate such as F, G,

Some fruits are apples. or H.*

Some fruits are oranges. 1. Universal instantiation (UI): Therefore, some apples are oranges.

Since the premises are true and the conclusion (x)ℱx (x)ℱx

false, the argument is clearly invalid. The defect in ℱy ℱa the proof occurs on line 4. This line asserts that the

something that is both an F and an O is the very 2. Universal generalization (UG):

same something that is both an F and an A. In other ℱy ℱa words, the restriction that the name introduced by not allowed: ( x ) ℱ x ( existential instantiation be a new name not occur-x ) ℱ x

requires that the existential name not occur in the line (∃x)ℱx (∃x)ℱx not allowed: that indicates the conclusion to be derived, and the ℱ a ℱ y second restriction requires that this name be a new ring earlier in the proof is violated. 3. Existential instantiation (EI): The first restriction on existential instantiation

name that has not occurred earlier in the proof. These Restriction: The existential name a must be a two restrictions can easily be combined into a single new name that does not appear in restriction that requires that the name introduced by any previous line (including the existential instantiation be a new name that has not conclusion line). occurred in any previous line, including the line adja-cent to the last premise that indicates the conclusion to

be derived.

One further restriction that affects all four of *Some textbooks use Greek letters such as φ (phi), χ (chi), and ψ (psi)

these rules of inference requires that the rules be in the place of ℱ to express these and other rules.

402Chapter8 Predicate Logic

4. Existential generalization (EG): In conformity with the restriction on existential

ℱa instantiation, the EI step is performed before the UI ℱy

step. The same letter is then selected in the UI step

( as was used in the EI step. In line 5, Bm is joined ∃ x ) ℱ x ( ∃ x ) ℱ x

disjunctively via addition to Am. This rule applies in

The not allowed version of universal generaliza- predicate logic in basically the same way that it does

tion recalls the already familiar fact that general- in propositional logic. Any statement or statement ization is not possible when the instantial letter is function we choose can be joined disjunctively to a a constant. In other words, the mere fact that the given line. individual a is an ℱ is not sufficient to allow us to Another example: conclude that everything in the universe is an ℱ. At present this is the only restriction needed for uni- 1. (∃x)Kx ⊃ (x)(Lx ⊃ Mx) versal generalization. In Sections 8.4 and 8.6, how- 2. Kc ● Lc / Mc ever, two additional restrictions will be introduced. 3. Kc 2, Simp The 4. (∃x)Kx 3, EG not allowed version of existential instantia-

naming process. Because variables ( 6. Lc ⊃ Mc 5, UI x , y , and z ) are 7. Lc ● Kc 2, Com not names, they cannot be used as instantial letters tion merely recalls the fact that this operation is a 5. (x)(Lx ⊃ Mx) 1, 4, MP

in existential instantiation. 8. Lc 7, Simp 9. Mc 6, 8, MP Let us now investigate some applications of these

rules. Consider the following proof: Since the instantiation (and generalization) rules

1. (x)(Hx ⊃ Ix) must be applied to whole lines, it is impossible to 2. (x)(Ix ⊃ Hx) / (x)(Hx K Ix) instantiate line 1. The only strategy that can be fol-3. Hx ⊃ Ix 1, UI lowed is to use some other line to derive the ante-4. Ix ⊃ Hx 2, UI cedent of this line and then derive the consequent 5. (Hx ⊃ Ix) ● (Ix ⊃ Hx) 3, 4, Conj via modus ponens. Once the consequent is derived 6. Hx K Ix 5, Equiv (line 5), it is instantiated using the same letter that 7. (x)(Hx K Ix) 6, UG appears in line 2.

Because we want to perform universal generaliza- The next example incorporates all four of the tion on the last line of the proof, we instantiate lines instantiation and generalization rules: 1 and 2 using a variable, not a constant. Notice that

the variable selected for lines 3 and 4 is the same 1. (x)(Px ⊃ Qx) ⊃ (∃x)(Rx ● Sx) letter that occurs in lines 1 and 2. While a new letter 2. (x)(Px ⊃ Sx) ● (x)(Sx ⊃ Qx) / (∃x)Sx ( 3. (x)(Px ⊃ Sx) 2, Simp y, or z ) could have been selected, this is never neces-

want to combine lines 3 and 4, that the 5. (x)(Sx ⊃ Qx) 4, Simp same variable 6. Py ⊃ Sy 3, UI be selected in deriving these lines. Another example: sary in such a step. It 4. (x)(Sx ⊃ Qx) ● (x)(Px ⊃ Sx) 2, Com is necessary, however, since we

7. Sy ⊃ Qy 5, UI

1. (x)[(Ax �� Bx) ⊃ Cx] 8. Py ⊃ Qy 6, 7, HS 2. (∃x)Ax / (∃x)Cx 9. (x)(Px ⊃ Qx) 8, UG 3. Am 2, EI 10. (∃x)(Rx ● Sx) 1, 9, MP 4. (Am �� Bm) ⊃ Cm 1, UI 11. Ra ● Sa 10, EI 5. Am �� Bm 3, Add 12. Sa ● Ra 11, Com 6. Cm 4, 5, MP 13. Sa 12, Simp 7. (∃x)Cx 6, EG 14. (∃x)Sx 13, EG

Section 8.2 Using the Rules of Inference 403

As with the previous example, line 1 cannot be 1. (x)Fx ⊃ Ga instantiated. To instantiate the two conjuncts in 2. Fx ⊃ Ga 1, UI (invalid � instantiation line 2, they must first be separated (lines 3 and 5). can be applied only to Because UG is to be used in line 9, lines 3 and 5 are whole lines)

instantiated using a variable. On the other hand, a 1. (x)Fx ⊃ (x)Gx constant is used to instantiate line 10 because the 2. Fx ⊃ Gx 1, UI (invalid �i nstantiation statement in question is a particular statement. can be applied only to Another example: whole lines)

1. [(∃x)Ax ● (∃x)Bx] ⊃ C j 1. Fc 2. ( ∃ x)(Ax ● Dx) 2. ( ∃x)Gx 3. ( ∃ x)(Bx ● Ex) / C j 3. Gc 2, EI (invalid � c appears in 4. Am ● Dm 2, EI line 1) 5. Bn ● En 3, EI

6. Am 4, Simp 1. Fm ⊃ Gm 7. Bn 5, Simp 2. (x)(Fx ⊃ Gx) 1, UG (invalid � the instantial 8. (∃x)Ax 6, EG letter must be a vari-9. (∃x)Bx 7, EG able; m is a constant)

10. (∃x)Ax ● (∃x)Bx 8, 9, Conj

1. (∃x)Fx

11. C j 1, 10, MP 2. (∃x)Gx

When line 2 is instantiated (line 4), a letter other than 3. Fe 1, EI j, which appears in line 1, is selected. Then, when line 4. Ge 2, EI (invalid � e appears in 3 is instantiated (line 5), another new letter is selected. line 3)

The conclusion is derived, as in earlier examples, via 1. Fs ● Gs modus ponens by deriving the antecedent of line 1. 2. (∃x)Fx ● Gs 1, EG (improper � generalization

The following examples illustrate invalid or can be applied only to

improper applications of the instantiation and gen- whole lines)

eralization rules: 1. ∼ ∼(x)Fx 1. Fy ⊃ Gy 2. ∼ ∼Fy 1, UI (invalid � lines involv-2. (x)(Fx ⊃ Gy) 1, UG (invalid � every instance ing negated quantifiers

of y must be replaced cannot be instantiated; with x) see Section�8.3)

Exercise 8.2

I. Use the eighteen rules of inference to derive the conclusions of the following symbol-

ized arguments. Do not use either conditional proof or indirect proof.

★(1) 1. (x)(Ax ⊃ Bx)

2. (x)(Bx ⊃ Cx) / (x)(Ax ⊃ Cx)

(2) 1. (x)(Bx ⊃ Cx)

2. (∃x)(Ax ● Bx) / (∃x)(Ax ● Cx)

(3) 1. (x)(Ax ⊃ Bx)

2. ∼ ∼Bm / (∃x)∼ ∼Ax

404Chapter8 Predicate Logic

★(4) 1. (x)(Ax ⊃ Bx)

2. (y)(Ay �� ∼ ∼By) / ( x)(Ax K Bx)

(5) 1. (x)[Ax ⊃ (Bx �� Cx)]

2. Ag ● ∼ ∼Bg / Cg

(6) 1. (x)[(Ax �� Bx) ⊃ Cx]

2. (∃y)(Ay ● Dy) / (∃y)Cy

★(7) 1. (x)[ Jx ⊃ (Kx ● Lx)]

2. (∃y)∼ ∼Ky / (∃z)∼ ∼Jz

(8) 1. (x)[Ax ⊃ (Bx �� Cx)]

2. (∃x)(Ax ● ∼ ∼Cx) / (∃x)Bx

(9) 1. (x)(Ax ⊃ Bx)

2. Am ● An / Bm ● Bn

★ (10) 1. (x)(Ax ⊃ Bx)

2. Am �� An / Bm �� Bn

(11) 1. (x)(Bx �� Ax)

2. (x)(Bx ⊃ Ax) / (x)Ax

(12) 1. (∃x)Ax ⊃ (y)(By �� ∼ ∼Ay)

2. An / ( ∃y)By

★ (13) 1. (x)[(Ax ● Bx) ⊃ Cx]

2. (∃x)(Bx ● ∼ ∼Cx) / (∃x)∼ ∼Ax

(14) 1. (x)(Ax ⊃ Bx)

2. (x)(Cx ⊃ Dx)

3. Ae �� Ce / ( ∃x)(Bx �� Dx)

(15) 1. (∃x)Ax ⊃ (x)(Bx ⊃ Cx)

2. Am ● Bm / Cm

★ (16) 1. (∃x)Ax ⊃ (x)Bx

2. (∃x)Cx ⊃ (∃x)Dx

3. An ● Cn / (∃x)(Bx ● Dx)

(17) 1. (x)(Ax ● Bx)

2. Cr �� ∼ ∼(x)Bx / ( ∃x)(Cx ● Ax)

(18) 1. (x)[Ax ⊃ (Bx K Cx)]

2. An ● Am

3. Cn ● ∼ ∼Cm / Bn ● ∼ ∼Bm

★ (19) 1. (∃x)Ax ⊃ (x)(Cx ⊃ Bx)

2. (∃x)(Ax �� Bx)

3. (x)(Bx ⊃ Ax) / (x)(Cx ⊃ Ax)

(20) 1. (∃x)Ax ⊃ (x)(Bx ⊃ Cx)

2. (∃x)Dx ⊃ (∃x)∼ ∼Cx 3. (∃x)(Ax ● Dx) / (∃x)∼ ∼Bx

Section 8.2 Using the Rules of Inference 405

(21) 1. (∃x)(Ax ● Bx) ⊃ (x)(Cx ● Dx)

2. (∃x)Ax ⊃ (x)(Bx ● Cx)

3. Ae / ( ∃x)(Dx ● Bx)

★(22) 1. (x)(Ax ● ∼ ∼Bx)

2. An ⊃ [∼ ∼(∃x)Cx ⊃ Bc]

3. Am ⊃ (x)(Cx ⊃ Dx) / ( ∃x)Dx

(23) 1. (x)(Ax ● Bx)

2. (x)Ax ⊃ (Ce �� ∼ ∼Be)

3. (x)Bx ⊃ [(∃x)Cx ⊃ (Dn �� ∼ ∼An)] / ( ∃x)Dx

(24) 1. (∃x)Ax ⊃ (x)[(Ax �� Ex) ⊃ Dx]

2. (∃x)Dx ⊃ (x)[Dx ⊃ (Cx �� ∼ ∼Bx)] 3. An ● Bn / ( ∃x)Cx

★(25) 1. (∃x)Ax ⊃ (x)(Ax ⊃ Bx)

2. (∃x)Cx ⊃ (x)∼ ∼(Dx ● Bx) 3. (∃x)(Ax ● Cx) / ( ∃x)∼ ∼Dx

II. Translate the following arguments into symbolic form. Then use the eighteen rules of

inference to derive the conclusion of each. Do not use conditional or indirect proof.

★1. Oranges are sweet. Also, oranges are fragrant. Therefore, oranges are sweet and

fragrant. (O, S, F )

2. Tomatoes are vegetables. Therefore, the tomatoes in the garden are vegetables.

(T, V, G )

3. Apples and pears grow on trees. Therefore, apples grow on trees. (A, P, G )

★4. Carrots are vegetables and peaches are fruit. Furthermore, there are carrots and

peaches in the garden. Therefore, there are vegetables and fruit in the garden. (C, V, P, F, G )

5. Beans and peas are legumes. There are no legumes in the garden. Therefore,

there are no beans in the garden. (B, P, L, G )

6. There are some cucumbers in the garden. If there are any cucumbers, there are

some pumpkins in the garden. All pumpkins are vegetables. Therefore, there are some vegetables in the garden. (C, G, P, V )

★7. All gardeners are industrious people. Furthermore, any person who is industri-

ous is respected. Therefore, since Arthur and Catherine are gardeners, it fol-lows that they are respected. (G, I, P, R )

8. Some huckleberries are ripe. Furthermore, some boysenberries are sweet. If

there are any huckleberries, then the boysenberries are edible if they are sweet. Therefore, some boysenberries are edible. (H, R, B, S, E )

9. If there are any ripe watermelons, then the caretakers performed well. Further-

more, if there are any large watermelons, then whoever performed well will get a bonus. There are some large, ripe watermelons. Therefore, the caretakers will get a bonus. (R, W, C, P, L, B )

406Chapter8 Predicate Logic

★10. If the artichokes in the kitchen are ripe, then the guests will be surprised. Fur-

thermore, if the artichokes in the kitchen are flavorful, then the guests will be pleased. The artichokes in the kitchen are ripe and flavorful. Therefore, the guests will be surprised and pleased. (A, K, R, G, S, F, P )

III. The following dialogue contains nine arguments. Translate each into symbolic form

and then use the eighteen rules of inference to derive the conclusion of each. Some of them are quite challenging.

Where�s the Beef ?

�Have you decided what to order?� Paul says to Mindy, as he folds his menu and puts it on the table.

�I think I�ll have the tofu stir-fry,� she replies. �And you?�

�I�ll have the rib steak,� Paul says. �You know the rib steak is really good here, and

so is the pork tenderloin.�

�I�m a vegetarian,� she says.

�Oh,� Paul says. He smiles sheepishly.

Mindy smiles too. �First date,� she says.

�Yeah,� says Paul. �What made you decide to become a vegetarian?�

�For one thing,� Mindy says, �I think a vegetarian diet is healthier. People who eat

meat increase their intake of cholesterol and carcinogens. Those who increase their intake of cholesterol run a higher risk of heart attack, and those who increase their in-take of carcinogens run a higher risk of cancer. Anyone who runs a higher risk of heart attack and cancer is less healthy. Thus, people who eat meat are less healthy.

�I might add that if people who eat meat are less healthy, then if parents are responsible,

then they will refrain from feeding meat to children. All parents love their children, and if they do, then they are responsible. But if parents refrain from feeding meat to children, then children will grow up to be vegetarians. And if that happens, then nobody will eat meat in the future. Thus, we can look forward to a future where everyone is a vegetarian.�

�Well, I won�t hold my breath on that,� Paul says, as he offers Mindy a slice of

bread from the basket on the table. �If children and teenagers fail to eat meat, then they become deficient in zinc. And if children become deficient in zinc, then they risk a weakened immune system. And if that happens, then they are less healthy. Also, if older adults fail to eat meat, then they become deficient in iron. And if older adults become deficient in iron, then they risk becoming anemic, and if that happens, they are less healthy. Therefore, if children and older adults fail to eat meat, then they are less healthy.�

Mindy smiles. �That�s what zinc tablets and iron supplements are for,� she

says. Anyway, there are also moral reasons for being a vegetarian. Consider this. Animals are sentient beings�they feel pain, fear, and joy�and they have an interest in preserving their lives. If animals are sentient, then if humans cause ani-mals to suffer, then they act immorally. And if animals have an interest in preserving their lives, then if humans exploit animals, then they act immorally. But if humans kill animals for food, then they cause animals to suffer or they exploit them. Therefore, if humans kill animals for food, then they act immorally.�

�I agree with you that animals should not be made to suffer,� Paul responds. �But if

animals are raised under humane conditions, and some of them are, then they will not

Section 8.2 Using the Rules of Inference 407

be caused to feel pain or distress. And if animals are not caused to feel pain, then we are morally justified in eating them. Thus, we are morally justified in eating some animals.�

�I disagree,� says Mindy.

�That�s because you think that animals have rights,� Paul says. �If animals have

rights, then they have moral judgment. And if animals have moral judgment, then they respect the rights of other animals. But every animal pursues its own self-interest to the exclusion of other animals, and if that�s so, then it does not respect the rights of other animals. Therefore, animals have no moral judgment and they also have no rights.�

�Well,� Mindy replies, �by that line of reasoning infants and adults with mental

challenges have no rights. But everyone recognizes that they do have rights. And if infants have rights, then some people who lack the capacity for moral judgment have rights. But if this is true, then animals have rights, and if they do, then surely they have the right to life. But if animals have the right to life, then if humans are moral, then they must respect that right and they cannot kill animals for food. Thus, if humans are moral, they cannot kill animals for food.�

�The question of infants and adults with mental challenges raises an interesting

point,� Paul says. �I think what it comes down to is this: Something is considered to have rights if and only if it looks human. Infants and adults with mental challenges look human, so they are considered to have rights. But animals do not look human, so they are not considered to have rights.�

�That sounds awfully arbitrary,� says Mindy. �But I think what it really comes down

to is power. Something is considered to have rights if and only if it has as much power as humans. Animals do not have as much power as humans, so animals are not consid-ered to have rights. But that seems terribly wrong to me. It shouldn�t be a question of power. Anyway, now that our food has arrived, how�s your steak?�

Paul takes a bite. �It�s great,� he says. �And how�s your stir-fry?�

�Great,� Mindy says. She laughs. �And so is my conscience.�

IV. Create two valid arguments relating to national parks. Then translate them into the

symbolism of predicate logic and use the rules of inference to derive the conclusion of each.

8.3 Quantifier Negation Rule

The rules of inference developed thus far are not suf- the conclusion cannot be derived. What is needed is ficient to derive the conclusion of every argument in a rule that will allow us to remove the negation signs. predicate logic. For instance, consider the following: This rule, which we will proceed to develop now, is

∼ ∼(∃x)(Px ● called the quantifier negation rule. ∼ ∼ Qx)

∼ As a basis for developing the quantifier nega-∼ (x)( ∼ ∼ Rx �� Qx)

(∃x)∼ ∼Px tion rule, consider the following statements:

Both premises have tildes preceding the quantifiers. As Everything is beautiful.

long as they remain, neither statement can be instan- It is not the case that everything is beautiful. tiated; and if these statements cannot be instantiated,

408Chapter8 Predicate Logic

Prominent Alfred North Whitehead 1861�1947

LogiciansBertrand Russell 1872�1970

A lfred North Whitehead and Bertrand Russell collaborated in writing Principia Mathemat-ica, which is generally considered the most important logical endeavor of the twentieth century. It represents an attempt to reduce all of mathemat-

ics to logic. Published in three volumes between

1910 and 1913, the manuscript was so huge that

it required a �four wheeler� to transport it to the

printer. The combined work comprises over 1,900

pages, nearly all of them filled with highly complex

and technical notation. The American philosopher

Willard Van Orman Quine described the Principia

as �one of the great intellectual monuments of all Hulton Archive/Getty Images Bettmann/Getty Images

time.� Bertrand Russell, one of the world�s best-known intel-

Whitehead was born the son of an Anglican minis- lectuals, was born in Wales to aristocratic parents. Both

ter in Ramsgate, England. He entered Trinity College, died when he was very young, and although they had Cambridge, with a scholarship in mathematics, and requested that their son be raised an agnostic, the young after graduating, he was elected a Fellow of Trinity. Russell was brought up by a staunchly Victorian grand-While there he published A Treatise on Universal mother who fed him a steady dose of religion. As he Algebra, for which he was elected to the prestigious grew older, he became an atheist (or at best an agnostic) Royal Society. Whitehead�s most distinguished student who considered religion little better than superstition. at Trinity was Bertrand Russell. After Russell graduated At Trinity College, Russell studied mathematics and phi-he and�Whitehead became close friends. At age thirty- losophy and befriended Ludwig Wittgenstein and G. E. one Whitehead married Evelyn Wade, and they had Moore, as well as Whitehead. After graduating he was three children, two sons and a daughter. elected fellow of Trinity College and later, fellow of the

In 1910, Whitehead left Cambridge for London, Royal Society. He was extremely prolific as a writer, and in

where he taught at University College London 1950 he won the Nobel Prize in literature. and later at the Imperial College of Science and Russell was married four times, engaged in numer-Technology. While in London he wrote books in the ous love affairs with prominent women, was imprisoned areas of physics and the philosophy of science. Then, for six months for opposing conscription in World in 1924, he was appointed professor of philosophy at War�I, and was later imprisoned for opposing nuclear Harvard University, where he wrote Process and Real- weapons. A staunch supporter of birth control, popula-ity. The book became a cornerstone of what would tion control, democracy, free trade, and world govern-later be called process philosophy and process theol- ment, he opposed communism, imperialism, and every ogy. Whitehead died at age eighty-six in Cambridge, form of mind control. He died at age ninety-seven at his Massachusetts. home in Wales.

Section 8.3 Quantifier Negation Rule 409

Something is beautiful. 7. ∼∼ ∼∼Ra ● ∼ ∼Qa 5, DM It is not the case that something is beautiful. 8. ∼ ∼Pa �� ∼∼ ∼∼Qa 6, DM

You should be able to see that these statements are 9. ∼ ∼Pa �� Qa 8, DN equivalent in meaning to the following statements, 10. ∼ ∼Qa ● ∼∼ ∼∼Ra 7, Com respectively: 11. ∼ ∼Qa 10, Simp

12. Qa �� ∼ ∼Pa 9, Com

It is not the case that something is not 13. ∼ ∼Pa 11, 12, DS beautiful. 14. ( ∃x)∼ ∼Px 13, EG

Something is not beautiful. Before either line 1 or line 2 can be instantiated, the It is not the case that everything is not tilde operators preceding the quantifiers must be beautiful. removed. In accordance with the quantifier nega-Everything is not beautiful. tion rule, tilde operators are introduced immediately

after the new quantifiers in the expressions on lines

If we generalize these English equivalencies sym- 3 and 4. bolically, we obtain: Another example:

These four expressions constitute the quantifier ∼ ( ∼ ( 1. (∃x)(Hxx ) ℱ x:: ∼ ∼ ( ∃ x ) ∼ ∼ℱ x● Gx) ⊃ (x)Ix 2. ∼ ∼ Im / (x)(Hx ⊃ ∼ ∼Gx) ∼ ( x ) ℱ x:: ( ∃ x ) ∼ ∼ℱ x 3. ( ∃ x) ∼ ∼ Ix 2, EG ∃ x ) ℱ x:: ∼ ∼ ( x ) ∼ ∼ℱ x 4. ∼ ∼ (x)Ix 3, QN ∼ ( ∃ x ) ℱ x:: ( x ) ∼ ∼ℱ x 5. ∼ ∼ ( ∃ x)(Hx ● Gx) 1, 4, MT 6. (x) ∼ ∼ (Hx ● Gx) 5, QN negation rule (QN). Since they are stated as logi-7. (x)( ∼ ∼ Hx �� ∼ ∼ Gx) 6, DM cal equivalences, they apply to parts of lines as 8. (x)(Hx ⊃ ∼ ∼ Gx) 7, Impl

well as to whole lines. They can be summarized The statement that m is not an I (line 2) intuitively as follows: implies that not everything is an I (line 4); but exis-

One type of quantifier can be replaced by tential generalization and quantifier negation are the other type if and only if immediately needed to get the desired result. Notice that lines 7 before and after the new quantifier: and 8 are derived via De Morgan�s rule and mate-

1. attached. Since these rules are rules of replacement, ent are deleted. they apply to parts of lines as well as to whole lines. 2. Tilde operators that were not originally The following example illustrates the same point present are inserted. with respect to the quantifier negation rule: Tilde operators that were originally pres- rial implication, even though the quantifier is still

let us return to the argument at the beginning of this To see how the quantifier negation rule is applied, 1. (∃x)Jx ⊃ ∼ ∼(∃x)Kx 2. (x) ∼ ∼ Kx ⊃ (x) ∼ ∼ Lx / (∃x)Jx ⊃ ∼ ∼(∃x)L x section. The proof is as follows: 3. ( ∃ x)Jx ⊃ (x) ∼ ∼ Kx 1, QN

4. (∃x)Jx ⊃ (x)∼ ∼Lx 2, 3, HS

1. ∼ ∼(∃x)(Px ● ∼ ∼Qx) 5. (∃x)Jx ⊃ ∼ ∼(∃x)Lx 4, QN 2. ∼ ∼(x)(∼ ∼Rx �� Qx) / (∃x)∼ ∼Px

3. (x)∼ ∼(Px ● ∼ ∼Qx) 1, QN The quantifier negation rule is applied to only the 4. (∃x)∼ ∼(∼ ∼Rx �� Qx) 2, QN consequent of line 1, yielding line 3. Similarly, 5. ∼ ∼(∼ ∼Ra �� Qa) 4, EI the quantifier negation rule is then applied to only 6. ∼ ∼(Pa ● ∼ ∼Qa) 3, UI the consequent of line 4, yielding line 5.

410Chapter8 Predicate Logic

Exercise 8.3

I. Use the quantifier negation rule together with the eighteen rules of inference to derive

the conclusions of the following symbolized arguments. Do not use either conditional proof or indirect proof.

★(1) 1. (x)Ax ⊃ (∃x)Bx

2. (x)∼ ∼Bx / (∃x)∼ ∼Ax

(2) 1. (∃x)∼ ∼Ax �� (∃x)∼ ∼Bx

2. (x)Bx / ∼ ∼(x)Ax

(3) 1. ∼ ∼(∃x)Ax / (x)(Ax ⊃ Bx)

★(4) 1. (∃x)Ax �� (∃x)(Bx ● Cx)

2. ∼ ∼(∃x)Bx / (∃x)Ax

(5) 1. (x)(Ax ● Bx) �� (x)(Cx ● Dx)

2. ∼ ∼(x)Dx / (x)Bx

(6) 1. (∃x)∼ ∼Ax ⊃ (x)(Bx ⊃ Cx)

2. ∼ ∼(x)(Ax �� Cx) / ∼ ∼(x)Bx

★(7) 1. (x)(Ax ⊃ Bx)

2. ∼ ∼(x)Cx �� (x)Ax

3. ∼ ∼(x)Bx / (∃x)∼ ∼Cx

(8) 1. (x)Ax ⊃ (∃x)∼ ∼Bx

2. ∼ ∼(x)Bx ⊃ (∃x)∼ ∼Cx / (x)Cx ⊃ (∃x)∼ ∼Ax

(9) 1. (∃x)(Ax �� Bx) ⊃ (x)Cx

2. (∃x)∼ ∼Cx / ∼ ∼(∃x)Ax

★ (10) 1. ∼ ∼(∃x)(Ax ● ∼ ∼Bx)

2. ∼ ∼(∃x)(Bx ● ∼ ∼Cx) / (x)(Ax ⊃ Cx)

(11) 1. ∼ ∼(∃x)(Ax ● ∼ ∼Bx)

2. ∼ ∼(∃x)(Ax ● ∼ ∼Cx) / (x)[Ax ⊃ (Bx ● Cx)]

(12) 1. (x)[(Ax ● Bx) ⊃ Cx]

2. ∼ ∼(x)(Ax ⊃ Cx) / ∼ ∼(x)Bx

★ (13) 1. (x)(Ax ● ∼ ∼Bx) ⊃ (∃x)Cx

2. ∼ ∼(∃x)(Cx �� Bx) / ∼ ∼(x)Ax

(14) 1. (∃x)∼ ∼Ax ⊃ (x)∼ ∼Bx

2. (∃x)∼ ∼Ax ⊃ (∃x)Bx 3. (x)(Ax ⊃ Cx) / (x)Cx

(15) 1. ∼ ∼(∃x)(Ax �� Bx)

2. (∃x)Cx ⊃ (∃x)Ax

3. (∃x)Dx ⊃ (∃x)Bx / ∼ ∼(∃x)(Cx �� Dx)

Section 8.3 Quantifier Negation Rule 411

★(16) 1. (∃x)(Ax ● Bx) ⊃ (x)(Cx ● Dx)

2. (x)[(Ax �� Ex) ● (Bx �� Fx)] 3. ∼ ∼(x)Dx / ( x)(Ex �� Fx)

(17) 1. (∃x)(Ax ● Bx) �� (∃x)(Cx �� Dx)

2. (∃x)(Ax �� Cx) ⊃ (x)Ex

3. ∼ ∼En / (∃x)Dx

(18) 1. (∃x)Ax ⊃ [(∃x)Bx �� (x)Cx]

2. (∃x)(Ax ● ∼ ∼Cx)

3. ∼ ∼(x)Cx ⊃ [(x)Fx ⊃ (x)∼ ∼Bx] / ( ∃x)∼ ∼Fx

★(19) 1. (∃x)(Ax ● Bx) ⊃ (x)(Bx ⊃ Cx)

2. Bn ● ∼ ∼Cn / ∼ ∼(x)Ax

(20) 1. (∃x)Ax ⊃ ∼ ∼(∃x)(Bx ● Ax)

2. ∼ ∼(x)Bx ⊃ ∼ ∼(∃x)(Ex ● ∼ ∼Bx) 3. An / ∼ ∼(x)Ex

II. Translate the following arguments into symbolic form. Then use the quantifier nega-

tion rule and the eighteen rules of inference to derive the conclusion of each. Do not use either conditional proof or indirect proof.

★1. If all the physicians are either hematologists or neurologists, then there are no

cardiologists. But Dr. Fabian is a cardiologist. Therefore, some physicians are not neurologists. (P, H, N, C )

2. Either Dr. Alanzo is an internist or all the pathologists are internists. But it

is not the case that there are any internists. Therefore, Dr. Alanzo is not a pathologist. (I, P )

3. If some surgeons are allergists, then some psychiatrists are radiologists. But no

psychiatrists are radiologists. Therefore, no surgeons are allergists. (S, A, P, R)

★4. Either some general practitioners are pediatricians or some surgeons are

endocrinologists. But it is not the case that there are any endocrinologists. Therefore, there are some pediatricians. (G, P, S, E )

5. All physicians who did not attend medical school are incompetent. It is not the

case, however, that some physicians are incompetent. Therefore, all physicians have attended medical school. (P, A, I )

6. It is not the case that some internists are not physicians. Furthermore, it is not

the case that some physicians are not doctors of medicine. Therefore, all inter-nists are doctors of medicine. (I, P, D)

★7. All pathologists are specialists and all internists are generalists. Therefore,

since it is not the case that some specialists are generalists, it is not the case that some pathologists are internists. (P, S, I, G )

8. If some obstetricians are not gynecologists, then some hematologists are radi-

ologists. But it is not the case that there are any hematologists or gynecologists. Therefore, it is not the case that there are any obstetricians. (O, G, H, R)

412Chapter8 Predicate Logic

9. All poorly trained allergists and dermatologists are untrustworthy specialists.

It is not the case, however, that some specialists are untrustworthy. Therefore, it is not the case that some dermatologists are poorly trained. (P, A, D, U, S)

★10. It is not the case that some physicians are either on the golf course or in the

hospital. All of the neurologists are physicians in the hospital. Either some phy-sicians are cardiologists or some physicians are neurologists. Therefore, some cardiologists are not on the golf course. (P, G, H, N, C )

III. Create two valid arguments relating to vacations. At least one of the premises of each

argument must begin with the words �It is not the case that. . . .� Then translate the argu-ments into the language of predicate logic and derive the conclusion of each.

8.4 Conditional and Indirect Proof

Many arguments with conclusions that are either In this argument the antecedent of the conclusion is difficult or impossible to derive by the conven- a complete statement consisting of a statement func-tional method can be handled with ease by using tion, Hx, preceded by a quantifier. This complete state-either conditional or indirect proof. The use of ment is assumed as the first line in the conditional these techniques on arguments in predicate logic sequence. The instantiation and generalization rules is basically the same as it is on arguments in prop- are used within an indented sequence (both condi-ositional logic. Arguments having conclusions tional and indirect) in basically the same way as they expressed in the form of conditional statements are in a conventional sequence. When the consequent or disjunctions (which can be derived from con- of the conclusion is derived, the conditional sequence ditional statements) are immediate candidates for is completed, and it is then discharged in a conditional conditional proof. For these arguments, the usual statement having the first line of the sequence as its strategy is to put the antecedent of the conditional antecedent and the last line as its consequent. statement to be obtained in the first line of an The next example differs from the previous one in indented sequence, to derive the consequent as the that the antecedent of the conclusion is a statement last line, and to discharge the conditional sequence function, not a complete statement. With arguments in a conditional statement that exactly matches the such as this, only the statement function is assumed one to be obtained. Here is an example of such a as the first line in the conditional sequence. The proof: quantifier is added after the sequence is discharged.

1. (x)(Hx ⊃ Ix) / (∃x)Hx ⊃ (∃x)Ix 1. (x)[(Ax �� Bx) ⊃ C x] / (x)(Ax ⊃ C x)

2. (∃x)Hx ACP 2. Ax ACP 3. Ha 2, EI 3. Ax �� Bx 2, Add 4. Ha ⊃ Ia 1, UI 4. (Ax �� Bx) ⊃ C x 1, UI 5. Ia 3, 4, MP 5. C x 3, 4, MP 6. (∃x)Ix 5, EG 6. Ax ⊃ C x 2�5, CP

7. (∃x)Hx ⊃ (∃x)Ix 2�6, CP 7. (x)(Ax ⊃ C x) 6, UG

Section 8.4 Conditional and Indirect Proof 413

This example leads to an important restriction on the argument are switched. The proof, which is per-

the x in line 2 of this proof is said to be free because it 1. (x)(Rx ⊃ Sx) / (x)Rx ⊃ (x)Sx is not bound by any quantifier. (In contrast, the use of universal generalization. You may recall that fectly legitimate, is as follows:

lines 1 and 7 are bound by quantifiers.) The restric-x�s in 2. (x)Rx ACP 3. Rx 2, UI tion is as follows: 4. Rx ⊃ Sx 1, UI

UG: ℱy 5. Sx 3, 4, MP Restriction: UG must not be used ( 6. (x)Sx 5, UG x ) ℱ x within the scope of an

indented sequence if 7. (x)Rx ⊃ (x)Sx 2�6, CP

the instantial variable Notice in this proof that UG is used within the scope y is free in the first of a conditional sequence, but the restriction is not line of that sequence. violated because the instantial variable x is not free

The proof just given does not violate this restric- in the first line of the sequence. tion, because UG is not used within the scope of the Let us now consider some examples of indirect indented sequence at all. It is used only after the proof. We begin an indirect sequence by assuming sequence has been discharged, which is perfectly the negation of the statement to be obtained. When acceptable. If, on the other hand, UG had been a contradiction is derived, the indirect sequence is applied to line 5 to produce a statement reading discharged by asserting the denial of the original (x)Cx, the restriction would have been violated assumption. In the examples that follow, the nega-because the instantial variable x is free in the first tion of the conclusion is assumed as the first line line of the sequence. of the sequence, and the quantifier negation rule is

To understand why this restriction is necessary, then used to eliminate the tilde. When the result-

consider the following defective proof: ing statement is then instantiated, a new letter, m,

1. (x)Rx is selected that�has not appeared anywhere in a pre-⊃ (x)Sx / (x)(Rx ⊃ Sx)

2. Rx ACP vious line. The same letter is then selected for the

3. (x)Rx universal instantiation of line 1: 2, UG (invalid)

4. (x)Sx 1, 3, MP 1. (x)[(Px ⊃ Px) ⊃ (Qx ⊃ Rx)] / (x)(Qx ⊃ Rx) 5. Sx 4, UI 2. ∼ ∼(x)(Qx ⊃ Rx) AIP

6. Rx ⊃ Sx 2�5, CP 3. (∃x)∼ ∼(Qx ⊃ Rx) 2, QN 7. (x)(Rx ⊃ Sx) 6, UG 4. ∼ ∼(Qm ⊃ Rm) 3, EI

If 5. (Pm ⊃ Pm) ⊃ (Qm ⊃ Rm) 1, UI Rx means � x is a rabbit� and Sx means � x is a snake,� 6. ∼ ∼ (Pm ⊃ Pm) 4, 5, MT then the premise translates �If everything in the uni-7. ∼ ∼ ( ∼ ∼ Pm �� Pm) 6, Impl verse is a rabbit, then everything in the universe is a 8. ∼∼ ∼∼ Pm snake.� This statement is true because the anteced-● ∼ ∼ Pm 7, DM 9. Pm ent is false; that is, it is not the case that everything ● ∼ ∼ Pm 8, DN 10. ∼∼ ∼∼ (x)(Qx ⊃ Rx) 2�9, IP in the universe is a rabbit. The conclusion, on the 11. (x)(Qx ⊃ Rx) 10, DN other hand, is false , because it asserts that all rabbits are snakes. The argument is therefore invalid. If the The next example has a particular statement for restriction on UG had been obeyed, UG could not its conclusion:

have been used on line 3 and, as a result, the illicit 1. (∃x)Ax �� (∃x)Fx conclusion could not have been derived. 2. (x)(Ax ⊃ Fx) / (∃x)Fx It is interesting to see what happens when 3. ∼ ∼ ( ∃ x)Fx AIP the premise and the conclusion of this defective 4. ( ∃ x)Fx �� ( ∃ x)Ax 1, Com

414Chapter8 Predicate Logic

5. (∃x)Ax 3, 4, DS To see that this argument is indeed invalid, let Ax 6. Ac 5, EI stand for �x is an apple� and Fx for �x is a fruit.� The 7. Ac ⊃ Fc 2, UI first premise then reads �Either an apple exists or a 8. Fc 6, 7, MP fruit exists� (which is true), and the second premise 9. (x)∼ ∼Fx 3, QN reads �All apples are fruits� (which is also true). The

10. ∼ ∼Fc 9, UI conclusion, however, reads �Everything in the uni-11. Fc ● ∼ ∼Fc 8, 10, Conj verse is a fruit,� and this, of course, is false.

12. ∼∼ ∼∼(∃x)Fx 3�11, IP As in propositional logic, conditional and indi-13. (∃x)Fx 12, DN rect sequences in predicate logic may include each

Since indirect proof sequences are indented, they other. The following proof uses an indirect sequence within the scope of a conditional sequence. are subject to the same restriction on universal generalization as are conditional sequences. The 1. (x) [(Px �� Qx) ⊃ (Rx ● Sx)] / (∃x)(Px �� Sx) ⊃ (∃ x)Sx following proof, which is similar to the previous 2. (∃x)(Px �� Sx) ACP one, violates this restriction because the instantial 3. ∼ ∼(∃x)Sx AIP

The violation (line 4) allows a universal statement 5. Pa �� Sa 2, EI 6. ∼ ∼ Sa 4, UI to be drawn for the conclusion, whereas only a variable 4. (x)∼ ∼Sx 3, QN x is free in the first line of the sequence.

particular statement is legitimate (as in the prior 7. Sa �� Pa 5, Com 8. Pa 6, 7, DS example): 9. Pa �� Qa 8, Add

2. (x)(Ax 1. ( 10. (Pa �� Qa) ⊃ (Ra ● Sa) 1, UI ∃ x)Ax �� ( ∃ x)Fx 11. Ra ● Sa 9, 10, MP ⊃ Fx) / (x)Fx 12. Sa ● Ra 11, Com 3. ∼ ∼ Fx AIP 13. Sa 12, Simp 4. (x) ∼ ∼ Fx 3, UG (invalid) 14. Sa ● ∼ ∼ Sa 6, 13, Conj 5. ∼ ∼ ( ∃ x)Fx 4, QN 15. ∼∼ ∼∼ ( ∃ x)Sx 3�14, IP 6. ( ∃ x)Fx �� ( ∃ x)Ax 1, Com 16. ( ∃ x)Sx 15, DN 7. ( ∃ x)Ax 5, 6, DS 17. ( ∃ x)(Px �� Sx) ⊃ ( ∃ x)Sx 2�16, CP 8. Ac 7, EI

9. Ac ⊃ Fc 2, UI The conditional sequence begins, as usual, by assum-

11. ∼ 10. Fc 8, 9, MP ing the antecedent of the conditional statement to be ∼

12. FcFc 4, UI derived. The objective, then, is to derive the conse-∼ ∼ Fc 10, 11, Conj ●

13. quent. This is accomplished by the indirect sequence, ∼∼ ∼∼ Fx 3�12, IP

14. Fx 13, DN which begins with the negation of the consequent 15. (x)Fx 14, UG and ends (line 14) with a contradiction.

Exercise 8.4

I. Use either indirect proof or conditional proof to derive the conclusions of the follow-

ing symbolized arguments.

★(1) 1. (x)(Ax ⊃ Bx)

2. (x)(Ax ⊃ Cx) / (x)[Ax ⊃ (Bx ● Cx)]

Section 8.4 Conditional and Indirect Proof 415

(2) 1. (∃x)Ax ⊃ (∃x)(Bx ● Cx)

2. (∃x)(Cx �� Dx) ⊃ (x)Ex / (x)(Ax ⊃ Ex)

(3) 1. (∃x)Ax ⊃ (∃x)(Bx ● Cx)

2. ∼ ∼(∃x)Cx / (x)∼ ∼Ax

★(4) 1. (x)(Ax ⊃ Cx)

2. (∃x)Cx ⊃ (∃x)(Bx ● Dx) / (∃x)Ax ⊃ (∃x)Bx

(5) 1. (x)(Ax ⊃ Bx)

2. (x)[(Ax ● Bx) ⊃ Cx] / (x)(Ax ⊃ Cx)

(6) 1. (∃x)Ax ⊃ (x)Bx

2. An ⊃ ∼ ∼Bn / ∼ ∼An

★(7) 1. (x)[(Ax �� Bx) ⊃ Cx]

2. (x)[(Cx �� Dx) ⊃ Ex] / (x)(Ax ⊃ Ex)

(8) 1. (∃x)(Ax �� Bx) ⊃ ∼ ∼(∃x)Ax / (x)∼ ∼Ax

(9) 1. (x)(Ax ⊃ Bx)

2. (x)(Cx ⊃ Dx) / (∃x)(Ax �� Cx) ⊃ (∃x)(Bx �� Dx)

★(10) 1. (x)(Ax ⊃ Bx)

2. Am �� An / (∃x)Bx

(11) 1. (x)[(Ax �� Bx) ⊃ Cx]

2. (x)[(Cx �� Dx) ⊃ ∼ ∼Ax] / (x)∼ ∼Ax

(12) 1. (∃x)Ax ⊃ (x)(Bx ⊃ Cx)

2. (∃x)Dx ⊃ (x)∼ ∼Cx / (x)[(Ax ● Dx) ⊃ ∼ ∼Bx]

★(13) 1. (∃x)Ax ⊃ (x)(Bx ⊃ Cx)

2. (∃x)Dx ⊃ (∃x)Bx / (∃x)(Ax ● Dx) ⊃ (∃x)Cx

(14) 1. (∃x)Ax �� (∃x)(Bx ● Cx)

2. (x)(Ax ⊃ Cx) / (∃x)Cx

(15) 1. (∃x)Ax ⊃ (∃x)(Bx ● Cx)

2. (∃x)Cx ⊃ (x)(Dx ● Ex) / (x)(Ax ⊃ Ex)

★(16) 1. (x)[(Ax �� Bx) ⊃ Cx]

2. (∃x)(∼ ∼Ax �� Dx) ⊃ (x)Ex / (x)Cx �� (x)Ex

(17) 1. (x)Ax K (∃x)(Bx ● Cx)

2. (x)(Cx ⊃ Bx) / (x)Ax K (∃x)Cx

(18) 1. (x)(Ax K Bx)

2. (x)[Ax ⊃ (Bx ⊃ Cx)]

3. (∃x)Ax �� (∃x)Bx / (∃x)Cx

★(19) 1. (x)[Bx ⊃ (Cx ● Dx)] / (x)(Ax ⊃ Bx) ⊃ (x)(Ax ⊃ Dx)

(20) 1. (x)[Ax ⊃ (Bx ● Cx)]

2. (x)[Dx ⊃ (Ex ● Fx)] / (x)(Cx ⊃ Dx) ⊃ (x)(Ax ⊃ Fx)

416Chapter8 Predicate Logic

(21) 1. (∃x)(Ax �� Bx)

2. (∃x)Ax ⊃ (x)(Cx ⊃ Bx) 3. (∃x)Cx / (∃x)Bx

★ (22) 1. (x)Ax �� (x)Bx / ( x)(Ax �� Bx)

(23) 1. (∃x)(Ax �� Ex) ⊃ (x)(Bx ● ∼ ∼Cx)

2. (∃x)(Bx �� Fx) ⊃ (x)(Cx �� Dx) / (x)(Ax ⊃ Dx)

(24) 1. (x)(Ax ⊃ Bx)

2. ∼ ∼(∃x)Ax ⊃ (∃x)(Cx ● Dx) 3. (∃x)(Dx �� Ex) ⊃ (∃x)Bx / ( ∃x)Bx

★ (25) 1. (∃x)Ax ⊃ (x)(Bx ⊃ Cx)

2. (∃x)Dx ⊃ (x)(Ex ⊃ ∼ ∼Bx) 3. ∼ ∼(∃x)(Cx ● ∼ ∼Ex) / ( ∃x)(Ax ● Dx) ⊃ ∼ ∼(∃x)Bx

II. Translate the following arguments into symbolic form. Then use conditional or indirect

proof to derive the conclusion of each.

★ 1. All ambassadors are wealthy. Furthermore, all Republicans are clever. There-

fore, all Republican ambassadors are clever and wealthy. (A, W, R, C )

2. All senators are well liked. Also, if there are any well-liked senators, then

O�Brien is a voter. Therefore, if there are any senators, then O�Brien is a voter. (S, W, V )

3. If all judges are wise, then some attorneys are rewarded. Furthermore, if there

are any judges who are not wise, then some attorneys are rewarded. Therefore, some attorneys are rewarded. ( J, W, A, R)

★ 4. All secretaries and undersecretaries are intelligent and cautious. All those who

are cautious or vigilant are restrained and austere. Therefore, all secretaries are austere. (S, U, I, C, V, R, A)

5. All ambassadors are diplomats. Furthermore, all experienced ambassadors are

cautious, and all cautious diplomats have foresight. Therefore, all experienced ambassadors have foresight. (A, D, E, C, F )

6. If there are any senators, then some employees are well paid. If there is anyone

who is either an employee or a volunteer, then there are some legislative assis-tants. Either there are some volunteers or there are some senators. Therefore, there are some legislative assistants. (S, E, W, V, L )

★ 7. If there are any consuls, then all ambassadors are satisfied diplomats. If no con-

suls are ambassadors, then some diplomats are satisfied. Therefore, some dip-lomats are satisfied. (C, A, S, D)

8. If there are any voters, then all politicians are astute. If there are any politicians,

then whoever is astute is clever. Therefore, if there are any voters, then all poli-ticians are clever. (V, P, A, C )

9. Either no senators are present or no representatives are present. Furthermore,

either some senators are present or no women are present. Therefore, none of the representatives who are present are women. (S, P, R, W )

Section 8.4 Conditional and Indirect Proof 417

★10. Either some governors are present or some ambassadors are present. If anyone

is present, then some ambassadors are clever diplomats. Therefore, some dip-lomats are clever. (G, P, A, C, D)

III. Create two valid arguments relating to hobbies. Then translate the arguments into the

symbolism of predicate logic; use conditional proof to derive the conclusion of one and indirect proof to derive the conclusion of the other.

8.5 Proving Invalidity

In the previous chapter we saw that natural deduc- In creating a substitution instance, beginning tion could not be used with any facility to prove inva- with the conclusion is often easiest. The conclusion lidity in propositional logic. The same thing can be is translated as �Some C are not A.� Thus, to make said about natural deduction in predicate logic. But this statement false, we need to find an example of in predicate logic there is no simple and automatic a class (for C ) that is included in another class (for technique such as truth tables or Venn diagrams to A). Cats and animals will serve this purpose. A little fall back on. However, there are two methods for ingenuity provides us with the following substitu-proving invalidity in predicate logic that are just as tion instance:

effective as these other techniques, even though they Some animals are not mammals. may not be as convenient. One is the method used All cats are mammals. in Chapter 1 to prove the invalidity of various kinds Therefore, some cats are not animals. of syllogisms�namely, the counterexample method. The other is what we will call the finite universe When we produce such a substitution instance, method. Both appeal to the basic idea underlying the premises must turn out to be indisputably true most proofs of invalidity: Any argument is proved in the actual world, and the conclusion indisput-invalid if it is shown that it is possible for it to have ably false. Statements involving the names of true premises and a false conclusion. Both methods animal classes are convenient for this purpose, are aimed at disclosing a situation that fulfills this because everyone agrees about cats, dogs, mam-requirement. mals, fish, and so on. Also, several different sub-

stitution instances can usually be produced that

suffice to prove the argument invalid. Finally, any

Recall from Chapter 1 that applying the and a true conclusion (or any arrangement other counter-than true premises and false conclusion) proves Counterexample Method substitution instance that results in true premises

tion instance of a given invalid argument form (or, nothing. example method consists of finding a substitu-

equally well, a given invalid symbolized argument) Here is an example of an invalid symbolized argu-ment that includes a singular statement: that has actually true premises and a false conclu-

sion. For an example of its use, consider the follow- (x)(Ax ⊃ Bx) ing invalid symbolized argument: ∼ ∼Ac / ∼ ∼Bc

(∃x)(Ax ● ∼ ∼Bx) This argument form commits the fallacy of denying the (x)(Cx ⊃ Bx) / (∃x)(Cx ● ∼ ∼Ax) antecedent. Producing a substitution instance is easy:

418Chapter8 Predicate Logic

All cats are animals. To see how this method works, we need to under-Lassie is not a cat. stand what happens to the meaning of universal and Therefore, Lassie is not an animal. particular statements when the universe is shrunk in

second premise, it is again necessary to pick some-In selecting the name of an individual for the size. Accordingly, let us imagine that the universe con-tains only one thing instead of the billions of things that it actually contains. Let us name that one thing �Abigail.� thing that everyone agrees on. Since everyone knows The statement �Everything in the universe is perfect� is that Lassie is a dog, this name serves our purpose. then equivalent to �Abigail is perfect� (because Abigail But if we had selected some other name, such as is all that there is), and the statement �Something in the Trixie or Ajax, this would hardly suffice, because universe is perfect� is also equivalent to �Abigail is per-there is no general agreement as to what these names fect� (because Abigail is that �something�). denote. To represent this equivalence symbolically, we Here is a slightly more complex example: need a new metalogical symbol that asserts that the expressions on either side of it necessarily have the (x)[Ax ⊃ (Bx �� Cx)] same truth value given a universe of a designated size. (x)[Bx ⊃ (Cx ● Dx)] / (x)(Ax ⊃ Dx) Although this equivalence bears a close resemblance A little ingenuity produces the following substitu-to logical equivalence, it is not identical to it because tion instance: logical equivalence holds independently of any altera-

All dogs are either sharks or animals. tions in the universe. The concept that we need to All sharks are animals that are fish. represent is a kind of conditional logical equivalence. Therefore, all dogs are fish. Accordingly, we will select the double colon super-

The counterexample method is effective with pose here, scribed with a �c� (for �conditional�). For our pur-:: has the same effect as c

logic. Since its application depends on the ingenuity one member: of the user, however, it is not particularly well suited for complex arguments. For those, the finite uni-c (x)Px :: Pa verse method is probably a better choice. most fairly simple invalid arguments in predicate former symbol, we have for a universe consisting of :: does. Using the

(∃ cx)Px :: Pa

Finite Universe Method Proceeding, if we imagine that the universe con-

tains exactly two things�let us name them �Abigail�

The finite universe method can be used to estab- and �Beatrice��the statement �Everything in the lish the invalidity of any invalid argument expressed universe is perfect� is equivalent to �Abigail is per-in terms of a single variable. It depends on the fect and Beatrice is perfect.� On the other hand, the idea that a valid argument remains valid no matter statement �Something in the universe is perfect� is how things in the actual universe might be altered. equivalent to �Abigail is perfect or Beatrice is per-Accordingly, if we are given a valid argument, then fect� (because �some� means at least one). In other that argument remains valid if it should happen that words, the universal statement is equivalent to a con-the universe is contracted so that it contains only a junction of singular statements, and the particular single member. On the other hand, if it should turn statement is equivalent to a disjunction of singular

conclusion in a universe consisting of only one or a c (x)Px :: (Pa few members, then that argument has been proved out that an argument has true premises and false statements. In symbols: ● Pb)

invalid. c (∃x)Px :: (Pa �� Pb)

Section 8.5 Proving Invalidity 419

If the universe is increased to three�let us call the Ga ⊃ Ha new member �Charmaine��we have Ha / Ga

(x)Px :: (Pa ● Pb ● Pc) Testing with an indirect truth table, we have c

(∃ cx)Px :: (Pa �� Pb �� Pc) Ga ⊃ Ha / Ha / / Ga

This equivalence continues indefinitely as more and F T T T F more members are added to the universe. Because it is possible for the premises to be true

Extending this treatment to the more typical and the conclusion false, the argument is invalid.

have, for a universe of three: (x)( Jx ⊃ Kx) c (x)(Px kinds of universal and particular statements, we Another example:

⊃ Qx) :: [(Pa ⊃ Qa) ● (Pb ⊃ Qb) ● (Pc ⊃ Qc)] (∃x)Jx / (x)Kx

(∃ cx)(Px ● Qx) :: [(Pa ● Qa) �� (Pb ● Qb) �� (Pc ● Qc)] For a universe having one member, the indirect

For expressions involving combinations of quanti- truth table is as follows:

fied statements, each of the component statements Ja ⊃ Ka / Ja / / Ka

groups are linked together by means of the connec- Since it is impossible for the premises to be true and tive appearing in the original statement. Here are is translated separately and the resulting statement T T F T F

two examples for a universe of three: the conclusion false for this universe, we try a uni-verse having two members, a and b : c [(x)Px ⊃ ( ∃ x)Qx] :: [(Pa ● Pb ● Pc) ⊃ (Qa �� Qb �� Qc)] ( Ja ⊃ Ka) ● ( Jb ⊃ Kb) / Ja vJb / / Ka ● Kbc [(x)(Px ⊃ Qx) �� ( ∃ x)(Rx ● Sx)] :: T T T T F T F T T F T F F

{[(Pa ⊃ Qa) ● (Pb ⊃ Qb) ● (Pc ⊃ Qc)] Since it is possible for the premises to be true and �� [(Ra ● Sa) �� (Rb ● Sb) �� (Rc ● Sc)]} the conclusion false for this universe, the argument

The method for proving an argument invalid is invalid.

into singular statements, as per the preceding exam- (∃x)Hx ⊃ (x)(Fx ⊃ Gx) ples, and then testing the result with an indirect consists in translating the premises and conclusion Here is an example involving compound statements: truth table (see Section 6.5). First a universe of one is (∃x)Fx /(∃x)Hx ⊃ (x)Gx tried. If it is possible for the premises to be true and The indirect truth table for a universe having one the conclusion false in this universe, the argument member is as follows:

is immediately identified as invalid. If, on the other Ha ⊃ (Fa ⊃ Ga) / Fa / / Ha ⊃ Ga

a universe of two is then tried. If, in this second uni- A contradiction results, so we try a universe having verse, it is possible for the premises to be true and hand, a contradiction results from this assumption, T T T F F T T F F

the conclusion false, the argument is invalid. If not, a two members. The resulting indirect truth table proves the argument invalid: universe of three is tried, and so on.

Consider the following symbolized argument: (Ha �� Hb) ⊃ [(Fa ⊃ Ga) ● (Fb ⊃ Gb)]

T T T T T T F T F

(x)(Gx ⊃ Hx)

( / Fa �� Fb ∃ x)Hx / ( ∃ x)Gx

For a universe having one member�call this member T T F

�Abigail��the argument translates into / / (Ha �� Hb) ⊃ (Ga ● Gb) T F T F F

420Chapter8 Predicate Logic

The next example involves singular statements: argument fails in a universe consisting of one, two,

or any number of members, it is invalid.

( ∃x)Mx ● (∃x)Nx While this method is intended primarily for prov-Md / Nd ing arguments invalid, theoretically it can also be used

is an was proved to the effect that an argument that does not M . For this argument, the assumption that the n fail in a universe of 2 The second premise asserts that something named to prove arguments valid. Several years ago a theorem d universe contains only one member entails that this members, where n designates the one member is named number of different predicates, is valid.* According to d . Here is the indirect truth

table for such a universe: this theorem, establishing the validity of an argument Md containing two different predicates requires a universe Nd / Md / / Nd ● having four members, establishing the validity of an

T T F T F argument containing three different predicates requires

When the universe is expanded to include two a universe having eight members, and so on. For most members, we are free to give any name we wish to arguments, however, a universe having four members is the second member. Let us call it e. The resulting unwieldy at best, and a universe having eight members indirect truth table, which follows, shows that the approaches the impossible (although a computer could argument is invalid. Notice that the second premise handle it easily). Thus, while this method is usually and the conclusion remain the same: quite convenient for proving invalidity, practical limita-

(Md �� Me) ● (Nd �� Ne) / Md / / Nd tions impede its usefulness in establishing validity.

T T T F T T T F

The basic concept behind this method of prov- *See Wilhelm Ackermann, Solvable Cases of the Decision Problem

ing invalidity rests on the fact that a valid argument (Amsterdam: North-Holland Publishing, 1954), Chapter 4. This theo-is valid in all possible universes. Consequently, if an rem, incidentally, holds only for monadic predicates.

Exercise 8.5

I. Use the counterexample method to prove that the following symbolized arguments are

invalid.

★(1) 1. (x)(Ax ⊃ Bx)

2. (x)(Ax ⊃ ∼ ∼Cx) / (x)(Cx ⊃ ∼ ∼Bx)

(2) 1. (∃x)(Ax ● Bx)

2. (x)(Cx ⊃ Ax) / (∃x)(Cx ● Bx)

(3) 1. (x)(Ax ⊃ Bx)

2. Bc / Ac

★(4) 1. (∃x)(Ax ● Bx)

2. (∃x)(Ax ● Cx) / (∃x)[Ax ● (Bx ● Cx)]

(5) 1. (x)[Ax �� (Bx �� Cx)] / (x)Ax �� [(x)Bx �� (x)Cx]

(6) 1. (x)[Ax ⊃ (Bx �� Cx)]

2. (x)[(Bx ● Cx) ⊃ Dx] / (x)(Ax ⊃ Dx)

Section 8.5 Proving Invalidity 421

★(7) 1. (∃x)Ax

2. (∃x)Bx

3. (x)(Ax ⊃ ∼ ∼Cx) / (∃x)(Bx ● ∼ ∼Cx)

(8) 1. (x)[(Ax �� Bx) ⊃ Cx]

2. (x)[(Cx ● Dx) ⊃ Ex] / (x)(Ax ⊃ Ex)

(9) 1. (x)[(Ax ● Bx) ⊃ Cx]

2. (x)[(Ax ● Cx) ⊃ Dx] / (x)[(Ax ● Dx) ⊃ Cx]

★(10) 1. (∃x)(Ax ● Bx)

2. (∃x)(Cx ● ∼ ∼Bx)

3. (x)(Ax ⊃ Cx) / (∃x)[(Cx ● Bx) ● ∼ ∼Ax]

II. Use the finite universe method to prove that the following symbolized arguments are

invalid.

★(1) 1. (x)(Ax ⊃ Bx)

2. (x)(Ax ⊃ Cx) / (x)(Bx ⊃ Cx)

(2) 1. (x)(Ax �� Bx)

2. ∼ ∼An / (x)Bx

(3) 1. (∃x)Ax �� (∃x)Bx

2. (∃x)Ax / (∃x)Bx

★(4) 1. (x)(Ax ⊃ Bx)

2. (∃x)Ax / (x)Bx

(5) 1. (x)[Ax ⊃ (Bx �� Cx)]

2. (∃x)Ax / (∃x)Bx

(6) 1. (∃x)Ax

2. (∃x)Bx / (∃x)(Ax ● Bx)

★(7) 1. (x)(Ax ⊃ Bx)

2. (∃x)Bx ⊃ (∃x)Cx / (x)(Ax ⊃ Cx)

(8) 1. (∃x)(Ax ● Bx) K (∃x)Cx

2. (x)(Ax ⊃ Bx) / (x)Ax K (∃x)Cx

(9) 1. (∃x)(Ax ● ∼ ∼Bx)

2. (∃x)(Bx ● ∼ ∼Ax) / (x)(Ax �� Bx)

★(10) 1. (∃x)(Ax ● Bx)

2. (∃x)(∼ ∼Ax ● ∼ ∼Bx) / (x)(Ax K Bx)

III. Translate the following arguments into symbolic form. Then use either the

counterexample method or the finite universe method to prove that each is invalid.

★1. Violinists who play well are accomplished musicians. There are some

violinists in the orchestra. Therefore, some musicians are accomplished. (V, P, A, M, O)

2. Pianists and harpsichordists are meticulous. Andr� Watts is a pianist. There-

fore, everyone is meticulous. (P, H, M )

422Chapter8 Predicate Logic

3. If there are any oboists, there are some bassoonists. If there are any clarinetists,

there are some flutists. Amelia is both an oboist and a clarinetist. Therefore, some bassoonists are flutists. (O, B, C, F )

★ 4. All timpanists are haughty. If some timpanists are haughty, then some

percussionists are overbearing. Therefore, all timpanists are overbearing. (T, H, P, O)

5. All cellists and violinists are members of the string section. Some violinists

are not cellists. Also, some cellists are not violinists. Therefore, everyone is a member of the string section. (C, V, M )

IV. Create an invalid argument relating to parties�the fun kind, not the political sort. Then

translate the argument into the symbolism of predicate logic and use both the counterex-ample method and the finite universe method to prove the argument invalid.

8.6 Relational Predicates and�Overlapping

Quantifiers

Even the logical machinery developed thus far is not is a predicate that is used to establish a connection adequate for deriving the conclusions of a number between or among individuals. of arguments. Consider, for example, the following: Relations occur in varying degrees of com-

All dogs are animals. Therefore, whoever plexity, depending on the number of individuals owns a dog owns an animal. related. The simplest, called binary (or dyadic) rela-

tions, establish a connection between two individ-

If there are any butterflies, then if all butter- uals. Some examples are the relation of being taller flies are free, they are free. There are butter- than, as expressed in the statement �Steve is taller

The first argument involves a relation�the relation flies in the garden. Therefore, if all butterflies than David,� and the relation of being a friend, as are free, something in the garden is free. expressed in �Sylvia is a friend of Olivia.� Trinary (or triadic ) relations establish a connection among of ownership�and we have yet to see how relations three individuals: for example, the relation of being can be dealt with. The second argument, while not between, as in �St. Louis is between Chicago and involving any relations, involves a quantifier that New Orleans,� and the relation of reading some-overlaps another quantifier. In this section the appa-thing to someone, as in �George read Othello to ratus of predicate logic will be extended to cover Madeline.� Quaternary (or tetradic ) relations link examples such as these. four individuals together�for example, the relation The predicates we have used thus far are called of reading something to someone at a certain time, monadic predicates, or one-place predicates, as in �George read Othello to Madeline on Thurs-because they are used to assign an attribute to indi-day.� The complexity increases until we have what vidual things. A relational predicate (or relation) are called n-ary (or n-adic ) relations, which link n

Section 8.6 Relational Predicates and�Overlapping Quantifiers 423

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). things together. In this section, we will restrict our The last four statements involve overlapping attention to binary relations. quantifiers. We may read these symbols as follows:

(x)( y) For all x and for all y . . .

Translating Relational Statements (∃x)(∃y) There exists an x such that there exists a y such that . . . Relations are symbolized like other predicates (x)( ∃ y) For all x there exists a y such that . . . except that two lowercase letters, representing the ( ∃ x)( y) There exists an x such that for all y . . . two related individuals, are written to the imme-Applying this phraseology to the last statement given, diate right of the uppercase letter representing the for example, we have �There exists an x such that for all relation. Here are some examples of relational state-y , x is different from y ��which is simply another way ments involving specifically named individuals: of saying �Something is different from everything.�

Symbolic When two quantifiers of the same sort appear

Statementtranslation adjacent to each other, the order in which they are

Andres is married to Carla. listed is not significant. In other words, the state-Mac ment ( x )( y ) Dxy is logically equivalent to ( y )( x ) Dxy , Deborah loves physics. Ldp and ( ∃ x )( ∃ y ) Dxy is logically equivalent to ( ∃ y )( ∃ x ) The Willis Tower is taller Dxy. A little reflection on the meaning of these state-than the Empire State ments should justify this equivalence. But when dif-Building. Twe ferent quantifiers appear adjacent to each other, the Donald is the father of Jim. Fdj order does make a difference, sometimes even when

the statement function is nonrelational. Accordingly,

Notice that the order in which the lowercase letters (x)(∃y)Dxy is not logically equivalent to (∃y)(x)Dxy. are listed often makes a difference. If the third state- This fact can be seen more clearly in terms of a different ment were translated Tew, the symbolic statement example. If Lxy means �x loves y� and we imagine the would read �The Empire State Building is taller universe of discourse restricted to people, then (x)(∃y) than the Willis Tower,� which is false. Quantifiers Lxy means �Everyone loves someone (or other),� while are attached to relational predicates in the same way (∃y)(x)Lxy means �There is someone whom everyone they are to monadic predicates. Some examples of loves.� Clearly these two statements are not equivalent. relational statements involving quantifiers are as Relational predicates can be combined with follows: ordinary predicates to translate statements having

varying degrees of complexity. In the examples that

Symbolic follow, Px means �x is a person.� The meaning of the

Statementtranslation other predicates should be clear from the context:

Thomas knows everything. (x)Kt x Any heavyweight can defeat any lightweight. Thomas knows something. (∃x)Kt x (x)[Hx ⊃ ( y)(Ly ⊃ Dxy)]

Everything is different from Some heavyweights can defeat any lightweight. everything. (x)( y)Dxy ( ∃x)[Hx ● ( y)(Ly ⊃ Dxy)]

Something is different from No heavyweight can defeat every lightweight. something. (∃x)(∃y)Dxy (x)[Hx ⊃ (∃y)(Ly ● ∼ ∼Dxy)]

Everything is different from or something (or other). ∼ ∼(∃x)[Hx ● ( y)(Ly ⊃ Dxy)] (x)( ∃ y)Dxy

Something is different from Everyone cares for someone (or other). everything. (x)[Px ⊃ (∃y)(Py ( ∃ x)( y)Dxy● Cxy)]

424Chapter8 Predicate Logic

Someone does not care for anyone. If anything is good and all good things are

(∃x)[Px ● ( y)(Py ⊃ ∼ ∼Cxy)] safe, then it is safe.

Anyone who cares for someone is cared for (x){[Gx ● ( y)(Gy ⊃ Sy)] ⊃ Sx}

himself. If anything is good and some good things

(x){[Px ● (∃y)(Py ● Cxy)] ⊃ (∃z)(Pz ● Czx)} are dangerous, then it is dangerous.

Not everyone respects himself. (x){[Gx ● (∃y)(Gy ● Dy)] ⊃ Dx}

(∃x)(Px ● ∼ ∼Rxx) Since the �it� at the end of these statements refers

or to one of the �good� things mentioned at the begin-

∼ ∼(x)(Px ⊃ Rxx) ning, the quantifier that binds the x in Gx must also

Anyone who does not respect himself is not bind the x in Sx and Dx. The set of braces in the sym-respected by anyone. bolic expressions ensures this.

(x)[(Px ● ∼ ∼Rxx) ⊃ ( y)(Py ⊃ ∼ ∼Ryx)] Another point to notice regarding statements

The same general rule applies in translating such as these is that the quantified expression inside

these statements as applies in translating any other the brackets is expressed in terms of a new variable. statement in predicate logic: Universal quantifiers This procedure is essential to avoiding ambiguity. go with implications and existential quantifiers If instead of y, x had been used, the variable in this go with conjunctions. Every one of the eight sym- expression would appear to be bound by two differ-bolic expressions given here follows this rule. For ent quantifiers at the same time. example, in the first statement, both quantifiers are In other statements, the one or more individuals universal and both operators are horseshoes. In the mentioned at the end are not necessarily the same second statement, the main quantifier is existential ones mentioned at the beginning. In such cases the and the subordinate quantifier universal; accord- quantifier that binds the individuals at the beginning ingly, the main operator is a dot and the subordi- should not bind those at the end. Compare the next nate operator is a horseshoe. Among these state- pair of statements with those we have just considered.

ments, the sixth is the most complex. The symbolic If anything is good and all good things are translation of this statement reads, �For all x , if x is safe, then something is safe. a person and there exists a y such that y is a person [( ∃ x)Gx ● ( y)(Gy ⊃ Sy)] ⊃ ( ∃ z)Sz and x cares for y , then there exists a z such that z is If anything is good and some good things a person and z cares for x .� It should be clear that are dangerous, then something is dangerous. this is simply another way of expressing the origi-[( ∃ x)Gx ● ( ∃ y)(Gy ● Dy)] ⊃ ( ∃ z)Dz nal English statement. In these cases the �something� at the end is not nec-Another important rule to keep in mind when essarily one of the �good� things mentioned at the translating statements of this kind is that every variable beginning. Accordingly, the quantifier that binds must be bound by some quantifier. If a variable is left dangling outside the scope of its intended quantifier, the x in Gx does not range all the way to the end of the statement. Furthermore, the quantifier in ques-the translation is defective. For example, if the second tion is now an statement were translated ( ∃ x ) Hx ● ( y )( Ly ⊃ Dxy ), then existential quantifier. In the previous x the in Dxy would not be bound by the existential pair of statements the quantifier had to be universal quantifier. As a result, the translation would be defec-because it ranged over the main operator, which was a horseshoe. In the new pair, however, no quanti-tive. Instead, in our example, brackets provide for the fier ranges over the implication symbol. As a result, existential quantifier to range over Dxy . the sense of these statements has shifted to mean �If The same techniques used to translate these eight statements are also used to translate certain state-something is good . . . .� Note that, although a different variable is used to ments involving monadic predicates throughout. express each of the three different components in Consider the following:

Section 8.6 Relational Predicates and�Overlapping Quantifiers 425

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). the pair of statements just given, this is not required. With each successive instantiation the outermost Because in this case no quantifier ranges over any quantifier drops off. Generalization restores the other quantifier, it would be perfectly appropriate to quantifiers in the reverse order. use the same variable throughout. This proof demonstrates our earlier observation

The next pair of statements involve relational that the order of the quantifiers is not significant

predicates. As in the previous pair, no single quan- when the same kind of quantifier is used through-tifier ranges over the entire statement, because the out. We also observed that the order does make a dif-individuals mentioned at the end are not necessarily ference when different quantifiers appear together. the same ones mentioned at the beginning: Accordingly, the statement (x)(∃y)Pxy is not logi-

If everyone helps himself, then everyone will cally equivalent to (∃y)(x)Pxy. As the instantiation be helped. and generalization rules now stand, however, it is

late statements involving relational predicates and This completes our explanation of how to trans-If someone helps himself, then someone will (x)(Px quite possible, with a proof similar to the one just ⊃ Hxx) ⊃ (x)[(Px ⊃ ( ∃ y)Hyx)] given, to establish the logical equivalence of these two expressions. Therefore, to keep this from hap-be helped. pening we now introduce a new restriction on uni-( ∃ x)(Px ● Hxx) ⊃ ( ∃ x)( ∃ y)(Px ● Hyx) versal generalization: UG: ℱ yRestriction: UG must not be ( x ) ℱ x used if the instantial overlapping quantifiers. You may, if you wish, pro-y ceed directly to Exercise 8.6 part I before completing variable is free in the remainder of this section. any preceding line

illustrates its application: 2, EI 4. ( y)May 3, UG (invalid) ∼ 1. ∼ (x)( ∃ y)Pxy 5. ( ∃ x)( y)Mxy 4, EG ∃ 2. ( x) ∼ ∼ ( ∃ y)Pxy 1, QN The proof fails on line 4 because the instantial vari-3. ( ∃ x)( y) ∼ ∼ Pxy 2, QN able y occurs free in line�3, which was obtained by As the tilde operator is moved past a quantifier, the EI. To see that line 4 is invalid, suppose that the quantifier in question is switched for its correlative. universe of discourse is restricted to people, and With the exception of a restriction on universal gen-that Mxy means � x is the mother of y .� Then line 1 eralization, which we will introduce presently, the asserts that for every person y there exists a person instantiation and generalization rules are also used in x such that x is the mother of y �which means that basically the same way as they are with single quanti-every person has a mother. On line 2 we select one fiers. Example: Let us now see how the various quantifier rules apply derive (∃x)(y)Mxy from(y)(∃x)Mxy: to overlapping quantifiers. The quantifier negation 1. ( y)( ∃ x)Mxy rule is applied in basically the same way as it is with 2. ( ∃ x)Mxy 1, UI single quantifiers. The following short sequence 3. May Using the Rules of Inference To see how this restriction applies, let us attempt to obtained by EI.

of these people at random, and on line 3 we give

1. ( the name Abigail to the mother of that person. ∃ x)( ∃ y)Pxy

2. (∃y)Pay 1, EI Then on line 4 we draw the conclusion that Abigail 3. Pab 2, EI is the mother of everyone in the universe. This infer-4. (∃x)Pxb 3, EG ence is clearly invalid. The new restriction on UG 5. (∃y)(∃x)Pxy 4, EG prevents this kind of inference from being drawn.

426Chapter8 Predicate Logic

In summary, we now have two restrictions on Notice that line 5 is derived by UG. Also notice

universal generalization. The first concerns only that the instantial variable x occurs in line 2, conditional and indirect sequences and prevents which was derived by EI. However, x is not free UG from occurring within the scope of such a in line 2. Thus, the second restriction on UG is sequence when the instantial variable is free in obeyed. the first line. The second restriction concerns The next example concludes with a line in which only arguments involving overlapping quantifiers. an individual is related to itself. Since there are no With these two restrictions in hand, we may now restrictions on universal instantiation, the proce-proceed to examine the use of natural deduction dure leading up to this line is perfectly legitimate. in arguments involving relational predicates and Notice in line 4 that tautology is used with relational overlapping quantifiers. The example that follows predicates in the same way that it is with monadic does not include any relational predicates, but it predicates:

does involve overlapping quantifiers: 1. (∃y)(x)(Exy �� Eyx) / (∃z)E zz

1. ( ∃x)Ax ⊃ (∃x)Bx / (∃y)(x)(Ax ⊃ By) 2. (x)(E xa �� Eax) 1, EI

2. Ax ACP 3. Eaa �� Eaa 2, UI 3. (∃x)Ax 2, EG 4. Eaa 3, Taut 4. (∃x)Bx 1, 3, MP 5. (∃z)Ezz 4, EG

5. Bc 4, EI Sometimes the order in which instantiation steps

6. Ax ⊃ Bc 2�5, CP are performed is critical. The following proof pro-

8. ( ∃ 7. (x)(Ax ⊃ Bc) 6, UG vides an example: y)(x)(Ax ⊃

By) 7, EG

Conditional and indirect proof are used in the 1. (x)(∃y)(Fxy ⊃ Gxy)

same way with relational predicates and over- 2. (∃x)( y)Fxy / (∃x)(∃y)Gxy lapping quantifiers as they are with monadic 3. ( y)Fmy 2, EI predicates and nonoverlapping quantifiers. The 4. (∃y)(Fmy ⊃ Gmy) 1, UI conditional proof just given begins, as usual, by 5. Fmo ⊃ Gmo 4, EI assuming the antecedent of the conclusion. When 6. Fmo 3, UI line 7 is reached, we must be careful that neither 7. Gmo 5, 6, MP

of the restrictions against universal generaliza- 9. ( 8. (∃y)Gmy 7, EG ∃x)(∃y)Gxy 8, EG

tion is violated. While the instantial variable x is free in the first line of the conditional sequence, Line 2 must be instantiated before line 1 because the line 7 does not lie within that sequence, so the step introduces a new existential name. For the same first restriction is obeyed. And while line 5 was reason, line 4 must be instantiated before line 3. obtained by EI, The next proof involves an indirect sequence. x is not free in that line. Thus, the second restriction is obeyed as well. Such sequences often make use of the quantifier

The next proof involves a relational predicate. negation rule, as this proof illustrates:

The proof shows that while (x)(∃y) Dxy is not logi- 1. (∃x)(∃y)( Jxy �� Kxy) ⊃ (∃x)Lx cally equivalent to (∃y)(x)Dxy, it can be derived 2. (x)( y)(Lx ⊃ ∼ ∼Ly) / (x)( y)∼ ∼Jxy from that statement: 3. ∼ ∼(x)( y)∼ ∼Jxy AIP

1. ( 4. (∃x)∼ ∼( y)∼ ∼Jxy 3, QN ∃ y)(x)Dxy / (x)( ∃ y)Dxy

2. (x)Dxm 5. (∃x)(∃y)∼∼ ∼∼Jxy 4, QN 1, EI

3. Dxm 6. (∃x)(∃y) Jxy 5, DN 2, UI

4. ( 7. (∃y) Jmy 6, EI ∃ y)Dxy 3, EG

5. (x)( 8. Jmn 7, EI ∃ y)Dxy 4, UG

Section 8.6 Relational Predicates and�Overlapping Quantifiers 427

9. Jmn �� Kmn 8, Add variable that is captured by no previously introduced

10. ( ∃y)( Jmy �� Kmy) 9, EG quantifier and that no other variables be captured 11. ( ∃x)(∃y)( Jxy �� Kxy) 10, EG by the newly introduced quantifier. The following 12. ( ∃x)Lx 1, 11, MP examples illustrate both correct and incorrect appli-13. Lo 12, EI cations of this rule: 14. ( y)(Lo ⊃ ∼ ∼Ly) 2, UI

15. Lo ⊃ ∼ ∼Lo 14, UI 1. (∃x)Pxy 16. ∼ ∼Lo 13, 15, MP 2. (x)(∃x)Pxx 1, UG (invalid�the new x has 17. Lo ● ∼ ∼Lo 13, 16, Conj been captured by the

18. ∼∼ ∼∼(x)( y)∼ ∼Jxy 3�17, IP existential quantifier) 19. (x)( y)∼ ∼Jxy 18, DN

1. (∃x)Pxy

egy is to derive the antecedent of the conditional has been captured by the old existential with the aim of deriving the consequent via Because line 1 cannot be instantiated, the only strat- 2. (∃x)(∃x)Pxx 1, EG (invalid�the new xmodus

ponens quantifier) . This is accomplished on line 11 via indirect proof. Notice on line 9 that addition is used with 1. (∃x)Pxy relational predicates in the same way that it is with 2. (∃y)(∃x)Pxy 1, EG (valid )

universal instantiation and the two generalization ∃y)L xy 1, UI 3. L xa 2, EI rules. First, when UI is used to introduce variables 4. ( ∃ x)L xx 3, EG (invalid�the quanti-into a proof, it is important that these variables end fier has captured the x monadic predicates. A final word of caution is called for regarding 1. (x)(∃y)L xy 2. (

up free and that they not be captured in the process immediately adjacent by other quantifiers. The following examples illus-to the L) trate both correct and incorrect applications of this rule: 1. (x)( ∃ y)L xy

2. (∃y)L xy 1, UI

1. (x)(∃y)Pxy 3. L xa 2, EI 2. (∃y)Pyy 1, UI (invalid�the instan- 4. (∃z)L xz 3, EG (valid�the x remains

tial variable y has been free)

captured by the existential 1. (x)(y)Kxy quantifier) 2. ( y)Kxy 1, UI

1. (x)(∃y)Pxy 3. Kxx 2, UI 2. (∃y)Pxy 1, UI (valid�the instantial vari- 4. (x)Kxx 3, UG (valid )

able x is free)

1. (x)( To see that the fourth example is indeed invalid, ∃ y)Pxy

2. ( let Lxy stand for �x is larger than y,� and let the ∃ y)Pzy 1, UI (valid�the instantial vari-

able z is free) variables range over the real numbers. The state-

ment (x)(∃y)Lxy then means that there is no small-

An analogous caution applies to the two gen- est number�which is true. But the statement (∃x)

eralization rules. When UG and EG are used, it is Lxx means that there is a number that is larger than important that the instantial letter be replaced by a itself�which is false.

428Chapter8 Predicate Logic

Exercise 8.6

I. Translate the following statements into symbolic form.

★ 1. Charmaine read Paradise Lost. (Rxy: x read y)

2. Whoever reads Paradise Lost is educated. (Rxy: x reads y; Ex: x is educated)

3. Vashon is a friend of either Shandra or Ronnell. (Fxy: x is a friend of y)

★ 4. If Vashon has any friends, then Marlene is one of them. (Fxy: x is a friend of y)

5. Dr. Jordan teaches only geniuses. (Txy: x teaches y; Gx: x is a genius)

6. Dr. Nelson teaches a few morons. (Txy: x teaches y; Mx: x is a moron)

★7. Every person can sell something or other. (Px: x is a person; Sxy: x can sell y)

8. Some people cannot sell anything.

9. No person can sell everything.

★ 10. Some people can sell anything.

11. The Royal Hotel serves only good drinks. (Sxy: x serves y; Gx: x is good;

Dx: x is a drink)

12. The Clark Corporation advertises everything it produces. (Axy: x advertises y;

Pxy: x produces y)

★ 13. Shavron can drive some of the cars in the lot. (Dxy: x can drive y; Cx: x is a car;

Lx: x is in the lot)

14. Ricardo can drive any car in the lot.

15. Sofia invited only her friends. (Ixy: x invited y; Fxy: x is a friend of y)

★ 16. Christopher invited some of his friends.

17. Some people break everything they touch. (Px: x is a person; Bxy: x breaks y;

Txy: x touches y)

18. Some people speak to whoever speaks to them. (Px: x is a person; Sxy:

x�speaks�to y)

★ 19. Every person admires some people he or she meets. (Px: x is a person; Axy:

x�admires y; Mxy: x meets y)

20. Some people admire every person they meet.

21. Some policemen arrest only traffic violators. (Px: x is a policeman; Axy: x arrests

y; Tx: x is a traffic violator)

★22. Some policemen arrest every traffic violator they see. (Px: x is a policeman;

Axy: x arrests y; Tx: x is a traffic violator; Sxy: x sees y)

23. If there are cheaters, then some cheaters will be punished. (Cx: x is a cheater;

Px: x will be punished)

Section 8.6 Relational Predicates and�Overlapping Quantifiers 429

24. If there are any cheaters, then if all the referees are vigilant they will be

punished. (Cx: x is a cheater; Rx: x is a referee; Vx: x is vigilant; Px: x will be punished)

★25. Every lawyer will represent a wealthy client. (Lx: x is a lawyer; Rxy: x will rep-

resent y; Wx: x is wealthy; Cx: x is a client)

26. Some lawyers will represent any person who will not represent himself.

(Lx: x�is a lawyer; Px: x is a person; Rxy: x represents y)

27. Some children in the third grade can read any of the books in the library.

(Cx: x is a child; Tx: x is in the third grade; Rxy: x can read y; Bx: x is a book; Lx: x�is in the library)

★28. All children in the fourth grade can read any of the books in the library.

29. If there are any safe drivers, then if none of the trucks break down they will

be hired. (Sx: x is safe; Dx: x is a driver; Tx: x is a truck; Bx: x breaks down; Hx: x�will be hired)

30. If there are any safe drivers, then some safe drivers will be hired.

II. Derive the conclusion of the following symbolized arguments. Use conditional proof or

indirect proof, as needed.

★(1) 1. (x)[Ax ⊃ ( y)Bxy]

2. Am / ( y)Bmy

(2) 1. (x)[Ax ⊃ ( y)(By ⊃ Cxy)]

2. Am ● Bn / Cmn

(3) 1. (∃x)[Ax ● ( y)(By ⊃ Cxy)]

2. (∃x)Ax ⊃ Bj / (∃x)Cxj

★(4) 1. (x)(∃y)(Ax ⊃ By) / (x)Ax ⊃ (∃y)By

(5) 1. (∃x)Ax ⊃ (∃y)By / (∃y)(x)(Ax ⊃ By)

(6) 1. (x)( y)(Ax ⊃ By)

2. (∃x)( y)(Ay ⊃ Cx) / (x)(∃y)[Ax ⊃ (By ● Cy)]

★(7) 1. (∃x)[Ax ● (y)(Ay ⊃ Bxy)] / (∃x)Bxx

(8) 1. (∃x)[Ax ● (y)(By ⊃ Cxy)]

2. (x)(∃y)(Ax ⊃ By) / (∃x)(∃y)Cxy

(9) 1. (∃x)( y)(Axy ⊃ Bxy)

2. (x)(∃y)∼ ∼Bxy / ∼ ∼(x)( y)Axy

★(10) 1. (x)(∃y)Axy ⊃ (x)(∃y)Bxy

2. (∃x)( y)∼ ∼Bxy / (∃x)( y)∼ ∼Axy

(11) 1. (∃x){Ax ● [(∃y)By ⊃ Cx]}

2. (x)(Ax ⊃ Bx) / (∃x)Cx

(12) 1. (∃x)( y)[(Ay ● By) ⊃ Cxy]

2. ( y)(Ay ⊃ By) / ( y)[Ay ⊃ (∃x)Cxy]

430Chapter8 Predicate Logic

★ (13) 1. (∃x){Ax ● ( y)[(By �� Cy) ⊃ Dxy]}

2. (∃x)Ax ⊃ (∃y)By / (∃x)(∃y)Dxy

(14) 1. (x){Ax ⊃ [(∃y)(By ● Cy) ⊃ Dx]}

2. (x)(Bx ⊃ Cx) / (x)[Ax ⊃ (Bx ⊃ Dx)]

(15) 1. (∃x)( y)(Ayx ⊃ ∼ ∼Axy) / ∼ ∼(x)Axx

★ (16) 1. (x)(Ax ⊃ Bx)

2. (∃x)Bx ⊃ ∼ ∼(∃x)(∃y)Cxy / (∃x)Ax ⊃ ∼ ∼Cmn

(17) 1. (∃x)( y)(Axy ⊃ Byx)

2. (x)(∃y)(Byx ⊃ ∼ ∼Axy) / ∼ ∼(x)( y)Axy

(18) 1. (x)[Ax ⊃ (∃y)(By ● Cxy)]

2. (∃x)[Ax ● ( y)(By ⊃ Dxy)] / (∃x)(∃y)(Cxy ● Dxy)

★ (19) 1. (∃x)( y)Ayx �� (x)(y)Bxy

2. (∃x)( y)(Cy ⊃ ∼ ∼Byx) / (x)(∃y)(Cx ⊃ Axy)

(20) 1. (x)( y)[Axy ⊃ (Bx ● Cy)]

2. (x)( y)[(Bx �� Dy) ⊃ ∼ ∼Axy] / ∼ ∼(∃x)(∃y)Axy

III. Translate the following arguments into symbolic form. Then derive the conclusion of

each, using conditional proof or indirect proof when needed.

★ 1. Any professional can outplay any amateur. Garcia is a professional but he

cannot outplay Meyers. Therefore, Meyers is not an amateur. (Px: x is a profes-sional; Ax: x is an amateur; Oxy: x can outplay y)

2. Whoever is a friend of either Michael or Paul will receive a gift. If Michael has

any friends, then Eileen is one of them. Therefore, if Ann is a friend of Michael, then Eileen will receive a gift. (Fxy: x is a friend of y; Rx: x will receive a gift)

3. A horse is an animal. Therefore, whoever owns a horse owns an animal. (Hx: x

is a horse; Ax: x is an animal; Oxy: x owns y)

★ 4. O�Brien is a person. Furthermore, O�Brien is smarter than any person in the

class. Since no person is smarter than himself, it follows that O�Brien is not in the class. (Px: x is a person; Sxy: x is smarter than y; Cx: x is in the class)

5. If there are any honest politicians, then if all the ballots are counted they will

be reelected. Some honest politicians will not be reelected. Therefore, some ballots will not be counted. ( Hx: x is honest; Px: x is a politician; Bx: x is a ballot; Cx: x is counted; Rx: x will be reelected)

6. Dr. Rodriguez can cure any person who cannot cure himself. Dr. Rodriguez is a

person. Therefore, Dr. Rodriguez can cure himself. (Px: x is a person; Cxy: x can cure y)

★ 7. Some people are friends of every person they know. Every person knows some-

one (or other). Therefore, at least one person is a friend of someone. (Px: x is a person; Fxy: x is a friend of y; Kxy: x knows y)

Section 8.6 Relational Predicates and�Overlapping Quantifiers 431

8. If there are any policemen, then if there are any robbers, then they will arrest

them. If any robbers are arrested by policemen, they will go to jail. There are some policemen and Macky is a robber. Therefore, Macky will go to jail. (Px: x is a policeman; Rx: x is a robber; Axy: x arrests y; Jx: x will go to jail)

9. If anything is missing, then some person stole it. If anything is damaged, then

some person broke it. Something is either missing or damaged. Therefore, some person either stole something or broke something. (Mx: x is missing; Px: x is a person; Sxy: x stole y; Dx: x is damaged; Bxy: x broke y)

★10. If there are any instructors, then if at least one classroom is available they

will be effective. If there are either any textbooks or workbooks, there will be instructors and classrooms. Furthermore, if there are any classrooms, they will be available. Therefore, if there are any textbooks, then some instructors will be effective. (Ix: x is an instructor; Cx: x is a classroom; Ax: x is available; Ex: x is effective; Tx: x is a textbook; Wx: x is a workbook )

IV. Create five statements involving the relational predicate �friend of.� Then translate each

statement into the language of predicate logic.

8.7 Identity

Many arguments in ordinary language involve a spe- Simple Identity Statements

requires special treatment. Consider, for example, the The simplest statements involving identity are cial relation called identity, and translating this relation

following argument: those asserting that a named individual is identi-cal to another named individual. Here are some examples: The only friend I have is Elizabeth. Elizabeth

is not Nancy. Nancy is a Canadian. Therefore,

there is a Canadian who is not my friend. Samuel Clemens is Mark Twain. s = m

The peculiar feature of this argument is that it LL Cool J is James Todd Smith. c = j involves special statements about ind ividuals. To Dr. Jekyll is Mr. Hyde. j = h

translate such statements, we adopt a symbol from The first statement asserts that Samuel Clemens is arithmetic, the equal sign (=), to represent the iden- identically the same person as Mark Twain, the second tity relation. We can use this symbol to translate a that LL Cool J is the same person as James Todd Smith, large variety of statements, including simple iden- and the third that Dr. Jekyll is the same person as tity statements, existential assertions about indi- Mr. Hyde. In other words, the statements claim that viduals, statements involving �only,� �the only,� �no the names �Samuel Clemens� and �Mark Twain� des-. . . except,� and �all except,� and statements involv- ignate the same person, �LL Cool J� and �James Todd ing superlatives, numerical claims, and definite Smith� designate the same person, and so on. descriptions. After seeing how the identity relation To translate a negated identity statement, we is used to translate such statements, we will see how simply draw a slash through the identity symbol. natural deduction is used to derive the conclusions Thus, �Beethoven is not Mozart� is translated of arguments involving identity. b ≠ m. The expression b ≠ m is just an abbreviated

432Chapter8 Predicate Logic

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). way of writing a stated attribute and anything having that attribute ∼ ∼ ( b = m ). Here are some additional examples: is identical to the designated individual. Here are

some examples:

William Wordsworth is not John Keats. w ≠ j

Only Nolan Ryan has struck out 5,000 batters.

Natalie Portman is not Rosie O�Donnell. n ≠ rSn ● (x)(Sx ⊃ x = n) Jake Tapper is not Kaitlan Collins. j ≠ k The only opera written by Beethoven is Fidelio.

The kinds of statements we will consider next are Of ● Bf ● (x)[(Ox ● Bx) ⊃ x = f ]

more complicated, and to facilitate their translation No nation except Australia is a continent. a set of conventions governing conjunctions, dis- Na ● Ca ● (x)[(Nx ● Cx) ⊃ x = a] junctions, and simple identity statements will now The only presidents who were Whigs were be introduced. Many of our translations will involve Taylor and Fillmore. lengthy strings of conjunctions, such as Pm ● Km ● Pt ● Wt ● Pf ● Wf ● (x)[(Px ● Wx) ⊃ (x = t �� x = f )] Pn ● Kn. Instead of introducing parentheses and

them as a string of conjuncts. Lengthy disjunctions has struck out 5,000 batters, and if anyone has struck out 5,000 batters, then he is identical to Nolan Ryan.� brackets into these expressions, we may simply write The first translation may be read as �Nolan Ryan may be treated the same way. In simple identity state-

ments, the identity symbol controls only the letters to The last part of the translation ensures that no other

n was written by Beethoven, and if anything is ● e = p ● s = t , and instead of writing P ⊃ ( a = m ), an opera written by Beethoven, then it is identical we may write P ⊃ a = m . Let us now use these con-to Fidelio .� Analogous remarks pertain to the other ventions to translate some special kinds of statements two statements. The third statement is equivalent to writing ( translation may be read as �Fidelio is an opera, and c = n ) ● ( e = p ) ● ( s = t ), we may write c = Fidelio its immediate left and right. Accordingly, instead of person has struck out 5,000 batters. The second involving identity.

�The only nation that is a continent is Australia.�

�Only,� �The Only,� and �No . . . �All Except�

Except� Statements beginning with �all except� are similar to Section 4.7 explained that the words �only,� �the those beginning with �the only� in that they, too, assert only,� and �no . . . except� signal an ordinary cat- something about a designated individual (or individu-egorical proposition when the word that follows is als). For example, the statement �All presidents except a plural noun or pronoun. For example, the state- Washington had a predecessor� means that Washing-ment �Only relatives are invited� means simply ton did not have a predecessor but all other presidents �All invited people are relatives,� and �The only did. Thus, the general form of such statements is that a animals in this canyon are skunks� means �All ani- designated individual lacks a stated attribute and that mals in this canyon are skunks.� However, when anything not identical to the designated individual has

except� designates an individual, something more is All painters except Jackson Pollock make sense. intended. Thus the statement �Only Nixon resigned the word that follows �only,� �the only,� or �no . . . the stated attribute. Examples: the presidency� means (1) that Nixon resigned the Pj ● ∼ ∼Mj ● (x)[(Px ● x ≠ j ) ⊃ Mx] presidency and (2) that if anyone resigned the presi- All continents except Antarctica are heavily dency, that person is Nixon. Thus, the general form populated. of such statements is that a designated individual has Ca ● ∼ ∼Ha ● (x)[(Cx ● x ≠ a) ⊃ Hx]

Section 8.7 Identity 433

All states except Alaska and Hawaii are property of at most n items, those that assert a property contiguous with a sister state. of at least n items, and those that assert a property of

Sa ● ∼ ∼Ca ● Sh ● ∼ ∼Ch ● (x)[(Sx ● x ≠ a ● x ≠ h) ⊃ Cx] exactly n items.

The first translation may be read as �Jackson The first group does not assert that there actually are

Pollock is a painter who does not make sense, and any items that have the stated property but only that, if every painter not identical to Jackson Pollock makes there are any with the stated property, then the maxi-sense.� mum number is n. Accordingly, for �at most� statements

we use universal quantifiers. Here are some examples:

Superlatives There is at most one god.

Statements containing superlative adjectives are yet (x)( y)[(Gx ● Gy) ⊃ x = y] another kind of statement that can be translated by There is at most one U.S. representative from using the identity relation. These are statements Alaska. asserting that, of all the members of a class, some- (x)( y)[(Ux ● Ax ● Uy ● Ay) ⊃ x = y]

est, and so on. To translate these statements, first (x)( y)(z)[(Sx ● Sy ● Sz) ⊃ (x = y �� x = z �� y = z)] give the designated item the class attribute, and then thing is the largest, tallest, smallest, heaviest, light- There are at most two superpowers.

how exceeded by the designated item. Here are some (x)( y)(z)[(Cx ● Kx ● Cy ● Ky ● Cz ● Kz) ⊃ (x = y �� x = z �� y = z)] examples: say that, if anything else has that attribute, it is some- There are at most two cities in Kuwait.

The largest planet is Jupiter. It can be seen from these examples that to translate

Pj �at most n� is to say that, if there are n + 1 items that ● (x)[(Px ● x ≠ j ) ⊃ Ljx]

The deepest lake is Ozero Baykal. have the stated property, then at least one of them

The highest peak in North America is Mt. Denali. Lo is identical to at least one of the �others.� The result ● (x)[(Lx ● x ≠ o) ⊃ Dox] is to limit the number of such items to n . Thus, to translate �at most one,� we need two quantifiers; to

Pd ● Nd ● (x)[(Px ● Nx ● x ≠ d) ⊃ Hdx] translate �at most two,� we need three quantifiers;

planet, and if anything is a planet and not identical such translations become rather lengthy, this discus-to Jupiter, then Jupiter is larger than it.� The second sion is limited to statements about one or two items may be read as �Ozero Baykal is a lake, and if any-at most. The first translation may be read as �Jupiter is a statements about any number of items, but because and so on. We could use this procedure to translate

thing is a lake and not identical to Ozero Baykal, Unlike �at most� statements, statements that then Ozero Baykal is deeper than it.� assert something about at least n items do claim

that the items actually exist. Thus, to translate

One of the more interesting uses of the identity symbol tifiers. The number of quantifiers must be equal to the number of items asserted. Examples: Numerical Statements �at least� statements we need to use existential quan-is to translate certain kinds of numerical statements,

such as �There are three people in this room.� In par- There is at least one city in Monaco. ticular, the identity symbol allows us to translate such (∃x)(Cx ● Mx) statements without the use of numerals. There are There are at least two women in Hamlet. three types of numerical statements: those that assert a (∃x)(∃y)(Wx ● Hx ● Wy ● Hy ● x ≠ y)

434Chapter8 Predicate Logic

Prominent Kurt G�del 1906�1978Logicians

K urt G�del, generally considered to be the most adequate to support important logician of the contemporary period, arithmetic contains at was born in what is today Brno, Czechoslovakia, least one assertion that is to a father who managed a textile factory and a mother neither provable nor dis-

who was educated and cultured. After excelling at provable from the axioms. the gymnasium in Brno, G�del entered the University A second incom pleteness of Vienna, where he studied mathematics, physics, theorem showed that and philosophy. On completing his undergraduate the consistency of such a degree he commenced graduate work in mathemat- system cannot be proved ics, earning his doctorate at age twenty-four. Four within the system itself. years later G�del began teaching at the university As a philosopher, as a Privatdozent. However, when the Nazis annexed G�del was a Platonic Austria they abolished his teaching position in favor of realist and a Leibnizian Pictorial Press Ltd /Alamy Stock Photo one that required a political test, and one year later rationalist. He thought they found him qualified for military service. that abstract concepts (such as number and figure)

In 1940, under threat of being drafted into the represented objects in an ideal realm that were per -

German army, G�del and his wife Adel (whom he had fect, changeless, and eternal. As a result, he thought married two years earlier) left for the United States, that mathematics is a science that describes this ideal where he accepted a position at the famous Institute realm, and not, as many think, a mere invention of the for Advanced Studies, in Princeton, New Jersey. human mind. Following Leibniz, G�del conceived the Soon after arriving, he became best of friends with visible world as fundamentally beautiful, perfect , and Albert Einstein, with whom he took daily walks. G�del thoroughly ordered. To complete this perfect world became a permanent member of the institute in he developed his own ontological argument for the 1946, and five years later he received the first Albert existence of God. Einstein Award. In 1974, he was awarded the National Tragically, from an early age, G�del was trou-Medal of Science. bled by emotional afflictions, including depression,

G�del is most famous for developing incomplete- and as he grew older he suffered from psychotic

ness theorems that relate to the efforts by logicians to delusions. In the middle of the winter he would reduce arithmetic to logic. Ordinary arithmetic rests open wide all the windows of his house because he on a set of axioms (called the Peano axioms), and thought that malevolent forces intended to poison for many years logicians thought (or hoped) that all him with gas. He also feared they wanted to poison of the theorems of arithmetic could be reduced to his food, so he ate only his wife�s cooking. When those axioms. Such a system would be complete in Adel became incapacitated from illness, G�del that every theorem would be linked to the axioms by a stopped eating altogether. At age seventy-one logical proof sequence. Quite to the surprise of these he died from malnutrition, at which point his body logicians, G�del showed that every axiom system weighed only sixty-five pounds.

Section 8.7 Identity 435

There are at least three satellites of Neptune. of words of the form �the such-and-such� that iden-

( ∃x)(∃y)(∃z)(Sx ● Sy ● Sz ● x ≠ y ● x ≠ z tifies an individual person, place, or thing. Here are

● y≠ z) some examples:

The first of these examples merely asserts that The author of Evangeline some city is in Monaco. Thus, it is translated with- The capital of Nebraska

out any inclusion of the identity relation. When The mother of John F. Kennedy

tive identity statements to ensure that the items low, the second the city of Lincoln, and the third Rose Fitzgerald Kennedy. Definite descriptions referred to are distinct. Thus, in the second state-are like names in that they identify only one ment, if translation must incorporate one or more nega- The first designates Henry Wadsworth Longfel-the stated number is greater than 1, however, the

Hamlet a situation or relationship that only that one thing . To ensure that there are at least two dis-satisfies. tinct women, we must conjoin the assertion that x Statements incorporating definite descriptions and would actually be only one woman (at least) in thing, but unlike names they do so by describing x and y should be identical, then there

conjoin the assertion that none of them is identi- nate interpretations of such statements can lead to conflicts in truth value. Suppose, for example, we cal to either of the other two. are given the statement �The queen of the United A statement about something about at least three items, we must have given rise to disputes in logic, because alter-y are not identical. Similarly, when we assert

in the lot and at most three cars in the lot. Thus, a question, most logicians today accept an interpreta-tion of definite descriptions originally proposed by statement about exactly n items requires n existen-Bertrand Russell. According to this interpretation, tial quantifiers to ensure the existence of the items, a statement that incorporates a definite descrip-one or more negated identity statements to ensure tion asserts three things: an item of a certain sort their distinctness (assuming n is greater than 1), and exists, there is only one such item, and that item a universally quantified statement to limit the group has the attribute assigned to it by the statement. If to at most n items. Here are some examples: we accept this interpretation, the statement about There is exactly one city in Grenada. the queen of the United States is false, because no such person exists. n ment to be true, because every queen is a woman, items and a statement about at most n items. For or should we consider it to be false, because there is example, the statement �There are exactly three cars no queen of the United States? In response to this in the lot� means that there are at least three cars to be the conjunction of a statement about at least States is a woman.� Should we consider this state-exactly n items can be seen (∃x){Cx ● Gx ● ( y)[(Cy ● Gy) ⊃ x = y]}

There are exactly two houses of Congress. Here are some additional examples with their

(∃x)(∃y){Hx ● Hy ● x ≠ y ● (z) [Hz translations: ⊃

(z = x �� z = y)]} The inventor of the phonograph was an

There are exactly two sopranos in La Boheme. American.

(∃x)(∃y){Sx ● L x ● Sy ● Ly ● x ≠ y ● (∃x)[Ixp ● ( y)(Iyp ⊃ y = x) ● Ax]

(z) [(Sz ● L z) ⊃ (z = x �� z = y)]} The author of Middlemarch was a Victorian

freethinker.

Definite Descriptions (∃x)[Wxm ● ( y)(Wym ⊃ y = x) ● Vx ● Fx]

The last form of phraseology considered here is the The painter of The Starry Night was Van Gogh. definite description. A ( ∃x)[Pxs ● ( y)(Pys ⊃ y = x) ● x = v] definite description is a group

436Chapter8 Predicate Logic

The first translation may be read as �There is The second rule is a rule of replacement; it someone who invented the phonograph, and if expresses what is called the symmetric property anyone invented the phonograph, then that person of the identity relation. It states, very simply, that is identical to the first, and the first person is an the letters on either side of the equal sign can be American.� The second may be read as �There is switched. An immediate use of this rule is to prove someone who wrote Middlemarch, and if anyone that x ≠ y is logically equivalent to y ≠ x. Recall wrote Middlemarch, then that person is identical that x ≠ y is simply an abbreviation for ∼ ∼(x = y). to the first, and the first person is a Victorian free- If we apply the rule to the latter expression, we thinker.� The third may be read as �There is some- obtain ∼ ∼(y = x), which, in its abbreviated form, one who painted The Starry Night, and if anyone is y ≠ x. painted The Starry Night, then that person is iden- The third rule expresses the intuitively obvi-tical to the first, and the first person is identical to ous idea that, if something is true of x and x Van Gogh.� is identical to y, then that something is true

This completes our explanation of how to trans- of y. This rule is the basis of what is called the

late statements involving the identity relation. At transitive property of identity, which allows this point, you may, if you wish, proceed to Exercise us to infer from x = y and y = z that x = z. If 8.7 part I before completing the remainder of this we suppose that the ℱ in this rule stands for section. the expression �x =,� that x is y, and that y is z,

Using the Rules of Inference second line reads then the first line of the rule reads x = y, the y = z, and the conclusion is x = z.

Now that we have seen how to translate statements This inference is used often in the derivation of involving the identity relation, let us use natu- the conclusions of arguments. ral deduction to derive the conclusions of argu- In general, the rules of inference used earlier ments that include statements of this sort. Before apply to arguments containing identity statements doing so, however, some special rules governing in the same way they apply to any other arguments. the identity relation must be introduced. These Also, conditional proof and indirect proof are used rules, which are collectively designated �Id,� are in the same way. as follows: The following argument illustrates the first

Id: (1) Prem (2) x = y :: y = x (3) ℱx No biologists are identical to Isabel. There-5 expression of the rule for identity.

x xx 5 y fore, Isabel is not a biologist.

ℱy If we use Bx to translate �x is a biologist,� and i for

These rules are expressed in terms of indi- 1. Isabel, this argument becomes

vidual variables ( x (x)(Bx ⊃ x ≠ i ) / ∼ ∼Bi and y ), so obviously they hold for any individual variables. But since any indi- The fact that the conclusion contains i suggests that vidual constant (such as a or b) is a substitution we instantiate line 1 with respect to that letter. The instance of any individual variable, these rules proof is as follows:

hold for individual constants as well. The first 1. (x)(Bx ⊃ x ≠ i ) / ∼ ∼Bi rule expresses the idea that anything is identical 2. Bi ⊃ i ≠ i 1, UI to itself; it asserts what is called the reflexive prop-3. i = i Id erty of the identity relation. The rule allows us to 4. ∼ ∼ (i ≠ i ) 3, DN insert a self-identity statement after any premise 5. ∼ ∼ Bi 2, 4, MT (that is, on any line in a proof).

Section 8.7 Identity 437

Line 3 comes merely from the first expression of The translation is as follows:

the identity rule, which allows us to insert any 1. Ps ● Is ● (x)[(Px ● Ix) ⊃ x = s] self-identity statement after any premise. Thus, 2. Pc ● Lc no numeral is included in the justification for that 3. ( ∃ x)(Px ● Ix ● ∼ ∼ Lx) / c ≠ s line. Also note that line 4 is simply another way of i writing ∼∼ ∼∼ ( = i ). Cursory inspection reveals no easy way to obtain the Now let us return to the argument given at the conclusion. This suggests indirect proof:

beginning of this section: 1. Ps ● Is ● (x)[(Px ● Ix) ⊃ x = s]

If we use The only friend I have is Elizabeth. Elizabeth 2. Pc ● Lc 3. ( ∃ x)(Px ● Ix ● ∼ ∼L x) / c ≠ s is not Nadia. Nadia is a Canadian. Therefore, 4. c = s AIP there is a Canadian who is not my friend. 5. Pa ● Ia ● ∼ ∼ La 3, EI Fx to translate � x is my friend� and Cx to 6. (x)[(Px ● Ix) ⊃ x = s] ● Ps ● Is 1, Com translate � x is a Canadian,� this argument may be 7. (x)[(Px ● Ix) ⊃ x = s] 6, Simp translated as follows: 8. Pa ● Ia ⊃ a = s 7, UI

9. Pa ● Ia 5, Simp

1. Fe ● (x)(Fx ⊃ x = e) 10. a = s 8, 9, MP 2. e ≠ n 11. s = c 4, Id 3. Cn / (∃x)(Cx ● ∼ ∼Fx) 12. a = c 10, 11, Id

Inspecting the second line, we see a negated identity 14. 13. ∼ ∼La ● Pa ● Ia 5, Com ∼ ∼La 13, Simp

statement involving e and n. This suggests that we 15. ∼ ∼Lc 12, 14, Id instantiate the universal statement in the first line 16. Lc ● Pc 2, Com with respect to n. The proof is as follows: 17. Lc 16, Simp

1. Fe 18. Lc ● ∼ ∼Lc 15, 17, Conj ● (x)(Fx ⊃ x = e) 19. c ≠ s 4 �18, IP 2. e ≠ n

3. Cn/ (∃x)(Cx ● ∼ ∼Fx) As usual, the existential statement is instanti-4. (x)(Fx ⊃ x = e) ● Fe 1, Com ated first, then the universal. Line 11 is derived 5. (x)(Fx ⊃ x = e) 4, Simp by commuting line 4 by the second rule of iden-6. Fn ⊃ n = e 5, UI tity, and line 12 is derived from lines 10 and 11 7. n ≠ e 2, Id by applying the third rule of identity. Line 15 is 8. ∼ ∼Fn 6, 7, MT derived by substituting c in the place of a in line

10. (∃ 9. Cn ● ∼ ∼Fn 3, 8, Conj 14 according to the third rule of identity. The x)(Cx ● ∼ ∼

Fx) 9, EG indirect sequence is discharged in line 19 in the

Line 7 is justified by the second rule for iden- normal way. tity. Also, since n ≠ e is simply an abbreviation for In arguments involving identity, especially ∼ ∼(n = e), line 8 follows directly from lines 6 and 7. more-complicated ones, it is often difficult or

Here is another example: impossible to see by mere inspection how to

The only person who invested is Ms. Snyder. obtain the conclusion. A good general procedure Cathy is one of the people who lost money. is to begin with instantiation. Always instantiate Some people who invested did not lose the existential statements first, then the univer-money. Therefore, Cathy is not Ms. Snyder. sals. When instantiating the universal statements,

438Chapter8 Predicate Logic

the letters) used to instantiate the Translation hints existential statement(s). If there are Only i is F. Fi (x)[Fxx 5 i ]

of the letters appearing in the singu- No F except i is G. Fi Gi (x)[(Fx Gx) x 5 i ] lar statements. If the conclusion is no existential statements, pick one The only F that is G is i. Fi Gi (x)[(Fx Gx) x 5 i ] still not apparent, try indirect proof. All F except i are G. Fi ,Gi (x)[(Fx x � i ) Gx] In general, whenever the conclu- i is the F that is most so-and-so. Fi (x)[(Fx x � i ) i is more sion is a complicated statement, it is so-and-so than x]

Developing facility in proving argu- There are at least two F�s. ( x)( y)[Fx Fy x � y] ments involving identity requires best to start out with indirect proof. There is at most one F. (x)(y)[(Fx Fy) x 5 y]

a little practice, but adequate skill There are exactly two F�s. ( x)( y){Fx Fy x � y ( z )[ Fz ( z 5 x v z 5 y )]} should not take too long to acquire.

The F is G. ( x)[Fx (y)(Fyy 5 x) Gx]

Exercise 8.7

I. Translate the following statements.

Simple identity statements

★ 1. Dr. Seuss is Theodore Geisel. (s, g)

2. Auguste Renoir is not Claude Monet. (r, m)

3. Marilyn Monroe is Norma Jean Baker. (m, b)

★4. Hermann Hesse is not Andr� Gide. (h, g)

Statements involving �only,� �the only,� and �no . . . except�

★5. Only Linus Pauling has won two unshared Nobel Prizes. (Wx: x has won two

Nobel Prizes; p: Linus Pauling)

6. Only Don Larsen has pitched a perfect World Series game. (Px: x has pitched a

perfect World Series game; l: Don Larsen)

7. The only national park in Maine is Acadia. (Nx: x is a national park; Mx: x is in

Maine; a: Acadia)

★8. The only nation having a maple-leaf flag is Canada. (Nx: x is a nation; Mx: x

has a maple-leaf flag; c: Canada)

9. The only U.S. presidents who were Federalists were Washington and Adams.

(Ux: x is a U.S. president; Fx: x is a Federalist; g: Washington; a: Adams)

10. No state except Hawaii is surrounded by water. (Sx: x is a state; Wx: x is sur-

rounded by water; h: Hawaii)

★11. No sport except hockey uses a puck. (Sx: x is a sport; Px: x uses a puck; h:�hockey)

SectionSection 8.7 8.7 Identity Identity 439439

Superlative statements

★12. Hydrogen is the lightest element. (Ex: x is an element; Lxy: x is lighter than y;

h: hydrogen)

13. The smallest planet in our solar system is Mercury. (Px: x is a planet in our

solar system; Sxy: x is smaller than y; m: Mercury)

14. Harvard is the oldest American university. (Ax: x is American; Ux: x is a uni-

versity; Oxy: x is older than y; h: Harvard)

★15. Death Valley is the lowest region in North America. (Rx: x is a region; Nx: x is

in North America; Lyx: x is lower than y; d: Death Valley)

Statements involving �all except�

★16. Every city except Istanbul is situated on a single continent. (Cx: x is a city; Sx:�x

is situated on a single continent; i: Istanbul)

17. Every U.S. president except Ford won a national election. (Ux: x is a U.S.

president; Wx: x won a national election; f : Ford)

18. All metals except mercury are solids at room temperature. (Mx: x is a metal;

Sx: x is a solid at room temperature; m: mercury)

★19. Every pitcher except Cy Young has won fewer than 500 games. (Px: x is a

pitcher; Wx: x has won fewer than 500 games; c: Cy Young)

Numerical statements

★20. There is at most one city in Belize. (Cx: x is a city; Bx: x is in Belize)

21. There are at most two national parks in South Dakota. (Nx: x is a national park;

Sx: x is in South Dakota)

22. There is at most one national holiday in July. (Nx: x is a national holiday; Jx: x

is in July)

★23. There are at most two cities in Malta. (Cx: x is a city; Mx: x is in Malta)

24. There is at least one quarterback on a football team. (Qx: x is a quarterback;

Fx: x is on a football team)

25. There are at least two atoms in a water molecule. (Ax: x is an atom; Wx: x is in

a water molecule)

★26. There are at least three carbon allotropes. (Cx: x is a carbon allotrope)

27. There is exactly one U.S. Supreme Court. (Ux: x is a U.S. Supreme Court)

28. There is exactly one natural satellite of the earth. (Sx: x is a satellite of the earth;

Nx: x is natural)

★29. There are exactly two bright stars in Gemini. (Sx: x is a star; Bx: x is bright;

Gx:�x is in Gemini)

440Chapter8 Predicate Logic

Statements containing definite descriptions

★30. The author of Vanity Fair was born in India. (Wxy: x wrote y; Bx: x was born in

India; v: Vanity Fair)

31. The wife of Othello is Desdemona. ( Wxy: x is the wife of y; o: Othello;

d:�Desdemona)

32. The man who composed The Nutcracker was Russian. (Mx: x is a man; Cxy:

x composed y; Rx: x was Russian; n: The Nutcracker)

★33. The artist who painted the Allegory of Spring was Botticelli. (Ax: x is an artist;

Pxy: x painted y; a: the Allegory of Spring; b: Botticelli)

34. The capital of Georgia is not Savannah. (Cxy: x is the capital of y, g: Georgia;

s: Savannah)

Assorted statements

★35. The smallest state is Rhode Island. (Sx: x is a state; Sxy: x is smaller than y;

r: Rhode Island)

36. There is at least one newspaper published in St. Louis. (Nx: x is a newspaper;

Sx: x is published in St. Louis)

37. Lady Gaga is Stefani Germanotta. (l: Lady Gaga; s: Stefani Germanotta)

★ 38. The only American president elected to a fourth term was Franklin

D. Roosevelt. (Ax: x is an American president; Ex: x was elected to a fourth term; r: Franklin D. Roosevelt)

39. There are at least two cities in Qatar. (Cx: x is a city; Qx: x is in Qatar)

40. Only George Blanda has played twenty-six seasons of professional football

games. (Px: x has played twenty-six seasons of professional football games; b: George Blanda)

★ 41. Hamlet had at most one sister. (Sxy: x is a sister of y; h: Hamlet)

42. No major league baseball player has hit seventy-three home runs in a single

season except Barry Bonds. (Mx: x is a major league baseball player; Hx: x has hit seventy-three home runs in a single season; b: Barry Bonds)

43. There are at most two senators from New Hampshire. (Sx: x is a senator; Nx:

x is from New Hampshire)

★44. Gustav Mahler is not Anton Bruckner. (m : Gustav Mahler; b: Anton

Bruckner)

45. The explorer who discovered the North Pole was Admiral Peary. (Ex: x is an

explorer; Dxy: x discovered y; n: the North Pole; a: Admiral Peary)

46. Hinduism is the oldest religion. (Rx: x is a religion; Oxy: x is older than y;

h: Hinduism)

★47. There are exactly two tenors in Carmen. (Tx: x is a tenor; Cx: x is in Carmen)

Section 8.7 Identity 441

48. Every defeated presidential candidate except Donald Trump has conceded the

election. (Dx: x is a defeated presidential candidate; Cx: x has conceded the election; t: Donald Trump)

49. The person who discovered relativity theory was an employee in the Swiss

patent office. (Px: x is a person; Dxy: x discovered y; Ex: x is an employee in the Swiss patent office; r : relativity theory)

★50. There are at least three stars in Orion. (Sx: x is a star; Ox: x is in Orion)

II. Derive the conclusion of the following symbolized arguments. Use conditional proof or

indirect proof, as needed.

★(1) 1. (x)(x = a)

2. (∃x)Rx / Ra

(2) 1. Ke

2. ∼ ∼Kn / e ≠ n

(3) 1. (x)(x = c ⊃ Nx) / Nc

★(4) 1. (∃x)(x = g)

2. (x)(x = i) / g= i

(5) 1. (x)(Gx ⊃ x = a)

2. (∃x)(Gx ● Hx) / Ha

(6) 1. (x)(Ax ⊃ Bx)

2. Ac ● ∼ ∼Bi / c ≠ i

★(7) 1. (x)(x = a)

2. Fa / Fm ● Fn

(8) 1. (x)(x = r)

2. Hr ● Kn / Hn ● Kr

(9) 1. (x)(Lx ⊃ x = e)

2. (x)(Sx ⊃ x = i)

3. (∃x)(Lx ● Sx) / i= e

★(10) 1. (x)(Px ⊃ x = a)

2. (x)(x = c ⊃ Qx)

3. a = c / (x)(Px ⊃ Qx)

(11) 1. (x)(y)(Txy ⊃ x = e)

2. (∃x)Txi / Tei

(12) 1. (x)[Rx ⊃ (Hx ● x = m)] / Rc ⊃ Hm

★(13) 1. (x)(Ba ⊃ x ≠ a)

2. Bc / a ≠ c

(14) 1. (∃x)Gx ⊃ (∃x)(Kx ● x = i) / Gn ⊃ Ki

(15) 1. (x)(Rax ⊃ ∼ ∼Rxc)

2. (x)Rxx / c ≠ a

442Chapter8 Predicate Logic

★ (16) 1. (x)[Nx ⊃ (Px ● x = m)]

2. ∼ ∼Pm / ∼ ∼Ne

(17) 1. (x)(Fx ⊃ x = e)

2. (∃x)(Fx ● x = a) / a = e

(18) 1. (x)[Ex ⊃ (Hp ● x = e)]

2. (∃x)(Ex ● x = p) / He

★ (19) 1. (x)(∃y)(Cxy ⊃ x = y)

2. (∃x)(y)(Cxy ● x = a) / Caa

(20) 1. (x)[Fx ⊃ (Gx ● x = n)]

2. Gn ⊃ (∃x)(Hx ● x = e) / Fm ⊃ He

III. Derive the conclusion of the following arguments. Use conditional proof or indirect

proof, as needed.

★ 1. Some of Jane Collier�s novels are interesting. The only novel Jane Collier wrote

is The Cry. Therefore, The Cry is interesting. (Nx: x is a novel; Wxy: x wrote y; Ix: x is interesting; j: Jane Collier; c: The Cry)

2. John F. Kennedy was the youngest U.S. president. Woodrow Wilson was a

U.S. president. Woodrow Wilson is not John F. Kennedy. Therefore, John F. Kennedy was younger than Woodrow Wilson. (Ux: x is a U.S. president; Yxy: x is younger than y; j: John F. Kennedy; w: Woodrow Wilson)

3. The artist who painted the Mona Lisa was a Florentine. Leonardo da Vinci is

the artist who painted the Mona Lisa. Therefore, Leonardo da Vinci was a Florentine. (Ax: x is an artist; Pxy: x painted y; Fx: x was a Florentine; m: the Mona Lisa; l: Leonardo da Vinci)

★ 4. The novel on the table was written by Margaret Mitchell. The only novel

Margaret Mitchell wrote is Gone with the Wind. Therefore, the novel on the table is Gone with the Wind. (Nx: x is a novel; Tx: x is on the table; Wxy: x wrote y; m: Margaret Mitchell; g: Gone with the Wind )

5. The author of King Lear was an English actor. John Milton was English but not

an actor. Therefore, John Milton is not the author of King Lear. (Wxy: x wrote y; Ex: x is English; Ax: x is an actor; k: King Lear; m: John Milton)

6. The dog that bit the letter carrier is a large terrier. Ajax is a small dog. There-

fore, Ajax did not bite the letter carrier. (Dx: x is a dog; Bx: x bit the letter car-rier; Lx: x is large; Tx: x is a terrier; a: Ajax)

★7. Every member except Elena sang a song. Every member except Nicole gave a

speech. Elena is not Nicole. Therefore, Elena gave a speech and Nicole sang a song. (Mx: x is a member; Sx: x sang a song; Gx: x gave a speech; e: Elena; n:�Nicole)

8. The only person who ordered fish is Astrid. The only person who suffered

indigestion is Ms. Wilson. Some person who ordered fish also suffered indi-gestion. Therefore, Astrid is Ms. Wilson. (Px: x is a person; Ox: x ordered fish; Sx: x suffered indigestion; a: Astrid; m: Ms. Wilson)

Section 8.7 Identity 443

9. The highest mountain is in Tibet. Therefore, there is a mountain in Tibet that

is higher than any mountain not in Tibet. (Mx: x is a mountain; Hxy: x is higher than y; Tx: x is in Tibet)

★10. The tallest building in North America is One World Trade Center. The tallest

building in North America is located in New York City. If one thing is taller than another, then the latter is not taller than the former. Therefore, One World Trade Center is located in New York City. (Bx: x is a building in North America; Txy: x is taller than y; Nx: x is located in New York City; o: One World Trade Center)

11. There are at least two philosophers in the library. Robert is the only French

philosopher in the library. Therefore, there is a philosopher in the library who is not French. (Px: x is a philosopher; Lx: x is in the library; Fx: x is French; r:�Robert)

12. The only dogs that barked were Fido and Pluto. Fido is not Pluto. Every dog

except Fido ran on the beach. Therefore, exactly one barking dog ran on the beach. (Dx: x is a dog; Bx: x barked; Rx: x ran on the beach; f: Fido; p: Pluto)

★13. There are at least two attorneys in the office. All attorneys are professionals.

There are at most two professionals in the office. Therefore, there are exactly two professionals in the office. (Ax: x is an attorney; Ox: x is in the office; Px: x is a professional)

14. There are at most two scientists in the laboratory. At least two scientists in the

laboratory are Russians. No Russians are Chinese. Therefore, if Norene is a Chinese scientist, then she is not in the laboratory. (Sx: x is a scientist; Lx: x is in the laboratory; Rx: x is Russian; Cx: x is Chinese; n: Norene)

15. Every candidate except Miranda was elected. The only candidate who was elected

is Regina. Miranda is not Regina. Therefore, there were exactly two candidates. (Cx: x is a candidate; Ex: x was elected; m: Miranda; r: Regina)

★16. Every student except Charles and Norman passed the course. The only

student who was dismissed was Norman. Every student retook the course if and only if he or she was not dismissed and did not pass. Charles is not Norman. Therefore, exactly one student retook the course. (Sx: x is a student; Px: x passed the course; Dx: x was dismissed; Rx: x retook the course; c: Charles; n: Norman)

IV. Create an argument relating to cities that involves an identity expression. Then translate

the argument into the symbolism of predicate logic and derive the conclusion using conditional proof or indirect proof, as needed.

444Chapter8 Predicate Logic

Summary

Predicate Logic: Combines the use of these symbols:

■ ● The five operators of propositional logic: ∼ ∼ , , ��, ⊃, K ■ Symbols for predicates: G_, H_, K_, etc.

■ Symbols for universal quantifiers: (x), ( y), (z)

■ Symbols for existential quantifiers: (∃x), (∃y), (∃z)

■ Symbols for individual variables: x, y, z

■ Symbols for individual constants: a, b, c, . . . u, v, w

Statements:

■ ● Singular statements combine predicate symbols with constants: Ga , HcKc, etc.

■ Universal statements are usually translated as conditionals: (x)(Px ⊃ Qx), etc.

■ ● Particular statements are usually translated as conjunctions: ( ∃ x )( PxQx), etc.

Using the Rules of Inference (modus ponens, modus tollens, etc.):

■ Rules are used in basically the same way as in propositional logic.

■ Using the rules usually requires that quantifiers be removed or inserted:

● Universal instantiation (UI): Removes universal quantifiers.

● Universal generalization (UG): Introduces universal quantifiers.

● Existential instantiation (EI): Removes existential quantifiers.

● Existential generalization (EG): Introduces existential quantifiers.

■ Restrictions:

● For EI, the existential name that is introduced must not appear in any previous line, includ-

ing the conclusion line.

● The instantiation and generalization rules can be applied only to whole lines�not parts

of lines.

Quantifier Negation Rule:

■ Is used to remove or insert tildes preceding quantifiers.

■ The instantiation rules cannot be applied if a tilde precedes the quantifier.

■ One type of quantifier can be replaced by the other type if and only if immediately before and

after the new quantifier:

● Tildes that were originally there are deleted, and

● Tildes that were not originally there are inserted.

Conditional Proof and Indirect Proof:

■ Used in basically the same way as in propositional logic.

■ Restriction:

● UG must not be used within an indented sequence if the instantial variable is free in the first

line of that sequence.

Summary445

Proving Invalidity:

■ Counterexample method:

● Produce a substitution instance of a symbolized argument that has indisputably true prem-

ises and an indisputably false conclusion.

■ Finite universe method:

● Reduce the universe of discourse to a finite number until an indirect truth table proves the

resulting argument invalid.

● Universal statements are rendered as conjunctions.

● Particular statements are rendered as disjunctions.

● Singular statements are kept as they are.

Relational Predicates:

■ Symbols for relational predicates: G_ _, H_ _, K_ _ _, etc.

■ Are used to translate relational statements.

■ Example: �Paul is taller than Cathy�: Tpc

Overlapping Quantifiers:

■ Quantifiers that fall within the scope of another quantifier.

■ ● Example: ( ∃ x )[ Mx (y)(Wy ⊃ Txy)]

■ Restriction:

● UG must not be used if the instantial variable is free in any preceding line obtained

by EI.

Identity:

■ The symbol for the identity relation is the equal sign: =

■ Used to assert that one thing is identical to another.

■ Used to translate statements involving �only,� �the only,� �no . . . except,� and �all except�

when these expressions are used with individuals.

■ Used to translate superlative statements, numerical statements, and definite descriptions.

■ The rule for identity (Id) allows the application of the other rules of inference.

446Chapter8 Predicate Logic

9Analogy and Legal and Moral Reasoning

9.1 Analogical Reasoning

9.2 Legal Reasoning

9.3 Moral Reasoning

9.1 Analogical Reasoning

Learning how to argue with analogies is empowering. Suppose you are addressing a group of expectant couples about the importance of prepar-ing for parenthood. Failing to prepare, you tell them, would be like jumping into a raging river when you don�t know how to swim. It would be like hiking across a desert without taking any water. These analogies invite your audience to see the situation in a new way. They plant a graphic image in the mind of the audience that has a powerful effect.

Analogical reasoning may be the most fundamental and the most common of all rational processes. It stands at the basis of many ordinary decisions in life. For example, a woman wanting to buy a can of soup might deliberate between Campbell�s and Brand X and, after recalling that other varieties of Campbell�s were good, decide in favor of that brand. A man contemplating a haircut might recall that a friend got an especially good cut at the Golden Touch and as a result decide to go there himself. A woman selecting a plant for her garden might observe that gardenias grow well at the house next door and conclude that gardenias would grow well at her house, too. A man thinking of reading a novel by Stephen King might recall that King�s last three novels were thrilling and conclude that his latest novel would also be thrilling.

447

Analogical reasoning is reasoning that depends earlier albums are called the primary analogues,

on a comparison of instances. If the instances are and the latest album is called the secondary ana-sufficiently similar, the decision reached in the logue. With these definitions in mind, we may now end is usually a good one; but if they are not suf- examine a set of principles that are useful for evalu-ficiently similar, the decision may not be good. ating most arguments from analogy. They include When such a reasoning process is expressed in (1) the relevance of the similarities shared by the pri-words, the result is an argument from analogy mary and secondary analogues, (2) the number of (see Chapter 1). Simple arguments from analogy similarities, (3) the nature and degree of disanalogy, have the following structure: (4) the number of primary analogues, (5)�the diver-

Entity A has attributes a, b, c, and z. sity among the primary analogues, and (6) the spec-Entity B has attributes a, b, c. ificity of the conclusion.

Therefore, entity B probably has attribute z also. 1. Relevance of the similarities. Suppose a certain

usually weak. Valeria argues that both cars have a padded Analogical arguments are closely related to gen-steering wheel, tachometer, fabric upholstery, eralizations. In a generalization, the arguer begins tinted windows, premium sound system, and with one or more instances and proceeds to draw white paint. Valeria�s argument is weak because a conclusion about all the members of a class. The these similarities are irrelevant to fuel economy. arguer may then apply the generalization to one On the other hand, if Valeria bases her conclu-or more members of this class that were not noted sion on the fact that both cars have the same-earlier. The first stage of such an inference is induc-size engine, her argument is relatively strong, tive, and the second stage is deductive. For example, because engine size is relevant to fuel economy. important way to z (that is, are relevant to z), the favor of a Chevrolet because she wants good fuel argument is usually strong. If they are not so con-economy and her friend Mateo�s new Chevy gets nected (that is, are irrelevant to z ), the argument is good fuel economy. To support her decision, If the attributes a, b, and c are connected in some ing the purchase of a new car. She decides in person�let us call her Valeria�is contemplat-the man thinking of reading the latest Stephen King

by King were thrilling, all King novels are thrilling the same-size engine, Valeria notes further sim-ilarities between the car she intends to buy and novel might argue that because the last three novels 2. Number of similarities. Suppose, in addition to

instances to a conclusion about another individual addition, Valeria notes that she and Mateo both drive about half on city streets and half on instance without appealing to any intermediate gen-freeways, her argument is strengthened even eralization. Such an argument is purely inductive. further. Thus, the arguer might conclude directly from read-3. Nature and degree of disanalogy . On the ing three earlier King novels that the latest one will be thrilling. other hand, if Valeria�s car is equipped with a turbocharger, if Valeria loves to make In any argument from analogy, the items that jackrabbit starts and screeching stops, if are compared are called analogues . Thus, if one Mateo�s car has variable valve timing but Valeria�s will be thrilling (deductive argument). In an argu- gear ratio, and tires. These additional similari-ties, all of which are relevant to fuel economy, ment from analogy, on the other hand, the arguer tend to strengthen Valeria�s argument. If, in proceeds directly from one or more individual to the latest novel, he might then conclude that it too Mateo�s car: curb weight, aerodynamic body, (inductive argument). Applying this generalization should argue that Rihanna�s latest album is prob-

ably good because her two previous albums were does not, and if Valeria constantly drives on con-good, the three albums are the analogues. The two gested freeways while Mateo drives on relatively

448Chapter9 Analogy and Legal and Moral Reasoning

clear freeways, Valeria�s argument is weak- tires to different pressures, drive at different ened. These differences are called disanalogies. speeds on different grades, and have different Depending on how they relate to the conclusion, attitudes toward using the oil additive, then it is disanalogies can either weaken or strengthen an less likely that the good fuel economy they enjoy argument. If we suppose instead that Mateo�s car is attributable to any factor other than the model has the turbocharger, that Mateo loves jackrab- and year of their car.

strengthened, because these disanalogies tend to sion is simply that her car will get �good� fuel reduce Mateo�s fuel economy. bit starts, and so on, then Valeria�s argument is 6. Specificity of the conclusion. Valeria�s conclu-

economy. If she now changes her conclusion to

4. Number of primary analogues. Thus far, Valeria state that her car will get fuel economy �at least

has based her conclusion on the similarity as good� as Mateo�s, then her argument is weak-between the car she intends to buy and only one ened. Such a conclusion is more specific than other car�Mateo�s. Now suppose that Valeria the earlier conclusion and is easier to falsify. has three additional friends, that all of them Thus, if her fuel economy were only one-tenth drive cars of the same model and year as Mateo�s, of a mile per gallon less than Mateo�s, her con-and that all of them get good fuel economy. clusion would turn out to be false. Now suppose These additional primary analogues strengthen that Valeria changes her conclusion to state that Valeria�s argument because they lessen the like- her car will get exactly the same fuel economy lihood that Mateo�s good fuel economy is a freak as Mateo�s. This conclusion is even more specific incident. On the other hand, suppose that two of than the �at least as good� conclusion, and it is these additional friends get the same good fuel easier still to falsify. Thus, such a conclusion ren-economy as Mateo but that the third gets poor ders Valeria�s argument much weaker than her fuel economy. As before, the first two cases tend original argument.

to strengthen Valeria�s argument, but the third Summarizing these results, we have, now tends to weaken it. This third case is called a counteranalogy because it supports a conclu-The more relevant the similarities between pri-sion opposed to that of the original analogy. mary and secondary analogues, the stronger

5. Diversity among the primary analogues. Suppose the argument.

get good fuel economy) all buy their gas at the primary and secondary analogues, the stron-same station, have their cars tuned up regularly now that Valeria�s four friends (all of whom The greater the number of similarities between by the same mechanic, put the same friction- ger the argument. reducing additive in their oil, inflate their tires Differences between primary and secondary to the same pressure, and do their city driving analogues are disanalogies, which can either on uncongested, level streets at a fuel-maximiz- strengthen or weaken the argument. ing 28 miles per hour. Such factors would tend The greater the number of similar primary ana-to reduce the probability of Valeria�s conclu- logues, the stronger the argument; dissimi-sion, because it is possible that one or a combi- lar primary analogues are counteranalogies,

economy and that this factor (or combination of The more diverse the primary analogues, the them) is absent in Valeria�s case. On the other nation of them is responsible for the good fuel which weaken the argument. hand, if Valeria�s friends buy their gas at differ- stronger the argument. ent stations, have their cars tuned up at different The more specific the conclusion, the weaker the intervals by different mechanics, inflate their argument.

Section 9.1 Analogical Reasoning 449

Let us now use these six principles to evaluate whereas you have a knack for killing them, your

an argument from analogy. Suppose that you want argument is weakened. to shade a bedroom window from the intense west- To this set of facts, suppose we add the obser-ern sun. To accomplish this purpose, you decide to vation that four of your neighbors have Italian plant some trees, but the trees must be narrow ones cypresses and that all of their trees are approximately that will grow approximately 25 feet tall. Recalling 25 feet tall. This fact tends to strengthen your argu-that a friend has some Italian cypresses that are ment. Also, if all four neighbors planted their trees almost exactly 25 feet tall, you conclude that this in different years, use different kinds of fertilizers variety of tree will do the job. Thus far, your argu- on them, and give them different amounts of water, ment is based on a single similarity, the species of your argument is further strengthened. Finally, the tree. Because this similarity is very relevant to should you broaden your conclusion to state that height, your argument is fairly strong. Now if, in your trees will grow between 22 and 28 feet tall, your addition, it turns out that your friend lives in the argument is strengthened even further. But if you same city and has the same kind of soil as you have, narrow your conclusion to state that your trees will your argument is strengthened. However, if it hap- grow to between 241∕2 and 251∕2 feet tall, your argu-pens that your friend has a knack for growing trees ment is weakened.

9.2 Legal Reasoning

Many of the arguments used by lawyers in the United the additional lost profits. When the repair company States and Canada to support a case at trial are ana- refuses your demand, you file suit. logical arguments. The reason is that the legal sys- The lawyer who argues your case will probably tems of these countries were derived many years ago make reference to a much earlier case, Hadley v. Bax-from the English system, and an essential feature of endale, decided in Gloucester, England, in 1854. In the English system is its dependence on precedent. that case, the operation of a mill was halted because According to the requirement of precedent, similar of a broken crankshaft, and the owner of the mill cases must be decided similarly. Thus, in arguing a (Hadley) employed a shipping company (Baxen-case, a lawyer will often attempt to show that the case dale) to return the crankshaft to the manufacturer. is analogous to some earlier case that was decided in Hadley informed Baxendale that the broken crank-a favorable way. Sometimes, even today, these prec- shaft had interrupted the operation of the mill, and edents go back to early English law. Baxendale replied that the broken crankshaft would

For example, suppose you own a factory and one be in the hands of the manufacturer within one

of your machines, a drill press, breaks down, causing day. As it turned out, the shipment was delayed by the entire operation to come to a halt. Urgently you several days, and Hadley incurred lost profits from call a repair company and explain the whole situa- the additional time the mill was shut down. Hadley tion. The spokesperson for the company promises to demanded reimbursement from Baxendale, and have the drill press back in operation within two days. when Baxendale refused, Hadley filed suit. The court Unfortunately, there are delays, and two weeks elapse ruled that Baxendale should pay only if he foresaw before the drill press is repaired. In the meantime, when he agreed to do the work that delays would your company loses an additional $10,000 in profits. result in lost profits for Hadley. Because you relied on the spokesperson�s assurance Applying this result to the drill-press case, your that the drill press would be fixed in two days, you lawyer will argue that because the repair company demand that the repair company reimburse you for was informed that delays in repairing the drill press

450Chapter9 Analogy and Legal and Moral Reasoning

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). would result in lost profits, it should reimburse you and the principles developed at the beginning of this for the losses incurred. section can be used to evaluate them. The first prin-

In the early days, much of English law resulted ciple, which deals with the relevance of the similari-

directly from judges� rendering decisions in particu- ties, has an obvious application to legal arguments. lar cases. This judge-made law is otherwise called Suppose, for example, that two cases are similar in common law. However, today many of our laws are that each involves a burning building, the building not the direct product of judges but rather of leg- in each case is located in Chicago, and in each case islative bodies. These laws, called statutes, are codi- ten people died. The comparison is pointless if the fied in books that are periodically revised. Can it be issue in the first case is whether an insurance com-said that precedent plays a central role in decisions pany is liable to pay damages whereas the issue in involving statutory law in the same way that it does the second is whether somebody is guilty of arson. in decisions involving common law? The answer is The second and third principles, which deal a definite yes. Statutory laws are invariably phrased with the number of similarities and the�nature and in relatively general language, and precedent is degree of disanalogy, are also applicable. If a preced-required to determine how the statutes should be ing case resembles the case at hand in virtually all interpreted and applied. important details, then the preceding case is said to

Consider, for example, a �law� that we are all be �on point� and, given certain other requirements

familiar with, the First Amendment to the U.S. (discussed shortly), will dictate the outcome of the Constitution, which provides for freedom of speech case to be decided. As for the nature and degree of and religious expression. Suppose that you decide, disanalogy, if�a� preceding case deals with fraud by a in reliance on the First Amendment, to pass out stockbroker while the case to be decided deals with religious pamphlets on a downtown street corner. fraud by a real estate broker, the precedent may be Suppose further that most of the people you hand more or less controlling, depending on such circum-your pamphlets to merely glance at them and then stances as applicable statutory law. throw them on the street and that the gathering litter As applied to legal analogies, the fourth principle makes the area look like a garbage dump. To prevent (number of primary analogues) usually determines the litter, the police tell you that you can hand out how well established a legal precedent is. If fifty your pamphlets only in the vicinity of public garbage previous courts have followed a certain rule, the cans, so that those who do not want the pamphlets precedent is stronger (everything else being equal) can properly dispose of them. You object that such than if only two courts have followed that rule. a restriction violates your First Amendment rights, The fifth principle (diversity among the primary and you take the issue to court. analogues) is also applicable. If a certain rule turns

In presenting your case, your lawyer will argue up in a broad range of cases, it provides a stronger

that the case is analogous to other cases where the precedent (other things being equal) than a rule that state attempted to limit not the content of religious turns up in a narrow range of cases. For example, the expression, but the time, place, and manner of its principle underlying the inadmissibility of hearsay expression. Next, your lawyer will note that in such testimony is well established because it appears in a cases�the state is allowed to restrict the expression broad cross section of cases. only if the restriction is �narrowly tailored� to serve Finally, the sixth principle states that the more a �significant government interest.� Finally, your specific a conclusion is, the weaker the argument lawyer will attempt to show that your case is anal- becomes. Suppose, in an earlier case, a person is ogous to cases in which the government failed to awarded half a million dollars for injuries inflicted prove that the restriction was so tailored. by a surgeon who, during an appendix operation,

As these examples illustrate, many of the argu- left a stethoscope in the body opening. The argu-

ments that occur in law are arguments from analogy, ment that another person injured in the same way

Section 9.2 Legal Reasoning 451

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). by another doctor should be awarded exactly the The first thing that any lawyer would do with this same amount of money is relatively weak in com- case is to recall the definition of �burglary.� Accord-parison with the argument that the person should be ing to the traditional definition, �burglary� means awarded approximately the same amount of money. �the trespassory breaking and entering of a dwelling

Analogical arguments that occur in law differ house of another at night with the intent to commit

from the fairly simple analogies considered earlier a felony therein.� Under many modern statutes, in at least two important ways. In the analogy deal- �dwelling house� has been replaced by �structure� ing with fuel economy, the modes of similarity that and �at night� has been deleted. linked the analogues (number of engine cylinders, There is little doubt that Blake has broken into a curb weight, and the like) were clearly defined by structure. The question is, is a soft-drink machine conventional thinking, and their relevance to the the kind of structure intended by the statute? A conclusion had been established by scientific prin- second question is, did Blake intend to commit a ciples. In law, however, clarity of this sort is seldom felony when he broke into the soft-drink machine? to be found. Modes of similarity between cases are To answer these questions, a lawyer would consult often the result of highly creative thinking by law- additional statutes and relevant cases. Let us sup-yers and judges, and the relevance of these simi- pose that our jurisdiction has a statute that defines larities to the proposed conclusion is nearly always felony theft as theft of $400 or more. Also, let us debatable. These differences render analogical argu- suppose that the cash box of this particular machine ments in law considerably more elusive than those contained $450 but most similar machines can hold dealing with subjects such as fuel economy. no more than $350.

The other difference between simple analogical As for cases, let us suppose that our jurisdiction

arguments and those found in law is that the pri- has two controlling precedents. In People v. Harris, mary analogues in law (earlier cases) do not all have Harris broke into a warehouse with the intent of equal weight. The reason is that our courts operate in stealing its contents. The warehouse contained different jurisdictions. The system of federal courts microwave ovens valued at $10,000, and Harris was is totally separate from the system of state courts. As found guilty of burglary. In People v. Sawyer, Sawyer a result, a case that might have considerable weight broke into a newspaper vending machine with the as a precedent in one system might have little or no intent to steal the cash in the cash box. The maxi-weight in the other. In addition, within these sys- mum capacity of the cash box was $20, and Sawyer tems there are different circuits, and a case that is was found not guilty of burglary. controlling in one circuit may not be in another. Relying on these facts and preceding cases, the Lastly, within these systems, the courts exist on dif- district attorney might present the following argu-

there are the district courts, courts of appeals, and In breaking into the soft-drink machine, Blake the U.S. Supreme Court. A case that is binding on a ferent levels. For example, within the federal system ment to the judge (or jury): lower court may not be binding on a higher court. is guilty of burglary for the following reasons.

Let us now illustrate the process of legal reason- In People v. Harris, the defendant was found

ing with a hypothetical example. Suppose a certain guilty of burglary for breaking into a ware-vagrant�let us call him Blake�attempted to earn house. A soft-drink vending machine is like

some pocket change by breaking into a soft-drink for resale. Furthermore, like a warehouse, a warehouse in that it contains goods stored vending machine and stealing the cash. The police people�s livelihoods depend on the goods apprehended Blake in the act, and the district stored therein. Also, the vending machine attorney charged him with burglary. Blake had contained over $400, and Blake intended one prior conviction for breaking into a soft-drink to steal this money. Stealing over $400 is a machine. felony. Therefore, Blake intended to commit

452Chapter9 Analogy and Legal and Moral Reasoning

a felony when he broke into the machine. quantities of products for resale. In People v. Also, by placing his hand into the machine, Sawyer, the defendant was found not guilty of he �entered� it. A soft-drink machine differs burglary in breaking into a newspaper vending from a newspaper machine in that it can hold

machine. Finally, Blake was familiar with soft-

goods valued much more than a stack of

newspapers. Thus, all the requirements are drink machines (he broke into one once before), met for conviction. and he therefore knew that their cash boxes are

Defense counsel, on the other hand, might argue as usually limited to holding less than $400. There-follows: fore, when he broke into this particular machine,

Blake intended to steal less than $400, and

In breaking into the soft-drink machine, Blake therefore he did not intend to commit a felony. is not guilty of burglary, for the following rea-

sons. The original crime of burglary extended In deciding whether Blake is guilty, the judge or jury only to dwelling houses because people live in will have to evaluate these analogical arguments by dwelling houses and it is important that their determining which analogies are the strongest.

Sometimes lawyers and judges are confronted with

lives be protected. Modern law has extended cases for which there is no clear precedent. Such cases the kind of structure requisite for burglary to are called cases of first impression, and the attempt include warehouses because people work and to deal with them by appeal to analogous instances live in them. A soft-drink machine is unlike a involves even more creativity than does the attempt warehouse in that a person is not capable of to deal with ordinary cases. In deciding cases of first working and living in it. Also, for the crime of impression, judges often resort to moral reasoning, burglary to be committed, the burglar must and they grope for any analogy that can shed light enter the structure. A soft-drink machine dif- on the issue. The reasoning process in such a deci-

sion often involves a sequence of analogies followed

fers from a warehouse in that a person is not by disanalogies and counteranalogies. These analogies capable of entering it with his whole body. present the issue in a continually shifting light, and the

On the contrary, a soft-drink vending experience of coming to grips with them expands the

machine is very similar to a newspaper vend- outlook of the reasoner. New perspectives are created, ing machine in that it contains relatively small attitudes are changed, and worldviews are altered.

9.3 Moral Reasoning

As in law, arguments from analogy are also useful who take drugs or drink alcohol during pregnancy in deciding moral questions. Such arguments often and cause damage to their fetus. The article said

of a dialogue that includes several arguments from country have traces of alcohol or illegal drugs in their analogy, consider the following one between two occur in the context of a dialogue. As an example that 11 percent of all babies currently born in this fictional characters, Jason and Talisa. The dialogue blood. I think such women should be put in jail!� deals with the morality and legality of fetal abuse. �I can see that you�re pretty worked up about

this, Talisa, but don�t you think you�re being a

�Jason, I read an article in this morning�s newspaper bit prudish? After all, what�s so bad about a little about fetal abuse�you know, expectant mothers alcohol or a few drugs?�

Section 9.3 Moral Reasoning 453

�I can�t believe you said that!� Talisa scolds. �A fetus is very much like a complete human

�The alcohol and drugs cause permanent brain being, though,� Talisa replies. �It�s made up of damage to the fetus and leave the child maimed for human flesh and blood, it has the same genetic life. Many of these kids will grow up mentally chal- code as a human being, and as it develops, it lenged, and many will become criminals and end has the same organization of internal organs as up in jail. The burden on society will be incredible! a human being and it even looks like a human Women who treat their unborn child in this way are being. Isn�t this enough?� utterly immoral, and they ought to be sent to jail.� �And because of these similarities you think

�Let�s take these claims one at a time,� Jason the fetus has rights?�

responds. �First, why do you think feeding alco- �Yes, I do.� hol or drugs to a fetus is immoral, Talisa?� �Well, I think your analogy is nice,� Jason replies,

�Because I think the fetus has rights.� �but I don�t think the similarities you cite are rele-�But wait a minute,� counters Jason. �Accord- vant to your conclusion. The reason human beings

ing to the laws in many states, the mother can have rights and can enter into contracts is because abort the fetus. That must mean the fetus has no they are rational and have the power of speech. right to life; and if it has no right to life, surely it They demonstrate these powers by proving theo-has no rights at all.� rems in geometry and engaging in conversations. Mom has a Talisa scratches her head. I�ve never seen a fetus do either of these things.� contract with �It seems to me you�re over- �No,� replies Talisa, �but neither do people her fetus. looking something,� she says. when they�re asleep. Do you think a sleeping

�When a mother intends person is not human? Would you take a sleeping

to abort her fetus, it�s one thing. But when she person and sweep him out the door like a clump intends to bring it to term, it�s quite another thing. of dust because he wasn�t at that moment dem-It�s almost as if she has extended a contract to the onstrating rational powers?� fetus. Look, suppose I tell a house painter, �You �Of course not,� Jason replies. �But unlike a paint my house, and� I�ll pay you $1,000.� If the sleeping person, a fetus has never demonstrated painter fulfills his end of the bargain but I refuse rational powers.�

to pay, clearly I am doing something wrong. In the �But neither has a newborn When does a same way, the mother who intends to bear her baby,� Talisa argues. �When fetus become child in effect tells the fetus, �You agree to grow human? do you think a newborn baby and develop in�a normal way, and I�ll provide you becomes human? Does it happen with a healthy environment.� If the fetus grows�and at the moment it�s born? Was it not a human being develops normally and then the mother feeds it one hour before birth? Isn�t a tomato growing on drugs, she reneges on�the deal.� a vine as much a tomato one day before it ripens

�I agree with you about the house painter,� as it is when it�s ripe? Isn�t an apple growing on a

Jason answers. �But it seems to me that you can tree as much an apple when it�s green as it is after only enter into contracts with human beings. You it turns red?� don�t enter into contracts with dogs and cats, �That�s a good analogy,� says Jason. and you don�t enter into contracts with fetuses, �Also, even if a fetus can�t be proved beyond either.� any uncertainty to be fully human, don�t you

454Chapter9 Analogy and Legal and Moral Reasoning

think it should be given the benefit of the doubt? organism. But the fetus has a unique genetic When a person falls through the ice on a frozen code. Also, it�s meant to develop and separate lake, don�t the rescuers who pull the person out from the body of its host, unlike a liver or a lung. rush him to a hospital even though no signs of Therefore, a fetus is not merely a part of the life remain? Isn�t the person, whether dead or mother�s body.� alive, treated as if he were still alive? Similarly, �Well, perhaps you�re right,� Jason says. �But shouldn�t a fetus, who shows no signs of rational- I have a better analogy. The fetus is like a seed ity, be treated as if it had rationality?� planted in the ground or a group of cells grow-

�Perhaps,� Jason answers, �but I have trou- ing in a petri dish. Imagine you were walking

ble attributing humanity or personhood to a down the street and a little seed shouted up at mere lump of flesh. This wart on the back of you, �Help! Get me out of this soil! It�s too acidic!� my hand is made of human flesh and blood, What would you do? Would you reach down, lift and its cells carry the same genetic code as the seedling up, and transplant it elsewhere? Of I do. Of course it�s not shaped like a human course not. You�d say to the seed, �Be quiet! It�s being, but neither is a fetus in the early lucky you found any soil at all to grow in.� Isn�t stages of development. Does this mean that that what you�d say?� I should consider this wart to be human or a �Maybe I would,� she says, �but not if I person?� expected the seed to grow into a human being.

�So what you�re saying,� Talisa replies, �is And incidentally, don�t ever accuse me of

that the fetus is just a part of its mother�s body, anthropomorphizing! I think a far better analogy like that wart is part of yours?� is that of someone who perpetrates a battery on

�Exactly,� he says. �And that means the fetus another person. I� was reading about a famous

can have no rights or interests distinct from case where a man approached a woman in her those of the mother, any more than someone�s sleep and gave her a kiss on the cheek. The man liver can have rights distinct from the person was someone the woman despised, and when who has the liver. When an alcoholic abuses his she woke up and found out what had happened, liver by excessive alcohol consumption, we don�t she was so furious that she sued the man for say that he is immoral and incarcerate him for battery�and she won. A battery, you know, is liver abuse. Similarly, we don�t say that smokers any intentionally produced harmful or offensive are immoral for abusing their lungs and put them contact. The case establishes the principle that in jail for lung abuse. Accordingly, we shouldn�t the target of a battery need not be aware of it call pregnant women immoral and put them in at the time it happens. Now if a kiss can be a jail for fetal abuse.� battery, how much more so is being drowned in

A fetus has its �Your analogy has one fatal drugs and booze? The fetus might not be able to own genes. flaw,� Talisa answers with a sue till after it�s born, but the battery was never-

smile. �If the fetus is merely theless committed before birth.�

part of its mother�s body, then it would have �Very clever,� Jason responds. �However, I exactly the same genetic code as the mother. don�t think you can say someone�s asleep if that Every ordinary body part�heart, liver, lungs, person has never been awake. A fetus has never and so on�carries the same genes as the host been awake.�

Section 9.3 Moral Reasoning 455

�I see you�re still harping on whether the fetus from that day on. That�s what fetal abuse does,

has human thought processes,� replies Talisa. as far as I�m concerned.�

�Well, consider this. After the child is born, every- �Getting a little emotional, aren�t we, Talisa?� one agrees it�s a full-fledged human being. When �Maybe. But I don�t think morality is some-that child enters the world with drugs in its body, thing devoid of feeling. Morality concerns we can appropriately ask where those drugs came life and life is feeling, and the two can�t be from. The answer, of course, is the mother. Thus, separated.� the mother has given drugs not to a mere fetus but �Well, even if I agreed with you that fetal to a real child. The situation would be exactly the abuse is immoral, that doesn�t mean I think it same as if a common drug dealer had given drugs should be made illegal. I know our criminal laws

to the kid. Thus, the mother are closely tied to public morality, but that�s true

Mom is like a should be treated the same only up to a point. I think making fetal abuse a drug dealer.

way as any drug dealer.� crime might be going too far.�

�I think you�ve got me there,� Jason admits. �Why is that?� asks Talisa.

�But your analogy holds only for kids who are �Because it�s so invasive,� Jason responds. actually born with drugs or alcohol in their �Like a surgeon performing an unwanted proce-system. What about fetuses who are fed drugs dure, the state would be invading the most or alcohol weeks before birth but who are deliv- private aspects of a woman�s life. Also, it would ered without any trace of them?� be patronizing. The state would be treating

�Suppose,� replies Talisa, �a doctor who is the pregnant woman like an errant, recalcitrant

treating a pregnant woman periodically injects child who�s done something wrong and needs to cocaine into the fetus using a long hypodermic be punished.�

needle. Not only would everyone consider the �But there is a long his-The state pro-doctor fiendish and immoral, but the doctor tory of state intervention in tects citizens would be liable for civil and criminal sanctions. Is from deviants. the lives of people who are the mother who feeds cocaine to her fetus any a danger to themselves or different from this doctor?� others,� Talisa observes. �For example, people

�Interesting point,� says Jason. �I expect, how- who attempt suicide are sometimes institution-

ever, that the doctor would be subject to liability alized to prevent them from killing themselves. for acting without the mother�s consent�not for Also, mental deviants are often committed to acting without the fetus�s consent.� mental institutions for their own benefit. The

�I hope you�re wrong,� says Talisa. �But you theory is that, if these people were in their right

know how I feel about these fetal abusers? I see minds, they would want the state to do exactly a child in a playpen near a window. Outside the that. Women who abuse their fetuses are very window other kids are playing on the grass. The similar to mental deviants. If these women were child in the playpen is excited and happy when it in their right minds, they would want the state to sees them and can�t wait until it�s big enough to intercede and prevent them from injuring their go out and join them. Then one day the mother fetuses.� comes in with a baseball bat and smashes the �Still,� replies Jason, �I think a law crimi-child�s legs, and the child is never able to walk nalizing fetal abuse would be unenforceable.

456Chapter9 Analogy and Legal and Moral Reasoning

It would be just like the law against alcohol during with vinegar. Such programs Honey catches prohibition. That law was eventually repealed might reduce the problem of more flies than because, among other things, it was impos- vinegar. fetal abuse to a point where sible to enforce. A law against fetal abuse would it becomes tolerable.� suffer the same fate. It would require that police �I think any amount of fetal abuse is intoler-officers snoop into the homes and bedrooms of able,� Talisa replies. every pregnant woman reasonably suspected of �Society is like a water supply,� Jason says. abusing her fetus. Such action would violate one �Public-health officials test it for contaminants of our most cherished values, the right to pri- and find microscopic amounts of this and micro-vacy, and it would require massive additions to scopic amounts of that. If these amounts are within our police force.� acceptable limits, they pronounce the water supply

�Yes,� answers Talisa, �but the problem is potable. Yes, the thought of any fetal abuse is dis-

so significant that I think something must be turbing, but we can never eliminate it entirely. The done. You can look at the whole of society as best we can hope for is to reduce it to a point a single human body grown large. Women where it won�t cripple society as a whole.� who abuse their unborn children are like spiders �Perhaps you�re right,� concludes Talisa, �but who inject venom into the body of society. This the thought of that child in the playpen still venom creates a sore that lasts for seventy years bothers me.� or more. Now I ask you, what person threatened

by such a spider would not go to any lengths This dialogue contains numerous analogies and to prevent its bite? What rational person would counteranalogies. The modes of similarity between stand idle, let the spider inject its venom, and then the many primary analogues and the secondary tolerate the resulting canker for seventy years?� analogues (the fetus, women who commit fetal

�Yes,� agrees Jason, �but unfortunately, abuse, laws against fetal abuse, and so on) are never

getting rid of fetal abuse is harder than simply relevant beyond dispute, and the nature and degree brushing off a spider. I think the solution is to of disanalogy rarely has undisputed consequences.

However, the dialogue serves to illustrate the subtle

stress educational programs and provide pre- effect that analogies have on our thought processes natal care for every expectant mother.�After all, and the power they have to alter our attitudes and you catch more flies with honey than you do perspectives.

Exercise 9

I. Evaluate the arguments from analogy contained in the dialogue between Jason and

Talisa. First, identify the primary and secondary analogues in each analogy and coun-teranalogy. Next, list the similarities between the primary and secondary analogues and note their relevance to the conclusion. Add any additional similarities you can think of. Finally, identify the disanalogies associated with each analogy or counteranalogy and evaluate their effectiveness in destroying the original analogy.

II. Work the following problems.

Section 9.3 Moral Reasoning 457

★1. Martin has long admired Julian�s near-perfect body, and he notes that Julian

works out on a Peloton exercise machine. Martin concludes that if he buys a Peloton for himself, he will be able to duplicate Julian�s results. How do the fol-lowing facts bear on Martin�s argument?

a. Peloton recently changed its corporate headquarters from New York to

New Jersey.

b. Martin and Julian are siblings.

c. Julian is 25 years old, but Martin is 32.

d. Martin is 40 pounds overweight, but Julian was only 10 pounds overweight

when he started using his Peloton.

e. Whenever Martin makes up his mind to do something, he always sticks

with it longer than Julian does.

f. Martin knows of five other men who work out on a Peloton, and these men

all look as good as Julian.

g. These men are all between the ages of 24 and 26.

h. Martin discovers four additional men between the ages of 31 and 33 who

work out on a Peloton, and these men look nearly as good as Julian.

i. Martin changes his conclusion to state that he will look almost as good as Julian.

j. Martin changes his conclusion to state that he will look better than Julian.

2. LeBron needs to have his rugs cleaned, and his friend Veronica reports that

Ajax Carpet Service did an excellent job on her rugs. From this, LeBron con-cludes that Ajax will do an equally good job on his rugs. How do the following facts bear on LeBron�s argument?

a. Veronica hired Ajax several times, and Ajax always did an excellent job.

b. Veronica�s rugs are wool, whereas LeBron�s are rayon.

c. Veronica�s carpets never had any stains on them before they were cleaned,

but LeBron�s have several large stains.

d. Veronica always had her rugs cleaned in mid-October, whereas LeBron

wants his done just a week before Easter.

e. LeBron knows of six additional people who have had their carpets cleaned

by Ajax, and all six have been very pleased.

f. All six own rugs made of different materials.

g. All six additional people were born in Massachusetts.

h. Ajax has recently undergone a change in management.

i. The Environmental Protection Agency recently banned the cleaning solu-

tion Ajax has used for many years.

j. LeBron changes his conclusion to state that Ajax will get his carpets approx-

imately as clean as it has gotten Veronica�s.

458Chapter9 Analogy and Legal and Moral Reasoning

3. Kristin is president of a corporation that operates a chain of clothing stores,

and she faces the task of hiring a manager to replace a man who retired from one of the stores. The former manager increased sales by 15 percent every year for the past five years. Kristin concludes that Carlos Sanchez, a recent graduate of Wharton School of Business, will duplicate the former manager�s perfor-mance. How do the following facts bear on Kristin�s argument?

a. The manager who retired was a graduate of Wharton.

b. The manager who retired liked tennis and drove a Jaguar, whereas Sanchez

dislikes tennis and drives a BMW.

c. Unlike the manager who retired, Sanchez formerly managed a shoe store,

where he increased sales 20 percent for each of the two years he was there.

d. A labor dispute has recently erupted in the store Sanchez will manage.

e. The manager who retired was an alcoholic, whereas Sanchez is a moderate

drinker.

f. The government has approved a 10 percent increase in federal income taxes

that takes effect at the beginning of the year.

g. Three additional stores owned by Kristin�s company are managed by

recent Wharton graduates, and all three managers have increased sales by 18� percent for each of the past three years.

h. These three stores are located in the city�s three wealthiest suburbs.

i. The store Sanchez will manage is located in a neighborhood that has recently

begun to decline.

j. Kristin changes her conclusion to state that Sanchez will increase sales by at

least 10 percent for the first year.

★ 4. Conrad has planned a one-day fishing trip in Alaska. He intends to fish off

Rocky Point, where he fished last year. Because he caught five fish in a one-day outing last year, Conrad concludes that he will catch five fish this year. How do the following facts bear on Conrad�s argument?

a. Last year Conrad used a wooden boat, but this year he will use a fiberglass boat.

b. Conrad used herring for bait last year, but this year he will use anchovies.

c. Conrad fished with four friends last year, and each of them caught more

than five fish on that day.

d. This year Conrad will fish on July 15, but last year he fished on August 1.

e. Last year these four friends caught an average of five fish per day from

July�15 to August 1.

f. These four friends are women.

g. These four friends used various kinds of bait�herring, squid, anchovies,

and artificial lures.

h. It is now July 12. Yesterday, ten people fished off Rocky Point, and none

caught any fish.

Section 9.3 Moral Reasoning 459

i. A fleet of commercial fishing boats has been fishing the area near Rocky

Point for the first time.

j. Conrad changes his conclusion to state that he will catch at least one fish.

5. Susan is considering a job as public relations specialist with the Chamber of

Commerce. Her friend Terry took such a job one year ago, and within nine months her annual salary was $90,000. Susan argues that if she takes this job, then her annual salary will be at least $90,000 within nine months. How do the following facts bear on Susan�s argument?

a. Susan is beautiful and outgoing, whereas Terry is average looking and

introverted.

b. Susan is a Democrat, but Terry is a Republican.

c. Like Terry, Susan is 5 feet 4 inches tall and has blonde hair.

d. Susan�s parents are elementary school teachers, but Terry�s parents are

prominent businesspeople.

e. One year ago, Cindy, April, Elizabeth, and Lanelle accepted jobs as public

relations specialists with the Chamber of Commerce, and today all are earn-ing $81,000.

f. April is a Republican, Cindy is a Democrat, Elizabeth is a Libertarian, and

Lanelle is a member of the Peace and Freedom Party.

g. Cindy, April, Elizabeth, and Lanelle are beautiful and outgoing.

h. Susan has five years of experience as a public relations specialist, whereas

Terry, when she was hired, had only two years of experience in that area.

i. There is a shortage of public relations specialists in the local job pool.

j. Susan changes her conclusion to state that after nine months she will be

earning at least $85,000.

6. Joaquin is searching for a puppy that will grow up to be friendly with his

children. His friend Antonia has an Airedale terrier that is good with her chil-dren. From this, Joaquin concludes that an Airedale puppy would make a good choice. How do the following facts bear on Joaquin�s argument?

a. Antonia�s dog is a female, but Joaquin plans to get a male.

b. Tim, Ed, and Irene have male Airedales that are friendly with their children.

c. Tim�s, Ed�s, and Irene�s dogs all came from the same litter.

d. Fran, Penny, and Bob have Airedales that snap at their children.

e. Fran�s, Penny�s, and Bob�s Airedales all came from different litters.

f. The puppy that Joaquin plans to get was born in the month of June, just as

Antonia�s was.

g. The puppy that Joaquin plans to get is of the same subspecies as Antonia�s dog.

h. The puppy that Joaquin plans to get had a littermate that was vicious and

had to be euthanized. Antonia�s dog had no such littermates.

460Chapter9 Analogy and Legal and Moral Reasoning

i. Joaquin plans to give his dog special training, but Antonia�s dog received no

such training.

j. Joaquin changes his conclusion to state that if he gets an Airedale puppy, it

will grow up to at least tolerate children.

★7. Laura is considering taking Calculus I from Professor Rogers. Her friend Gina

took Professor Rogers for this class and got an A. Laura concludes that she, too, will get an A from Professor Rogers. How do the following facts bear on Laura�s argument?

a. Gina earned straight A�s in high school math, whereas Laura got B�s.

b. Laura took precalculus in high school, but Gina took no precalculus class.

c. Laura is a Buddhist, whereas Gina is a Protestant.

d. Kevin, Toni, and Samantha took precalculus in high school, and all of them

received A�s in Professor Rogers�s Calculus I class.

e. Kevin, Toni, and Samantha all have different majors.

f. Kevin, Toni, and Samantha attended Franklin High School.

g. Laura attended Rosa Parks High School.

h. Kevin, Toni, and Samantha took precalculus from different instructors.

i. Laura and Gina earned equal scores on the college math entrance exam.

j. Laura changes her conclusion to state that she will receive an A+ in Professor

Rogers�s class.

8. Andrew is thinking about buying stock in E-Tron, a new company that

sells electronic equipment online. Six months ago, he bought shares in E-Boot, a new company that sells shoes online, and the price of the stock doubled in two months. Andrew argues that if he buys E-Tron, the stock will double in two months. How do the following facts bear on Andrew�s argument?

a. Stocks in Internet companies have been in a steep decline for the past two

weeks, whereas they were rising six months ago.

b. E-Tron will be run by the same management team that runs E-Boot.

c. During the past year, the stock of five other new companies that sell online

doubled within two months of their initial offering.

d. These five companies market Swiss chocolates, tires, appliances, furniture,

and luggage.

e. A survey was taken of E-Boot customers, and 90 percent said they would

not consider buying electronic equipment online.

f. Two other companies that market jewelry and lingerie online have done

poorly.

g. Comcast, the most widely used Internet service provider, just increased its

monthly service charge by 50 percent.

Section 9.3 Moral Reasoning 461

h. E-Boot is incorporated in New Jersey, whereas E-Tron is incorporated in

Delaware.

i. E-Boot introduced its products with a major ad campaign, whereas E-Tron

plans no such campaign.

j. Andrew changes his conclusion to state that E-Tron stock will triple within

the next two months.

9. According to the doctrine of adverse possession, a person occupying a piece

of land in a way that is open, notorious, and hostile to the owner�s rights can claim ownership of the land after a certain number of years of continu-ous occupancy. In this connection, Dr. Wacko, a mad scientist, hauled tele-scopes and radio antennas to the top of Mica Peak every night for twenty years to detect extra terrestrial life. At the end of the period, he petitioned the court for a decree stating that he owned Mica Peak under the theory of adverse possession. Tom Bell, the owner of Mica Peak, attempts to defeat Dr. Wacko�s petition.

Construct two arguments, one supporting Dr. Wacko�s position, the other supporting Tom Bell�s position. The statutory period for adverse possession in this jurisdiction is twenty years, and there are two controlling cases:

Crick v. Hoskins: Crick occupied a house in Centerville that was owned by Hoskins.

She lived in the house continuously, mowed the lawn weekly, and parked her car in front. After twenty years she petitioned the court for a decree stating that she owned the property under the theory of adverse possession, and the court granted the decree against the objections of Hoskins.

Raymond v. McBride: Raymond set up a large tent on a piece of rural property

owned by McBride and lived in it for nine months each year. During the remain-ing three months, heavy snow made the property inaccessible. After twenty years Raymond petitioned the court for a decree stating that he owned the property under the theory of adverse possession, and the court denied Ray-mond�s petition.

★10. The Fourth Amendment to the U.S. Constitution prohibits unreason-

able searches and seizures. A search is defined as a �violation of a person�s reasonable expectation of privacy by the police.� Legal issues dealing with searches often turn on the question of whether the person who was searched had a reasonable expectation of privacy at the time and place the search was conducted. In this connection, consider the following set of facts. Maxie was lining up drug contacts from his cell phone while he was driving on the freeway. Without obtaining a search warrant, federal agents intercepted one of his messages and used it to locate Maxie�s car and arrest Maxie. Maxie argues that the federal agents violated his reasonable expectation of privacy, and the government denies it.

Your task is to construct two arguments, one supporting Maxie�s position,

the other supporting the position of the federal agents. There are three control-ling cases in this jurisdiction:

462Chapter9 Analogy and Legal and Moral Reasoning

U.S. v. Taylor: Federal agents suspected Taylor of illegal distribution of firearms.

Without obtaining a search warrant, they planted a listening device on the phone in Taylor�s home and used the evidence obtained to arrest Taylor. The court ruled that their action violated Taylor�s reasonable expectation of privacy and disallowed the evidence.

U.S. v. Weber: Without obtaining a search warrant, federal agents operating inside

U.S. borders intercepted a radio transmission from a plane flying over Mexico. The message mentioned drugs, and it disclosed the location of a landing strip inside the United States. Using this information, the agents met the plane and arrested its pilot for drug smuggling. The court ruled that their action did not violate the pilot�s reasonable expectation of privacy.

U.S. v. Robinson: Federal agents were using a telescope to conduct surveillance

on the resident of a distant apartment building. Inadvertently they noticed some teenagers smoking crack in a car parked in the lot near the building, and they proceeded to arrest them. The court ruled that their action did not violate the teenagers� reasonable expectation of privacy.

11. The First Amendment to the U.S. Constitution states that Congress shall make no

law abridging the freedom of speech or the right to peaceable assembly. This �law� applies to states (and cities) as a result of the adoption of the Fourteenth Amend-ment. In reliance on the First Amendment, a group of gay and lesbian individu-als apply for a permit to march on Gay Freedom Day. Similar parades in other cities have been calm and orderly, but the police chief denies a parade permit to this group.

Construct two arguments, one supporting the position of the group, the

other supporting the position of the police chief. There are three controlling cases in this jurisdiction:

Lester v. City: Police seized an art collection displayed in a public park, because

it depicted heterosexuals in nude poses. The court ruled that the display was protected by the First Amendment.

Byron v. City: Police denied a parade permit to a Nazi group that wanted to

march on Hitler�s birthday. The court ruled that the denial violated the group�s rights under the First Amendment.

Stone v. City: Police closed down a theater showing an erotic gay film, because

they claimed that the film promoted the spread of AIDS. The court ruled that the action of the police was allowed under the First Amendment.

12. Constructive eviction is a legal doctrine by which a landlord who substantially

interferes with a tenant�s use and enjoyment of the premises will be considered to have evicted the tenant. Such a landlord cannot collect rent from the tenant. In this connection, Isabel signed a lease for an apartment owned by Carolyn. A clause in the lease stated that all tenants must keep noise to a minimum during evening hours. After moving in, Isabel found that another tenant played his stereo all night long at ear-splitting levels. Carolyn declined to enforce the noise clause in the lease. After one week without sleep, Isabel moved out, and Carolyn now sues Isabel for rent.

Section 9.3 Moral Reasoning 463

Construct two arguments, one supporting Isabel�s position, the other

supporting Carolyn�s. There are two controlling cases in this jurisdiction:

Garvin v. Linder: Linder rented an apartment from Garvin, and the lease stated

that Garvin would provide heat during the winter. When winter arrived, Garvin provided heat for only one hour per day, and the inside temperature some-times dropped to 50�F. Linder moved out, and Garvin sued Linder for rent. The court ruled in favor of Linder.

Quincy v. Fulton: Fulton rented a tenth-floor apartment from Quincy. Shortly

thereafter, Quincy decided to have repairs made on the elevator, causing it to be shut down for a month. Because Fulton had arthritis, he had great difficulty climbing ten flights of stairs. Fulton moved out, and Quincy sued Fulton for rent. The court ruled in favor of Quincy.

★13. A negligent person who causes an injury to another person is liable for the

latter�s injuries. Because it is sometimes very difficult to determine the causal extent of a negligent action, courts have developed the theory of proximate cause, which limits the scope of liability. In this connection, Liz Shaffer negli-gently failed to maintain the brakes on her car, and as a result her car crashed into one driven by Sonali Gupta. Sonali was taken to the hospital for bumps and bruises, but while she was there, doctors mistakenly amputated her perfectly healthy leg. Sonali sues Liz for loss of her leg.

Construct two arguments, one supporting Sonali�s position, the other sup-

porting Liz�s position. The only issue is whether Liz proximately caused the loss of the leg. There are two controlling cases in this jurisdiction:

Sacco v. Lane: Lane negligently used gasoline to light his barbecue in a strong

wind. The flames from the barbecue ignited nearby trees and then spread to ten houses in the neighborhood, burning them to the ground. The court ruled that Lane was liable for damage to the houses.

Hunt v. Gomez: Hunt was a passenger in a taxi driven by Gomez. Gomez was drunk

and negligently let Hunt out at the wrong corner. While Hunt was walking home, a worker dropped a brick from a building that was under construction, injuring Hunt. Hunt sued Gomez for injury resulting from the falling brick. The court ruled in favor of Gomez.

14. Lynn Dodd, age twenty, has become a member of New Age Enlightenment, a

religious cult group. The group has convinced Lynn to sell all her worldly posses-sions and give the proceeds to the group. The group forbids any contact between Lynn and her relatives and friends. On one occasion Lynn�s parents encountered Lynn on a street corner, and they tried to persuade her to leave the group and return home, but she refused. The parents fear that some form of mind control is being used to keep Lynn in the group, and they plan a rescue operation.

Write a dialogue, similar to the one presented in this chapter, examining the

morality and legality of such an operation. Include in your dialogue analogies between the proposed rescue operation and the following: rescuing an adult who has been hypnotized by captors, kidnapping someone against his or her will, rescuing a child who has been kidnapped, spiriting away a novice who has joined a Jesuit seminary or Franciscan convent, coercing an alcoholic to enter

464Chapter9 Analogy and Legal and Moral Reasoning

a detoxification center, forcing a child to attend a certain school, preventing someone from committing suicide, forcing a child to attend a certain church, rescuing an eighteen-year-old from a Nazi youth group, removing one�s daugh-ter from the Girl Scouts, and any other pertinent analogy that comes to mind.

15. During fifteen years of marriage, James and Darnella Knox had not succeeded in

having a baby. As a last resort, they tried in vitro fertilization. Doctors surgically removed nine eggs from Darnella and fertilized them with James�s sperm. The embryos were then placed in a freezer until the proper time for them to be implanted in Darnella�s womb. In the meantime, James and Darnella came to realize that their marriage was going nowhere, and they decided to get a divorce. The couple has had no trouble arriving at a settlement concerning the house, cars, bank accounts, and other possessions, but they cannot agree on the disposition of the frozen embryos. James is adamant about not want-ing to have any children by Darnella, and he wants the embryos destroyed. Darnella, on the other hand, realizing that she is getting older, thinks that the frozen embryos offer her the best chance of having a baby, and she wants them implanted.

Write a dialogue, similar to the one in this chapter, examining the moral

and/or legal ramifications of this dispute. Include in your dialogue analogies between this situation and the following: killing a fetus, killing a baby, forcing your wife to have an abortion, having an abortion against the wishes of your husband, owning a dog or cat, owning a child, determining custody of a child upon divorce, dividing a bank account, determining ownership of a unique and valuable painting produced jointly by two people, implanting an embryo in a surrogate mother, and any other analogies that come to mind, including analogies that show that the embryos have or do not have rights and analogies that illustrate the investment that James and Darnella have in the embryos.

Summary

Argument from Analogy:

■ An inductive argument that rests on a similarity between two things.

■ The two things that are compared are called analogues.

● Primary analogue: An analogue mentioned in the premise.

● Secondary analogue: The analogue mentioned in the conclusion.

■ Principles for evaluating arguments from analogy:

● Relevance of the similarities

● Number of similarities

Summary465

● Nature and degree of disanalogy

● Number of primary analogues

● Diversity among the primary analogues

● Specificity of the conclusion

Legal Reasoning:

■ Requirement of precedent: Similar cases are decided similarly.

■ The six principles for evaluating arguments from analogy apply to legal arguments.

Moral Reasoning:

■ Analogies are useful in constructing arguments about moral issues.

466Chapter9 Analogy and Legal and Moral Reasoning

10Causality and Mill�s Methods

10.1 �Cause� and Necessary and Sufficient Conditions

10.2 Mill�s Five Methods

10.3 Mill�s Methods and Science

10.1 �Cause� and Necessary and�Sufficient

Conditions

Being acquainted with the different kinds of causality and how Mill�s methods work is empowering. For example, suppose you read in the syllabus for a course you are taking, �If you don�t take the midterm, you won�t pass the class.� Based on that, you reason that since you did take the midterm, you will automatically pass the class. Thus, there is no need to take the final. Your reasoning confuses one kind of causality with another, and it is likely to have bad consequences.

A knowledge of causal connections plays a prominent role in our effort to con-trol the environment in which we live. We insulate our homes because we know insulation will prevent heat loss, we vaccinate our children because we know vaccination will protect them from polio and diphtheria, we practice the piano and violin because we know that by doing so we may become proficient on them, and we cook our meat and fish because we know that doing so will make them edible.

When the word �cause� is used in ordinary English, however, it is seriously

affected by ambiguity. For example, when we say that sprinkling water on the

467

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). flowers will cause them to grow, we mean that water necessary condition, and if we are trying to produce is required for growth, not that water alone will a certain phenomenon, we usually search for a do the job�sunshine and the proper soil are also cause that is a sufficient condition. For example, required. On the other hand, when we say that going in attempting to prevent the occurrence of smog for a swim on a hot summer day will cause us to cool around cities, scientists try to isolate a necessary off, we mean that the swim by itself will do the job; condition or group of necessary conditions that, if but we understand that other things will work just as removed, will eliminate the smog. And in their effort well, such as taking a cold shower, entering an air- to produce an abundant harvest, farmers search for conditioned room, and so on. a sufficient condition that, given sunshine and rain-

To clear up this ambiguity affecting the mean- fall, will increase crop growth.

ing of �cause,� adopting the language of suffi- Another important point is that whenever an cient and necessary conditions is useful. When event occurs, at least one sufficient condition is pres-we say that electrocution is a cause of death, we ent and all the necessary conditions are present. The mean �cause� in the sense of sufficient condition. conjunction of the necessary conditions is the suf-Electrocution is sufficient to produce death, but ficient condition that actually produces the event. there are other methods equally effective, such as For example, the necessary conditions for lighting a poisoning, drowning, and shooting. On the other match are heat (produced by striking) and oxygen. hand, when we say that the presence of clouds is Combining these two necessary conditions gives a cause of rain, we mean �cause� in the sense of the sufficient condition. In other words, striking necessary condition. Without clouds, rain cannot the match in the presence of oxygen is sufficient to occur, but clouds alone are not sufficient. Certain ignite it. In cases where the sufficient condition is combinations of pressure and temperature are also also a necessary condition, there is only one neces-required. sary condition, which is identical with the sufficient

Sometimes �cause� is used in the sense of neces- condition.

sary and sufficient condition, as when we say that We can now summarize the meaning of cause in the action of a force causes a body to accelerate or the sense of a sufficient condition and a necessary that an increase in voltage causes an increase in condition:

electrical current. For a body to accelerate, nothing A is a sufficient condition for B: A�s occur-more and nothing less is required than for it to be rence requires B�s occurrence. acted on by a net force; and for an electrical cur-A is a necessary condition for B: B�s occur-rent to increase through a resistive circuit, nothing rence requires A�s occurrence. more and nothing less is required than an increase

in voltage. According to these statements, if A occurs and B

Thus, as these examples illustrate, the word does not occur, then A is not a sufficient condition

�cause� can have any one of three different meanings: for B, and if B occurs and A does not occur, then A is

1. Sufficient condition not a necessary condition for B. Thus:

2. Necessary condition A is not a sufficient condition for B: A is pres-

3. Sufficient and necessary condition ent when B is absent.

Sometimes the context provides an immediate clue A is not a necessary condition for B: A is to the sense in which �cause� is being used. If we absent when B is present. are trying to prevent a certain phenomenon from These results will serve as important rules in the happening, we usually search for a cause that is a material that follows.

468Chapter10 Causality and Mill�s Methods

10.2 Mill�s Five Methods

In his System of Logic, the nineteenth-century phi- all of them ordered was vanilla ice cream for dessert. losopher John Stuart Mill compiled five methods In other words, all of the dinners were in agreement for identifying causal connections between events. only as to the ice cream. Such a situation suggests These he called the method of agreement, the that the ice cream caused the sickness. The method method of difference, the joint method of agree- of agreement consists of a systematic effort to ment and difference, the method of residues, and the find a single factor (such as the ice cream) that is method of concomitant variation. In the years that common to several occurrences for the purpose of have elapsed since the publication of Mill�s Logic, identifying that factor as the cause of a phenomenon the five methods have received a good deal of philo- present in the occurrences (such as the sickness). sophical criticism. Today, most logicians agree that The method of agreement identifies a cause in the methods fall short of the claims made for them the sense of a necessary condition. To see how this by Mill, but the fact remains that the methods func- method works, let us develop the restaurant example tion implicitly in many of the inductive inferences a bit further.

we make in ordinary life. Five people eat dinner in a restaurant. Jose In addition to criticizing these methods, modern has salad, french fries, a hamburger, ice logicians have introduced many variations that have cream, and mixed vegetables; Sage has multiplied their number beyond the original five. salad, french fries, soup, ice cream, fish, and Some of these variations have resulted from the fact mixed vegetables; Zoe has a hamburger, that Mill himself failed to distinguish causes that soup, and ice cream; Arlo has fish, mixed are sufficient conditions from causes that are nec-vegetables, ice cream, salad, and soup; and essary conditions. The account that follows intro-Maya has mixed vegetables, fish, ice cream, duces this distinction into the first three methods, french fries, and salad. Afterwards, all of but apart from that, it remains faithful to Mill�s own them became sick from something they ate. presentation. What food caused the sickness?

Method of Agreement If we let A through G represent salad, soup, french

fries, a hamburger, fish, ice cream, and mixed veg-

Suppose that five people eat dinner in a certain res- etables, respectively, we can construct a table that taurant, and a short while later all five become sick. reflects what these five people ate (Table 10.1). In Suppose further that these people ordered an assort- this table, an asterisk means that a certain food was ment of items from the menu, but the only food that eaten, and a dash means that it was not eaten.

Table 10.1

Possible Necessary ConditionsPhenomenon

OccurrenceABCDEFG(sickness)

1 (Jose) * 2 * * 2 * * *

2 (Sage) * * * 2 * * * *

3 (Zoe) 2 * 2 * 2 * 2 *

4 (Arlo) * * 2 2 * * * *

5 (Maya) * 2 * 2 * * * *

Section 10.2 Mill�s Five Methods 469

Now, since the method of agreement identifies Furthermore, because the conclusion yields a

a cause in the sense of a necessary condition, we cause in the sense of a necessary condition, it does begin by eliminating from this table the condi- not assert that anyone who ate the ice cream would tions that are not necessary for the occurrence of get sick. Many people have a natural immunity to the phenomenon. In doing so, we use the rule that food poisoning. What the conclusion says is that a condition is not necessary for the occurrence of patrons who did not eat the ice cream would not get a phenomenon if that condition is absent when sick�at least not from the food. Thus, the method the phenomenon is present. Thus, occurrence of agreement has a certain limited use. Basically 1 eliminates condition B and condition E; occur- what it says, in reference to the example, is that the rence 2 eliminates D; occurrence 3 eliminates A, ice cream is a highly suspect factor in the sickness of C, E (again), and G; occurrence 4 eliminates C the patrons, and if investigators want to track down (again) and D (again); and occurrence 5 eliminates the cause of the sickness, this is where they should B (again) and D (again). This leaves only F (the ice begin. cream) as a possible necessary condition. The con- An example of an actual use of the method of clusion is therefore warranted that the ice cream agreement is provided by the discovery of the ben-caused the sickness of the diners. eficial effects of fluoride on teeth. It was noticed

This conclusion follows only probably for two several decades ago that people in certain commu-

reasons. First, it is quite possible that some condi- nities were favored with especially healthy teeth. tion was overlooked in compiling conditions A In researching the various factors these communi-through G. For example, if the ice cream was served ties shared, scientists discovered that all had a high with contaminated spoons, then the sickness of the level of natural fluoride in their water supply. The diners could have been caused by that condition scientists concluded from this evidence that fluoride and not by the ice cream. Second, if more than one causes teeth to be healthy. of the foods were contaminated (for example, both

been caused by this combination of foods and not soup and french fries), then the sickness could have Method of Difference by the ice cream. Thus, the strength of the argu- For an example of how this method works, let us ment depends on the nonoccurrence of these two modify our earlier case of people becoming sick possibilities. from eating food in a restaurant. Instead of five

It is also important to realize that the conclusion people, suppose that identical twins, who have iden-

yielded by Table 10.1 applies directly only to the five tical susceptibilities to food poisoning, go to that diners represented in the five occurrences, and not restaurant for dinner. They both order identical to everyone who may have eaten in the restaurant. meals except that one orders ice cream for dessert Thus, if some food other than those listed in the while the other does not. The ice cream is the only table�for example, spaghetti�was contaminated, way that the two meals differ. Later, the twin who then only if they avoided both spaghetti and ice ordered the ice cream gets sick, whereas the other cream could the other diners be assured of not get- twin does not. The natural conclusion is that the ice ting sick. But if, among all the foods in the restaurant, cream caused the sickness. only the ice cream was contaminated, the conclusion The method of difference consists in a system-would extend to the other patrons as well. This last atic effort to identify a single factor that is present in point illustrates the fact that a conclusion reached by an occurrence in which the phenomenon in question the method of agreement has limited generality. It is present, and absent from an occurrence in which applies directly only to those occurrences listed, and the phenomenon is absent. The method is confined only indirectly, through a second inductive infer- to investigating exactly two occurrences, and it iden-ence, to others. tifies a cause in the sense of a sufficient condition.

470Chapter10 Causality and Mill�s Methods

Prominent John Stuart Mill 1806�1873Logicians

J ohn Stuart Mill, who would become the most important English-speaking philosopher of the nineteenth century, was born in London to the philosopher and historian James Mill and his wife, Harriet Barrow. From a very early age the father took

charge of his son�s education, imposing on him an

unusually strict course of studies. The boy began study-

ing Greek at age three; he began Latin, geometry, and

algebra at eight; and by fourteen he had read most of

the Latin and Greek classics. Also, by that time he had

developed a thorough knowledge of history, scholastic

and Aristotelian logic, and higher-level mathematics.

At seventeen he accepted a position with the East Library of Congress Prints and Photographs Division[LC-USZ62-76491] India Company, a private corporation that ruled vir-

tually all of India. He remained there for twenty-five and Jeremy Bentham. As perfected by John Stuart years, eventually becoming chief examiner, which put Mill, this theory holds that we should always perform him largely in charge of the company. the action that produces the greatest happiness for

When he was twenty, Mill suffered an acute the greatest number of people. This work was impor-

attack of depression that lasted for several months. tant, among other reasons, for creating an ethical Although he managed to continue his work, every- theory that was not based on any theology or reli-thing seemed worthless to him at that time. This con- gion. Mill also wrote extensively on women�s rights, dition was most likely brought on by the severity of and when he was later elected to a three-year term in his educational upbringing, which had attended to Parliament, he became the first member to urge that his intellect but left his emotional life a desert. Grad- women receive the right to vote. ually he pulled himself out by reading the poetry of Mill�s greatest work was his System of Logic, which William Wordsworth. Four years later, Mill met Har- went through eight editions during his lifetime. In that riet Taylor, with whom he became emotionally but book, Mill argues that deductive argumentation, far not sexually intimate (she was married at that time). from being the paradigmatic form of reasoning most Taylor would exert a profound influence on him for philosophers take it to be, is largely bereft of value the rest of his life. After the death of her husband, because it never yields any new knowledge. Also, it nineteen years after she and Mill first met, they were is essentially dependent on induction. Thus, it is really married. induction that should be studied as the primary engine

Today, John Stuart Mill is widely known for his essay of scientific discovery.

�On Liberty� and his book Utilitarianism. The essay is In 1858, Harriet Taylor died from a lung infection a sustained argument in favor of freedom of speech while the couple were visiting Avignon, France. Nearly and the maximization of individual liberty against inconsolable, Mill bought a house there so he could be the strictures of government. Utilitarianism further near her grave. He died in Avignon fifteen years later, develops the ethical theory promoted by James Mill and his body was buried alongside hers.

Section 10.2 Mill�s Five Methods 471

For a clearer illustration of how this method works, respects, the result can often be generalized to cover let us add a few details to the twin example: these others as well. At the very least, if others in the

A pair of twins, Jane and Jan, have dinner in restaurant became sick, the fact that the ice cream a restaurant. The twins have identical suscep- is what made Jane sick suggests that� this is where tibilities to food poisoning. Jane orders soup, investigators should begin when they try to explain salad, chicken, carrots, rice, and ice cream. what made the others sick. Jan orders soup, salad, chicken, carrots, rice, The conclusion yielded by the method of dif-and no ice cream. Later, Jane gets sick from ference is only probable, however, even for the one something she ate, but Jan does not. What occurrence to which it directly applies. The problem food caused Jane�s sickness? is that it is impossible for two occurrences to be lit-

erally identical in every respect but one. The mere

If we let A stand for the specific susceptibility fact that two occurrences occupy different regions to food poisoning shared by Jane and Jan, and B of space, that one is closer to the wall than the other, through G stand for soup, salad, chicken, carrots, amounts to a difference. Such differences may be rice, and ice cream, respectively, we can produce insignificant, but therein lies the possibility for error. Table 10.2. Again, an asterisk indicates that a cer- It is not at all obvious how insignificant differences tain condition is present, and a dash indicates it is should be distinguished from significant ones. Fur-absent. thermore, it is impossible to make an exhaustive list

As with the method of agreement, we proceed of all the possible conditions; but without such a list

to eliminate certain conditions, but in this case we there is no assurance that significant conditions have use the rule that a condition is not sufficient for the not been overlooked. occurrence of a phenomenon if it is present when the The objective of the method of difference is to phenomenon is absent. Accordingly, occurrence 2 identify a sufficient condition among those that are eliminates A, B, C, D, E, and F. This leaves only G as present in a specific occurrence. Sometimes, how-the sufficient condition for the phenomenon. Thus, ever, the absence of a factor can count as something G (ice cream) is the cause of Jane�s sickness. positive that must be taken into account. Suppose,

Since the result yielded by the method of differ- for example, that both of the twins who dined in the

ence applies to only the one occurrence in which the restaurant are allergic to dairy products, but they can phenomenon is present (in this case, Jane), it is often avoid an allergic reaction by taking Lactaid tablets. less susceptible to generalization than is the method Suppose further that both twins ordered ice cream of agreement, which usually applies to several occur- for dessert, but only Jan took the tablets. After the rences. Thus, the mere fact that the ice cream may meal, Jane got sick. We can attribute Jane�s sickness have caused Jane to become sick does not mean that to the absence of the Lactaid. it caused other patrons who ate ice cream to get sick. We can illustrate this situation by modifying Perhaps these other people have a higher resistance Table 10.2 as follows. Let A stand for the allergy to to food poisoning than does Jane or Jan. But given dairy products, let B through G stand for the same that the others are similar to Jane and Jan in relevant foods as before, and let H stand for Lactaid tablets.

Table 10.2

Possible Sufficient ConditionsPhenomenon

OccurrenceABCDEFG(sickness)

1 (Jane) * * * * *� * * *

2 (Jan) * * * * * * 2 2

472Chapter10 Causality and Mill�s Methods

Then ~H will stand for the absence of Lactaid (the of them gets sick. The conclusion is warranted that symbol ~ means �not�). These results are reported the ice cream is what made the first three diners in Table 10.3. sick.

Using the same rule for elimination as with The joint method of agreement and dif-

Table 10.2, we see that occurrence 2 eliminates Aference consists of a systematic effort to identify through G as sufficient conditions. This leaves ~H a single condition that is present in two or more (the absence of Lactaid) as the cause of Jane�s sick- occurrences in which the phenomenon in ques-ness. In this case, the ice cream is not identified as tion is present and that is absent from two or more the cause, because Jan (occurrence 2) ate ice cream occurrences in which the phenomenon is absent. In (G) but did not get sick. addition, the condition must never be present when

The method of difference has a wide range of the phenomenon is absent nor absent when the

applicability. For example, a farmer might fertil- phenomenon is present. Stated briefly, the occur-ize one part of a field but not the other part to test rence/nonoccurrence of the condition matches the the benefit of using fertilizer. If the fertilized part occurrence/nonoccurrence of the phenomenon. of the crop turns out to be fuller and healthier than When the phenomenon is present, so is the condi-the nonfertilized part, then the farmer can conclude tion, and when the phenomenon is absent, so is the that the improvement was caused by the fertilizer. condition. This condition is then taken to be the On the other hand, a cook might leave some ingre- cause of the phenomenon in the sense of a neces-dient out of a batch of biscuits to determine the sary and sufficient condition. To see more clearly importance of that ingredient. If the biscuits turn how this method works, let us add some details to out dry and crunchy, the cook can attribute the dif- the�example:

ference to the absence of that�ingredient. Six people eat dinner in a restaurant. Leah

has soup, a hamburger, ice cream, french

Joint Method of Agreement fries, and mixed vegetables. Adan has

salad, soup, fish, mixed vegetables, and ice

and Difference cream. Dylan has salad, a hamburger, french

To illustrate the joint method, we can once again fries, and ice cream. Sara has french fries, modify the example of the diners who got sick by a hamburger, and salad. Luna has fish and eating food in a restaurant. In place of the original mixed vegetables. Liam has french fries, a five diners and the pair of twins, suppose that six hamburger, and soup. Later Leah, Adan, people eat dinner in the restaurant. Among the first and Dylan get sick from something they three, suppose that a variety of meals are eaten but ate, but Sara, Luna, and Liam do not. What

food�made the first three diners sick?

that only ice cream is consumed by all, and later

all three get sick. And among the other three, sup- Let A through G stand for salad, soup, a hamburger, pose that a variety of meals are eaten but that none fish, ice cream, french fries, and mixed vegetables of these diners eats any ice cream, and later none (see Table 10.4).

Table 10.3

Possible Sufficient ConditionsPhenomenon

OccurrenceABCDEFG~H(sickness)

1 (Jane) * * * * * * * * *

2 (Jan) * * * * * * * 2 2

Section 10.2 Mill�s Five Methods 473

Table 10.4

Possible Necessary or Sufficient ConditionsPhenomenon

OccurrenceABCDEFG(sickness)

1 (Leah) 2 * * 2 * * * *

2 (Adan) * * 2 * * 2 * *

3 (Dylan) * 2 * 2 * * 2 *

4 (Sara) * 2 * 2 2 * 2 2

5 (Luna) 2 2 2 * 2 2 * 2

6 (Liam) 2 * * 2 2 * 2 2

In the first three occurrences the phenomenon is that is identical to the column beneath the phe-

present, so we proceed by eliminating possible nec- nomenon. The column beneath E is *** � � �. The essary conditions. Using the rule that a condition column beneath the phenomenon is also � � � ***. is not necessary if it is absent when the phenom- Thus, E is the cause in the sense of a necessary and enon is present, occurrence 1 eliminates A and D, sufficient condition. occurrence 2 eliminates C and F, and occurrence 3 Since the joint method yields a cause in the sense eliminates B, D (again), and G. This leaves only E of both a necessary and sufficient condition, it is usu-as the necessary condition. Next, in the last three ally thought to be stronger than either the method of occurrences the phenomenon is absent, so we use agreement by itself, which yields a cause in the sense the rule that a condition is not sufficient if it is pres- of a necessary condition, or the method of difference ent when the phenomenon is absent. Occurrence 4 by itself, which yields a cause in the sense of a suffi-eliminates A, C, and F; occurrence 5 eliminates D cient condition. However, when any of these methods and G; and occurrence 6 eliminates B, C (again), and is used as a basis for a subsequent inductive gener-F (again). This leaves only E as the sufficient condi- alization, the strength of the conclusion is propor-tion. Thus, condition E (ice cream) is the cause in tional to the number of occurrences that are included. the sense of both a necessary and sufficient condi- Thus, an application of the method of agreement that tion of the sickness of the first three diners. included, say, 100 occurrences might offer stronger

Students sometimes have difficulty understand- results than an application of the joint method that

ing the joint method because its definition is a included, say, only six occurrences. By similar reason-bit complicated. But this method is really easier ing, multiple applications of the method of difference to grasp than either the method of agreement or might offer stronger results than a single application the method of difference. To understand the joint of the joint method. method, keep in mind that claims about some- As with the other methods, the conclusion thing being a necessary and sufficient condition yielded by the joint method is only probable because are equivalent to �if and only if� claims. Thus, in some relevant condition may have been overlooked reference to Table 10.4, the phenomenon occurs in producing the table. If, for example, both salad if and only if E occurs. If E occurs then so does and soup were contaminated, then the ice cream the phenomenon; if E does not occur, then neither could not be identified as a necessary condition for does the phenomenon. So, to identify a cause in the sickness, and if one of the last three diners was the sense of a necessary and sufficient condition, naturally immune, then the ice cream could not be look for a column beneath the possible conditions identified as a sufficient condition. Obviously the

474Chapter10 Causality and Mill�s Methods

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). attempt to extend the results of this example to other The method of residues as illustrated in this example patrons of the restaurant who may have gotten sick may be diagramed as follows: is fraught with other difficulties.

joint method suggests that it results from a mere A causes a. combination of the method of agreement with the Lastly, we note that even though the name of the A B C causes a b c. method of difference, this is not the case. Such a B causes b. combination would consist of one occurrence in Therefore, C causes c. which the phenomenon is present, one occurrence

differs from the former occurrence as to only one combination of possible conditions producing condition, and one occurrence in which the phe-in which the phenomenon is absent and which In reference to the draft example, A B C is the

a b c , the total draft. When A (the broken window)

nomenon is present and which agrees with the first is separated out, it is found to cause a (a portion of

ing to Mill, the joint method requires �two or more� found to cause b (another portion of the draft). The occurrences in which the phenomenon is present, occurrence in only one condition. However, accord- the draft). When B (the crack) is separated out, it is

conclusion is that C (the broken damper) causes c

nomenon is absent.* His further description of the follows only probably because it is quite possible that method accords with the foregoing account. and �two or more� occurrences in which the phe- (the remaining portion of the draft). The conclusion

a fourth source of the draft was overlooked. Here is

another example:

Method of Residues After realizing a loss of $100,000, a depart-

This method and the one that follows are used to ment store�s chief accountant could suggest

without regard for the specific kind of connection. clerks, increases in utility rates, and damage to merchandise caused by a flood. These Both methods may be used to identify conditions that identify a causal connection between two conditions only three causes: an excessive number of

are sufficient, necessary, or both sufficient and nece- expenses were estimated at $25,000, $30,000, and $10,000, respectively. Since no ssary. The method of residues consists of sepa-other ordinary sources could be found, the rating from a group of causally connected conditions accountant attributed the remaining $35,000 and phenomena those strands of causal connection to shoplifting. that are already known, leaving the required causal connection as the �residue.� Here is an example: In this case, A B C D is the combination of con-

After occupying his new house Mr. Wiggins ditions consisting of the number of clerks, the found it drafty. He traced the source of the utility rate increases, the flood, and shoplifting; draft to three conditions: a broken window in and a b c d is the total loss of $100,000. After the garage, a crack under the front door, and A, B, and C are separated out, the conclusion is a broken damper in the fireplace. When the that D (shoplifting) caused d (the remaining loss). window was replaced he noticed an improve- Because the estimates might have been incorrect and ment, and a further improvement when because additional sources of financial loss might weather stripping was installed on the door. have been overlooked, the conclusion is only He concluded that the draft that remained probable. was caused by the broken damper in the Some procedures that, at least on the face of it, fireplace. appear to use the method of residues come closer to

being deductive than inductive. A case in point is

*A System of Logic, Book 3, Chapter 8, Section 4, Third Canon. the procedure used to determine the weight of the

Section 10.2 Mill�s Five Methods 475

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). cargo carried by a truck. First, the empty truck is put metabolism, with B representing blood pressure; on a scale and the weight recorded. Then the truck is and X Y Z is another set of� observable condi-loaded and the truck together with the cargo is put tions with Y representing certain brain waves. In on the same scale. The weight of the cargo is the dif- the second and third� rows, B+ and B− represent ference between the two weights. If, to this proce- increases and decreases in blood pressure, and Y+ dure, we add the rather unproblematic assumptions and Y− represent increases and decreases in the that weight is an additive property, that the scale is intensity of the brain waves. The conclusion asserts accurate, that the scale operator reads the indicator that either B causes Y, Y causes B, or B and Y have properly, that the truck is not altered in the loading a common cause. Determining which is the case process, and a few others, the conclusion about the requires further investigation; of course, if B occurs weight of the cargo follows deductively. earlier in time than Y, then Y does not cause B, and

To distinguish deductive from inductive uses of if Y occurs earlier in time than B, then B does not

the method of residues, we must take into account cause Y. such factors as the role of mathematics. If the con- The blood pressure example matches increases clusion depends on a purely arithmetical computa- in one condition with increases in another. For an tion, the argument is probably best characterized as example that matches increases in one condition deductive. If not, then it is probably inductive. with decreases in another, consider the following:

Method of Concomitant Variation A sociologist studying divorce noticed a

correlation between changes in the national

The method of concomitant variation identi- divorce rate and fluctuations in the gross fies a causal connection between two conditions by domestic product (GDP). As the GDP matching variations in one condition with variations increases, the divorce rate decreases, and in another. According to one formulation, increases when the GDP sags, the divorce rate goes are matched with increases and decreases with up. The sociologist concluded that the two decreases. Example: phenomena are causally connected.

In attempting to diagnose Mrs. Thompson�s This second version of the method of concomi-high blood pressure, a cardiologist noticed tant variation may be diagramed as follows: a correlation between fluctuations in blood

pressure and certain brain waves. As the A B C is coincident with X Y Z .

sity of the brain waves, and as the blood A B+ C is coincident with X Y− Z . blood pressure increased, so did the inten-

pressure decreased, the intensity of the brain A B− C is coincident with X Y+ Z .

waves decreased. The cardiologist concluded Therefore, B is causally connected to Y. that the two conditions were causally related.

The method of concomitant variation as illustrated In this example, A B C is a set of economic condi-

in this example may be diagrammed as follows: tions such as stock-exchange indexes, interest rates, A B C and commodity prices, with B representing gross is coincident with X Y Z . domestic product; and X Y Z is a set of social A B

+ C is coincident with X Y+ Z . conditions such as birthrates, employment rates,

Therefore, national divorce rate. The conclusion is that B is B is causally connected to Y. somehow causally connected to A B− C is coincident with X Y− Z . and immigration numbers, with Y representing the Y. However, in this

Here, A B C is a set of observable conditions example an even stronger conclusion could probably such as cholesterol level, liver function, and basal be drawn�namely, that decreases in the GDP cause

476Chapter10 Causality and Mill�s Methods

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). increases in the divorce rate, and not conversely. The argument is clearly weak. Because it is virtually That changes in economic prosperity (which is indi- inconceivable that either phenomenon could cause cated by GDP) should affect the divorce rate is quite a change in the other, or that changes in both could plausible, but the converse is less so. have a common cause, it is most likely that the cor-

At this point, we should note that the existence relation is purely coincidental.

of a mere correlation between two phenomena is The method of concomitant variation is useful never sufficient to identify a causal connection. In when it is impossible for a condition to be either addition, the causal connection suggested by the wholly present or wholly absent, as was required for correlation must at least make sense. Consider the the use of the first three of Mill�s methods. Many con-following example: ditions are of this sort�for example, the temperature

of the ocean, the price of gold, the incidence of crime,

After an in-depth study, researchers dis- the size of a mountain glacier, a person�s cholesterol covered a correlation between the price of level, and so on. If some kind of correlation can be pork-belly futures on the Chicago Mercantile detected between variations in conditions such as Exchange and earthquake activity in Japan. these, the method of concomitant variation asserts As the number and intensity of the quakes that the two are causally connected. The method has increased, the futures prices also increased, been used successfully in the past to help establish and vice versa. The researchers concluded the existence of causal connections between smoking that the two phenomena were causally and lung cancer, nuclear radiation and leukemia, and connected. alcohol consumption and cirrhosis of the liver.

10.3 Mill�s Methods and Science

Mill�s methods closely resemble certain scientific mice had identical genes, were the same methods that are intended to establish causal con- age, were placed in identical cages in the nections and correlations. For example, the method same location, were subjected to the same of difference is virtually identical to the method of environmental conditions, and were fed the

biology, pharmacology, and psychology. A con- One mouse was injected with the suspected carcinogen, and the other was not. After two trolled experiment is one that involves two groups of controlled experiment employed in such fields as same food over the same period of time.

subjects: an experimental group and a control group. months, the injected mouse developed can-cerous tumors but the other mouse did not. The experimental group includes the subjects that

receive a certain treatment, and the control group To summarize the results of this experiment we can includes the subjects that do not receive the treat-construct a table just like the one for the method ment but are otherwise subjected to the same condi-of difference. If we let A through F represent the tions as the experimental group. conditions common to the two mice (genes, age, The simplest type of controlled experiment cage, etc.) and G represent the injection, we have involves an experimental group and a control group Table 10.5. Using the rule that a condition is not each consisting of just one member. Example: sufficient if it is present when the phenomenon is A Two mice were used in a controlled experi-absent, occurrence 2 eliminates through F , leav-ing ment to determine whether a certain G (the suspected carcinogen) as the cause of the phenomenon. substance was carcinogenic. The two

Section 10.3 Mill�s Methods and Science 477

Table 10.5

Possible Sufficient ConditionsPhenomenon

OccurrenceABCDEFG(tumors)

1 (Injected) * * * * * * * *

2 (Not injected) * * * * * * 2 2

The principal drawback of this experiment is the joint method is to determine which condition,

that the suspected carcinogen was given to only among a selected class of conditions, is a cause of a one mouse. As a result, the experiment offers rela- phenomenon. The joint method applies rules for suf-tively weak evidence that any mouse injected with ficient conditions and necessary conditions to (one this substance would develop tumors. Perhaps the hopes) reduce the class of conditions to just one. injected mouse had some hidden defect that would Returning to the expanded experiment with the have caused tumors even without the injection. Sci- mice, let us suppose that only forty of the mice in entists address this problem by increasing the size the experimental group developed tumors, and of the experimental group and the control group. none of those in the control group did. From Mill�s Thus, suppose the same experiment were performed standpoint, this would be equivalent to forty uses on 100 mice, of which 50 were injected and 50 were of the�method of difference yielding positive results not. Suppose that after two months all 50 injected and ten yielding negative results. On the basis of mice developed tumors and none of the control such an outcome, the experimenter might conclude subjects did. Such a result would constitute much that the likelihood of�the suspected carcinogen pro-stronger evidence that the injected substance was ducing tumors in mice was 80 percent (40 � 50). carcinogenic. Controlled experiments such as the one involving

This expanded experiment can be considered to 100 mice are often conducted on humans, but with

be a case of multiple uses of Mill�s method of differ- humans it is never possible to control the circum-ence. Since the method of difference always involves stances to the degree that it is with mice. Humans just two occurrences, mouse #1 could be matched cannot be put in cages or fed exactly the same food for with mouse #51 for one use of the method, mouse #2 any length of time, they are not genetically identical with mouse #52 for a second use of the method, and to other humans, and so forth. Also, humans react to so on. Although this expanded experiment might their environment in ways that cannot be anticipated look like a case of the joint method, it is not (at least or controlled. To correct this deficiency, statistical not a successful one). Any successful application of methods are applied to the results to enable the draw-the joint method includes occurrences in which cer- ing of a conclusion. tain conditions are absent when the phenomenon For example, an experiment could be conducted is present. Those conditions are then eliminated as to test the effectiveness of some new drug on chil-possible necessary conditions. The expanded exper- dren with attention deficit/hyperactivity disor-iment with the mice includes no such occurrences. der (ADHD). Fifty children with ADHD could be

This fact points to an important difference selected, with twenty-five being placed at random in

between Mill�s joint method and his method of dif- the experimental group and twenty-five in the con-ference. The purpose of the method of difference is trol group. The children in the experimental group to determine whether a preselected condition is the would be given the drug, probably in a classroom cause of a phenomenon. This preselected condition situation, and the children in the control group is present in one of the two occurrences and absent would be given a placebo (sugar pill). This could be from the other. On the other hand, the purpose of done on a �double blind� basis, so that neither the

478Chapter10 Causality and Mill�s Methods

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). children nor the people conducting the experiment this sense, the experiment resembles the method of would know in advance who was getting the drug. concomitant variation. The objective is to control the conditions as much In some kinds of scientific research on humans, as possible. Then the negative behavior of each of controlling the circumstances to any significant the fifty children would be recorded over a period, degree is nearly impossible. Such research includes say, of an hour. Each time a child was out of his or investigations that cover a long period or investiga-her seat, disturbed others, fidgeted, failed to follow tions in which the health of the subjects is at issue instructions, and so forth, the incident would be and legal or moral restrictions come into play. noted. Example:

expected to follow what is called a normal prob-The results of such an experiment would be A nutritionist wanted to determine the effect of fifteen vitamin and mineral supplements ability distribution (bell-shaped curve). (Further on atherosclerosis (artery disease). Because discussion of normal probability distribution is the effect was expected to require several presented in Chapter 12.) One curve would repre-years to develop, the nutritionist ran an ad in sent the experimental group, another curve the con-a health publication for individuals who had trol group. Assuming the drug is effective, the two ingested these substances on a regular basis curves would be displaced from one another, but for five years. Eighty people answered the they would probably overlap in part, as indicated ad and reported which substances they had in Figure 10.1. This means that certain children taken and how much of each. These people in the experimental group exhibited more nega- were then examined. Some showed no tive behaviors than certain children in the control evidence of the disease, while others mani-group, but most of the children in the experimen- fested the disease in varying degrees. After tal group exhibited fewer negative behaviors than analyzing the data, the nutritionist concluded most of the children in the control group. Then that certain vitamins and minerals offer pro-statistical methods would be applied to the two tection against atherosclerosis.

curves to determine the effectiveness of the drug. The procedure followed by the nutritionist is very Such an experiment is similar to Mill�s method of similar to Mill�s joint method of agreement and dif-difference except that instead of the phenomenon ference. The objective is to identify a cause among (negative behavior) being wholly present in some a preselected class of possible conditions, some of occurrences, it is present in varying degrees. In which are accompanied by the phenomenon (ath-

erosclerosis), and some of which are not. The major

difference between the nutritionist�s procedure and

Figure 10.1 Mill�s joint method is that the nutritionist took

account of varying degrees in which the conditions

and the phenomenon were present in the eighty

Experimental Control occurrences. In this sense, again, the nutritionist�s

group group procedure was similar to Mill�s method of con-

comitant variation. However, the nutritionist may

also have taken into account various combinations

of conditions, a consideration that lies beyond the

power of the joint method by itself.

Number of children The procedure illustrated in this example is

not an experiment but a study. More precisely, it

is a retrospective study because it examines sub-

Number of negative behaviors jects who have already fulfilled the requirements

Section 10.3 Mill�s Methods and Science 479

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). of the examination�as opposed to a prospective then the correlation coefficient would be −1.00. study, where the subjects are expected to fulfill the This would reflect a perfect negative correlation. A requirements in the future. For a prospective study, correlation coefficient of 0.00 would mean that no the nutritionist could select a group of subjects and correlation exists, which would be the case if the follow their future vitamin and mineral intake for IQ scores corresponded randomly with the GPAs. five years. Such a study, however, would be more A�correlation coefficient of, say, +0.60 would mean costly, and if the nutritionist instructed the sub- that in general higher IQ scores corresponded posi-jects to ingest certain vitamins and minerals and tively with higher GPAs. avoid others, the study could involve legal or moral The outcome of this study can also be indicated implications. graphically, with IQ score representing one coordi-

Another kind of study found widely in the social nate on a graph, and GPA the other coordinate. The

sciences uses what is called the correlational method, result for each student would then correspond to a which closely replicates Mill�s method of concomi- point on the graph, such as the one in Figure�10.2. tant variation. Example: The graph of all the points is called a scatter diagram.

A line that �best fits� the data� can� then be drawn

A psychology professor was interested in through the scatter diagram. This line is called the the relationship between IQ scores and regression line, and it can be used to produce a linear grade point average. The professor ran- equation, y = bx + a, that describes the approximate

domly selected 100 graduating seniors, relation between IQ score (x) and GPA (y). This obtained their GPAs from the registrar, and equation can then be used to predict future GPAs. asked them to take an IQ test. All of them For example, suppose that an entering freshman complied. Then the professor compared the student has an IQ of 116. This student�s future GPA scores of each student with his or her could be predicted to be approximately 116 b + a . GPA. The professor found that, in general, Hundreds of studies such as this one have been higher IQ scores corresponded with conducted to detect correlations between such fac-higher GPAs. tors as TV violence and aggression in children,

The results of this study indicate a positive cor- self-esteem and intelligence, motivation and learn-

relation between IQ score and GPA. If the study ing time, exposure to classical music and retention showed that students with higher IQs had, in gen- ability, and marijuana use and memory. However, eral, lower GPAs, this would indicate a negative cor- such studies often fall short of establishing a causal relation. Once a study of this sort is done, the next connection between the two factors. For example, step is to compute the a positive correlation between exposure to TV vio-correlation coefficient , which is a number between lence and aggression in children does not necessarily + 1.00 and − 1.00 that expresses the degree of correlation. If it turned out that the stu-

GPA, the student with the next highest IQ score had y proportionately the next highest GPA, and so on, so dent with the highest IQ score also had the highest Figure 10.2

that the IQ�GPA graph was a straight line, then the Correlation a correlation coefficient would be + 1.00. This would + y = bx coefficient + 0.60 reflect a perfect positive correlation. Regression line A On the other hand, if it turned out that the stu-GP dent with the highest IQ score had the lowest GPA, the student with the next highest IQ score had pro-portionately the next lowest GPA, and so on, so that, x once again, the IQ�GPA graph was a straight line, IQ score

480Chapter10 Causality and Mill�s Methods

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). mean that exposure to TV violence causes aggres- unit of time. Recognizing that the downward force sion. It might be the case that children who tend on the sphere is proportional to the pitch of the to be violent are naturally attracted to violent TV plane, Galileo derived the law that the acceleration shows. Or perhaps a third factor is the cause of both. imparted to a body is directly proportional to the But a positive correlation can at least suggest a causal force acting on it. Galileo also noticed that whatever connection. the angle of the plane, the distance covered by the ball

Once a correlation is established, a controlled increased exponentially with the time in which it was

experiment can often be designed that will iden- allowed to roll. For example, if the time was doubled, tify a causal connection. For example, in the case the distance was quadrupled. From this correlation of TV violence, a group of children could be ran- he derived the law that the distance traveled by a fall-domly divided into an experimental group and a ing body is proportional to the square of the time that control group. The experimental group could then it falls. be exposed to violent TV shows for a certain period, In the seventeenth century, Robert Boyle con-and the control group would be exposed to nonvio- ducted experiments involving the pressure and lent shows for the same period. Later, the behavior volume of gases. Boyle constructed an apparatus of the children in both groups could be observed, that allowed him to compute the volume of a gas with every act of aggression noted. If the experimen- as he varied the pressure. He noted that as the pres-tal group displayed more acts of aggression than the sure increased, the volume decreased, and as the control group, the conclusion might be drawn that pressure decreased, the volume increased. From TV violence causes aggression. this correlation, he derived the law that the volume

Experimental procedures resembling Mill�s of a gas is inversely proportional to its pressure. A

method of concomitant variation have also been century later, Jacques Alexandre Charles observed a used in the physical sciences to identify causal con- correlation between the temperature of a gas and its nections. For example, in the early part of the nine- volume. As the temperature increased, the volume teenth century, Hans Christian Oerstead noticed increased, and vice versa. This correlation provided that a wire carrying a current of electricity could the basis for Charles�s law for gases. deflect a nearby compass needle. As the current was Detecting correlations has also been important in increased, the amount of deflection increased, and astronomy. Early in the twentieth century, Henrietta as the current was decreased, the amount of deflec- Swan Leavitt recognized a correlation between the tion decreased. Also, if the current were reversed, average brightness of certain variable stars, called the deflection would be reversed. Oerstead con- Cepheids (pronounced sef ʹ-ee-ids), and their cluded that a current of electricity flowing through a periods of variation. Cepheids fluctuate in bright-wire causes a magnetic field to be produced around ness over periods ranging from a day to several the wire. months, and Leavitt discovered that longer periods

Other applications of the method of concomi- correspond with greater average brightness, and

tant variation have been used not so much to detect shorter periods with lesser average brightness. Once the existence of a causal connection (which may the distance of a few nearby Cepheids was deter-have already been known) as to determine the pre- mined, the distance of any Cepheid could then be cise nature of a causal law. For example, in the latter computed from its average brightness and its period part of the sixteenth century, Galileo performed of variation. This correlation provided the first experiments involving spheres rolling down inclined method available to astronomers for measuring the planes. As the plane was incrementally lifted upward, distance between our planet and galaxies other than the sphere covered greater and greater distances in a our own.

Section 10.3 Mill�s Methods and Science 481

Exercise 10

I. Identify the kind of causality intended by the following statements. Is the cause

a sufficient condition, necessary condition, or both sufficient and necessary condition?

★1. Throwing a brick through a window causes the window to break.

2. Heating an iron rod causes it to expand.

3. Slashing an inflated automobile tire with a knife will cause it to go flat.

★4. Touching the �photo� button on a cell phone causes a fixed image to appear

on the screen.

5. Pulling the trigger of a gun causes it to fire.

6. Wetting litmus paper with an acid causes it to turn red.

★7. Pouring water on a wood fire causes it to be extinguished.

8. Eating contaminated food causes one to become ill.

9. Stretching a spring causes it to exert an opposing force.

★10. Flipping the wall switch to the �up� position causes the overhead lights to

go�on.

11. Spilling red wine on a white napkin causes a stain to appear.

12. Planting grass seed in the ground causes grass to grow.

★13. Getting proper exercise causes a person to be healthy.

14. Robbing a bank causes a person to be a criminal.

15. Getting at least five votes on an issue before the U.S. Supreme Court causes the

issue to become law.

★16. Signing a personal check causes it to be negotiable.

17. Rubbing a knife on a whetstone causes it to be sharpened.

18. Increasing the temperature of a glass of ice water causes the ice cubes to melt.

★19. Picking up a bat causes a baseball player to hit a home run.

20. Heating the gas in a ridged, closed container causes the pressure of the gas

to�rise.

II. The following problems pertain to Mill�s methods. Answer each as directed.

★1. To determine the effectiveness of an oil additive, a testing firm purchased

two cars of the same make, year, and model, and drove each a distance of 30,000�miles using the same kind of gasoline, the same kind of oil, the same driver, under the same road conditions. The oil in one engine included the additive, whereas the oil in the other engine did not. At the end of the test, the engines of both cars were dismantled, and it was found that the engine

482Chapter10 Causality and Mill�s Methods

that contained the additive had less wear. The testing firm concluded that the oil additive caused the reduced wear. Construct a table that supports this conclusion. Which one of Mill�s methods did the testing firm use? What sense of causality is involved in the conclusion?

2. An eighth-grade teacher had five students who read very well. These stu-

dents were distinguished by the following features. Ethan came from a large family, had professional parents who were not wealthy, had training in phonics, read novels, lived close to a library, and watched educational TV. Ryan came from a family that was not large, had wealthy professional parents, watched educational TV, did not live close to a library, did not read novels, and had training in phonics. Nova did not read novels, lived close to a library, had training in phonics, had wealthy professional parents, came from a large family, and did not watch educational TV. Mia had training in phonics, read novels, lived close to a library, watched educational TV, came from a large family, and had wealthy parents who were not professionals. Amir read novels, did not live close to a library, had professional parents who were not wealthy, had training in phonics, watched educational TV, and came from a large family. What can the teacher conclude caused these children to be good readers? Construct a table that supports this conclusion. Which one of Mill�s methods did the teacher use? What sense of causality is involved in the conclusion?

3. An administrator for the Internal Revenue Service noticed that tax revenues

for a certain year were down by 14 percent. Of this amount, the administrator attributed 6 percent to an economic slowdown that year, 3 percent to higher interest rates that led to higher write-offs, and 2 percent to changes in the tax code. Unable to attribute the remaining 3 percent to any lawful factor, the administrator concluded that it resulted from increased cheating by the taxpayers. Which one of Mill�s methods did the administrator use?

★4. A campus security officer is investigating a rash of recent thefts reported by

students. None of the students knew for sure how or where they were vic-timized. The officer asked the students to fill out a form listing their move-ments on the day of the theft. Vivian, whose gold bracelet was stolen, visited the lecture hall, went swimming in the gym, and visited the activities center and the bookstore. Jamar had his laptop stolen, and he visited the activities center, the library, the cafeteria, and lifted weights in the gym. Martin lost his iPhone, and he visited the library, ate in the cafeteria, went swimming in the gym, and attended class in the lecture hall. Elene had her designer jacket stolen. She ate in the cafeteria, visited the bookstore, the activities center, and took a yoga class in the gym. Shandra visited the activities center and the lecture hall. She had a $20 bill stolen from her wallet. Hank, who had his car keys stolen, visited the library, lecture hall, cafeteria, and the bookstore. He also did Pilates in the gym. Daniela, whose credit card was stolen from her wallet, visited the cafeteria and lecture hall, played cards in the activi-ties center, and swam in the pool at the gym. Shandra later amended her report to say that she did cycling in the gym, and Daniela amended hers to

Section 10.3 Mill�s Methods and Science 483

say that she went to the library. The security officer concluded that someone in the gym had access to the lockers and stole all these articles from vari-ous lockers. Construct a table that supports this conclusion. Which one of Mill�s methods did the officer use? What sense of causality is involved in the conclusion?

5. A manufacturer of fishing equipment conducted a test of its products by

giving them to eight fishermen who used them with the following results. Elon and Ann used a Hot Spot lure and caught no fish. Deon and Lily used a Trusty rod with a Husky reel, and Lily caught fish but Deon did not. Axel used a Sure Catch lure, a Best Bet rod, a Spiffy reel, and caught fish. Luis and Gael used a Husky reel, but only Luis caught fish. Ava used a Best Bet rod with a Husky reel and caught fish. Lily and Ava used a Sure Catch lure, but only Ava used monofilament line. Luis used a Trusty rod, monofilament line, and a Sure Catch lure. Ann and Gael used Best Bet rods, and both Gael and Deon used Dynamite lures. Finally, Ann used a Husky reel, Axel and Deon used monofilament line, and Elon used a Spiffy reel on a Trusty rod. What conclusion can the manufacturer draw about the power of this equipment to catch fish? Construct a table that supports this conclusion. Which one of Mill�s methods did the manufacturer use? What sense of causality is involved in the conclusion?

6. Alfonso was contemplating the purchase of a new oven for his kitchen, but he

could not decide between an electric or a gas unit. Thus, he decided to con-duct an experiment using the ovens�one gas, one electric�of two friends who live in adjoining apartments. He mixed up two identical batches of brownies under identical conditions of relative humidity and identical alti-tudes, put them into identical baking dishes, and cooked them at the same temperature for exactly the same time. The brownies cooked in the electric oven came out moist, while the brownies cooked in the gas oven came out relatively dry. Alfonso concluded that baking in the electric oven caused the brownies to be moist. Construct a table that supports this conclusion. Which one of Mill�s methods did Alfonso use? What sense of causality is involved in the conclusion?

★7. Mrs. Wilkins sometimes has trouble sleeping. In order to determine the cause,

she decided to take note of her pre-bedtime behavior over the course of a week. On Monday she drank chamomile tea, had a late dinner, took a hot bath, read from a book, took a walk, and slept well. On Tuesday she had a late dinner, read from a book, and slept poorly. On Wednesday she drank a glass of wine, got a massage, took a walk, and slept poorly. On Thursday she had a late dinner, drank a glass of wine, read from a book, took a hot bath, got a massage, and slept well. On Friday she read from a book, drank chamomile tea, got a massage, had a late dinner, took a walk, and slept well. On Saturday she drank a glass of wine, got a massage, took a walk, drank chamomile tea, took a hot bath, and slept well. On Sunday she read from a book, took a hot bath, drank chamomile tea, and slept poorly. What can Mrs. Wilkins conclude is the cause of sleeping well? Construct a table that supports this conclusion.

484Chapter10 Causality and Mill�s Methods

Which one of Mill�s methods did Mrs. Wilkins use? What sense of causality is involved in the conclusion?

8. From a comparison of statistics a criminologist detected what she thought was

a correlation between fluctuations in the employment rate and in crimes of theft. For every 2 percent increase in the employment rate, the rate of theft decreased by 1 percent, and for every 2 percent decrease in the employment rate, the theft rate increased by 1 percent. The criminologist concluded that unemployment causes crimes of theft. Which one of Mill�s methods did the criminologist use?

9. A man developed an allergic reaction to an unknown food. His doctor asked

him about foods that often cause allergic reactions, and the man replied that he had eaten coconut, chocolate, nuts, milk products, shellfish, peppers, eggs, and wheat products prior to suffering the reaction. The doctor told him to eliminate all of these foods from his diet; when he had done so, the reaction disappeared. The doctor then told him to introduce each of these foods back into his diet, one at a time. The man did so, and the reaction reappeared only when he ate milk products. The doctor concluded that milk products caused the allergic reaction. Construct a table that supports this conclusion. Which one of Mill�s methods did the doctor use? What sense of causality is involved in the conclusion?

★10. A psychiatrist had six adult women patients who suffered blurred ego bound-

aries. These patients were distinguished by the following characteristics. Meg and Sue had been subjected to corporal punishment as children and both had siblings. Dot and Jane were adopted, had a male parent figure, and experienced sexual abuse. Lynn and Flo had siblings and a male parent figure, but only Flo had a domineering mother. Also, Lynn and Meg had experienced sexual abuse and were adopted, but only Meg had a domineering mother. Jane and Dot were uprooted often as children. Flo and Sue had experienced sexual abuse, but only Sue was raised in day-care centers. Jane and Dot each had a domi-neering mother, and each had been subjected to corporal punishment. Flo was uprooted often as a child and had been subjected to corporal punishment. Dot and Lynn were raised in day-care centers, Sue had a male parent figure, and Jane had siblings. What can the psychiatrist conclude is the cause of the blurred ego boundaries of these women? Construct a table that supports this conclusion. Which one of Mill�s methods did the psychiatrist use? What sense of causality is involved in the conclusion?

11. A metallurgist added eight substances in various combinations to seven

samples of molten aluminum for the purpose of producing aluminum alloys. The metallurgist was interested in producing an alloy that had a special resistance to corrosion. The samples were distinguished by the following characteristics. Samples 1, 2, and 3 contained both tin and copper, but only samples 1 and 2 were resistant. Samples 6 and 7 contained both silver and iron, but only sample 7 was resistant. Samples 4 and 5 contained sodium, but only sample 5 was resistant. In addition, samples 1 and 7 contained zinc, silicon, and nickel. Sample 2 contained sodium, iron, and silicon, and

Section 10.3 Mill�s Methods and Science 485

sample 5 contained silicon and copper. Finally, sample 3 contained nickel, and sample 4 contained zinc. Assuming no interaction occurs among these eight additives, what conclusion can the metallurgist draw about these addi-tives? Construct a table that supports this conclusion. Which one of Mill�s methods did the metallurgist use? What sense of causality is involved in the conclusion?

12. A doctor has five patients who suffer from an unusual form of cancer. The

patients are distinguished by the following living conditions. Davis, Jones, and Ellis live in a smoggy area near high-voltage power lines, and Smith and Frank smoke cigarettes and live downwind from a company that produces chemi-cal defoliants for the military. Frank also lives near the nuclear power plant. Davis, Smith, and Ellis eat red meat every day and live near the nuclear power plant. Jones smokes cigarettes and lives downwind from the chemical defoli-ant company. Smith lives in a smoggy area, and Davis and Ellis live down-wind from the defoliant company. What can the doctor conclude is the cause of the cancer? Construct a table that supports this conclusion. Which one of Mill�s methods did the doctor use? What sense of causality is involved in the conclusion?

★13. Two of Santiago�s rose bushes became infected with aphids. Santiago pro-

ceeded to spray one of the bushes with soapy water but left the other bush untouched. The two bushes are both American Beauty, are five years old, have virtually identical locations, receive the same amount of water and sunlight, are planted in the same kind of soil, and receive the same degree of cultivation and the same amount of bone meal rose food. Within three days the aphids disappeared from the bush that was sprayed, but they continued to thrive on the other bush. Santiago concluded that the soapy water killed the aphids on the bush that was sprayed. Construct a table that supports Santiago�s conclusion. Which one of Mill�s methods did Santiago use? What sense of causality is involved in the conclusion?

14. A sociologist conducted a study to determine if a correlation exists between

grade point average at the time of graduation and income ten years after graduation. The sociologist visited the alumni office of a local university and obtained the names and addresses of those who had graduated ten years earlier. The sociologist then contacted those graduates and asked them to disclose their income and give permission for the university to disclose their graduating GPAs. Two hundred students replied to this inquiry. The sociolo-gist then applied statistical methods to compute the correlation coefficient, which turned out to be +0.2. How does this study relate to Mill�s methods? What does a correlation coefficient of +0.2 say about the correlation between GPA and income? Is such a study called a prospective study or a retrospective study? How do the two differ?

15. A cosmetics manufacturer tested a new cleansing cream on forty mice. The

mice were virtually the same genetically and were the same age. Twenty of the mice were randomly placed in the experimental group and the other twenty were placed in the control group. A patch containing the cream

486Chapter10 Causality and Mill�s Methods

was applied to the skin of each mouse in the experimental group, and an identical patch containing a harmless substance was applied to the skin of each mouse in the control group. Then, all forty mice were kept in the same room, at the same temperature, and under the same lighting conditions and were fed the same food. After five days, two of the mice in the experimental group developed a rash where the patch was applied, but none of the other mice did. The manufacturer concluded that there is a 10 percent likelihood that the cleansing cream will cause a rash on the skin of mice. How does this experiment relate to Mill�s methods? What sense of causality is involved in the conclusion?

III. Identify the cause suggested by the information presented in the following tables. Is the

cause a sufficient condition, necessary condition, or both sufficient and necessary con-dition? Can you identify the method?

★ 1.

Possible Conditions

OccurrenceABCDEPhenomenon

1 * 2 * * * *

2 * * * * 2 *

3 * 2 * * * *

4 * * 2 * * *

5 2 * * * 2 *

Possible Conditions

OccurrenceABCDEPhenomenon

1 * * * * * *

2 * 2 * * * 2

Possible Conditions

OccurrenceABCDEFPhenomenon

1 * 2 2 * 2 2 2

2 2 * * * 2 2 *

3 * 2 * * * 2 *

4 2 * 2 2 2 * 2

5 * 2 2 2 * 2 2

6 * * * 2 * 2 *

Section 10.3 Mill�s Methods and Science 487

★4.

Possible Conditions

OccurrenceABCDEPhenomenon

1 * * * * 2 *

2 2 2 * 2 * *

3 * 2 2 2 * 2

4 2 * 2 * * *

5 * * 2 2 2 2

6 2 * * 2 2 2

Possible Conditions

OccurrenceABCDEPhenomenon

1 2 2 * 2 * 2

2 2 * * 2 * *

3 * * 2 * * *

4 2 * 2 2 * 2

5 * 2 * * 2 2

6 * * * 2 2 *

IV. Identify three causal connections that are not mentioned in this book: One must be a

sufficient condition, one must be a necessary condition, one must be a sufficient and necessary condition. Then explain why each causal connection fits the classification you have given it.

Summary

Three Kinds of Cause:

■ Sufficient condition

■ Necessary condition

■ Sufficient and necessary condition

Mill�s Methods:

■ Method of agreement: Identifies necessary conditions.

■ Method of difference: Identifies sufficient conditions.

■ Joint method of agreement and difference: Identifies sufficient and necessary conditions.

488Chapter10 Causality and Mill�s Methods

■ Method of residues: Identifies generic causal connections.

■ Method of concomitant variation: Identifies generic causal connections.

Mill�s Methods and Science:

■ The methods of agreement, difference, and concomitant variation resemble the methods used

by scientists to identify causal connections.

■ Controlled experiment and the method of difference.

■ Experiment versus study (retrospective, prospective).

■ Correlation versus causal connection.

■ Scientists: Oerstead, Galileo, Boyle, Charles, Leavitt.

Summary489

11Probability

11.1 Theories of Probability

11.2 The Probability Calculus

11.1 Theories of Probability

Knowing about mathematical probability is empowering. For example, suppose you are considering getting vaccinated. Every vaccine, including this one, has side effects, and the probability of suffering a side effect is expressed mathematically. The only way of contracting the disease is by close contact with others who have it, and the probability of this happening is expressed mathemati-cally. You can use these figures to reach an informed decision on whether to get the vaccine.

Probability is a topic that is central to the question of induction, but like causal-ity, it has different meanings. Consider the following statements:

The probability of picking a spade from a standard deck of cards is one-fourth.

The probability that a twenty-year-old man will live to age seventy-five is .63.

There is a high probability that Margaret and Sonia will get married.

In each statement the word �probability� is used in a different sense. This dif-ference stems from the fact that a different procedure is used in each case to determine or estimate the probability. To determine the probability of picking a spade from a standard deck of cards, a purely mathematical procedure is used.

490

Given that there are fifty-two cards in a deck and on a single roll of a die is 1 to 5 because there is only thirteen are spades, thirteen is divided by fifty-two one 1 and 5 outcomes that are not ones. Conversely, the to obtain one-fourth. A different procedure is used odds of not rolling a one on a single roll is 5 to 1. Given to determine the probability that a twenty-year-old the acceptance of these odds, if you were to place a $1 man will live to age seventy-five. For this, one must bet on rolling a one, and you won the bet, you would sample a large number of twenty-year-old men and collect $5. If you were to place a $5 bet on not rolling a count the number that live fifty-five more years. Yet one, and you won the bet, you would collect $1. a different procedure is used to determine the prob- For an example involving cards, the odds of draw-ability that Margaret and Peter will get married. This ing a red queen on a single draw is 2 to 50 (or 1 to 25) probability can only be estimated roughly, and doing because there are two red queens and fifty cards that so requires that we become acquainted with Marga- are not red queens. Given the acceptance of these ret and Peter and with how they feel toward each odds, if you were to place a $2 bet that you would other and toward marriage. These three procedures draw a red queen on a single draw, and you won the give rise to three distinct theories about probability: bet, you would collect $50. If you were to place a $50 the classical theory, the relative frequency theory, bet that you would not draw a red queen, and you and the subjectivist theory. won the bet, you would collect $2.

The classical theory of probability traces For an example involving horse racing, suppose

its origin to the work of the seventeenth-century that five horses are running a race; three are yours, mathematicians Blaise Pascal and Pierre de Fermat two are your friend�s, and there is an equal chance in determining the betting odds for a game of of any of the horses winning. The odds that one of chance. The theory is otherwise called the a priori your horses will win is 3 to 2, and the odds that one theory of probability because the computations are of your friend�s horses wins is 2 to 3. made independently of any sensory observation of Given that you and your friend accept these odds, actual events. According to the classical theory, the if you bet $3 that one of your horses wins, and you win probability of an event A is given by the formula the bet, your friend must pay you $2. On the other

hand, if your friend bets $2 that one of her horses

P(A) 5 wins, and she wins the bet, then you must pay her $3. f_n If we are given the odds of an event, we can derive

where f is the number of favorable outcomes and n the probability of that event, and if we are given the is the number of possible outcomes. For example, in probability of an event, we can derive the odds of that computing the probability of drawing an ace from a event. For example, if the odds of an event are 1 to 3, the standard deck of cards, the number of favorable out- number of favorable outcomes is 1 and the number of comes is four (because there are four aces) and the unfavorable outcomes is 3. Thus, the number of possi-number of possible outcomes is fifty-two (because ble outcomes is 1 1 3, or 4, so the probability is � or .25. there are fifty-two cards in the deck). Thus, the Conversely, if we are given a probability of .25, we probability of that event is 4/52 or 1/13 (or .077). can derive the odds. First, we convert .25 to a frac-

It is important not to confuse the probability of tion (�). Next, we find the number of unfavorable

an event�s happening with the odds of its happening. outcomes by subtracting the number of favorable For events governed by the classical theory, the odds outcomes (1) from the number of possible outcomes

that an event A will happen is given by the formula (4), giving 3. So, the odds of the event are 1 to 3. odds(A) Sometimes in converting a decimal to a fraction it 5 f : u is best to approximate. For example, if we are given a

where probability of .34, it may be best to convert this figure f is the number of favorable outcomes, and u is the number of unfavorable outcomes. The expression to .33, which is approximately ⅓. Then, the number f:u of favorable outcomes is 1, the number of possible is read f to u . For example, the odds of rolling a one

Section 11.1 Theories of Probability 491

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). outcomes is 3, so the number of unfavorable outcomes outcomes may not be possible in the practical sense, is 3 2 1, or 2. Thus, the odds of the event is 1 to 2. but they are logically possible in that they do not

Two assumptions are involved in computing prob- involve any contradiction. Because these outcomes

abilities and odds according to the classical theory: (1) are so unusual, however, it is reasonable to think that that all possible outcomes are taken into account and for all practical purposes the two assumptions hold (2) that all possible outcomes are equally probable. and that therefore the classical theory is applicable. In the card example, the first assumption entails that There are many events, however, for which the two only the fifty-two ordinary outcomes are possible. In assumptions required by the classical theory obviously other words, it is assumed that the deck has not been do not hold. For example, in attempting to determine altered, that the cards will not suddenly self-destruct or the probability of a sixty-year-old woman dying of reproduce, and so forth. In the racing example, the first a heart attack within ten years, it would be virtually assumption entails that no other horses are running in impossible to take account of all the possible outcomes. the race, and that none of the horses will simply vanish. She might die of cancer, pneumonia, or an especially

The second assumption, which is otherwise virulent case of the flu. She might be incapacitated

called the principle of indifference, entails for by a car accident, or she might move to Florida and the card example that there is an equal likelihood of buy a house on the beach. Furthermore, none of these selecting any card. In other words, it is assumed that outcomes is equally probable in comparison with the the cards are stacked evenly, that no two are glued others. To compute the probability of events such as together, and so on. For the horse-race example, the these we need the relative frequency theory. second principle entails that each of the horses has The relative frequency theory of probability an equal chance of winning. originated with the use of mortality tables by life

Whenever these two assumptions can be made insurance companies in the eighteenth century. In

about the occurrence of an event, the classical theory contrast with the classical theory, which rests upon can be used to compute its probability or the odds of a priori computations, the relative frequency theory its happening. Here are some additional examples: depends on actual observations of the frequency

with which certain events happen. The probability

P (a fair coin turning up odds 5 1:1 of an event A is given by the formula heads) 5 1/2

P (drawing a face card) odds 5 12:40 = 3:10 f_P(A) 5 5 12/52 5 3/13 n�

P (a single die coming odds 5 1:5 where f is the number of observed favorable out-up �3�) 5 o 1/6 comes and n is the total number of observed out-P (a single die coming o odds 5 1:1 comes. For example, to determine the probability up even) 5 3/6 5 1/2 that a fifty-year-old man will live five more years, Strictly speaking, of course, the two assump- a sample of 1,000 fifty-year-old men could be

tions underlying the classical theory are�never per- observed. If 968 were alive five years later, the proba-fectly reflected in any actual situation. Every coin bility that a fifty-year-old man will live an additional is slightly off balance, as is every pair of dice. As a five years is 968/1000 or .968. result, the probabilities of the various outcomes Similarly, if one wanted to determine the prob-are never exactly equal. Similarly, the outcomes are ability that a certain irregularly shaped pyramid with never strictly confined to the normal ones entailed different-colored sides would, when rolled, come to by the first assumption. When tossing a coin, there rest with the green side down, the pyramid could be is always the possibility that the coin will land on rolled 1,000 times. If it came to rest with its green edge, and in rolling dice there is the analogous pos- side down 327 times, the probability of this event sibility that one of them might break in half. These happening would be computed to be .327.

492Chapter11 Probability

The relative frequency method can also be used attic of a deceased great-grandparent and you happen

to compute the probability of the kinds of events to come across an old violin. When you peer through that conform to the requirements of the classical one of the openings in the face of the violin you read theory. For example, the probability of a coin coming �Strad 1691.� You conclude with glee that the instru-up heads could be determined by tossing the coin ment is probably a Stradivarius. Then you do some 100�times and counting the heads. If, after this many investigating and find that �Strad 1691� means noth-tosses, 46 heads have been recorded, one might assign ing at all, so you conclude that the violin is probably a probability of .46 to this event. This leads us to an not a Stradivarius. After further research you discover important point about the relative frequency theory: that a true Stradivarius will have a circular logo bear-The results hold true only in the long run. It might be ing the initials of Antonio Stradivari beneath a cross. necessary to toss the coin 1,000 or even 10,000 times When you look through the other opening, you see to get a close approximation. After 10,000 tosses one �A+S� inside a circle. So, the violin is probably a Strad would expect to count close to 5,000 heads. If in fact after all. Then you check with an expert who tells you only 4,623� heads have been recorded, one would that the logo in a genuine Stradivarius has the initials probably be justified in concluding that the coin is off inside a double circle, but yours is set within a single balance or that something was irregular in the way it circle. In the end, you conclude that the violin you had been tossed. found is probably not a genuine Strad.

Strictly speaking, neither the classical method nor As for how different people can assign different

the relative frequency method can assign a probabil- probabilities to the same thing, consider the stock ity to individual events. From the standpoint of these market. Every trade involves a buyer and a seller. approaches only certain kinds or classes of events have The buyer believes at a certain moment in time that probabilities. But many events in the actual world are a certain stock is probably a good investment, but unique�for example, Margaret�s marrying Sonia or the seller believes at the same moment that the same Native Prancer�s winning the fourth race at Churchill stock is probably not a good investment. If the buyer Downs. To interpret the probability of these events we and seller did not assign different probabilities, there turn to the subjectivist theory. would be no trade. In fact, if looked at in this way,

The subjectivist theory of probability rests on the entire economy (and practically every deci-

the degrees of belief that people have about the occur- sion in life) rests on people assigning probabilities rence of some event or the truth of some proposition. according to the subjectivist theory. You buy a cer-Such beliefs are grounded in the knowledge we have tain kind of laundry soap or can of tuna because you about the world in which we live. Because this knowl- believe that the product is probably a good value, or edge is incomplete and fluctuates from day to day as probably more environmentally friendly, or prob-we acquire new knowledge and forget the knowledge ably more healthy than some alternate brand. we once had, the probabilities we assign to things are The subjectivist theory of probability differs in subject to change. Also, since the knowledge acquired important ways from the classical theory and the rel-by one person differs from the knowledge acquired by ative frequency theory. Whereas the classical theory another (one person might know a lot about econom- assigns probabilities independently of the actual ics, another a lot about football), the probability that state of the world, the subjectivist theory requires one person assigns to something will often differ from empirical knowledge. In fact, the more extensive the probability that another assigns to it. The subjec- the knowledge, the more accurate the probability tivist theory is sometimes called the �epistemic theory assigned. The relative frequency theory deals with of probability,� after the Greek word episteme, which similar events that are repeatable in some relevant means knowledge. way. The great majority of events that the subjec-

For an example of how a probability can change tivist theory addresses are not repeatable. Also, the

from day to day, suppose you are rummaging in the first two theories yield probabilities that are highly

Section 11.1 Theories of Probability 493

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). precise, whereas the probabilities given by the sub- stock is probably worth less. Extending this idea, the jectivist theory are often imprecise. Yet, the subjec- probability that any product or service is worth a cer-tivist theory is vastly more applicable than either of tain amount of money is arguably settled once that the other theories. Only the subjectivist theory can product or service is offered for sale in a free market. assign a probability to the claim that a certain person The idea that probabilities can rest on people�s would make a good spouse, or the assertion that beliefs requires a proviso: The beliefs must be ratio-someone committed a crime, or the claim that one nal (or �coherent,� according to some accounts). But football team will defeat another. what is the meaning of rational? For an example of

The lack of precision affecting the subjectivist a belief that is not rational, suppose that a friend is

theory can sometimes be mitigated by averaging the willing to give you 7 to 4 odds that Ajax will win in probabilities that a large number of people assign the fifth race. This corresponds to a probability of 7/ to a given event. This is exactly what happens when (7 + 4) or 7/11. At the same time your friend will people make bets on the outcome of a sporting event. give you 5 to 4 odds that Ajax will lose in that race, For example, in horse racing, as the bets are placed, which corresponds to a probability of 5/9. Obviously the odds of each horse winning are quickly com- your friend believes that Ajax will both win and lose puted. From these odds we can calculate what might the same race. Such a belief is not rational. Given the be called the �true� probability that each horse will 7 to 4 odds that Ajax will win, the best odds your win. Similarly, the claim that the stock in some ini- friend can give that Ajax will lose is 4/7. It has been tial public offering is probably worth $10 per share proved that any assignment of probabilities that sat-is quickly settled once the stock begins trading. If isfies the rules of the probability calculus is rational. no investors are willing to pay that amount, then the

11.2 The Probability Calculus

The three theories discussed thus far�the classi- events have been given a probability, the ground-cal theory, the relative frequency theory, and the work has been laid for computing the probabili-subjectivist theory�provide separate procedures ties of compound arrangements of events. This is for assigning a probability to an event (or class done by means of what is called the probability of events). Sometimes one theory is more readily calculus. In this respect, the probability calculus applicable, sometimes another. But once individual functions analogously to the set of truth-functional

Classical theory

To determine

the probability Relative Probability Probability of

of simple frequency theory calculus compound events

events, use:

Subjectivist theory

494Chapter11 Probability

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). rules in propositional logic. Just as the truth- For example, the probability of tossing two heads on

values of compound propositions from the indi- 5 P(H and H ) 1/2 3 1/2 5 1/4 vidual truth values of the simple components, the functional rules allow us to compute the truth a single throw of two coins is 1 2

rules of the probability calculus allow us to com- This result may be checked very easily by listing all pute the probability of compound events from the the possible outcomes and comparing that number individual probabilities of the simple events. with the number of favorable outcomes: Two preliminary rules of the probability calcu-lus are (1) the probability of an event that must nec- Coin 1Coin 2

ability of an event that necessarily cannot happen is H T taken to be 0. For example, the event consisting of it essarily happen is taken to be 1, and (2) the prob- H H either raining or not raining (at the same time and T H place) has a probability of 1, and the event consist- T T ing of it both raining and not raining (at the same

time and place) has a probability of 0. These events Only one of the four possible outcomes shows both correspond to statements that are tautological and coins turning up heads. self- contradictory, respectively. Contingent events, Similarly, we may compute the probability of roll-on the other hand, have probabilities greater than ing two sixes with a pair of dice:

0 but less than 1. For example, the probability that P(S and S ) 5 1/6 3 1/6 5 1/36 1 the Dow Jones Industrial Average will end a certain 2 week at least five points higher than the previous Again, we may check the results by listing all the week would usually be around 1/2, the probabil- possible outcomes: ity that the polar ice cap will melt away next year

is very close to 0, and the probability that a traffic 1-1 2-1 3-1 4-1 5-1 6-1

close to 1. Let us now consider six additional rules 1-3 2-3 3-3 4-3 5-3 6-3 of the probability calculus. accident will occur somewhere tomorrow is very 1-2 2-2 3-2 4-2 5-2 6-2

1-4 2-4 3-4 4-4 5-4 6-4

1-5 2-5 3-5 4-5 5-5 6-5

1. Restricted Conjunction Rule 1-6 2-6 3-6 4-6 5-6 6-6

The restricted conjunction rule is used to com- Since only one of the thirty-six possible outcomes pute the probability of two events occurring together shows two sixes together, the probability of this when the events are independent of each other. Two event is 1/36. events are said to be independent when the occur-rence of one has no effect on the probability of

is replaced before the second is drawn, and playing the probability of two events occurring together whether or not the events are independent. When two sequential games of roulette. The probability of the events are independent, the general conjunc-two such events A and B occurring together is given tion rule reduces to the restricted conjunction rule. by the formula Some examples of events that are not independent two heads from two tosses of a coin, drawing two The general conjunction rule is used to compute hearts from a standard deck of cards when the first the other one occurring. Examples include getting 2. General Conjunction Rule

P(A and B) = P(A) � P(B) (that is, that are dependent) are drawing two spades

Section 11.2 The Probability Calculus 495

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). first card is drawn, the number of cards available for pute the probability of either of two events occur-ring when the events are mutually exclusive �that the second draw is reduced, and after one of the seats is, when they cannot both occur. Examples of such is taken on the plane, the number of seats remaining events include picking either an ace or a king from for subsequent choices is reduced. In other words, a standard deck of cards on a single draw or rolling the first card drawn is not replaced, and selecting The restricted disjunction rule is used to com-two or more window seats on an airplane. After the from a standard deck of cards on two draws when 3. Restricted Disjunction Rule

in both cases the second event is dependent on the either a six or a one on a single roll of a die. The first. The formula for computing the probability of probability is given by the formula two such events occurring together is

P(A or B) = P(A) + P(B)

P(A and B) 5 P(A) 3 P(B given A)

For example, the probability of drawing either a

The expression P(B given A) is the probability that king or a queen (of any suit) from a standard deck of B will occur on the assumption that A has already cards on a single draw is occurred. Let us suppose, for example, that A

from a standard deck of cards when the first card For another example, consider an urn containing six is not replaced before the second is drawn. If and P(K or Q) = 4/52 + 4/52 = 8/52 = 2/13 B designate the events of drawing two kings

event A occurs, then only three kings remain, and probability of selecting either a black or red ball on black balls, four white balls, and two red balls. The

the deck is also reduced to fifty-one cards. Thus, a single draw is P ( B given A ) is 3/51. Since the probability of event A is 4/52, the probability of both events happening P(B or R) = 6/12 + 2/12 = 8/12 = 2/3

is the product of these two fractions, or 12/2652 nec-When the event in question is one that must ( 5 1/221). essarily occur, the probability is, of course, 1. Thus, For another illustration, consider an urn contain-the probability of obtaining either heads or tails on a ing five red balls, six green balls, and seven yellow single toss of a coin is balls. The probability of drawing two red balls (with-out replacement) is computed as follows: P(H or T ) = 1/2 + 1/2 = 1

P(R and R ) 5 5/18 3 4/17 5 20/306 5 10/153 The restricted disjunction rule may be combined 1 2

with the restricted conjunction rule to compute the

If a red ball is selected on the first draw, this leaves probability of getting either a five or a six on each of four red balls from a total of seventeen. Thus, the two consecutive rolls of a single die:

already been drawn is 4/17. P[(F or S) and (F or S) ] = P(F or S) � P(F or S) 1 2 12 = probability of drawing a second red ball if one has

urn with the same contents, but let us For another example, consider the same (1/6 + 1/6) � (1/6 + 1/6) = 1/3 � 1/3 = 1/9 compute the probability of drawing first a green ball and then a yellow ball (without replacement): Since getting a five and getting a six on a

single die are mutually exclusive events,

If a green ball is selected on the first draw, this affects junction rule. The same is true of P(F or S) . 2 Then, since two rolls of a die are independent events, P(G and Y ) = 6/18 � 7/17 = 42/306 = 7/51 P(F� or S) 1 is evaluated using the restricted dis-

only to the extent of reducing the total number of the conjunction of the two disjunctive events is eval-the selection of a yellow ball on the second draw balls to seventeen. uated by the restricted conjunction rule.

496Chapter11 Probability

The ting at least one six when rolling a pair of dice. The general disjunction rule is used to compute computation is 4. General Disjunction Rule For another example, consider the probability of get-

not they are mutually exclusive. The rule holds for P(S or S ) = 1/6 + 1/6 − (1/6 � 1/6) 1 2 any two events, but since its application is simpli-the probability of either of two events whether or

= 2/6 − 1/36

fied when the events are independent, we will con- = 11/36 fine our attention to events of this kind. Examples of

independent events that are not mutually exclusive The general disjunction rule may be combined include obtaining at least one head on two tosses of with the restricted disjunction rule to compute the a coin, drawing at least one king from a standard probability of getting either a three or a five when deck of cards on two draws when the first card is rolling a pair of dice. This is the probability of get-replaced before the second card is drawn, and get- ting either a three or a five on the first die or either ting at least one six when rolling a pair of dice. The a three or a five on the second:

two such events is P [(T or F ) or (T or F ) ] 5 P(T or F ) + P(T or F ) − 1 2 1 2 [P(T or F ) formula for computing the probability of either of

1 � P(T or F ) ] 2

P(A or B) = P(A) + P(B) − P(A and B) = (1/6 + 1/6) + (1/6 +

puted using the restricted conjunction rule, and the (1/6 + 1/6)] = 2/6 + 2/6 − 4/36 general disjunction formula reduces to If the events are independent, 1/6) − [(1/6 + 1/6) � P ( A and B ) is com-

= 20/36

P(A or B) = P(A) + P(B) − [P(A) � P(B)] = 5/9

follows. When The general disjunction rule may be proved as Since getting a three or getting a five on a single throw A and B are nonexclusive, A occurs are mutually exclusive events, P ( T or F ) is equal to the either with or without B , and B occurs either with or 1 sum of the separate probabilities. The same is true for without A . Thus, P ( T or� F ) . 2 = P(A) P(A and B) + P(A and not-B) The general disjunction rule may be combined

P(B) = P(B and A) + P(B and not-A) with the general conjunction rule to compute

Thus, But the probability of drawing first a red ball and A or B occurs in exactly three possible then a black ball on pairs of draws from either of ways: A and not-B , B and not-A , and A and B . two urns (without replacement). Suppose that the first urn contains two red balls, two black = P(A or B) P(A and not-B) + P(B and not-A) balls, and one green ball, and that the second urn + P(A and B) contains three red balls, one black ball, and one

Thus, when white ball. The probability, giving two draws per P ( A and B ) is subtracted from

P(A or B). [Note: P(A and B) = P(B and A).] P[(R and B) or (R and B) ] For an example of the use of the general dis-1 2 = P urn, is ( A ) + P ( B ), the difference is equal to

junction rule, let us consider the probability of P(R and B) + P(R and B) − [P(R and B) 1 21 � P(R and B) ] getting heads on either of two tosses of a coin. We 2 = (2/5 � 2/4) + (3/5 � 1/4) − [(2/5 � 2/4) have � (3/5 � 1/4)]

P(H or H ) = 1/2 + 1/2 − (1/2 � 1/2) = 4/20 + 3/20 − (4/20 � 3/20) 1 2

= 1 − 1/4 = 7/20 − 12/400 = 3/4 = 8/25

Section 11.2 The Probability Calculus 497

The plex, consider an urn containing two white, two negation rule is useful for computing the black, and two red balls. To compute the probability 5. Negation Rule For an example that is only slightly more com-

probability of an event when the probability of the of getting either a white or black ball on two draws event not happening is either known or easily com-(without replacement) we first compute the prob-puted. The formula is as follows: ability of the event not happening. This is the prob-P(A) = 1 − P(not-A) ability of getting red balls on both draws, which is

restricted disjunction rule the probability of A or Now, by the negation rule the probability of drawing not-The formula can be proved very easily. By the P(R and R ) = 2/6 � 1/5 = 2/30 = 1/15 1 2

A is either a white or black ball is

P(A or not-A) = P(A) + P(not-A) P(W or B) = 1 − 1/15

But since = A or not-A happens necessarily, P ( A or 14/15

not-A) = 1. Thus, 6. Bayes�s Theorem

1 = P(A) + P(not-A) Bayes�s theorem , named after the eighteenth-

Rearranging the terms in this equation gives us the century English clergyman Thomas Bayes, is a negation rule. For an example of the use of this rule, useful rule for evaluating the conditional probabil-consider the probability of getting heads at least ity of two or more mutually exclusive and jointly once on two tosses of a coin. The probability of the exhaustive events. The conditional probability of event not happening, which is the probability of get- an event is the probability of that event happening ting tails on both tosses, is immediately computed given that another event has already happened, and by the restricted conjunction rule to be 1/4. Then, it is expressed P(A given B). You may recall that an

applying the negation rule, we have expression of this form occurred in the formulation P(H of the general conjunction rule. When the number or H ) = 1 − 1/4 1 2 of mutually exclusive and jointly exhaustive events

= 3/4 is limited to two, which we will designate A and A ,

suppose we are given an urn containing two black independent events. Let us [P(A ) � P(B given A )] 1 [P(A ) � P(B given A )] 1 1 2 2 balls and three white balls. To compute the prob-This limited formulation of Bayes�s theorem may ability of getting at least one black ball on two draws be proved as follows. Applying the general conjunc-(without replacement), we first compute the prob-tion rule to the events A and B , we have 1 ability of the event not happening. This event con-P(A and B) = P(A ) � P(B given A ) sists in drawing two white balls, which, by the gen-the probabilities of disjunctive events that are depen-P(A given B) = 1 dent . In presenting the general disjunction rule we P(A ) � P(B given A ) 1 1 confined our attention to The negation rule may also be used to compute 1 2 Bayes�s theorem is expressed as follows: 1 1 1

eral conjunction rule, has the probability Applying the same rule to the same events written in

the reverse order, we have

P(W and W ) = 3/5 � 2/4 = 6/20 1 2

Now, applying the negation rule, the probability P(B and A ) = P(B) � P(A given B) 1 1

of getting at least one black ball on two draws is Now, since P(ABP 1 and ) is equal to (B and A1), we P(B may set the right-hand side of these two equations or B ) = 1 − 6/20 1 2 equal to each other:

= 14/20

= 7/10 P(B) � P(A given B) = P(A ) � P(B given A ) 1 1 1

498Chapter11 Probability

Dividing both sides of this equation by P(B), we have Now, since there are a total of five urns, the prob-

B Now, it can be easily proved using a truth table that P(A ability of randomly selecting the beige urn is 1/5, given B) = P(A ) � P(B given A ) (*) 1 1 1 and the probability of randomly selecting one of P(B) the gray urns is 4/5. Finally, since each urn con-tains ten balls, the probability of drawing a red ball B is logically equivalent to the expression [( A and 1 from the beige urn is 8/10, and the probability of ) or (not-A and B )]. Furthermore, since A and A 1 1 2 drawing a red ball from the gray urn is 3/10. Thus, are mutually exclusive and jointly exhaustive, one we have and only one of them will happen. Thus, A 2 is simply 1/5 � another way of writing not-A . Accordingly, the 8/10 1 P(B given R) = former expression in brackets can be written [( A and (1/5 � 8/10) 1 (4/5 � 3/10) 1 B ) or ( A and B )], and since this expression is logically 2 8/50 8/50 5 5 5 8/20 = .40 equivalent to B , 8/50 1 12/50 20/50

P(B) = P[(A and B) or (A and B)] 1 2

restricted disjunction rule to the right-hand side of events as new information is acquired. For an exam-this equation, we have ple of how Bayes�s theorem is used for this purpose, consider the following example: Now, applying the general conjunction rule and the us to change our probability estimates of ordinary Bayes�s theorem is quite practical because it allows

P(B) = [P(A ) � P(B given A )] + [P(A ) 1 1 2

� P(B given A )] 2 Mr. Jones became seriously ill from ingesting

Finally, substituting the right-hand side of this equa- defectively manufactured tablets of the pain-tion in place of P(B) in the earlier equation marked killer novaprine. He intends to file a lawsuit (*), we have Bayes�s theorem: against the most probable manufacturer of

P(A given B) = 1 the tablets. After a brief investigation, he [P(AP(A finds that 70 percent of the novaprine sold in ) � P(B given A ) 1 1 this country is manufactured by Alpha Phar-)

1 � P(B given A )] 1 [P(A ) � P(B given A )] 1 2 2 maceuticals, and 30 percent is manufactured

For an illustration of the use of Bayes�s theorem, by Beta Pharmaceuticals.

suppose we are given one beige urn and four gray From this information the probability that Alpha urns in a dimly lit room, so we cannot distinguish Pharmaceuticals manufactured the tablets is .70, one color from the other. The beige urn contains in comparison with .30 for Beta Pharmaceuticals. eight red and two white balls, and each of the gray Thus, Mr. Jones tentatively decides to file his law-urns contains three red and seven white balls. Sup- suit against Alpha. However, on further investiga-pose that a red ball is drawn from one of these five tion Mr. Jones discovers the following:

urns. What is the probability that the urn was beige? Of the novaprine manufactured by Alpha, In other words, we want to know the probability that 10 percent is sold on the East Coast and the urn was beige given that a red ball was drawn. 90 percent on the West Coast, while 80 per-The events A and A in Bayes�s theorem correspond 1 2 cent of Beta�s novaprine is sold on the East to drawing a ball from either a beige urn or a gray Coast and 20 percent on the West Coast. urn. Substituting B and G for A and A , respectively, 1 2 Furthermore, Mr. Jones recalls that he pur-we have chased his novaprine on the East Coast.

P(B given R) = Clearly this new information affects Mr. Jones�s

P(B) � P (R given B) original probability estimate. We use Bayes�s the-

[P(B) � P(R given B)] 1 [P(G) � P (R given G)] orem to recompute the probability that Alpha

Section 11.2 The Probability Calculus 499

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). manufactured the defective tablets given that to be found, probably owing to the scarcity of reli-the tablets were purchased on the East Coast as able and substantive data�even the direction the follows: plane was flying at the moment it disappeared is an

P(A given E ) unknown. 5

[P(A) in finding the wreckage of Air France Flight 447, � P(E given A)] 1 [P(B) � P(E given B)] which disappeared over the ocean after departing P(A) � P(E given A) Searchers had better luck three years earlier

For Rio de Janeiro for Paris. The initial search, span-P ( A ) and P ( B ) we use the original probabilities of .70 and .30, respectively. The probability that the ning two years, was fruitless; but six days after tablets were purchased on the East Coast given that applying Bayes�s theorem, the flight data recorder they were manufactured by Alpha is .10, and the was found in 10,000 feet of water. Bayes�s theorem probability that they were purchased on the East was also used to locate German U-boats during Coast given that they were manufactured by Beta is World War II, a lost U.S. submarine in the 1960s, .80. Thus, we have and a missing hydrogen bomb following the crash

.70 3 .10 of a B-52 in Spain.

P(A given E ) =

(.70 � .10) 1 (.30 � .80) Additional Applications

.07

5 5 .07/.31 5 .23 Most of the examples considered thus far have .07 1 .24

used the classical theory to determine the prob-

The new information that Mr. Jones acquired ability of the component events. But as was men-

has significantly affected the probability that Alpha tioned earlier, the probability calculus can also be manufactured the defective tablets. The probability used in conjunction with the relative frequency has dropped from .70 to .23. With this new informa- theory and the subjectivist theory. If we apply the tion, it is more likely that Beta manufactured the tab- relative frequency theory to the mortality tables lets, so Mr. Jones should now file his lawsuit against used by insurance companies, we find that the Beta. The probability for Beta is 1 − .23, or .77. probability of a twenty-five-year-old man living

The earlier probability for Alpha (.70) is called an additional forty years is .82, and the prob-

the prior probability, and the later figure (.23) is ability of a twenty-five-year-old woman living called the posterior probability. If Mr. Jones should the same number of years is .88. To compute the acquire even more information, the posterior probability of such a man and woman both living probability becomes a new prior, and it is used as that long, we use the restricted conjunction rule the value for P(A) in a subsequent application of and obtain .82 � .88 = .72. For the probability Bayes�s theorem. that either of these people would live that long,

rem that garnered international attention was its .82 + .88 − (.82 � .88) = .98 use to search for the wreckage of Malaysia Airlines A relatively recent application of Bayes�s theo- we use the general disjunction rule and obtain Flight 370. Two hundred thirty-nine people were Let us suppose that these two people are married on board when it disappeared without a trace while and both would give 9 to 1 odds on�their staying en route from Kuala Lumpur to Beijing in March married for forty years. This translates into a prob-2014. In an effort to find the missing plane, all the ability of 9/(9 + 1) or .90. Using the restricted known data, including flight path, fuel, wind speed, conjunction rule, the probability of this event hap-ocean flow, as well as areas that had been previ- pening is the product of the latter figure and the ously searched, were plugged into the theorem. The probability of their both living that long, or .65. results were continually updated as additional data For an example involving the subjectivist theory, became available. However, the wreckage has yet if the Philadelphia Eagles are given 7 to 5 odds of

500Chapter11 Probability

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). winning the NFC championship, and the New Eng- The strength of an inductive argument depends land Patriots are given 3 to 2 odds of winning the not merely on whether the conclusion is probably AFC championship, the probability that at least one true but on whether the conclusion follows prob-of these teams will win is computed using the general ably from the premises. As a result, to evaluate the disjunction rule. The odds translate respectively into strength of an inductive argument it is not sufficient probabilities of 7/12 and 3/5, and so the probability merely to know the probability of the conclusion of the disjunction is 7/12 + 3/5 − (7/12 � 3/5) = on the assumption that the premises are true. One 5/6. The probability that the two teams will meet in must also know whether the probability of the con-the Super Bowl (that both will win their conference clusion rests on the evidence given in the premises. championship) is, by the restricted conjunction rule, If the probability of the conclusion does not rest on 7/12 � 3/5 = 21/60, or 7/20. The probability that this evidence, the argument is weak regardless of neither will play in the Super Bowl is, by the nega- whether the conclusion is probably true. The follow-tion rule, 1 − 5/6 = 1/6. ing argument is a case in point:

The probability calculus can also be used to eval-

uate the strength of inductive arguments. Consider All dogs are animals. Therefore, probably the following: a traffic accident will occur somewhere

tomorrow.

The Philadelphia Eagles are given 7 to 5

odds of winning the NFC championship. The The conclusion of this argument is probably true New England Patriots are given 3 to 2 odds independently of the premises, so the argument is of winning the AFC championship. There- weak. fore, the Eagles and the Patriots will meet in In this connection the analogy between deductive the Super Bowl. and inductive arguments breaks down. As we saw in

is, on the assumption that the odds are reported the content of its premises. But any inductive argu-correctly, the conclusion follows with a probability ment having a probably true conclusion is not strong of 7/20 or .35. Thus, the argument is not particu-unless the probability of the conclusion rests on the On the assumption that the premises are true, that is necessarily true is deductively valid regardless of Chapter 6, any argument having a conclusion that

larly strong. But if the odds given in the premises evidence given in the premises. should increase, the strength of the argument would A final comment is in order about the material increase proportionately. The premises of the fol-covered in this chapter. Probability is one of those lowing argument give different odds: subjects about which there is little agreement

The Philadelphia Eagles are given 7 to 2 among philosophers. Some philosophers defend odds of winning the NFC championship. The each of the theories we have discussed as provid-New England Patriots are given 8 to 3 odds ing the only acceptable approach, and there are of winning the AFC championship. There- numerous views regarding the fine points of each. fore, the Eagles and the Patriots will meet in However, some philosophers argue that there are the Super Bowl. certain uses of �probability� that none of these

theories can interpret. The statement �There is

In this argument, if the premises are assumed true, high probability that Einstein�s theory of relativity is correct� may be a case in point. In any event, the the conclusion follows with a probability of 7/9 � various theories about the meaning of probability, 8/11 = 56/99, or .57. Thus, the argument is at least as well as the details of the probability calculus, moderately strong. are highly complex subjects, and the brief account Lest this procedure be misinterpreted, how-given here has merely scratched the surface. ever, we should recall a point raised in Chapter�1.

Section 11.2 The Probability Calculus 501

Exercise 11

I. Compute probabilities or odds for the following simple events.

★1. What is the probability of rolling a five on a single roll of a die? What are the

odds for this event?

2. From a sample of 9,750 Ajax trucks, 273 developed transmission problems

within the first two years of operation. What is the probability that an Ajax truck will develop transmission problems within the first two years?

3. If the standard odds are 8 to 5 that the Chargers will beat the Lions, what is the

probability that this event will happen?

★4. From a sample of 7,335 seventy-five-year-old women, 6,260 lived an additional

five years. What is the probability that a seventy-five-year-old woman will live to age eighty?

5. What is the probability of picking a black jack from a standard deck of cards on

a single draw?

6. If the probability of the Red Sox beating the Tigers is 6/17, what are the odds

for this event?

★7. Given an urn containing three red balls, four green balls, and five yellow balls,

what is the probability of drawing a red ball on a single draw?

8. If the odds of the Broncos beating the Dolphins is 5 to 4, and you bet $10 on

the Broncos, how much do you stand to win?

9. Given an urn containing four red balls, three green balls, and five yellow balls,

what are the odds of drawing a red ball on a single draw?

★10. Suppose you give 1:6 odds that you can roll a �1� with a single fair die. If some-

one accepts your bet, how much could you expect to win after 100 rolls if you bet $1 on each roll?

11. Suppose four weather forecasters assign the following probabilities that it will

rain tomorrow: 1/4, 1/5, 2/5, 1/3. What is the probability that this event will happen?

12. Given a regular dodecahedron (a twelve-sided solid in which each face is a

regular pentagon that is congruent with each of the other faces): If this solid is rolled on a surface, what is the probability that a specifically named face will be in contact with the surface?

★13. Suppose three people enter a total of nine horses in a race. Lucas enters two,

Alexa enters three, and Victoria enters four. If each horse has an equal chance of winning, what is the probability that one of Victoria�s horses wins?

14. Over a ten-year period, in a given population, there were 5,491 births, and

thirteen of the babies had a certain genetic disorder. What is the probability that one of the babies born during this period had the disorder?

502Chapter11 Probability

15. Five bettors give the following odds that Black Mercury will win in the sixth

race: 7 to 5, 2 to 1, 8 to 3, 5 to 2, and 7 to 4. Based on these odds, what is the probability that Black Mercury will win?

II. Compute probabilities for the following compound events:

★1. What is the probability of getting either a six or a one from a single roll of a

die?

2. What is the probability of getting heads on three successive tosses of a coin?

3. What is the probability of drawing either a king or a queen from a standard

deck of cards on a single draw?

★4. What is the probability of drawing two aces from a standard deck of cards in

two draws

a. If the first card is replaced before the second is drawn? b. If the first card is not replaced before the second is drawn?

5. What is the probability of drawing at least one ace from a standard deck of

cards on two draws if the first card is replaced before the second is drawn?

6. What is the probability of getting at least one head on three tosses of a coin?

★7. What is the probability of getting at least one six on three rolls of a die?

8. If a pair of dice are rolled, what is the probability that the points add up to

a. five?

b. six?

c. seven?

9. Given two urns, one containing two red, three green, and four yellow balls, the

other containing four red, two green, and three yellow balls, if a single ball is drawn from each urn, what is the probability that

a. Both are red?

b. At least one is green? c. One is red, the other yellow? d. At least one is either red or yellow? e. Both are the same color?

★10. Given an urn containing three red, four green, and five yellow balls,

if two balls are drawn from the urn (without replacement), what is the probability that

a. Both are red?

b. One is green, the other yellow? c. One is either red or green? d. At least one is green? e. Both have the same color?

11. What is the probability of drawing either an ace or a king (or both) on three

draws (without replacement) from a standard deck of cards? (Hint: Use the negation rule.)

Section 11.2 The Probability Calculus 503

12. What is the probability of drawing an ace and a king on three draws (without

replacement) from a standard deck of cards? (Hint: Use the negation rule.)

★13. The probability of a twenty-year-old man living to age seventy is .74, and

the probability of a twenty-year-old woman living to the same age is .82. If a recently married couple, both age twenty, give 8 to 1 odds on their staying mar-ried for fifty years, what is the probability that

a. At least one will live to age seventy? b. They will celebrate their golden wedding anniversary?

14. Assign a numerical value to the strength of the following argument: The odds

are 5 to 3 that the Guardians will win the American League pennant and 7 to 5 that the Cardinals will win the National League pennant. Therefore, the Guardians and the Cardinals will meet in the World Series.

15. Assign a numerical value to the strength of the following argument: The

Wilson family has four children. Therefore, at least two of the children were born on the same day of the week.

★16. We are given three new urns each containing seven red, five green, and three

white balls, and two old urns each containing five red, three green, and seven white balls. The urns are identical except for an old or new date stamped beneath the base. If a single red ball is randomly drawn from one of these urns, was it most probably drawn from an old urn or a new urn? (Hint: For this exercise and the ones that follow, use Bayes�s theorem.)

17. Knowing that your friend Angella studies for her tests only 40 percent of the

time, you place an even bet with a third party that Angella will not study for her upcoming test. Later, you just happen to hear that Angella passed the test. Should you retract your bet? Assume that Angella passes nine out of ten tests when she studies, and three out of ten when she does not.

18. A physician has diagnosed a patient as having either hepatitis or liver cancer

(but not both). Statistics reveal that hepatitis occurs in the general population twice as frequently as liver cancer. Thus, the physician tentatively concludes that the patient probably has hepatitis. Later the physician conducts a test on the patient that turns out positive. On this test, nine out of ten cases of liver cancer trigger a positive outcome, and one out of six cases of hepatitis triggers a positive outcome. What is the new probability that the patient has liver cancer?

★19. Ms. Cummings, a bookstore owner, wishes to send out an advertisement to

potential customers. She estimates that 20 percent of the town�s residents are recreational readers, and she knows that all of the residents are either recre-ational readers or TV addicts (but not both). Also, a media expert has told her that two of ten TV addicts are newspaper subscribers, and from a survey of her own customers she has learned that seven of ten recreational readers are newspaper subscribers. If Ms. Cummings places an ad in the local newspaper, how effective will the ad be? (That is, what is the probability that a newspaper subscriber is a recreational reader?)

504Chapter11 Probability

20. Mr. Castelini, a grape merchant, inquires about a certain wine that he tastes at

a party. The host tells him that from a total of ten bottles of that wine, six came from the north vineyard and four came from the south vineyard. From this information, the merchant concludes that he should visit the north vineyard for a sample of the grapes. Later, he discovers that the wine in question came from a bottle labeled Classic Reserve, and 30 of 100 barrels of north vineyard wine were bottled with that label, while 130 of 200 barrels of south vineyard wine were bottled with that label. Given this new information, what is the probability that the wine came from the north vineyard?

III. Do some research and find an actual application of Bayes�s theorem that is not men-

tioned in this book. Then write a paragraph on your findings.

Summary

Theories of Probability:

■ Classical theory: Rests on a priori calculations.

■ Relative frequency theory: Depends on empirical experimentation.

■ Subjectivist theory: Rests on subjective beliefs of individual people.

Probability Calculus:

■ Used to calculate the probability of compound events.

■ Conjunction rule: Computes the probability of two events occurring together.

● Restricted conjunction rule: The events are independent.

● General conjunction rule: The events need not be independent.

■ Disjunction rule: Computes the probability of either of two events occurring.

● Restricted disjunction rule: The events are mutually exclusive.

● General disjunction rule: The events need not be mutually exclusive.

■ Negation rule: Computes the probability of an event not occurring.

■ Bayes�s theorem: Computes the conditional probability of two or more mutually exclusive and

jointly exhaustive events.

Summary505

12Statistical Reasoning

12.1 Evaluating Statistics

12.2 Samples

12.3 The Meaning of �Average�

12.4 Dispersion

12.5 Graphs and Pictograms

12.6 Percentages

12.1 Evaluating Statistics

Learning the basics of statistical reasoning is empowering. Suppose you are considering the purchase of a new car. The salesperson tells you that 97 percent of those who have purchased this model are happy they bought it. If you know something about statistics, you will ask, �Is that figure supported by a survey? Was the survey conducted randomly? What was the size of the sample? Was more than one survey taken?� Only by knowing the answers to these ques-tions can you be justified in believing what the salesperson has told you.

In our day-to-day experience, all of us encounter arguments that rest on statistical evidence. The advertising industry is an especially prolific source of such argu-ments. We are constantly told that we ought to sign up with a certain mobile phone service because it has 30 percent broader coverage, buy a certain kind of car because it gets 5�percent better fuel economy, and use a certain cold remedy because it is recommended by four out of five physicians. But the advertising industry is not the only source of these kinds of arguments. We often read in the newspapers that some union is asking for an increase in pay because its members earn less than the average, or that a certain region is threatened with floods because rainfall has been more than the average.

506

To evaluate such arguments, we must be able to the language but with the devices that unscrupulous

interpret the statistics on which they rest, but doing individuals use to deceive others. so is not always easy. Statements expressing aver- This chapter touches on five areas that are fre-ages and percentages are often ambiguous and can quent sources of such ambiguity and deception: mean a variety of things, depending on how the problems in sampling, the meaning of �average,� average or percentage is computed. These difficul- the importance of dispersion in a sample, the use ties are compounded by the fact that statistics pro- of graphs and pictograms, and the use of percent-vide a highly convenient way for people to deceive ages for the purpose of comparison. By becoming one another. Such deceptions can be effective even acquainted with these topics and with some of the though they fall short of being outright lies. Thus, to misuses that occur, we improve our ability to deter-evaluate arguments based on statistics one must be mine whether a conclusion follows probably from a familiar not only with the ambiguities that occur in set of statistical premises.

12.2 Samples

Much of the statistical evidence presented in support For example, when a physician draws a blood of inductively drawn conclusions is gathered from sample to test for blood sugar, there is no need to analyzing samples. When a sample is found to possess take a little bit from the finger, a little from the arm, a certain characteristic, it is argued that the group as and a little from the leg. Because blood is a circulat-a whole (the population) possesses that characteristic. ing fluid, it can be assumed that it is homogenous For example, if we wanted to know what the student in regard to blood sugar. body at a certain university thought about a proposal The randomness requirement demands more to adopt an academic honor code, we could take a poll attention when the population consists of discrete of 10 percent of the students. If the results of the poll units. Suppose, for example, that a quality-control showed that 80 percent of those sampled favored the engineer for a manufacturing firm needed to deter-code, we might draw the conclusion that 80 percent of mine whether the components on a certain con-the entire student body favored it. Such an argument veyor belt were within specifications. To do so, the would be classified as an inductive generalization. engineer removed every tenth component for mea-

The problem that arises with the use of samples surement. The sample obtained by such a procedure

has to do with whether the sample is representative would not be random if the components were not of the population. Samples that are not representative randomly arranged on the conveyor belt. As a result are said to be biased samples. Depending on what of some malfunction in the manufacturing process the population consists of, whether machine parts, it is quite possible that every tenth component was children�s toys, cans of soup, or human beings, dif- perfect and the rest were imperfect. If the engineer ferent considerations enter into determining whether happened to select only the perfect ones, the sample a sample is biased. These considerations include would be biased. A selection procedure that would (1) whether the sample is randomly selected, (2)�the be more likely to ensure a random sample would be size of the sample, and (3) psychological factors. to roll a pair of dice and remove every component

A random sample is one in which every corresponding to a roll of ten. Since the outcome of

member of the population has an equal chance a roll of dice is a random event, the selection would of being selected. The requirement that a sample also be random. Such a procedure would be more be randomly selected applies to practically all sam- likely to include defective components that turn up ples, but sometimes it can be taken for granted. at regular intervals.

Section 12.2 Samples 507

The randomness requirement presents even the Depression, at a time when many people could

greater problems when the population consists of afford neither a telephone nor a subscription to the human beings. Suppose, for example, that a public- Digest. These were the people who were overlooked opinion poll is to be conducted on the question in the poll, and they were also the ones who voted of excessive corporate profits. It would hardly do for Roosevelt. to ask such a question randomly of the people A different kind of sampling bias can occur in encountered on Wall Street in New York City. clinical trials undertaken to disclose the negative Such a sample would almost certainly be biased in side effects of a particular drug. If a participant in favor of the corporations. such a trial should suffer serious side effects, that

When the population to be sampled is relatively participant might leave the trial altogether and never

small and uniform, such as the matriculated students report any of the side effects to the doctors conduct-at a university, the members of a labor union, or the ing the trial. As more and more participants drop employees of a medium-sized corporation, there out for the same reason, the sample would become is a database consisting of the identification num- increasingly biased. At the conclusion of such a trial, bers of these people. A fairly random sample could if a regulatory agency should grant approval for the be obtained by contacting by email, or by texting, drug to be sold to the public, the decision to approve every individual having an ID number ending in the would be based on a biased sample. A similar kind numeral 2, for example. But when the population is of bias can occur if the duration of a clinical trial is large and diverse, such as the residents of a city or too short to allow for certain side effects to emerge. state or the citizens of a country, such a method is For example, cancer can take years to develop. If the usually not possible. If the residents of a county were clinical trial evaluating a particular drug is ended to be sampled by taking names from property tax after a relatively short time, the possibility that the rolls, for example, renters would be excluded. If the drug causes cancer might never be discovered. In residents of a state were to be sampled via driver�s effect, the sample consisting of the trial participants license numbers, those who could not afford a car would be biased. might not have driver�s licenses. If the citizens of a Size is also an important factor in determining country were to be sampled via passport informa- whether a sample is representative. Given that a tion, those who do not travel outside the country sample is randomly selected, the larger the sample, might not have passports, and these people would the more closely it replicates the population. In be missed. statistics, this degree of closeness is expressed in

A classic case of a poll that turned out to be biased terms of sampling error. The sampling error is

in spite of a good deal of effort and expense occurred the difference between the relative frequency with when the Literary Digest magazine undertook a poll which some characteristic occurs in the sample to predict the outcome of the 1936 presidential elec- and the relative frequency with which the same tion. The sample consisted of a large number of the characteristic occurs in the population. If, for magazine�s subscribers together with other people example, a poll were taken of a labor union and selected from the telephone directory. Because four 60 percent of the members sampled expressed similar polls had picked the winner in previous their intention to vote for Smith for president years, the results of this poll were highly respected. but in fact only 55 percent of the whole union As it turned out, however, the Republican candidate, intended to vote for Smith, the sampling error Alf Landon, got a significant majority in the poll, but would be 5 percent. If a larger sample were taken, Franklin D. Roosevelt won the election by a land- the error would be smaller. slide. The incorrect prediction is explained by the Just how large a sample should be is a function fact that the 1936 election occurred in the middle of of the size of the population and of the degree of

508Chapter12 Statistical Reasoning

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). sampling error that can be tolerated. For a sam- level, the margin of error is approximately equal pling error of, say, 5 percent, a population of 10,000 to 1.29/√n. Comparing these two expressions would require a larger sample than would a popu- shows that as the confidence level increases, so lation of 100. However, the ratio is not linear. To does the margin of error. obtain the same precision, the sample for the larger In most polls, the margin of error is based on population need not be 100 times as large as the a 95 percent (or higher) confidence level. But if a one for the smaller population. When the popula- much lower confidence level should be selected, tion is very large, the size of the sample needed to and if this fact is not disclosed, then the results of a ensure a certain precision levels off at a constant poll could be deceptive�even if the margin of error figure. For example, a random sample of 500 will is stated. The reason for this is that the margin of yield results that are just as accurate whether the error would be combined with a low likelihood population is 100,000 or 100 million. that it covered any actual discrepancy between

When the population is very large, and the sample the sample and the population. Also, of course,

is random and less than 5� percent of the population, any poll that simply fails to disclose the margin of sampling error can be expressed in terms of a math- error can be deceptive. For example, if a poll should ematical margin of error as per Table 12.1: show Adams leading Baker for U.S. Senate by 55 to

45 percent, this means virtually nothing if neither

Table 12.1 Sample Size and Margin of Error the size of the sample nor the margin of error is

Sample SizeMargin of Error given. If the margin of error should be as high as

2401 who leads Adams by 55 to 45 percent. 6 2% Even when the margin of error is stated, how-9604 10 percent, then it could be the case that it is Baker 6 1% 1067 63% ever, incorrect conclusions are often drawn when 600 64% comparing the results of one poll with the results of 384 65% another. Suppose a poll shows Adams leading Baker

196 appears in the newspaper, a second poll is taken that 6 7% shows Adams trailing Baker by 49�to 51 percent. Sup- 267 by 55 to 45 percent. After an article critical of Adams 6 6% 96 610% pose further that the margin of error for both polls

is 4 percent and the confidence level is 95 percent.

The figures in this table are based on a 95 From such polls people often draw the conclusion

percent level of confidence, which means that that Adams�s lead over Baker has disappeared, when we can be 95 percent certain that a sample will in fact no such conclusion is warranted. Taking into accurately reflect the whole population within the account the margin of error, it could be the case that margin of error for that sample. Thus, if from a in fact Adams�s lead over Baker has remained con-random poll of 9604�people, 53 percent say they stant at, say, 53 to 47 percent. prefer Jones for governor, then we can be 95 per- Statements of sampling error are often conspic-cent certain that between 52 and 54 percent of the uously absent from surveys used to support adver-whole population prefers Jones. The figures in the tising claims. Marketers of products such as patent table are based purely on mathematics and are not medicines have been known to take rather small dependent on any empirical measurement. For a samples until they obtain one that gives the �right� 95 percent confidence level, the margin of error result. For example, several polls of twenty-five is approximately equal to .98/√n, where n is the people might be taken inquiring about the pre-size of the sample. For a 99 percent confidence ferred brand of aspirin. Even though the samples

Section 12.2 Samples 509

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). might be randomly selected, one will eventually and the respondent. Suppose, for example, that a be found in which twenty of the twenty-five door-to-door survey were taken to determine how respondents indicate their preference for Alpha many people believe in God or attend church on brand aspirin. Having found such a sample, the Sunday. If the survey were conducted by priests marketing firm proceeds to promote this brand and ministers dressed in clerical garb, one might as the one preferred by four out of five of those expect a larger number of affirmative answers than sampled. The results of the other samples are, if the survey were taken by nonclerics. The simple of course, discarded, and no mention is made of fact is that many people like to give answers that sampling error. they think will please the questioner.

Psychological factors can also have a bearing To prevent this kind of interaction from affecting

on whether the sample is representative. When the outcome, scientific studies are often conducted the population consists of inanimate objects, such under �double-blind� conditions in which neither as cans of soup or machine parts, psychological the surveyor nor the respondent knows what the factors are usually irrelevant, but they can play a �right� answer is. For example, in a double-blind significant role when the population consists of study to determine the effectiveness of a drug, human beings. If the people composing the sample bottles containing the drug would be mixed with think that they will gain or lose something by other bottles containing a placebo (sugar tablet). the kind of answer they give, their involvement The contents of each bottle would be matched with will likely affect the outcome. For example, if the a code number on the label, and neither the person residents of a neighborhood were to be surveyed distributing the bottles nor the person recording for annual income with the purpose of determin- the responses would know what the code is. Under ing whether the neighborhood should be ranked these conditions the people conducting the study among the fashionable areas in the city, we would would not be able to influence, by some smile or expect the residents to exaggerate their answers. gesture, the response of the people to whom the But if the purpose of the study were to determine drugs are given. whether the neighborhood could afford a special Most of the statistical evidence encountered levy that would increase property taxes, we might in ordinary experience contains no reference to expect the incomes to be underestimated. such factors as randomness, sampling error, or

The kind of question asked can also have a psy- the conditions under which the sample was taken.

chological bearing. Questions such as �How often In the absence of such information, the person do you brush your teeth?� and �How many books faced with evaluating the evidence must use his do you read in a year?� can be expected to generate or her best judgment. If either the organization responses that overestimate the truth, while �How conducting the study or the people composing many times have you been intoxicated?� and �How the sample have something to gain by the kind many extramarital affairs have you had?� would of answer that is given, the results of the survey probably receive answers that underestimate the should be regarded as suspect. And if the ques-truth. Similar exaggerations can result from the tions that are asked concern topics that would nat-way a question is phrased. For example, �Do you urally elicit distorted answers, the results should favor a reduction in welfare benefits as a response probably be rejected. In either event, the mere fact to rampant cheating?� would be expected to receive that a study appears scientific or is expressed in more affirmative answers than simply �Do you mathematical language should never intimidate favor a reduction in welfare benefits?� a person into accepting the results. Numbers and

Another source of psychological influence is scientific terminology are no substitute for an

the personal interaction between the surveyor unbiased sample.

510Chapter12 Statistical Reasoning

12.3 The Meaning of �Average�

In statistics the word �average� is used in three dif- together. The problem for induction occurs when ferent senses: mean, median, and mode. In evaluat- a great disparity among these values occurs. This ing arguments and inferences that rest on averages, sometimes happens in the case of salaries. Consider, we often need to know in precisely what sense the for example, Table 12.3, which reports the salaries word is being used. of a hypothetical architectural firm. Since there are

The mean value of a set of data is the arithmeti- twenty-one employees and a total of $2,730,000 is

cal average. It is computed by dividing the sum of paid in salaries, the mean salary is $2,730,000/21, the individual values by the number of values in or $130,000. The median salary is $90,000 because the set. Suppose, for example, that we are given ten employees earn less than this and ten earn more, Table 12.2 listing the ages of a group of people: and the mode, which is the salary that occurs most

frequently, is $60,000. Each of these figures repre-

Table 12.2 sents the �average� salary of the firm, but in differ-

Number of People ent senses. Depending on the purpose for which the Age average is used, different figures might be cited as 1 16 the basis for an argument. 4 17 For example, if the senior engineer were to request

1 18 a raise in salary, the president could respond that the

3 the sense of median and mode) and that therefore she 23 or he does not deserve a raise. If the junior draftsmen 2 engineer�s salary is already well above the average (in 19

To compute the mean age, we divide the sum of the respond that they are presently earning the firm�s were to make the same request, the president could individual ages by the number of people: average salary (in the sense of mode), and that for

(1 3 16) 1 (4 3 17) 1 (1 3 18) draftsmen, to be earning the average salary is excel-

1 (2 3 19)1 (3 3 23) lent. Finally, if someone from outside the firm were

mean age 5 5 19 to allege that the firm pays subsistence-level wages, 11 the president could respond that the average salary of

The median of a set of data is the middle the firm is a hefty $130,000. All of the president�s

point when the data are arranged in ascending

order. In other words, the median is

the point at which there is an equal Table 12.3

number of data above and below. In Number of

there are five people above this age and President 1 $750,000 five below. Table 12.2, the median age is 18 because CapacityPersonnelSalary

The Senior architect 2 200,000 mode is the value that occurs

with the greatest frequency. Here, the Junior architect 2 160,000 mode is 17, because there are four Senior engineer 1 130,000 d mean

any other age. Senior draftsman 1 90,000 d median In this example, the mean, median, people of that age and fewer people of Junior engineer 4 110,000 and mode differ but are all fairly close Junior draftsman 10 60,000 d mode

Section 12.3 The Meaning of �Average� 511

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). responses would be true, but if the reader or listener he means average in the sense of median, it could were not sophisticated enough to distinguish the be the case that the four heavier passengers weigh various senses of �average,� he or she might be per- 200 pounds and the four lighter ones weigh 145, suaded by the arguments. for a combined weight of 1530 pounds. Similarly,

In some situations, the mode is the most useful if the passenger representative means average in

average. Suppose, for example, that you are in the the sense of mode, it could be that two passengers market for a three-bedroom house. Suppose further weigh 150 pounds and that the others have varying that a real estate agent assures you that the houses weights in excess of 200 pounds, for a combined in a certain complex have an average of three bed- weight of over 1700 pounds. Only if the represen-rooms and that therefore you will certainly want to tative means average in the sense of mean do you see them. If the agent has used �average� in the sense know that the combined weight of the passengers is of mean, it is possible that half the houses in the 9 � 150 or 1350 pounds. complex are four-bedroom, the other half are two- Finally, sometimes a median average is the most bedroom, and there are no three-bedroom houses at meaningful. Suppose, for example, that you are a man-all. A similar result is possible if the agent has used ufacturer of a product that appeals to people under average in the sense of median. The only sense of age 35. To increase sales you decide to run an ad in average that would be useful for your purposes is a national magazine, but you want some assurance mode: If the modal average is three bedrooms, there that the ad will be read by the right age group. If are more three-bedroom houses than any other kind. the advertising director of a magazine tells you that

On other occasions a mean average is the most the average age of the magazine�s readers is 35, you

useful. Suppose, for example, that you have taken know virtually nothing. If the director means aver-a job as a pilot on a plane that has nine passenger age in the sense of mean, it could be that 90 percent seats and a maximum carrying capacity of 1350 of the readers are over 35� and that the remaining pounds (in addition to yourself). Suppose further 10 percent bring the average down to 35. Similarly, that you have arranged to fly a group of nine pas- if the director means average in the sense of mode, sengers over the Grand Canyon and that you must it could be that 3 percent of the readers are exactly determine whether their combined weight is within 35 and that the remaining 97 percent have ages the required limit. If a representative of the group ranging from 35 to 85. Only if the director means tells you that the average weight of the passengers average in the sense of median do you know that half is 150 pounds, this by itself tells you nothing. If the readers are 35 or less.

12.4 Dispersion

Although averages often yield important informa- Let us first consider the range of a set of tion about a set of data, there are many cases in data, which is the difference between the largest which merely knowing the average, in any sense of and the smallest values. For an example of the the term, tells us very little. The reason for this is that importance of this parameter, suppose that after the average says nothing about how the data are dis- living for many years in an intemperate climate, tributed. For this, we need to know something about you decide to re locate�in an area that has a more dispersion, which refers to how spread out the data ideal mean temperature. On discovering that the are in regard to numerical value. Three important annual mean temperature of Oklahoma City is measures of dispersion are range, variance, and stan- 60�F you decide to move there, only to find that dard deviation. you roast in the summer and freeze in the winter.

512Chapter12 Statistical Reasoning

Figure 12.1* that the range in ages of the children is six. Unable

San Nicholas Island to decide between these two centers, you decide to Oklahoma City pay them a visit. Having done so, you see that the

F) Rumpus Center will meet your needs better than 8 ( 100 the Bumpus Center. The reason is that the ages

60 of the children in the two centers are distributed

ature differently.

20 The ages of the children in the two centers are as

Temper follows: �20

Rumpus Center: 1, 3, 3, 4, 4, 4, 5, 5, 7

J F M A M J J A S O N D Bumpus Center: 1, 1, 2, 2, 4, 6, 6, 7, 7

Month To illustrate the differences in distribution, these

* This example is taken from Darrell Huff, How to Lie with Statistics ages can be plotted on a certain kind of bar graph, (New York: Norton, 1954), 52. called a histogram , as shown in Figures 12.2a and 12.2b. Obviously the reason why the Rumpus

Center comes closer to meeting your needs is

Unfortunately, you had ignored the fact that that it has seven children within one year of your Oklahoma City has a temperature range of 130�, own child, whereas the Bumpus Center has only extending from a record low of −17� to a record one such child. This difference in distribution is high of 113�. In contrast, San Nicholas Island, off

the coast of California, has a mean temperature of

61� but a range of only 40�, extending from 47� in Figure 12.2 the winter to 87� in the summer. The temperature Rumpus Center

measure how every data point varies or deviates ranges for these two locations are approximated mean 4 median in Figure 12.1. y Even granting the importance of the range of 3 the data in this example, however, range really 2 equenc tells us relatively little because it comprehends Fr only two data points, the maximum and mini-1 mum. It says nothing about how the other data points are distributed. For this we need to know (a) 1 2 3 4 5 6 7 the variance , or the standard deviation , which Age from the mean. Bumpus Center For an example of the importance of these two mean 4 parameters in describing a set of data, suppose median you have a four-year-old child and you are look-y 3 ing for a day-care center that will provide plenty

of possible playmates about the same age as your 2 equencFr child. After calling several centers on the phone, 1 you narrow the search down to two: the Rumpus Center and the Bumpus Center. Both report that they regularly care for nine children, that the (b) 1 2 3 4 5 6 7 mean and median age of the children is four, and Age

Section 12.4 Dispersion 513

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). measured by the variance and the standard devia- the final column, we obtain the variance by divid-

for the two centers is quite easy, but first we must Variance = σ2 22 = = 2.44 introduce some symbols. The standard deviation tion. Computing the value of these parameters ing the sum (22) by n (9): 9

is represented in statistics by the lowercase Greek Finally, to obtain the standard deviation, we take the letter σ (sigma), and the variance, which is the square root of the variance: square of the standard deviation, is represented by σ2. We compute the variance first, which is defined Standard deviation = σ = 2.44 = 1.56 as follows:

σ 2 ∑ (x − μ)

5 Table 12.5n

In this expression (which looks far more com- x(x 2 μ)(x 2 μ)2

plicated than it is), ∑ (uppercase sigma) means 1 23 9

ages of the children, the lowercase Greek letter 2 22 4 μ �the sum of, � x is a variable that ranges over the 1 23 9

children. Thus, to compute the variance, we take (mu) is the mean age, and n is the number of 2 22 4 4 0 0 each of the ages of the children, subtract the mean age (4) from each, square the result of each, 6 12 4

the number of children (9). The first three steps 7 13 9 of this procedure for the Rumpus Center are add up the squares, and then divide the sum by 6 12 4 reported in Table 12.4. 7 13 9

Total = 52

Table 12.4 Next, we can perform the same operation on the

x(x 2 μ)(x 2 μ)2 ages of the children in the Bumpus Center. The fig-

1 ures are expressed in Table 12.5. 2 3 9 Now, for the variance, we have 3 2 1 1 52 σ 2 = = 5.78 3 2 1 1 9

4 0 0 And for the standard deviation, we have

4 0 0 σ = 5.78 = 2.40 4 0 0

5 11 1 These figures for the variance and standard devi-

5 ation reflect the difference in distribution shown 1 1 1 in the two histograms. In the histogram for the 7 1 3 9 Rumpus Center, the ages of most of the children are clumped around the mean age (4). In other words, Total = 22

they vary or deviate relatively slightly from the

mean, and this fact is reflected in relatively small fig-

First, the column for x (the children�s ages) is ures for the variance (2.44) and the standard devia-entered, next the column for (x − μ), and last the tion (1.56). On the other hand, in the histogram for column for (x − μ)2. After adding up the figures in the Bumpus center, the ages of most of the children

514Chapter12 Statistical Reasoning

Figure 12.3

(a) 57 59 61 63 65 67 69 71 73 (b) 57 59 61 63 65 67 69 71 73

vary or deviate relatively greatly from the mean, so under the curve represents a certain subset of these the variance (5.78) and the standard deviation (2.40) heights. The number of heights trails off toward zero are larger. at the extreme left and right ends of the curve, and

One of the more important kinds of distribution it reaches a maximum in the center. The peak of

used in statistics is called the normal probability the curve reflects the average height in the sense of distribution, which expresses the distribution of mean, median, and mode. random pheno mena in a population. Such phenom- The parameters of variance and standard devia-ena include (approximately) the heights of adult men tion apply to normal distributions in basically the or women in a city, the useful life of a certain kind same way as they do for the histograms relating to of tire or lightbulb, and the daily sales figures of a the day-care centers. Normal curves with a relatively certain grocery store. To illustrate this concept, sup- small standard deviation tend to be relatively narrow pose that a certain college has 2000 female students. and pointy, with most of the population clustered The heights of these students range from 57 inches close to the mean, while curves with a relatively to 73 inches. If we divide these heights into one-inch large standard deviation tend to be relatively flat-intervals and express them in terms of a histogram, tened and stretched out, with most of the population the resulting graph would probably look like the one distributed some distance from the mean. This idea in Figure 12.3a. is expressed in Figure 12.4. As usual, σ represents the

This histogram has the shape of a bell. When a standard deviation, and μ represents the mean.

continuous curve is superimposed on top of this For a final example that illustrates the importance histogram, the result appears in Figure 12.3b. This of dispersion, suppose that you decide to put your curve is called a normal curve, and it represents life savings into a business that designs and manu-a normal distribution. The heights of all of the factures women�s dresses. As corporation president students fit under the curve, and each vertical slice you decide to save money by restricting production

Figure 12.4

σ 1 =

σ = 1 σ = 2

μ μ μ

Section 12.4 Dispersion 515

average size in the sense of mean, median, and Women who can mode is 12, you decide to make only size 12 dresses. wear size 12 Unfortunately, you later discover that while size 12 is

indeed the average, 95 percent of women fall outside omen this interval, as Figure 12.5 shows.

The problem is that you failed to take into account

the standard deviation. If the standard deviation

were relatively small, then most of the dress sizes Number of w would be clustered about the mean (size 12). But in

fact the standard deviation is relatively large, so most 4 8 12 16 20 of the dress sizes fall outside this interval. Dress size

12.5 Graphs and Pictograms

Graphs provide a highly convenient and informative picture is not very bright. Although they convey practi-way to represent statistical data, but they are also cally no information, graphs of this kind are used quite susceptible to misuse and misinterpretation. Here often in advertising. A manufacturer of vitamins, for we will confine our attention to some of the typical example, might print such a graph on the label of the ways in which graphs are misused. bottle to suggest that a person�s energy level is supposed

First of all, if a graph is to represent an actual situa- to increase dramatically after taking the tablets. Such

tion, it is essential that both the vertical and horizontal ads often make an impression because they look sci-axes be scaled. Suppose, for example, that the profit entific, and the viewer rarely bothers to check whether level of a corporation is represented by a graph such the axes are scaled, or precisely what the curve is sup-as Figure 12.6. Such a graph is practically meaningless posed to signify. because it fails to show how much the profits increased A graph that more appropriately represents cor-over what period of time. If the curve represents a 10 porate profits is given in Figure 12.7 (the corpora-percent increase over twenty years, then, of course, the tion is fictitious).

Figure 12.7

Figure 12.6

Tangerine Technology, 2022

Profits 5

Profits (millions of dollars)

1 2 3 4

TimeQuarter

516Chapter12 Statistical Reasoning

Figure 12.8 The same strategy can be used with bar graphs.

10 Tangerine Technology, 2022 The graphs in Figure 12.9 compare sales volume for

two consecutive years, but the one on the right con-

veys the message more dramatically. Of course, if the

sales volume has decreased, the corporate directors

9 would probably want to minimize the difference, in

which case the design on the left is preferable.

An even greater illusion can be created with the

Profits (millions of dollars) compares two situations through drawings that differ 8 1 2 3 4 either in size or in the number of entities depicted. use of pictograms. A pictogram is a diagram that

Quarter Consider Figure 12.10, which illustrates the increase

in the milk production of a dairy between 2014

Inspection of the graph reveals that between and 2022.

January and December profits rose from $8�million This pictogram accurately represents the increase to $10 million, which represents a respectable because it unequivocally shows that the amount dou-25 percent increase. This increase can be made bled between the years represented. But the effect is to appear even more impressive by chopping off not especially dramatic. The increase in production the bottom of the graph and altering the scale can be exaggerated by representing the 2022 level with on the vertical axis while leaving the horizontal a milk storage container twice as tall (Figure 12.11). scale as is (Figure� 12.8). Again, strictly speak- Even though the actual production is stated ing, the graph accurately represents the facts, but adjacent to each drawing, this pictogram creates if the viewer fails to notice what has been done the illusion that production has much more than to the vertical scale, he or she is liable to derive doubled. While the drawing on the right is exactly the impression that the profits have increased by twice as high as the one on the left, it is also twice something like a hundred percent or more. Most as wide. Thus, it occupies four times as much of the graphs shown on CNBC, a widely watched room on the page. Furthermore, when the viewer�s business channel, are of this sort. three-dimensional judgment is called into play,

Figure 12.9

5 7 4

2 6

TangerineTangerine

1 TechnologyTechnology

Total sales (thousands of units)Total sales (thousands of units)

2021 2022 2021 2022

Section 12.5 Graphs and Pictograms 517

Figure 12.10Figure 12.11

20142022

Milk 1000 2000 MilkMilk

gallons gallons

the container on the right is perceived as having 2014Milk2022 eight times the volume of the one on the left. Thus, 1000 2000 when the third dimension is taken into account, Milk gallons gallons the increase in consumption is exaggerated by 600�percent.

12.6 Percentages

The use of percentages to compare two or more sit- years we could expect one less heart attack than with uations or quantities is another source of illusion in 100 people who do not take the drug. statistics. A favorite of advertisers is to make claims However, a 1 percent reduction in heart attacks such as �Zesty Cola has 20 percent fewer calories� will never sell anything. So the pharmaceutical com-or �The price of the new Vizio flat screen has been pany switches the numbers. The company figures that reduced by 15 percent.� These claims are virtually the drug reduced the incidence of heart attacks from meaningless. The question is, 20 percent less than 2 percent to 1 percent, which is a 50 percent reduc-what, and 15 percent reduced from what? If the tion. Now it can claim in its advertising that the drug basis of the comparison or reduction is not men- reduces heart attacks by 50 percent. This conveys the tioned, the claim tells us nothing. Yet such claims false impression that if 100 people take the drug for are often effective because they leave us with the two years, there will be 50 fewer heart attacks than impression that the product is in some way supe- with 100 people who do not take the drug. Clearly rior or less expensive. this is not what the clinical trial suggests.

One ploy sometimes used by pharmaceutical com- What the pharmaceutical company has done is

panies in selling drugs, such as statins, to reduce the to switch the base for the percentages. In the initial incidence of heart attacks, involves switching the base computation, the base is 100. In the revised compu-upon which percentages are computed. To see how tation, the base is two. One out of 100 is 1 percent; this works, suppose a company sets up a clinical trial one out of two is 50 percent. The same strategy is involving 200 people. One hundred are given the drug sometimes used by nonprofits engaged in fund- and 100 are given a placebo (sugar pill). By the end raising. Suppose, for example, that you are a uni-of the trial, suppose that one person in the treatment versity president involved with a funding drive to group suffered a heart attack and two people in the increase the university�s endowment. Suppose further placebo group had a heart attack. That means that 1 that the endowment currently stands at $15 million percent of the treatment group had a heart attack and and that the objective is to increase it to $20 million. 2 percent of the placebo group had a heart attack. This To guarantee the success of the drive, you engage the is a 1 percent reduction in heart attacks. Projected services of a professional fund-raising organization. into the future, if 100 people take the drug for two At the end of the allotted time the organization has

518Chapter12 Statistical Reasoning

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). increased the endowment to $16 million. They justify A different kind of fallacy occurs when a person their effort by stating that, since $16 million of the $20 attempts to add percentages as if they were cardi-million has been raised, the drive was 80 percent suc- nal numbers. Suppose, for example, that a baker cessful (16/20 � 100%). increases the price of a loaf of bread by 50 percent. To

In fact, of course, the drive was nowhere near that justify the increase the baker argues that it was neces-

successful. The objective was not to raise $20 mil- sitated by rising costs: The price of flour increased lion but only $5 million, and of that amount only by 10 percent, the cost of labor rose 20� percent, util-$1 million has actually been raised. Thus, at best ity rates went up 10 percent, and the cost of the lease the drive was only 20 percent successful. Even this on the building increased 10�percent. This adds up figure is probably exaggerated, though, because to a 50 percent increase. Again, the argument is fal-$1 million might have been raised without any lacious. If everything had increased by 20 percent, special drive. The trick played by the fund-raising this would justify only a 20 percent increase in the organization consisted in switching the numbers by price of bread. As it is, the justified increase is less which the percentage was to be computed. than that. The fallacy committed by such arguments

This same trick, incidentally, was allegedly used would probably be best classified as a case of missing

by Joseph Stalin to justify the success of his first five- the point (ignoratio elenchi). The arguer has failed to year plan.* Among other things, the original plan grasp the significance of his own premises. called for an increase in steel output from 4.2 million Statistical variations of the suppressed evidence tons to 10.3 million. After five years the actual output fallacy are also quite common. One variety con-rose to 5.9 million, whereupon Stalin announced sists in drawing a conclusion from a comparison that the plan was 57 percent successful (5.9/10.3 � of two different things or situations. For example, 100%). The correct percentage, of course, is much people running for political office sometimes cite lower. The plan called for an increase of 6.1 million figures indicating that crime in the community tons and the actual increase was only 1.7� million. has increased by, let us say, 20 percent during the Thus, at best, the plan was only 28 percent successful. past three or four years. What is needed, they con-

Similar devices have been used by employ- clude, is an all-out war on crime. But they fail to

ers on their unsuspecting employees. When busi- mention that the population in the community ness is bad, an employer may argue that salaries has also increased by 20�percent during the same must be reduced by 20� percent. Later, when busi- period. The number of crimes per capita, there-ness improves, salaries will be raised by 20 percent, fore, has not changed. Another example of the thus restoring them to their original level. Such an same fallacy is provided by the ridiculous argu-argument, of course, is fallacious. If a person earns ment that 90 percent more traffic accidents occur $10 per hour and that person�s salary is reduced by in clear weather than in foggy weather and that 20 percent, the adjusted salary is $8. If that figure therefore it is 90 percent more dangerous to drive is later increased by 20 percent, the final salary is in clear than in foggy weather. The arguer ignores $9.60. The problem, of course, stems from the fact the fact that the vast percentage of vehicle miles that a different base is used for the two percentages. are driven in clear weather, which accounts for the The fallacy committed by such arguments is a vari- greater number of accidents. ety of equivocation. Percentages are relative terms, A similar misuse of percentages is committed and they mean different things in different contexts. by businesses and corporations that, for whatever

*Stephen K. Campbell, less profit than they actually have. The technique Flaws and Fallacies in Statistical Thinking consists of expressing profit as a percentage of sales reason, want to make it appear that they have earned to Eugene Lyons�s Workers� Paradise Lost. volume instead of as a percentage of investment. (Englewood Cliffs, NJ: Prentice-Hall, 1974), 8. The original reference is

Section 12.6 Percentages 519

For example, during a certain year a corporation the case of the jewelry merchant who buys one might have a total sales volume of $100 million, a piece of jewelry each morning for $9 and sells it in total investment of $10�million, and a profit of $5 the evening for $10. At the end of the year the total million. If profits are expressed as a percentage of sales volume is $3650, the total investment $9, and investment, they amount to a hefty 50 percent; but the total profit $365. Profits as a percentage of sales as a percentage of sales, they are only 5�percent. To amount to only 10 percent, but as a percentage of appreciate the fallacy in this procedure, consider investment they exceed 4000 percent.

Exercise 12

I. Criticize the following arguments in light of the material presented in this chapter.

★1. To test the algae content in a lake, a biologist took a sample of the water at one

end. The algae in the sample registered 5 micrograms per liter. Therefore, the algae in the lake at that time registered 5 micrograms per liter.

2. To estimate public support for a new municipality-funded convention

center, researchers surveyed 100 homeowners in one of the city�s fashionable neighborhoods. They found that 89 percent of those sampled were enthusiastic about the project. Therefore, we may conclude that 89 percent of the city�s resi-dents favor the convention center.

3. A quality-control inspector for a food-processing firm needed assurance that

the cans of fruit in a production run were filled to capacity. He opened every tenth box in the warehouse and removed the can in the left front corner of each box. He found that all of these cans were filled to capacity. Therefore, it is prob-able that all of the cans in the production run were filled to capacity.

★4. When a random sample of 600 voters was taken on the eve of the presidential

election, it was found that 51 percent of those sampled intended to vote for the Democrat and 49 percent for the Republican. Therefore, the Democrat will probably win the popular vote.

5. To gauge the attitude of the voting public in regard to student loan cancelation,

1500 people were surveyed on college and university campuses. The names of fifteen campuses were drawn from a hat and, on each campus chosen, 100 people registered to vote were queried at different times and places on campus. Of this group, 83 percent said they favored canceling student debt. From these results we may conclude that approximately 83 percent of all people registered to vote favor canceling student debt.

6. To predict the results of a U.S. Senate race in New York State, two polls were

taken. One was based on a random sample of 600 voters, the other on a random sample of 2400 voters. Since the second sample was four times as large as the first, the results of the second poll were four times as accurate as the first.

★7. In a survey conducted by the manufacturers of Ultrasheen toothpaste, 65 percent

of the dentists randomly sampled preferred that brand over all others. Clearly, Ultrasheen is the brand preferred by most dentists.

520Chapter12 Statistical Reasoning

8. To determine the percentage of adult Americans who have never read the U.S.

Constitution, surveyors put this question to a random sample of 1000 adults. Only 13 percent gave negative answers. Therefore, since the sampling error for such a sample is 3 percent, we may conclude that no more than 16�percent of American adults have not read the Constitution.

9. To determine the percentage of patients who follow the advice of their personal

physician, researchers asked 200 randomly chosen physicians to put the ques-tion to their patients. Of the 2400 patients surveyed, 98 percent replied that they did indeed follow their doctor�s advice. We may therefore conclude that at least 96 percent of the patients across the nation follow the advice of their personal physician.

★10. Janet Ryan can afford to pay no more than $15 for a birthday gift for her eight-

year-old daughter. Since the average price of a toy at General Toy Company is $15, Janet can expect to find an excellent selection of toys within her price range at that store.

11. Anthony Valardi, who owns a fish market, pays $2 per pound to fishermen for

coho salmon. A certain fisherman certifies that the average size of the salmon in his catch of the day is 10 pounds, and that the catch numbers 100 salmon. Mr. Valardi is therefore justified in paying the fisherman $2000 for the whole catch.

12. Tamira intends to go shopping for a new pair of shoes. She wears size 8. Since

the average size of the shoes carried by the Bon Marche is size 8, Tamira can expect to find an excellent selection of shoes in her size at that store.

★13. Tim Cassidy, who works for a construction company, is told to load a pile of

rocks onto a truck. The rocks are randomly sized, and the average piece weighs 50 pounds. Thus, Tim should have no trouble loading the rocks by hand.

14. The average IQ (in the sense of mean, median, and mode) of the students in

Dr. Jacob�s symbolic logic class is 120. Thus, none of the students should have any trouble mastering the subject matter.

15. An insecticide manufacturer prints the following graph on the side of its

spray cans:

After one application

Number of bugs killed

Time

Section 12.6 Percentages 521

Obviously, the insecticide is highly effective at killing bugs, and it keeps work-ing for a long time.

★16. A corporation�s sales for two consecutive years are represented in a bar graph.

Since the bar for the later year is twice as high as the one for the previous year, it follows that sales for the later year were double those for the previous year.

17. Forced to make cutbacks, the president of a manufacturing firm reduced cer-

tain costs as follows: advertising by 4 percent, transportation by 5 percent, materials by 2 percent, and employee benefits by 3 percent. The president thus succeeded in reducing total costs by 14 percent.

18. During a certain year, a grocery store chain had total sales of $100 million and

profits of $10 million. The profits thus amounted to a modest 10 percent for that year.

★19. There were 30 percent more fatal traffic accidents in 2021 than there were in

2015. Therefore, it was 30 percent more dangerous to drive a car in 2021 than it was in 2015.

20. An efficiency expert was hired to increase the productivity of a manufactur-

ing firm and was given three months to accomplish the task. At the end of the period the productivity had increased from 1500 units per week to 1700. Since the goal was 2000 units per week, the effort of the efficiency expert was 85 percent successful (1700/2000).

21. A pharmaceutical company claims that its new cholesterol drug reduces the

likelihood of a heart attack by 50 percent. In a clinical trial consisting of 400 people, four people in the group not taking the drug suffered a heart attack, while only two people in the group taking the drug had a heart attack. This is a 50 percent reduction in the number of heart attacks.

II. Compute the answers to the following questions:

★1. What figures should be written in the left-hand column of Table 12.1, in the text,

to produce a similar table based on a confidence level of 99 percent? For a confi-dence level of 99 percent, the margin of error is approximately equal to 1.29/√n.

2. Given the following group of people together with their weights, what is the

average weight in the sense of mean, median, and mode?

Number of PeopleWeight

2 150

4 160

3 170

1 180

1 190

1 200

1 220

2 230

522Chapter12 Statistical Reasoning

3. Given the following group of people together with their salaries, what is the

average salary in the sense of mean, median, and mode?

Number of PeopleSalary

1 $190,000

2 170,000

1 140,000

3 80,000

1 60,000

2 40,000

5 30,000

★ 4. A day-care center cares for ten children. Their ages are 1, 1, 2, 2, 2, 3, 3, 4, 6, 6.

Construct a histogram that represents the distribution of ages. What is the mean age? What is the variance and standard deviation of these ages?

5. A small company has five employees who missed work during a certain month.

The number of days missed were 1, 1, 2, 4, 7. What is the mean number of days missed? What is the variance and standard deviation of this set of data?

6. An instructor gave a ten-question multiple-choice quiz to twelve students.

The scores were 10, 10, 9, 9, 8, 8, 8, 7, 7, 7, 7, 6. What is the mean score? What is the variance and standard deviation of these scores?

III. Answer �true� or �false� to the following statements:

★1. If a sample is very large, it need not be randomly selected.

2. If a population is randomly arranged, a sample obtained by selecting every

tenth member would be a random sample.

3. If a sample is randomly selected, the larger the sample is, the more closely it

replicates the population.

★4. To ensure the same precision, a population of 1 million would require a much

larger random sample than would a population of 100,000.

5. In general, if sample A is twice as large as sample B, then the sampling error for

A is one-half that for B.

6. When a sample consists of human beings, the purpose for which the sample is

taken can affect the outcome.

★7. The personal interaction between a surveyor and a respondent can affect the

outcome of a survey.

8. The mean value of a set of data is the value that occurs with the greatest frequency.

9. The median value of a set of data is the middle point when the data are arranged

in ascending order.

★ 10. The modal value of a set of data is the arithmetical average.

Section 12.6 Percentages 523

11. If one needed to know whether a sizable portion of a group were above or

below a certain level, the most useful sense of average would be mode.

12. Data reflecting the results of a random sample conform fairly closely to the

normal probability distribution.

★13. If a set of data conforms to the normal probability distribution, then the mean,

median, and mode have the same value.

14. Range, variance, and standard deviation are measurements of dispersion.

15. Statements about averages often present an incomplete picture, lacking infor-

mation about the dispersion.

★16. Data reflecting the size of full-grown horses would exhibit greater dispersion

than would data reflecting the size of full-grown dogs.

17. The visual impression made by graphs can be exaggerated by changing one of

the scales while leaving the other unchanged.

18. Data reflecting a 100 percent increase in housing construction could be

accurately represented by a pictogram of two houses, one twice as high as the other.

★19. If a certain quantity is increased by 10 percent and later decreased by 10� percent,

the quantity is restored to what it was originally.

20. Expressing profits as a percentage of sales volume presents an honest picture of

the earnings of a corporation.

IV. Do some research and find a way that people or corporations cheat with statistics. Write

a paragraph describing what you found.

Summary

Statistical Reasoning:

■ Includes any kind of argumentation based on statistical measurement.

■ Such arguments can be misleading for several reasons:

● Biased samples:

▶ Not randomly selected

▶ Too small (note sampling error)

▶ Psychological factors

● Different meaning for �average�:

▶ Mean (the numerical average)

▶ Median (the midpoint)

▶ Mode (the value that occurs with the greatest frequency)

524Chapter12 Statistical Reasoning

● Failure to take dispersion into account:

▶ Range (difference between largest and smallest values)

▶ Variance (how much every data point varies from the mean)

▶ Standard deviation (square root of the variance)

● Illusory graphs:

▶ Failure to scale the axes

▶ Chopping off the bottom of the graph

▶ Altering the scale of the axes

● Illusory pictograms:

▶ Illusions related to spatial dimensions

● Improper percentages:

▶ Failure to state the base of the percentage

▶ Slight-of-hand tricks with the base

▶ Numerical errors

Summary525

13Hypothetical/Scientific Reasoning

13.1 The Hypothetical Method

13.2 Hypothetical Reasoning: Four Examples from Science

13.3 The Proof of Hypotheses

13.4 The Tentative Acceptance of Hypotheses

13.1 The Hypothetical Method

Learning how to reason with hypotheses is empowering. Suppose it should happen that you can�t locate your wallet. You ask yourself, where might it be? Perhaps it�s in the bedroom, where you changed your clothes. Or on the floor of the closet. Possibly it�s in the car. Perhaps you absentmindedly put it in the refrigerator. You look in the car and find it between the front seats. The suggestions you posed to yourself are hypotheses, and they often form the basis for a natural and productive way of reasoning.

Hypothetical reasoning is most immediately applied to the production of explanations. You will recall from Chapter 1 of this book that an explanation is a kind of expression that purports to shed light on some event. Many explana-tions are grounded in our immediate experience or background knowledge. For example, if someone should ask you why the Titanic sank, you would answer that it sank because it hit an iceberg. Why did you fail the calculus test? Because I didn�t study. Why did President Kennedy die so young? Because he was assassinated. Why are you so jittery? Because I just drank three cups of coffee. No hypotheses are needed to produce explanations of this sort.

However, it often happens that the needed explanation is not so forthcom-

ing. Suppose you are faced with questions such as these: Why is the suicide

526

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). rate for young men four times the rate for young relativity are all hypotheses for making sense of women? Why has the incidence of autism exploded the data of observation. The problem for the sci-in the past twenty-five years? Why is the universe entist is that the underlying structure of nature is expanding more rapidly today than it was a mil- hidden from view, and the data of observation by lion years ago? Why do animals sleep? No appeal themselves are not sufficient to reveal this struc-to immediate experience or background knowledge ture. In response, the scientist constructs hypoth-will supply an answer. Clearly a different approach eses that provide ways of conceptualizing the data is needed. and that suggest specific questions to be answered

For a simpler example, suppose you try to open through the design of controlled experiments.

a window but find that it will not budge. Since it Analogously, every philosophical system can be is impossible to see what is jamming it, you make viewed as a grand hypothesis for interpreting the a conjecture. Perhaps the window has been painted content of experience. Plato�s theory of forms, shut from the outside. If that is the case, then run- Aristotle�s theory of substance, Leibniz�s monads, and ning a knife along the edge of the window from Kant�s theory about the mind are all hypotheses the outside should free it up. You do this, and the aimed at illuminating various aspects of experience. window opens. In solving the stuck-window prob- Just as the structure of nature is hidden from the lem, the informed conjecture that you made is a scientist, the meaning of experience is hidden from hypothesis, and the reasoning process you used to the philosopher, and ordinary common sense will produce it, draw an implication, and test the impli- not provide the answer. In response, the philosopher cation is called hypothetical reasoning. constructs hypotheses that can be used to shed light

Hypothetical reasoning is used by nearly all on the content of experience and to provide sugges-

of us in our day-to-day experience. The auto tions for further analysis. mechanic constructs hypotheses to determine why Whether it is applied in philosophy, science, or a vibration occurs at certain speeds, the motorist ordinary life, the hypothetical method involves four on the freeway or turnpike reasons hypothetically basic stages: to determine why the traffic is backed up bumper-

to-bumper even though it is not yet rush hour, the 1. Occurrence of a problem

physician hypothesizes about the cause of a dis- 2. Formulating a hypothesis

ease prior to prescribing medicine, the teacher 3. Drawing implications from the hypothesis

hypothesizes about the best way to present a com- 4. Testing the implications plicated subject in the classroom, and the pros-

ecuting attorney suggests hypotheses to the jury These four stages may be illustrated through the in arguing about the motive for a crime. In all of procedure used by a detective in solving a crime. these cases the evidence is not sufficient to indi- Suppose that a woman has been murdered in her cate exactly what is going on, what lies behind the apartment. Initially, everything in the apartment is scene, or what approach to take, so hypotheses are a potential clue: the empty wine glasses in the sink, constructed to make sense of the situation and to the small container of cocaine on the coffee table, direct future action. the automobile key found on the carpet, the strand

Hypothetical reasoning is used most explicitly of blond hair removed from the couch, and so on.

in philosophical and scientific inquiry. Every sci- To introduce an element of rationality into the situ-entific theory can be viewed as a hypothesis for ation, the detective formulates a hypothesis�let us unifying and rationalizing events in nature. The say the hypothesis that the key found on the carpet Ptolemaic and Copernican theories about the fits the murderer�s car. sun and planets, Dalton�s atomic theory, Darwin�s From this hypothesis several implications can be theory of evolution, and Einstein�s theory of drawn. Suppose that the key is the kind that fits only

Section 13.1 The Hypothetical Method 527

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). late-model Cadillacs. It follows that the murderer Such a fact would prove the hypothesis false because drives a late-model Cadillac. Furthermore, if the key it would falsify the implication that the murderer is the only one the murderer owns, it follows that drives a Cadillac. The argument form involved in the car may be parked nearby. A third implication such an inference is modus tollens:

is that the murderer�s name may be on record at the If H, then I local Cadillac dealership. To test these implications, not I the detective conducts a search of the streets in the not H vicinity and contacts the local Cadillac dealer for the

names of recent buyers. For the hypothesis to be proved true, the car that the This example illustrates three additional points key fits would have to be found and the owner would about hypotheses. The first is that a hypothesis is not have to confess to the crime. derived from the evidence to which it pertains but As we have just shown, hypotheses and hypo-rather is added to the evidence by the investigator. A thetical reasoning play an important role in many hypothesis is a free, albeit educated, creation of the of life�s endeavors�from philosophy, to science, to mind used to structure the evidence and unveil the the solution of ordinary day-to-day problems. An pattern that lies beneath the surface. It may be that important final question is, How does the hypotheti-the detective�s hypothesis is completely false. Perhaps cal reasoning that occurs in science differ from these the key fits a car that was lent to the victim for the other modes of hypothetical reasoning ? One answer weekend. A wide variety of other possibilities are relates to the kinds of tests that scientific hypotheses conceivable. are subjected to. Scientific tests always involve the The second point is that a hypothesis directs the measurement of something, and the results of that search for evidence. Without a hypothesis for guid-measurement are nearly always expressed in the lan-ance, all facts are equally relevant. The mineral con-guage of mathematics. tent of moon rocks and the temperature in the Sahara For example, physics and chemistry test hypotheses would be as relevant as the cars parked on the street by measuring physical quantities such as mass, force, outside the apartment. The hypothesis tells the inves-distance, time, motion, pressure, pH levels, heat, and tigator what to look for and what to ignore. so on. For this purpose, instruments are nearly always The third point concerns the proof of hypotheses. used�meter sticks, clocks, balance scales, micro-Let us suppose that the detective finds a late-model scopes, thermometers, and such. These instruments Cadillac parked outside the apartment building and automatically yield a mathematical expression of the that the key fits the ignition. Such a discovery might measurement�10 centimeters, 4 minutes, 20 grams. lend credibility to the hypothesis, but it would not Many of these same kinds of measurements are also in any sense prove it. Concluding that a hypothesis used in biology. For example, a biologist wanting to is proven true by the discovery that one of its impli-test the hypothesis that water temperature affects the cations is true amounts to committing the fallacy of color of fish would use a thermometer to measure the affirming the consequent (see Section 6.6). Where temperature of the water and a color spectrometer or H stands for a hypothesis and I for an implication, color chart to measure the color of the fish. such an argument has the invalid form Social scientists also use measurements to test

If H, then I hypotheses. Suppose a psychologist wanted to test I the hypothesis that playing violent video games H causes children to exhibit aggressive behaviors. The

psychologist might first develop a scale to measure

Let us suppose, on the other hand, that the murderer the violence levels of the games, and then con-turns himself or herself in to the police and that struct a chart of aggressive behaviors to measure the only car the murderer owns or drives is a Ford. the aggression levels of the children. Similarly, a

528Chapter13 Hypothetical/Scientific Reasoning

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). sociologist wanting to test the hypothesis that drug These types of test procedures differ from the use on a college campus is correlated with date kind of test undertaken in the stuck-window exam-rape might begin by accumulating statistics for ple, where a knife was used to cut through dried the phenomena in question, and then use statisti- paint. Most hypotheses formed in connection with cal methods to prove the correlation. Collecting ordinary life problems are like this one and do not the initial statistics for such research amounts to a require measurements as a test. But in science, the measurement of the degree to which the phenom- lack of precise measurements would undermine the enon occurs in the community. credibility of most test procedures.

13.2 Hypothetical Reasoning:

Four Examples from Science

Some of the clearest illustrations of the hypothetical photographic plates she substituted an electrometer, method of reasoning can be found in scientific dis- which was better suited to measuring the intensity of coveries. Four examples are the discovery of radium the rays, and she proceeded to conduct various experi-by Pierre and Marie Curie; the discovery of the ments with pure uranium to determine the source of planet Neptune by Adams, Leverrier, and Galle; the the rays that the metal emitted. When none of these discovery of atmospheric pressure by Torricelli; and experiments proved fruitful, she shifted her attention Pasteur�s research concerning the spontaneous gen- to the question of whether other metals or minerals eration of life. Following is a consideration of each emitted the same kind of rays as uranium. She tested of these examples with special attention to the four hundreds of metals, compounds, and ores, but the only stages of hypothetical inquiry. one that proved interesting was pitchblende, a certain

ore of uranium. Because pitchblende contained ura-

Radium nium, she anticipated that it would emit rays; however,

because it also contained a number of impurities, she

In 1896, the French physicist Henri Becquerel dis- expected the rays to be weaker than they were for pure covered that crystals containing uranium had the uranium. Instead, they turned out to be stronger. This power to expose photographic plates. He found that problem caught Madame Curie�s attention and pro-when these crystals were placed on top of an unex- vided the focus for her research in the months ahead. posed plate and left for a certain time, dark blotches In response to the problem, Madame Curie formu-appeared in their place when the plate was devel- lated the hypothesis that the impurities in the pitch-oped. Becquerel concluded that the crystals emitted blende somehow triggered the uranium to increase the certain rays that penetrated the opaque covering of emission of rays. One implication of this hypothesis the plates and reacted with the photosensitive mate- was that mixing pure uranium with the kinds of impu-rial underneath. Further investigation showed that rities found in pitchblende would cause an increase these rays were not as strong as X-rays, which could in the emission of rays. To test this implication, Curie be used to photograph bone structure, and so Bec- diluted pure uranium with various elements and mea-querel�s interest in them lapsed. sured the strength of the rays. The results were always

A year later, Marie Curie revived the question when the same: The emissions were always less than they

she adopted it as the topic of her doctoral research were for pure uranium. Because of these results, she at the University of Paris. In place of Becquerel�s abandoned the hypothesis.

Section 13.2 Hypothetical Reasoning: Four Examples from Science 529

Madame Curie then formulated a second hypoth- gravitational interactions between Uranus, Jupiter,

esis: The intensified emissions were caused directly and Saturn had been worked out mathematically. by some impurity in the pitchblende. The only other The latter task was accomplished by Pierre Laplace element besides uranium that was known to emit in his M�canique C�leste, and in 1820 Alexis Bou-rays, however, was thorium, and the pitchblende vard used this work to construct tables for all three that had been tested contained no thorium. Thus, planets. These tables predicted the orbital motions an immediate implication of the hypothesis was that of Jupiter and Saturn very accurately, but within a the increased rays were caused by an unknown ele- few years Uranus was found to have deviated from ment. A second implication was that this element its predicted path. A problem thus emerged: Why could be separated from the pitchblende through did the tables work for Jupiter and Saturn but not a process of refinement. At this point Marie Curie for Uranus? was joined by her husband, Pierre, and they began In response to this problem, several astrono-a combined effort to isolate the unknown element. mers entertained the hypothesis that an eighth

Because the element was present in only the most planet existed beyond the orbit of Uranus and that

minute quantities, separating a measurable amount the gravitational interaction between these two from the other impurities required a great deal of planets caused Uranus to deviate from its predicted effort. The Curies began by grinding up some pitch- position. Not until 1843, however, did John Couch blende and dissolving it in acid. Finally, after numer- Adams, a recent graduate of Cambridge, undertake ous stages of filtration and the addition of other the task of working out the mathematical impli-chemicals, they obtained a pinch of white powder. cations of this hypothesis. After two years� work, By weight, this material was found to be 900 times Adams produced a table of motions and orbital more radioactive than pure uranium, but since the elements that predicted the location of the hypo-primary component in the powder was barium, the thetical planet, and his computations were so unknown element still had not been isolated. accurate that if anyone with a telescope had both-

Rather than continue with additional stages of ered to look, they would have found the new

refinement, the Curies decided to attempt a spectro- planet within two degrees of its predicted position. graphic analysis of the powder. Such analysis, they Unfortunately, no one looked for it. hoped, would reveal the characteristic spectrum At about the same time that Adams completed line of the unknown element. This proposal, which his work on the problem, the French astronomer amounted to a third implication of the hypothesis, U. J. J. Leverrier, working independently of Adams, was put to the test. When the powder was burned reported a similar set of motions and orbital ele-in a spectrometer, a line appeared in the ultraviolet ments to the French Academy of Science. The close range that was different from that for any other ele- agreement between Adams�s and Leverrier�s predic-ment. From the combined evidence of the spectrum tions prompted a search for the planet, but because line and the intense radiation, the Curies announced a rather broad section of sky was swept, the planet in 1898 the discovery of a new element, which they was missed. called radium. After more processing and refine- Finally, Leverrier sent a copy of his figures to ment, enough of the material was finally obtained to Johann Galle at the Berlin Observatory, where a determine its atomic weight. set of star charts was being prepared. It was sug-

gested that the region corresponding to Leverrier�s

Neptune computations be observed and the results matched

against the charts. This was done, and a small star-

In 1781, William Herschel discovered the planet like object was found that was not on the charts. Uranus. However, the production of a table giving The next night the same object was sighted, and the motion of the new planet had to wait until the it was found to have moved. The new planet was

530Chapter13 Hypothetical/Scientific Reasoning

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). thus identified. It was named Neptune after most tolerate the creation of one in the glass vessel? Fur-astronomers outside France objected to the origi- thermore, why did the water always descend to the nal suggestion that it be called Leverrier. same level in the pipe?

The results of Berti�s experiment were commu-

Atmospheric Pressure nicated to Evangelista Torricelli in Florence, who

was at that time Galileo�s assistant. Galileo himself

Originated by Aristotle, the principle that nature thought that the water was supported in the pipe by abhors a vacuum was used for centuries to explain the power of the vacuum, but after Galileo�s death in the fact that in emptying a keg of wine an opening 1642, Torricelli formulated his own hypothesis: The had to be made at the top as well as at the bottom. water was supported in the pipe by the pressure of Because nature would not allow a vacuum to be the atmosphere. Torricelli reasoned that we live �at created inside the keg, the wine would not drain the bottom of an ocean of air� and that the pressure from the bottom until air was let in at the top. It of the air pushing against the bottom of the pipe sup-was thought that this principle held universally for ported the water at a certain height in the pipe. A all applications involving a vacuum, but in the six- point of equilibrium was reached, he thought, when teenth century it was found that suction pumps used the weight of the water remaining in the pipe equaled to drain water from mine shafts would not work if the weight of the air pushing down from above. the pump was situated over 30 feet above the water From this hypothesis Torricelli derived several level. This caused people to wonder whether nature�s implications. One was that the pressure of the abhorrence of a vacuum, while holding true for kegs atmosphere would support a column of mercury of wine, had certain limits for pumps. about 29 inches high in a tube sealed at the top.

In 1630, Giovanni Baliani of Genoa discovered This followed from the fact that the atmosphere

a similar limitation in regard to siphons. When he supports a column of water 33 feet high, that attempted to siphon water from a reservoir over mercury is 13.6 times as dense as water, and that a 60-foot hill, he found that the siphon would not 33/13.6 � 12 inches = 29 inches. A second impli-work. When the siphon was completely filled with cation was that such a tube filled with mercury water and the stoppers were removed from both could be used to measure fluctuations in atmo-ends, a vacuum seemed to be created in the upper- spheric pressure. This second implication won most parts of the pipe. Torricelli credit for formulating the theory of the

These findings were communicated to Gasparo barometer. Finally, Torricelli reasoned that if such

Berti in Rome, who, around 1641, attempted to a device were conveyed to a place where the air determine more scientifically whether a vacuum was more rarefied, such as on a mountaintop, the could actually be created. Berti designed an appara- column of mercury would descend. tus consisting of a spherical glass vessel attached to The first of these implications was tested by Tor-a pipe about 40 feet long. The apparatus was affixed ricelli�s associate, Vincenzo Viviani. Viviani obtained upright to the side of a tower and after the valve at the a 4-foot section of glass tube sealed at one end, lower end of the pipe was closed, water was poured enough mercury to completely fill it, and a dish to through the upper opening in the glass vessel. When hold more mercury. After pouring the mercury into both the pipe and the glass vessel were completely the tube, Viviani placed his thumb over the open filled, the opening in the vessel was sealed and the end, inverted the tube, and placed the open end in valve at the lower end of the pipe was opened. Imme- the dish of mercury. After he released his thumb, he diately water rushed from the bottom of the pipe, cre- watched the column of mercury descend to about ating a vacuum in the glass vessel. This experiment 29 inches above the level of mercury in the dish. crystallized a problem that had been developing for Thus was created the first barometer. Its successful many years: If nature abhorred a vacuum, why did it use in measuring atmospheric pressure came later.

Section 13.2 Hypothetical Reasoning: Four Examples from Science 531

The test of Torricelli�s third implication was and wasps, ticks, fireflies, and other insects devel-

taken up in 1647 by the French philosopher Blaise oped continually from the morning dew and from Pascal. Having received word of Torricelli�s experi- dry wood and hair. He also held that crabs and ments with the barometer, Pascal constructed one various mollusks developed from moist soil and for himself. He readily became convinced of the decaying slime. Extensions of this theory prevailed correctness of Torricelli�s hypothesis, and to dem- throughout the Middle Ages and well into modern onstrate its correctness in opposition to the vacuum times. In the seventeenth century it was widely held principle, he requested that his brother-in-law, that frogs were produced from the slime of marshes F. Perier, transport a barometer to the top of the Puy de and eels from river water, and the physician Van D�me, one of the highest mountains in Auvergne. Helmont thought that mice were produced from A year later, Perier was able to fulfill this request. the action of human sweat on kernels of wheat. All He began the experiment by setting up two barom- one needed to do, according to Van Helmont, was eters in the monastery at the foot of the mountain. toss a dirty shirt into a container of wheat, and in After noting that both columns of mercury rose 21�days the container would be teeming with mice. to an identical height, he disassembled one of the Even Descartes and Newton accepted the theory of barometers and instructed one of the friars to check spontaneous generation. Descartes held that various the mercury level in the other throughout the day. plants and insects originated in moist earth exposed Then Perier, accompanied by a group of witnesses, to sunlight, and Newton thought that plants were set off up the mountain with the other barometer. produced from emanations from the tails of comets. On reaching the summit, he assembled the second The first systematic effort to abolish the belief in barometer and discovered to the amazement of all spontaneous generation was made by the Italian phy-that the mercury level was more than 3 inches lower sician Francesco Redi. In response to the commonly than it had been at the foot of the mountain. As a held idea that worms were spontaneously generated double-check, the barometer was taken apart and in rotting meat, Redi hypothesized that the worms reassembled at five different spots on the summit. were caused by flies. An immediate implication was Each time, the results were the same. that if flies were kept away from the meat, the worms

At the midpoint of his descent, Perier reassembled would not develop. To test this hypothesis, Redi cut

the barometer once again. He found that the mercury up a piece of meat and put part of it in sealed glass level was about midway between where it was at the flasks and the other part in flasks open to the air. bottom and at the top of the mountain. Finally, on Flies were attracted to the open flasks, and in a short returning to the monastery, the friar who had been time worms appeared; but no worms developed in watching the barometer there was questioned about the flasks that were sealed. what he had observed. He reported that the mer- When Redi published his findings in 1668, they cury level had not changed since early that morning immediately affected the theory of spontaneous gen-when the group had departed. Pascal announced the eration. Within a few years, though, the microscope results of this experiment to the educated world, and came into common use, and it was discovered that the announcement succeeded in abolishing the prin- even though meat sealed in glass containers pro-ciple that nature abhors a vacuum. duced no worms, it did produce countless microor-

ganisms. The theory of spontaneous generation was

Spontaneous Generation thus reawakened on the microbial level.

By the middle of the nineteenth century the

The theory of spontaneous generation holds that theory had received considerable refinement. It living beings arise spontaneously from lifeless was thought that spontaneous generation resulted matter. The roots of the theory extend into ancient from the direct action of oxygen on lifeless organic times. Aristotle held that worms, the larvae of bees nutrients. Oxygen was thought to be essential to the

532Chapter13 Hypothetical/Scientific Reasoning

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). process because the technique of canning fruits and second hypothesis that life forms are carried by dust vegetables had come into practice, and people knew particles in the air. To test this second hypothesis, that boiling fruits and vegetables and sealing them Pasteur took a wad of cotton and drew air through it, in the absence of oxygen would cause them to be trapping dust particles in the fibers. Then he washed preserved. If they were left exposed to the air, how- the cotton in a mixture of alcohol and examined ever, microbes would develop in a short time. drops of the fluid under a microscope. He discov-

One of the defenders of spontaneous generation ered microbes in the fluid.

at that time was the Englishman John Needham, an Returning to his first hypothesis, Pasteur pre-amateur biologist. Needham conducted an experi- pared a nutrient solution and boiled it in a narrow-ment in which flagons containing oxygen and a necked flask. As the solution boiled, the air in the vegetable solution were buried in hot coals. The neck of the flask was forced out by water vapor, and coals would have been expected to kill any life in the as it cooled the water vapor was slowly replaced solution, but several days later the contents of the by sterilized air drawn through a heated platinum flagons were alive with microbes. Needham con- tube. The neck of the flask was then closed off with cluded that the oxygen acting alone on the nutrient a flame and blowpipe. The contents of the flask thus solution caused the generation of the microbes. In consisted of a sterilized nutrient solution and unpol-response to this experiment, Lazzaro Spallanzani, luted sterilized air�all that was supposedly needed an Italian physiologist, conducted a similar experi- for the production of life. Over time, however, no life ment. To ensure that the nutrient solution was life- developed in the flask. This experiment posed a seri-less, he boiled it for an hour. Later, no microbes ous threat to the theory of spontaneous generation. could be found. To this Needham objected that in Pasteur now posed the hypothesis that sterile nutri-boiling the solution for a full hour Spallanzani had ent solutions exposed to the air normally developed destroyed its �vegetative force.� In addition, Needham life forms precisely because these forms were deposited argued, he had polluted the small amount of by dust particles. To test this third hypothesis Pasteur oxygen in the containers by the fumes and heat. reopened the flask containing the nutrient solution, Thus, it was no wonder that microbes were not and, using a special arrangement of tubes that ensured spontaneously generated. that only sterilized air would contact the solution, he

To settle the issue once and for all, the French deposited a piece of cotton in which dust particles had

Academy of Science offered a prize for an experi- been trapped. The flask was then resealed, and in due mental endeavor that would shed light on the ques- course microbes developed in the solution. This exper-tion of spontaneous generation. This challenge drew iment proved not only that dust particles were respon-Louis Pasteur into the controversy. Spontaneous sible for the life but that the �vegetative force� of the generation presented a special problem for Pas- nutrient solution had not been destroyed by boiling, as teur because of his previous work with fermenta- Needham was prone to claim. tion. He had discovered that fermentations, such as Pasteur anticipated one further objection from the those involved in the production of wine and beer, proponents of spontaneous generation: Perhaps the required yeast; and yeast, as he also discovered, was capacity of oxygen to generate life was destroyed by a living organism. In view of these findings, Pasteur drawing it through a heated tube. To dispel any such adopted the hypothesis that life comes only from life. notions Pasteur devised yet another experiment. He An immediate implication was that for life forms to boiled a nutrient solution in a flask with a long, narrow develop in a sterile nutrient solution, they must first gooseneck. As the solution boiled, the air was forced be introduced into the solution from the outside. out, and as it cooled, the air returned very slowly

It was well known that life forms did indeed through the long neck, trapping the dust particles on

develop in sterile nutrient solutions exposed to the moist inside surface. No microbes developed in the the air. To account for this Pasteur adopted the solution. Then, after a prolonged wait, Pasteur sealed

Section 13.2 Hypothetical Reasoning: Four Examples from Science 533

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). the flask and shook it vigorously, dislodging the par- the prize that had been offered a year earlier. The ticles that had settled in the neck. In a short time, the experiments dealt a mortal blow to the theory of solution was alive with microbes. spontaneous generation, and although the theory

When Pasteur reported these experiments to was not abandoned immediately, by 1900 it had

the Academy of Science in 1860, he was awarded very little support.

13.3 The Proof of Hypotheses

The four instances of hypothetical reasoning in sci- only confirmed to varying degrees. The greater the ence that we have investigated illustrate the use of number of implications that are found to be cor-two different kinds of hypotheses. The hypotheses rect, the more certain we can be of the hypothesis. involved in the discovery of Neptune and radium If an implication is found to be incorrect, however, are sometimes called empirical hypotheses, and a theoretical hypothesis can be disproved. For exam-those relating to atmospheric pressure and spon- ple, if it should happen some day that life is pro-taneous generation are sometimes called theoreti- duced in a test tube from inorganic materials, Pas-cal hypotheses. Empirical hypotheses concern teur�s hypothesis that life comes only from life might the production of some thing or the occurrence of be considered to be�disproved. some event that can be observed. When radium had The problem with the distinction between empir-finally been obtained as a pure metal it was some- ical and theoretical hypotheses is that observation thing that could be seen directly, and when Neptune is theory-dependent. Consider, for example, a man was finally sighted through the telescope, it, too, had and a woman watching a sunrise. The man happens been observed. Theoretical hypotheses, on the to believe that the sun travels around the earth, as other hand, concern how something should be con- Ptolemy held, and the woman that the earth trav-ceptualized. When Galileo observed the water level els around the sun, as Copernicus and Galileo con-rising in a suction pump, he conceived it as being tended. As the sun rises, the man thinks that he sees sucked up by the vacuum. When Torricelli observed the sun moving upward, while the woman thinks it, however, he conceived it as being pushed up by she sees the earth turning. The point is that all of us the atmosphere. Similarly, when Needham observed have a tendency to see what we think is out there to life emerging in a sterile nutrient solution, he con- be seen. As a result, it is sometimes difficult to say ceived it as being spontaneously generated by the when something has or has not been observed. action of oxygen. But when Pasteur observed it, he In regard to the discovery of Neptune, the conceived it as being implanted there by dust par- unknown planet was observed two times in 1795 ticles in the air. by J. J. Lalande, fifty-one years before it was �discov-

The distinction between empirical and theo- ered� by Adams, Leverrier, and Galle. Lalande noted

retical hypotheses has certain difficulties, which that his observations of the position of the small we will turn to shortly, but it sheds some light on starlike object were discordant, so he rejected one as the problem of the verification or confirmation of erroneous. Because he thought he was observing a hypotheses. Empirical hypotheses are for all practi- star, he received no credit for discovering a planet. cal purposes proved when the thing or event hypoth- Analogous remarks extend to Galle�s observations esized is observed. Today, practically all of us would of the planet Neptune in 1846. If Leverrier�s compu-agree that the hypotheses relating to radium and tations had been erroneous, Galle might have seen Neptune have been established. Theoretical hypoth- what was really a comet. Thus, if we can never be eses, on the other hand, are never proved but are sure that we really see what we think we see, is it

534Chapter13 Hypothetical/Scientific Reasoning

Prominent Charles Sanders Peirce 1839�1914Logicians

C harles Sanders Peirce (pronounced �purse�) alive. Beginning in the was born in Cambridge, Massachusetts. late 1930s, his writ-His father, Benjamin, was the leading ings were collected mathematician of the day, and he took a special inter-and published, bring-

est in his son�s intellectual development. Under his ing him the atten-direction Charles was reading college-level material, tion he missed during including logic, at age twelve, and Benjamin would his lifetime. As the challenge the boy with highly complex problems that founder of pragmatism, Charles would solve on his own. Although his most America�s only unique significant education came from his father, Charles philosophy, Peirce is went on to attend Harvard University, where he now considered to received a BS in chemistry in 1863. Yet, he was not a �s People Collection be America�s great-successful student (he usually placed in the lower quar- NOAA est philos opher and ter of his class), partly because he showed disdain for probably its greatest his professors as inadequately qualified to teach him. logician as well. Peirce�s own version of pragmatism This arrogance is likely the main reason Peirce lived a took the form of a thoroughly scientific enterprise difficult life. He died in poverty at age seventy-four in according to which the meaning of a concept depends the then-isolated town of Milford,�Pennsylvania. on the practical effect it has on how we lead our lives

Throughout his life, Peirce suffered a painful and and conduct our inquiries.

debilitating neurological condition that may have been In logic, Peirce is best known for his work in the responsible for his detached, depressed, and some- logic of relations and quantification theory. The times violent behavior. It may also have been the reason theory of quantification that Peirce developed is he could not obtain a tenured university position. In the one that led ultimately to the writing of Principia 1879, he did secure a position as a lecturer at Johns Mathematica by Whitehead and Russell, and it closely Hopkins University, and in some ways, he proved him- resembles the system used today in most logic self a remarkable teacher�a collection of his students� courses. In the logic of relations, Peirce extended essays is considered a key work in the development of the work done by De Morgan by applying the con-logic. Yet, despite his teaching, Peirce was discharged cepts of Boolean algebra to relative terms such as from his faculty position. He had recently separated �friend of� or �enemy of.� This work has also signifi-from his wife and had lived with another woman before cantly impacted the way we treat this subject today. his divorce was finalized. This �immoral� behavior and In addition, Peirce was a leader in exploring the use other supposed transgressions were probably the chief of truth values, and he was the first to develop a factors leading to his dismissal. After that he never held three-valued logic incorporating true, false, and an another academic position. intermediate or unknown value. In propositional

Without an academic base, Peirce continued to logic Peirce succeeded in reducing all the con-

write copiously, producing some 80,000 pages of nectives to one, and in inductive logic he rejected material; yet he did not complete a single book in the subjectivist theory of probability in favor of an logic or philosophy, which limited his influence while objectivist theory based on observed regularities.

Section 13.3 The Proof of Hypotheses 535

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). ever possible for a hypothesis to be actually proved? Copernicus�s hypothesis that the earth and planets Perhaps it is better to interpret the proof of empirical move around the sun, instead of the sun and planets hypotheses as a high degree of confirmation. around the earth, as Ptolemy hypothesized. Can we

Conversely, with theoretical hypotheses, would consider this theoretical hypothesis to be proved?

we want to say that Torricelli�s hypothesis relat- If a motion picture camera were sent outside the ing to atmospheric pressure has not been proved? solar system and pictures were taken supporting the Granted, we cannot observe atmospheric pressure Copernican hypothesis, would we say that these pic-directly, but might we not say that we observe it tures constituted proof? We probably would. Thus, instrumentally? If barometers can be regarded as while the distinction between theoretical and empir-extensions of our sense organs, Torricelli�s hypothe- ical hypotheses is useful, it is more a distinction in sis has been proved. Another example is provided by degree than in kind.

13.4 The Tentative Acceptance

of Hypotheses

A certain amount of time is required for a hypoth- A hypothesis is inadequate to the extent that facts esis to be proved or disproved. The hypotheses relat- exist that the hypothesis cannot account for. The ing to the discovery of radium and Neptune required principle that nature abhors a vacuum was inad-more than a year to prove. Theoretical hypotheses equate to explain the fact that water would rise no in science often take much longer, and theoretical more than 30 feet in suction pumps and siphons. hypotheses in philosophy may never be confirmed Nothing in the hypothesis could account for this fact. to the satisfaction of the majority of philosophers. Similarly, Needham�s hypothesis that life is generated During the period that intervenes between the pro- by the direct action of oxygen on nutrient solutions posal of a hypothesis and its proof, confirmation, was inadequate to account for the fact that life would or disproof, the question arises as to its tentative not develop in Pasteur�s flask containing a sterilized acceptability. Four criteria that bear on this question nutrient solution and sterilized oxygen. are (1) adequacy, (2) internal coherence, (3) external In scientific hypotheses, a second kind of ade-consistency, and (4) fruitfulness. quacy is the accuracy with which a hypothesis

Adequacy is the extent to which a hypothesis accounts for the data. If one hypothesis accounts for

fits the facts it is intended to unify or explain. A a set of data with greater accuracy than another, then hypothesis is said to �fit� the facts when each fact that hypothesis is more adequate than the other. For can be interpreted as an instance of some idea or example, Kepler�s hypothesis that the orbits of the term in the hypothesis. For example, before the planets were ellipses rather than circles, as Coperni-Neptune hypothesis was confirmed, every fluc- cus had hypothesized, accounted for the position of tuation in the position of Uranus could be inter- the planets with greater accuracy than the Coperni-preted as an instance of gravitational interaction can hypothesis. Similarly, Einstein�s theory of relativ-with an unknown planet. Similarly, before Torricelli�s ity accounted for the precise time of certain eclipses hypothesis was confirmed, the fact that water would with greater accuracy than Newton�s theory did. For rise only 30� feet in suction pumps and siphons these reasons, Kepler�s and Einstein�s theories were could be interpreted as an instance of equilibrium more adequate than the competing theories. between the pressure of the water and the pressure Internal coherence is the extent to which of the atmosphere. the component ideas of a hypothesis are rationally

536Chapter13 Hypothetical/Scientific Reasoning

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). interconnected. The purpose of a hypothesis is to everything. Spinoza�s theory is thus more internally unify and interconnect a set of data and by so doing coherent than Descartes�s. to explain the data. Obviously, if the hypothesis itself External consistency occurs when a hypoth-is not internally connected, there is no way that it esis does not disagree with other, well-confirmed can interconnect the data. After Adams and Lever- hypotheses. Adams�s and Leverrier�s hypothesis of rier had worked out the mathematical details of the an eighth planet was perfectly consistent with the Neptune hypothesis, it exhibited a great deal of inter- nineteenth-century theory of the solar system, nal coherence. The hypothesis showed how all the and it was rendered even more attractive by the fluctuations in the position of Uranus could be ratio- fact that the seventh planet, Uranus, had been dis-nally linked in terms of the gravitational interaction covered only a few years earlier. Similarly, Marie of an eighth planet. Similarly, Torricelli�s hypothesis Curie�s hypothesis of the existence of a new ele-showed how the various fluid levels could be ratio- ment was consistent with Mendeleev�s periodic nally interconnected in terms of the equilibrium of table and with the general hypothesis that elements pressures. Internal coherence is responsible for the could emit penetrating rays. In 1890, Mendeleev�s features of elegance and simplicity that often attract table had certain gaps that were expected to be scientists to a hypothesis. filled in by the discovery of new elements, and two

An example of incoherence in science is pro- ray-emitting elements, thorium and uranium, had

vided by the theoretical interpretation of light, already been discovered. electricity, and magnetism that prevailed during The fact that a hypothesis is inconsistent with the first half of the nineteenth century. During that other, well-confirmed hypotheses does not, how-period each of these phenomena was understood ever, immediately condemn it to obscurity. It often separately, but the interconnections between them happens that a new hypothesis arises in the face were unknown. Toward the end of the century the of another, well-confirmed hypothesis and that English physicist James Clerk Maxwell showed the two hypotheses compete for acceptance in the how these three phenomena were interconnected future. Which hypothesis will win is determined by in terms of his theory of the electromagnetic field. an appeal to the other three criteria. For example, Maxwell�s theory was thus more coherent than the Torricelli�s hypothesis was inconsistent with the ones that preceded it. ancient hypothesis that nature abhors a vacuum, and

Similarly, in philosophy, Spinoza�s metaphysical Pasteur�s hypothesis was inconsistent with the equally

theory is more internally coherent than Descartes�s. ancient hypothesis of spontaneous generation. In the Descartes postulated the existence of two kinds of end, the newer hypotheses won out because they substance to account for the data of experience. He were more adequate, coherent, or fruitful than their introduced extended, material substance to explain competitors. For the same reason, the Copernican the data of the visible world, and nonextended, hypothesis eventually triumphed over the Ptolemaic, non-material substance to explain the phenomena the theory of oxidation won out over the old phlogis-of the invisible world, including the existence and ton theory, and Einstein�s theory of relativity won out activity of the human soul. But Descartes failed to over Newton�s theory. show how the two kinds of substance were inter- Fruitfulness is the extent to which a hypoth-connected. In the wake of this disconnection the esis suggests new ideas for future� analy sis and famous mind�body problem arose, according to confirmation. Torricelli�s hypothesis suggested the which no account could be given of how the human design of an instrument for measuring fluctuations body acted on the mind through the process of sen- in the pressure of the atmosphere. Similarly, Pas-sation or how the mind acted on the body through teur�s hypothesis suggested changes in the proce-the exercise of free choice. Spinoza, on the other dures used to maintain sterile conditions in hospi-hand, postulated only one substance to account for tals. After these latter changes were implemented,

Section 13.4 The Tentative Acceptance of Hypotheses 537

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). the death rate from surgical operations decreased mercury level in a barometer decreases when the dramatically. The procedure of pasteurization, instrument is carried to the top of a mountain. Do used to preserve milk, was another outgrowth of the demons decide to maintain these levels by free the hypothesis that life comes only from life. choice or according to some plan? Because there is

Newton�s theory of universal gravitation is an no answer to this question, the hypothesis exhibits

example of a hypothesis that proved especially fruit- internal disconnectedness, which leaves it open to ful. It was originated to solve the problem of falling the charge of being irrational. As for the fourth cri-bodies, but it also explained such things as the ebb terion, the demon hypothesis is unfruitful because and flow of the tides, the orbital motion of the moon it suggests no new ideas that experimenters can and planets, and the fluctuations in planetary motion put to the test. The hypothesis that nature abhors a caused by a planet�s interaction with other planets. vacuum is hardly any more fruitful, which accounts Einstein�s theory of relativity is another example. in part for why it was so suddenly abandoned in It was originated to account for certain features of favor of Torricelli�s hypothesis�it simply did not Maxwell�s theory of electricity and magnetism, but it lead anywhere. ushered in the atomic age forty years later. In summary, for any hypothesis to receive tenta-

The factors of coherence and fruitfulness together tive acceptance it must cover the facts it is intended

account for the overall rationality and explanatory to interpret and it must rationally interconnect power of a hypothesis. Suppose, for example, that these facts�in other words, it must be adequate someone formulated the hypothesis that the water and coherent. After that, it helps if the hypoth-level in suction devices is maintained by the action esis does not conflict with other, well-confirmed of demons instead of by atmospheric pressure. hypotheses. Finally, it is important that a hypoth-Such a hypothesis would be neither coherent nor esis capture the imagination of the community to fruitful. It would not be coherent because it would which it is posed. This it does by being fruitful� not explain why the maximum water level in these by suggesting interesting ideas and experiments to devices is consistently about 30 feet, why the mer- which members of the community can direct their cury level in barometers is much less, and why the attention in the years ahead.

Exercise 13

I. For each of the following situations, invent at least three hypotheses aimed at explaining

what went wrong or at solving the problem. Then, for each hypothesis, describe one or more experiments or inquiries that would confirm or disconfirm the hypothesis.

1. Someone you have been dating has a birthday coming up, so you call a florist and

order flowers for the occasion. Two days later, the person is cold and distant.

2. After settling in a new location, you buy some seeds for a flower garden. When

the flowers come up, they are stunted and of poor quality.

3. After mailing your paycheck to the bank, you write a check to pay your rent.

The rent check bounces.

4. On a windy day you return home, switch on the lights, and they don�t go on.

You live in a house and share expenses with a housemate.

538Chapter13 Hypothetical/Scientific Reasoning

5. After driving for miles on a dusty road, you park your car near the bank of a

river and take a hike through the woods. Soon you become lost. In one direc-tion you see a small stream, and in the other, a trail.

6. A close friend calls you on the phone and confides a personal secret. That night

you write the secret in your diary. A few days later your friend is furious with you because the friend overheard someone on campus talking about the secret.

7. On a rainy evening you return to your ground-floor apartment and discover

that you have been burglarized. You can�t believe it, because you are cer-tain you locked the door when you went out. You share the apartment with a roommate.

8. You are a philosophy professor at a university, and while reading a set of term

papers you discover that two of them are identical. Those two papers are also extremely well written.

9. You are a manager of a department store, and, although business has been brisk

lately, your chief accountant informs you that the store is losing money.

10. A short time before final exams you lose your class notes. You recall having

seen them four days earlier. Use your own personal activities and movements during the past four days as the basis for formulating your hypotheses as to the whereabouts of the notes.

II. For the four scientific discoveries presented in this chapter, identify the problem, the

hypotheses that were formulated, the implications that were drawn, and the test proce-dure that was used. Include an account of measurements that were taken and possible instruments that were used.

III. Write a short paper (3�5 pages) on one of the following scientific events. Discuss the

problem, one or more hypotheses that were formulated, the implications that were drawn, and the test procedures that were used. Then evaluate the hypothesis in terms of adequacy, internal coherence, external consistency, and fruitfulness.

1. Isaac Newton: corpuscular theory of light

2. Christian Huygens: wave theory of light

3. Johannes Kepler: orbit of Mars

4. Nicolaus Copernicus: theory of the solar system

5. Count von Rumford: theory of heat

6. Charles Darwin: theory of natural selection

7. John Dalton: theory of atoms

8. William Harvey: circulation of the blood

9. Louis Pasteur: theory of vaccination

10. J. J. Thomson: discovery of the electron

Section 13.4 The Tentative Acceptance of Hypotheses 539

11. Andr� Marie Amp�re: discovery of the electromagnet

12. Niels Bohr: structure of the atom

13. Marie Curie: discovery of polonium

14. Alexander Fleming: discovery of penicillin

15. Henri Becquerel: radioactivity of uranium

16. Dmitri Mendeleev and Clemens Winkler: discovery of germanium

17. Dorothy Hodgkin: discovery of the structure of the insulin molecule

18. Cecilia Payne-Gaposchkin: discovery of the composition of stars

19. Amedeo Avogadro: Avogadro�s law

20. Johann Balmer: theory of the spectrograph

21. Alfred Wegener: theory of continental drift

22. Rosalind Franklin: discovery of the helical structure of A-DNA

23. James Watson and Francis Crick: structure of the DNA molecule

24. John Bardeen: theory of superconductivity

25. Inge Lehman: discovery of the composition of the Earth�s core

26. Albert Einstein: theory of Brownian motion

27. Lisa Meitner: discovery of nuclear fission

28. Barbara McClintock: discovery of jumping genes

29. Edwin Hubble: recession of the galaxies

30. Jean Baptiste Lamarck: inheritance of acquired characteristics

IV. Write a short paper (2�3 pages) analyzing one or more of the hypotheses formulated by

Sherlock Holmes in one of the stories by Arthur Conan Doyle. Include a discussion of the problem, the hypothesis, the implications that were drawn, and the test procedures.

V. Answer �true� or �false� to the following statements:

★1. Hypothetical reasoning is useful when the evidence by itself does not provide

the solution to the problem.

2. Hypotheses are derived directly from the evidence.

3. Hypotheses serve the purpose of directing the search for additional evidence.

★4. If the implications of a hypothesis are true, then we may conclude that the

hypothesis is true.

5. If an implication of a hypothesis is false, then we may conclude that the hypoth-

esis is false, at least in part.

6. In the episode pertaining to the discovery of radium, all of the hypotheses

turned out to be true.

540Chapter13 Hypothetical/Scientific Reasoning

★ 7. In the Neptune episode, Adams and Leverrier deserve the credit for working

out the implications of the hypothesis.

8. Torricelli�s hypothesis was consistent with the hypothesis that nature abhors a

vacuum.

9. In Pasteur�s day, the theory of spontaneous generation held that life was pro-

duced by the direct action of oxygen on organic nutrients.

★10. The hypotheses relating to the discoveries of radium and Neptune may be clas-

sified as empirical hypotheses.

11. Torricelli�s and Pasteur�s hypotheses may be classified as theoretical hypotheses.

12. Theoretical hypotheses concern how something should be conceptualized.

★13. The problem with the distinction between empirical and theoretical hypoth-

eses is that observation is dependent on theory.

14. The adequacy of a hypothesis has to do with how well the ideas or terms in the

hypothesis are rationally interconnected.

15. The coherence of a hypothesis has to do with how well the hypothesis fits the facts.

★ 16. If a hypothesis is not externally consistent, then it must be discarded.

17. A hypothesis is fruitful to the extent that it suggests new ideas for future

analysis and confirmation.

18. If a theory is incoherent, it is deficient in rationality.

★ 19. The theoretical interpretations of light, electricity, and magnetism during the

first part of the nineteenth century illustrate a condition of inadequacy.

20. If a hypothesis gives rise to contradictory implications, it is incoherent.

Summary

Hypothetical Reasoning:

■ Used to solve a problem when the solution is not readily apparent.

■ Involves four basic stages:

● Occurrence of a problem

● Formulating a hypothesis

● Drawing implications from the hypothesis

● Testing the implications

■ Four examples taken from science:

● Discovery of radium by Marie Curie

● Discovery of Neptune by Adams and Leverrier

Summary541

● Discovery of atmospheric pressure by Torricelli

● Refutation of spontaneous generation by Louis Pasteur

■ Kinds of hypotheses:

● Empirical:

▶ Can be proved true or false.

● Theoretical:

▶ Can only be confirmed in varying degrees.

■ Criteria that bear on the tentative acceptance of hypotheses:

● Adequacy (Does the hypothesis fits the facts?)

● Internal coherence (Are the component ideas interconnected?)

● External consistency (Are there conflicts with other hypotheses?)

● Fruitfulness (Are new ideas suggested for future analysis?)

542Chapter13 Hypothetical/Scientific Reasoning

14Science and Superstition

14.1 Distinguishing Between Science and Superstition

14.2 Evidentiary Support

14.3 Objectivity

14.4 Integrity

14.5 Abusing Science

14.1 Distinguishing Between Science

and Superstition

Learning the difference between good reasoning and super-stition is empowering. Suppose you suggest to a friend that the two of you share an apartment for the upcoming school year. You would split the rent and utilities, and you could cook meals together. Your friend is initially interested, but before committing to the deal, decides to consult a set of tarot cards. Totally exasperated, you try to convince your friend that tarot cards are nonsense. What arguments could you use to prove your point?

The idea that the human mind is capable of operating on different levels in its effort to comprehend reality is as old as philosophy itself. Twenty-four centu-ries ago Plato drew a distinction between what he called opinion and knowledge. Opinion, he said, is a kind of awareness that is uncertain, confined to the particu-lar, inexact, and subject to change, whereas knowledge is certain, universal, exact, and eternally true. Every human being starts out in life by operating on the level of opinion, and only through great struggle and effort can he or she escape it and rise to the level of knowledge. This struggle is called education, and it opens the eye of the mind to realities that cannot even be imagined from the standpoint of opinion.

543

Today�s distinction between science and super- following account of evidentiary support encom-

stition is a modern equivalent of Plato�s distinction passes adequacy and fruitfulness, and the account of between knowledge and opinion. Everyone recog- integrity encompasses adequacy, internal coherence, nizes that science has revealed wonderful truths and external consistency. about the world of nature. It has put men on the Science and superstition are, in large measure, moon, wiped out life-threatening diseases, and polar opposites. Where scientific activity recognizes ushered in the computer age. Also, almost everyone the importance of evidentiary support, objectivity, recognizes that superstition is little better than fool- and integrity, superstition ignores them. Accord-ishness. It leads people to fear walking under lad- ingly, these criteria can be used as a kind of measur-ders, breaking mirrors, and spilling salt. Practically ing stick for sizing up the various beliefs people have everyone agrees that if some claim is grounded in about the world. To the extent that those beliefs are science, then it is probably worthy of belief, while if supported by evidence, are objective, and arise from it is grounded in superstition, then it should prob- research that reflects integrity, the closer they come ably be ignored. Where people do not agree, how- to the ideal of science, and the more justified they ever, is in what constitutes science and what con- are. Conversely, to the extent that our beliefs do not stitutes superstition. What one person calls science share in these characteristics, the closer they come another calls superstitious nonsense. to the �ideal� of superstition, and the less justified

Both science and superstition involve hypoth- they are.

eses, so the four criteria developed in Chapter 13 for Note, however, that to say a belief is justified is evaluating hypotheses are relevant to the distinction not to say it is true in any absolute sense. As we saw between science and superstition: adequacy, internal in Chapter 13, all beliefs that arise from science are coherence, external consistency, and fruitfulness. tentative at best. But such beliefs are the best ones But the distinction between science and superstition we can have for now. Also, to say that a belief is not also involves psychological and volitional elements. justified is not to say it is absolutely false. It is quite It involves such factors as how the observer�s subjec- possible that a belief grounded in superstition today tive states influence how he sees the world, and how could tomorrow be grounded in science. But such a his needs and desires play a role in the formation of belief is not worthy of assent today. An analogy can his beliefs. Accordingly, to explore the distinction be found in rolling dice. No sensible person would between science and superstition, we must intro- bet even money that a pair of dice will come up duce criteria that include these psychological and �snake eyes� on the next roll, even though he realizes volitional elements. The criteria suggested here are that tomorrow it might be discovered that the dice evidentiary support, objectivity, and integrity. The were loaded in favor of this outcome.

14.2 Evidentiary Support

In Chapter 13 we saw that hypotheses are mathematics. Such measurement is nearly always conjectures�informed conjectures�but still con- accomplished through some kind of instrumen-jectures. Before they can be believed by reasonable tation. This requirement that scientific evidence people they must be supported by evidence. We also includes measurements expressed mathematically saw that scientific evidence nearly always involves is useful in distinguishing science from superstition. the measurement of something such as force, For example, in the sixteenth century, Coperni-mass, levels of aggression, social affluence, and so cus formulated the hypothesis that the sun is the on, and the result is expressed in the language of center of our planetary system and that the earth

544Chapter14 Science and Superstition

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). revolves around the sun�in opposition to the pre- evidence is that it is too isolated to establish any vailing Ptolemaic hypothesis, which put the earth at causal connection. Thus, the garlic evidence ignores the�center. In�the years that followed, the telescope the thousands of people with cancer who have eaten was invented, and thousands of precise measure- garlic and have not been cured, and it ignores the ments confirmed the Copernican hypothesis and thousands of people who have experienced sponta-disconfirmed the Ptolemaic hypothesis. Without neous remission of cancer and have not eaten any these measurements, the Copernican hypothesis garlic. Also, there is no way to turn the clock back would never have been adopted. and try the experiment again.

In contrast, consider the superstition that break- One of the key features of scientific evidence

ing a mirror will bring bad luck. No evidence has gathering is that an experiment be replicable under ever been collected in support of this hypothesis. No controlled conditions. This means that the experi-

tests have been conducted and no experiments per- ment must be repeatable Scientific experiments formed. Possibly on one occasion or other someone by different scientists at must be replicable. did break a mirror and later lost money on the stock different times and places. exchange or was injured in a car accident, but surely Replicability helps ensure that the outcome of the it would be unreasonable to believe that breaking a experiment did not result from anything peculiar mirror caused the loss or the accident. Such reason- to one certain experimenter operating at a single ing constitutes a classic case of the post hoc ergo place and time. Also, the controlled conditions are propter hoc (false cause) fallacy. But in spite of the designed to eliminate the influence of extraneous lack of evidence, many people believe the mirror factors. Perhaps, in reference to the garlic example, hypothesis. the cure was effected not by the garlic but by some-

Before inquiring further into the need for evi- thing else that was eaten, or by any one of a thousand

dence, however, we must first investigate what counts other factors that occurred during this time, or by as evidence. Do books count as evidence? What about any combination of these factors.

Do books count as religious books such as the The evidence offered in support of superstitious evidence? Torah, the Bhagavad Gita, the hypotheses is rarely replicable, and when it is, the

sutras of Buddhism, the Bible, outcome almost always fails to support the hypoth-

and the Koran? If books are credible reports of mea- esis. For example, the belief in ghosts is usually surements expressed mathematically, then they can be supported by what one or more individuals claim accepted as the conveyance of evidence. But because to have seen on some unique occasion. This occa-religious books report no such measurements, they sion can never be repeated. And the belief in psy-count neither as evidence nor as the conveyance of chic phenomena such as extrasensory perception is evidence for scientific purposes. This does not imply, sometimes claimed to be supported by experiments of course, that the content of such books automati- involving Zener cards: cards imprinted with crosses, cally falls under the rubric of superstition. It simply circles, wavy lines, stars, and squares whose image means that it cannot be used as a source for confirm- an observer might �transmit� to a psychic receiver. ing or disconfirming scientific hypotheses. But when these experiments have been repeated

Another kind of evidence that is considered unre- under carefully controlled conditions, the outcome

liable is anecdotal evidence. Suppose that you have has never been other than what would have been cancer and a friend advises you that eating garlic can expected to occur through mere chance. cure it. You decide to take this advice, and after Another defect found in superstitious hypotheses is eating a clove of garlic every day for a year, the cancer that they are often framed so vaguely that it is virtually goes into remission. Did the garlic cure the cancer? impossible to provide any kind of unequivocal confir-Evidence of this sort is called anecdotal, and science mation. For example, according to feng shui (pro-usually rejects it. The trouble with anecdotal nounced �fung-shway�), an ancient Chinese system of

Section 14.2 Evidentiary Support 545

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). magic, bad luck travels in straight lines, whereas good hypothesis must be framed narrowly enough so that luck does not. As a result, one invites bad luck by living it forbids certain things from happening. In other in a house or apartment that has two (or, what is worse, words, the hypothesis must be falsifiable. In the three) doors lined up in a row. But what, exactly, is the years following its announcement, many philoso-meaning of bad luck? What is interpreted as bad luck phers criticized Popper�s falsifiability criterion today may turn out to be good luck tomorrow. If a because, strictly speaking, hypotheses are rarely sus-person loses $1,000 in the stock market today, that may ceptible of being disproved. But, as we saw in Chap-lead her to be more cautious in the future, and that ter�13, hypotheses can be disconfirmed (or rendered increased caution may save $10,000 later on. less plausible). Thus, we can retain Popper�s basic

In contrast, the hypotheses of science are often insight by requiring that any genuinely scientific

framed in the language of mathematics, or they can hypothesis be disconfirmable. This means that the at least be translated into some mathematical expres- hypothesis must be framed narrowly enough so that

Scientific hypotheses sion. This fact provides it is possible for evidence to count against it. New-are phrased precisely. for extremely accurate ton�s gravitational hypothesis, for example, satisfies

confirmations and is this criterion because the discovery of two large

largely responsible for the extraordinary success bodies that failed to attract each other would tend to science has enjoyed during the past 500 years. For disconfirm the hypothesis. But the dietary hypothe-example, in 1802, the French chemist Joseph Louis sis just mentioned fails the disconfirmability crite-Gay-Lussac formulated the hypothesis that if the tem- rion because no outcome could ever count against it. perature is raised 1 degree Celsius in a closed con- A problem closely associated with excessively tainer of gas�any gas�the pressure of the gas will broad hypotheses arises in connection with what increase by 0.3663 percent. The hypothesis has been are called ad hoc modifications of hypotheses. For tested thousands of times by chemists and students in an example, suppose that you are a sociologist con-chemistry labs, and it has been found to be correct. ducting research into alcoholism. You formulate a

Closely related to the problem of vagueness is hypothesis that alco-Ad hoc modifications can

the breadth with which a hypothesis is framed. If a holism is caused by ruin hypotheses. hypothesis is framed so broadly and comprehensively cultural factors that that even contradictory evidence serves to confirm present alcohol consumption in a favorable light. it, then the hypothesis is not really confirmed by When you gather evidence to support this hypoth-anything. Suppose, for example, that a healthcare esis, however, you find that relatively few people practitioner should invent a hypothesis involving diet: who come from such cultures are alcoholics. Thus, Practicing this diet is guaranteed to make you feel you modify the hypothesis to say that alcoholism is great, but before it has this effect it may make you feel caused by cultural factors but only when a genetic either rotten or the same as usual. After following this predisposition exists. But then you find that many diet for six months you report that you feel the same alcoholics drink to ease the pain of depression and as before. The practitioner replies that your experi- other psychological problems. Thus, you modify the ence confirms the hypothesis, because this is what the hypothesis once again to take this fact into account. diet is supposed to do. On the other hand, suppose Further research shows that parental drinking pat-that after six months you feel great, or perhaps rotten. terns play a role, so you add another modification. Again the practitioner will report that your experi- These changes are called ad hoc (�to this�) modifi-ence confirms the hypothesis. Hypotheses of this sort cations because they are introduced purely to cover are not genuinely scientific. some problem or anomaly that was not recognized

In 1919, the philosopher Karl Popper discovered when the hypothesis was first framed.

this very problem concerning hypotheses. In The problem with ad hoc modifications is that response, he argued that any genuinely scientific their purpose is to shore up a failure of evidentiary

546Chapter14 Science and Superstition

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). support in the original hypothesis. As more and Ockham�s razor says that we should look for natu-more modifications are added, the hypothesis ralistic explanations first. Perhaps the tears came becomes self-supporting; it becomes a mere descrip- from condensation or from rainwater that may have tion of the phenomenon it is supposed to explain. leached into the statue from small surface cracks; For example, suppose that we introduce a certain another possible explanation could be that they are hypothesis h to explain the occurrence of a certain the work of pranksters. Scientists have shown how phenomenon x among a group of entities A, B, C, D, the appearance of such tears can be replicated by E. As ad hoc modifications are added, we find that drilling a small hole in a statue�s head, pouring a A has x because of some unique attribute a, B has liquid into the hole, and then scratching away a bit x because of b, and so on. In the end, our hypoth- of lacquer or paint from the corner of the eyes. The esis states that anyone who has attributes a, b, c, d, liquid will percolate down through the porous sub-e exhibits x. But the set of attributes a, b, c, d, e is stance of the statue and escape through the scratches. simply a description of A, B, C, D, E. If we should ask Another closely related consequence of Ockham�s why entity A has x, the answer is that A has x because razor is that explanations based on known realities of a, where a is just a unique something that A has. are preferred to those based on the fantastic or the Applying this analysis to the alcoholism hypothesis, bizarre. For example, there is a region in Peru, high if we ask why a certain person (let us call him Smith) in the Andes mountains, known as the Nazca Desert. is an alcoholic, the answer is that Smith is an alco- The floor of that desert features large drawings of holic because he has a certain attribute s that causes geometrical figures and animals such as monkeys, him to be an alcoholic. The explanation is vacuous. hummingbirds, and lizards. The drawings were pro-

Another problem with ad hoc modifications is that duced more than a thousand years ago by scraping

they result in hypotheses that are so complicated that away the iron oxide�coated pebbles that cover the applying them becomes difficult. Science has always land. But given their huge size, they can be recog-favored simplicity over complexity. Given two hypoth- nized as coherent figures only from a position high eses that explain the same phenomenon, the simpler of above. One accepted explanation is that the Nazcas the two is always the preferable one. In part, this pref- produced the drawings using simple tools and sur-erence is aesthetic. The simpler hypothesis is more veying equipment to impress the rain gods that dwell �beautiful� than the more complex one. But the prefer- in the clouds. However, the best-selling author Erich ence for simplicity also results from the application of von D�niken proposes that the lines represent land-what has been called �Ockham�s razor.� This is a prin- ing strips for alien spacecraft, and it was those aliens ciple, introduced by the fourteenth-century philoso- who guided the Nazcas from overhead in produc-pher William of Ockham, that holds that theoretical ing the drawings. Given its far-fetched character, this entities are not to be multiplied needlessly. Why settle explanation would seem to violate Ockham�s razor. for a complicated theory when a simpler one works Returning to the question of evidentiary support, equally well? Besides, the simpler one is easier to apply. one of the surest ways to know that our hypotheses are

One consequence of Ockham�s razor is that nat- supported by evidence is that they lead to predictions

uralistic explanations are preferable, at least at the that turn out to be true. Each true prediction repre-outset, to supernatural ones. For example, over the sents a pillar of support for the hypothesis. But some years there have been numerous reports of stat- predictions are better than others, and the best ones

Naturalistic explanations ues depicting Jesus are those that reveal ways of viewing the world that are preferred. or the Virgin Mary would never have been dreamed of apart from the

that are said to weep hypothesis. If a hypothesis leads to predictions of this

tears of water, or oil, or blood. After viewing these sort, and if those predictions are confirmed by evi-statues, thousands of faithful come away con- dence, then the hypothesis has earned a very special vinced that they have seen a miracle. However, kind of support. Such a hypothesis reveals hidden

Section 14.2 Evidentiary Support 547

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). truths about nature that would never have been theory was quickly adopted. Within a few years, the recognized without it. theory led to the discovery of atomic energy.

A classic example of a prediction of this kind Hypotheses that yield striking, novel predictions

resulted from the hypothesis under lying Einstein�s are largely responsible for progress in science. And

general theory of rela- it is precisely these kinds of predictions, argues the

Science is progressive; tivity. One of the con- philosopher Imre Lakatos, that distinguish science superstition is not. sequences predicted by from pseudoscience. Of course, not every scien-

this hypothesis is that tific hypothesis leads to such startling predictions

light is affected by gravity. In particular, the hypoth- as Einstein�s, but this kind of hypothesis may at esis predicted that a light ray coming from a star least be integrally connected to broader, umbrella and passing by the sun would be bent in the direc- hypotheses that have led to such predictions. In tion of the sun. As a result, the position of the star contrast, the hypotheses underlying astrology have with respect to other stars would appear to be dif- been around for twenty-seven centuries, and they ferent from what it was usually observed to be. Of have produced not a single startling prediction that course, testing such a prediction under normal cir- has been verified and not a single new insight into cumstances would be impossible, because the light the course of human events. They have produced of the sun is so bright that it completely blocks out no master plan for future civilization and no hint the light from stars. But it could be tested during a about future discoveries in physics or medicine. solar eclipse. Such an opportunity arose on May 29, This lack of progress over centuries is one reason 1919, and scientists took advantage of it. The predic- that the philosopher Paul Thagard named astrol-tion turned out to be true and, as a result, Einstein�s ogy a pseudoscience.

14.3 Objectivity

Our beliefs about the world are objective to the emotions that give rise to superstitious beliefs are fear extent that they are unaffected by conditions pecu- and anxiety, and they are often reinforced by a disposi-liar to the experiencing subject. Such conditions can tion to fantasy and mental laziness. Much of the fear be either motivational or observational. For exam- and anxiety is generated ple, a belief that is motivated by the emotions of the Superstitions exist to by the fact that every-experiencing subject and that exists for the primary satisfy emotional needs. one dies. Death can purpose of satisfying those emotions tends to lack come suddenly, as in a objectivity. Also, a belief that is grounded in obser- freeway accident, a fall from a roof, or an avalanche, or vations peculiar to the experiencing subject, such as it can come as a result of cancer, heart failure, or stroke. visual hallucinations, lacks objectivity. Even though Short of death, everyone is subject to injury with its objectivity is an ideal that can never be completely attendant pain, and most people at some time experi-attained, practically everyone would agree that ence the mental suffering that accompanies rejection, beliefs are more trustworthy if their content is not loneliness, and failure. distorted by the experiencing subject. The scientist People have little control over these facts of life, constantly strives to avoid such distortions, but the and to relieve the anxiety they produce, many resort superstitious mind either revels in them or, in the to charms and amulets, the rosary beads dangling more tragic cases, succumbs to�them. from the rearview mirror or the talisman or medal

All superstitions exist at least in part to satisfy the worn around the neck. If nothing else will protect us

emotional needs of the experiencing subject. The chief from the terrors of life, perhaps these objects will.

548Chapter14 Science and Superstition

After all, science has failed to conquer disease and The disposition toward the magical and the fantas-death, and it offers to the believer nothing but tenta- tic is greatly reinforced by the media, particularly tive truths that may change tomorrow. Besides, sci- television and motion pictures. The media are slav-ence is too hard to understand. Becoming compe- ishly subservient to the entertainment desires of their tent requires years of serious study. To the person audience, so, given a widespread fascination with the facing an uncertain future, dejection, or loneliness, it magical, the media issue a constant stream of movies, may seem more reasonable to contact a local psychic miniseries, and �news� stories devoted to that subject. and buy a bit of immediate consolation than to trust These programs touch everything from vampires and in science. disembodied spirits to irrational conspiracies and the

A second element in the human condition that intervention of angels. This persistent attention to the

generates anxiety is freedom and the responsibility fantastical increases the public�s acceptance of super-freedom entails. The idea that you, and you only, are stitious explanations whenever realistic ones are not in charge of your destiny can be an extremely fright- readily available, or even in the face of realistic ening idea. Many people recoil from the thought and explanations. seek refuge in a leader or guru. They turn all their A disposition to mental laziness also assists in power of critical thinking over to this leader and the formation of superstitious beliefs. It is, in fact, blindly follow his or her instructions to the last extremely difficult to ensure that one�s beliefs are detail. When the leader orders them to believe any supported by evidence and that they pass the test form of nonsense, no matter how silly, they do so. of internal coherence. Sloppy logic is so easy it is The belief or practice ordered by the leader, they are no wonder people told, is essential to their protection. But, following Superstitions are sup- resort to it. Most of such orders can sometimes lead to tragedy, as it did ported by sloppy thinking. the informal fallacies in the Jonestown massacre in 1978 and the Heaven�s treated in Chapter�3 Gate suicides in 1997. can arise from sloppy thinking. After old Mrs. Chad-

A disposition to magical ways of thinking and wicke hobbled past the church, lightning struck the

mental laziness greatly facilitates the flight to steeple and burned the church to the ground. Obvi-superstition. Many people, if not most, are fasci- ously old Mrs. Chadwicke is a witch (false cause). nated by the mysterious, the arcane, and the occult, Furthermore, old Mrs.� Chadwicke wears a black and some would rather believe an explanation clothed cape and a black hood. It must be the case that all in magic than they would a scientifically grounded witches wear such clothing (hasty generalization).

one. The psycholo- And of course witches exist because everybody in

People are fascinated by gists Barry F. Singer the village believes in them (appeal to the people). the mysterious. and Victor A. Benassi Another kind of sloppy thinking involves an

performed a series of appeal to what might be called false coherence. A

experiments on their students in which they had a farmer discovers that one of his cows has been killed. magician pose as a �psychic� and perform demon- At the same time the farmer happens to read a story strations of psychic feats. Before the demonstrations in a local tabloid saying that a satanic cult is operat-began, the students were told repeatedly in the clear- ing in the vicinity. The cult practices its rites on the est language that the magician was only pretending thirteenth day of each month. The cow was killed to be a psychic, and that what they were about to wit- on the thirteenth. Thus, the farmer concludes that ness was really a series of conjurer�s tricks. Neverthe- the cow was killed by Satan worshippers. This line less, in spite of these warnings, a majority of students of thinking involves many loose ends, but that rarely concluded, in one experiment after another, that the deters people from drawing a conclusion. Becoming magician was really a psychic. Furthermore, many a clear, critical thinker is one of the primary goals of concluded that the magician was an agent of Satan. education, but unfortunately becoming educated is

Section 14.3 Objectivity 549

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). no less of a struggle for students today than it was for the belief that bear bile reduces fevers and improves students in Plato�s day. eyesight is most likely an imagined cure, but it has

Thus far, we have focused on emotions and dis- led to the horribly inhumane treatment of bears in

positions in the experiencing subject that lead to Asia. superstitious beliefs. We now turn to some of the Another well-documented effect that influences many ways that our observation of the world can be our observation of the world around us is called distorted. Such distortions constitute avenues in pareidolia. This is the effect by which we can look which conditions peculiar to the experiencing sub- at clouds, smoke, or the textured coatings on walls ject enter into the content of observation. When and ceilings and see animals, faces, trees, and so on. such distorted observations are combined with the We project familiar visual images onto vague, rela-emotions and dispositions mentioned earlier, super- tively formless sensory stimuli and �see� that image stitious beliefs are likely to arise. The distorted as if it were really there. Pareidolia is responsible for observations can occur in the same person who has a good deal of religious superstition. For example, a the emotions and dispositions or they can be con- few years ago a woman making burritos saw the face veyed secondhand. In either case, the combination of Jesus in the skillet burns on a tortilla. She built a leads to superstition. shrine to house the tortilla, and thousands of believ-

One well-documented phenonemon that influ- ers came to pray before it. More recently, a grilled

ences our observation of our own bodily states is cheese sandwich that believers thought bore the

the so-called placebo image of the Virgin Mary sold on eBay for $28,000.

Placebos can masquer- effect. Recall that a But incidents of this sort are not confined to the ade as cures. placebo is any kind of Christian world. Followers of Islam have seen the

�medicine� or proce- Arabic words for �Allah� and �Muhammad� in such

dure that provides no medicinal or therapeutic bene- media as fish scales, eggshells, and lamb�s wool. fit by itself but that can effect a cure when the patient Closely related to pareidolia is the concept of the is told that it has such benefit. For example, patients perceptual set, where �set� refers to our tendency to with knee pain have been told that an operation will perceive events and objects in a way that our prior cure them, and after they undergo a minor incision experience has led us to expect. The idea of percep-that, by itself, has no therapeutic effect, the pain often tual set is a product of Gestalt psychology, accord-disappears. Also, patients who suffer from nervous ing to which perceiving is a kind of problem solving. tension or depression have been told that a little col- When we are confronted with a problem, such as ored pill (which consists of nothing but sugar) will finding the solution to a riddle or puzzle, we enter cure them, and after they take the pill, the tension or into a state of mental incubation in which potential depression disappears. Obviously in these cases it is solutions are turned over in our minds. This state not the placebo alone that effects the cure but the pla- is followed by a flash of insight (assuming we are cebo together with the suggestion implanted in the able to solve the puzzle), after which the solution patients� minds by their doctors. seems obvious. When we consider the puzzle at a

The perceived efficacy of many treatments in later time, the solution leaps into our minds. Such

Eastern medicine can probably be attributed to the a solution is called a gestalt, from the German, for placebo effect. For example, the widely held belief in �form� or �configuration.� Analogously, every act of China that tiger bones cure ulcers, malaria, and perception involves solv-arthritis, and that rhinoceros horns cure fevers and We perceive what we ing the puzzle of organiz-convulsions and act as an aphrodisiac are almost expect to perceive. ing sensory stimuli into certainly grounded in superstition. But they have led meaningful patterns. Each to the decimation of the tiger and rhino populations such pattern is a perceptual gestalt, or set, and once wherever these animals are found in the wild. Also, such a set is formed, it serves to guide the processing

550Chapter14 Science and Superstition

darkness will often seem to move. One can prove the

existence of this effect for oneself by looking at a

bright star on a dark night, or by observing a small

stationary point of light in a dark room. Psycholo-

gists speculate that the autokinetic effect results

from small, involuntary motions of the eyeball of the

observer, and they have shown that the effect is

enhanced by the reports of other observers. If some-

one standing nearby says that she just saw the object

move, others will often confirm this report. The

autokinetic effect is thought to be responsible for

many claims of UFO sightings.

Can you see the white triangle? Hallucinations of various sorts can also distort the

Is the white triangle really �there�? content of perception. Two kinds of hallucination

(Taken from Kanizsa, 1979, 74.) that affect many people

Hallucinations distort in the drowsy moments

of future perceptions. As a result, we perceive what perception. between sleep and wake-we expect to perceive. fulness are hypnagogic and

In 1949, the psychologists Jerome S. Bruner and hypnopompic hallucinations (Hines, 1988, 61�62).

Leo J. Postman performed a famous experiment in The former occur just before drifting off to sleep, which subjects were shown replicas of ordinary play- when the brain�s alpha waves are switching to theta ing cards�but some of the cards had been altered by waves, and the latter occur just before awakening. reversing their color. For example, in some groups During these moments the subject may experience of cards, the three of hearts was black and the six extremely vivid, emotionally charged images that of spades was red. Of twenty-eight subjects, twenty- seem very real. These hallucinations are thought to seven initially saw the altered cards as normal ones. be responsible for the ghosts and other appearances One subject identified the black three of hearts as that people sometimes see in bedrooms. a three of spades on forty-four successive show- Collective hallucinations are another kind of per-ings. This experiment clearly shows that we perceive ceptual distortion that can occur in large crowds of what we expect to perceive and, indeed, this fact is people. Before such hallucinations can happen, the familiar to nearly everyone. For example, we expect crowd must be brought to a heightened emotional to receive a phone call, and while taking a shower state, which may be brought on by the expectation we think we hear the phone ringing, only to be told of seeing something important or miraculous. An by someone in the other room that the phone did occurrence of this sort may have happened on not ring. Or, while driving, we might approach a red October 13, 1917, when some 70,000 people gath-octagonal sign that reads ST_P (our view of the sign ered in the village of Fatima in Portugal expecting to being partly blocked by a tree branch between the see a miraculous sign from heaven. At midday, one S and the P). However, we bring the car to a stop, of the children who was supposedly in contact with because we perceived the sign to read STOP. In fact, the Virgin Mary cried out to the people to look at what our sense of vision received was three conso- the sun. They did so, whereupon thousands saw the nants (S, T, P), meaningless until processed through sun swirl amid the clouds and plunge toward the perception. earth. Of course, if the sun had actually moved, it

Yet another factor that influences our sense of would have triggered seismograph readings all over

vision is the autokinetic effect. According to this the globe. Also, many of the people there did not see effect a small, stationary light surrounded by anything unusual, but their reports were discounted.

Section 14.3 Objectivity 551

Nevertheless, even to this day many of the faithful many details are inevitably left out, the final picture take this observation of the swirling sun as recalled may range anywhere from a fairly accurate evidence of a miracle. representation to a total fabrication.

Finally, the operation of memory can distort the These effects represent only a few of the ways the

way we recall our observations. Human memory subjective state of the observer can influence human is not like the process whereby a computer recalls observation and memory. To avoid such distortions, information from its hard drive with total accuracy. scientific inquiry restricts human observation to cir-

Rather, it is a creative cumstances in which known aberrations of percep-

Confabulation produces process susceptible tion and recall are least likely to occur. In the natu-false recollections. to many influences. ral sciences, much if not most observation occurs

When images are through instruments, such as volt meters, Geiger

recalled from human memory, they are retrieved counters, and telescopes, the behavior of which is in bits and pieces. The brain then fills in the gaps well known and highly predictable. The results are through a process called confabulation. The brain then recorded on relatively permanent media such naturally and unconsciously tries to produce a as photographic film, or digital storage devices. In coherent account of what happened, but precisely the social sciences, techniques such as double-blind how the gaps are filled in depends on such things sampling and statistical analysis of data insulate the as one�s feelings at the time of recall, other people�s observer from the outcome of the experiment. Such suggestions about the event recalled, and one�s own procedures provide considerable assurance that the successive reports of what happened. Given that data are not distorted by the subjective state of the memory recall is selective to begin with and that experimenter.

14.4 Integrity

Our efforts to understand the world in which we live world as a psychic who could perform marvelous have integrity to the extent that they involve honesty feats such as bending spoons, keys, nails, and other in gathering and presenting evidence and honest, metal objects through the sheer power of his mind. logical thinking in responding to theoretical prob- Such objects would appear to bend when he merely lems that develop along the way. Most� forms of stroked them with his finger, or even without his superstition involve elements of dishonesty in gath- touching them at all. Scientists were called in to wit-ering evidence or a failure of logic in responding to ness these feats, and many came away convinced of theoretical problems. Such failures of logic can be their authenticity. But in fact Geller was just a clever found in the lack of response by the community of trickster who duped his audiences. Geller�s trick-practitioners to problems involving the adequacy, ery was exposed in large measure by the late James coherence, or external consistency of the hypotheses Randi, a stage magician himself. related to their practices. After watching videos of Geller�s performances,

The most severe lack of integrity arises when the Randi discovered how Geller performed his tricks, and

evidence is faked. One of the more striking exam- in no time he was able to perform every one of them

Practitioners of supersti- ples of faked evidence himself. Sometimes Geller would prepare a spoon or tion fake the evidence. is found in the case of key beforehand by bending it back and forth several

the Israeli entertainer times to the point where it was nearly ready to break.

Uri Geller. Beginning in the early 1970s, Geller pre- Later, by merely stroking it gently, he could cause sented himself in numerous venues throughout the it to double over. On other occasions Geller, or his

552Chapter14 Science and Superstition

Donald Singleton, a reporter for the New York Daily News, was familiar with Uri Geller�s alleged psychic ability to bend spoons and keys and identify handmade drawings that had been sealed inside two envelopes, one inside the other. He suspected that the latter trick was performed by holding the envelopes up to a strong light while the subject�s attention was diverted. Prior to writing a story on Geller, Singleton performed the following test:

I went to a locksmith and got a duplicate of the strongest, thickest key on my key ring.

I tried with all my might and I couldn�t bend it, even by pressing it against the corner

of a steel desk. Then I made a simple drawing (of an eye), wrapped it in aluminum foil

and put it into two envelopes.

I went to see Geller the next afternoon.

He tried for more than an hour, with me keeping the envelope in my sight every

second, to get the drawing. And he failed.

Then he made an effort to bend the key, again with me keeping it in view every

second. Again, nothing happened. Uri said that he was terribly disappointed, that this

simply had been an all-around bad day for him.

accomplices, would use sleight-of-hand maneuvers to burning coals and escape unharmed. The physicist substitute bent objects in the place of straight ones. In Bernard J. Leikind proved this, at least to his own yet another trick, Geller claimed to be able to deflect satisfaction, when he showed up at a Robbins semi-a compass needle by merely concentrating his atten- nar. Even though he had not attended the sessions tion on it. As he would wave his hands over the com- and he declined to think cool thoughts as per the pass, the needle would spin�and his hands had been instructions of the attendants, he found that he could thoroughly examined earlier for hidden magnets. perform the fire walk without even getting singed. But Geller had concealed a powerful magnet in his He explained his success by noting certain basic laws mouth, and as he bent over the compass, the needle of physics. In spite of their high temperature, wood would spin in tune with his head gyrations. coals contain a very low quantity of heat, and they

For another example of faked evidence, let us conduct heat very poorly. Also, the foot is in contact

look at fire walking. Practitioners of this art claim with the coals for only a second at a time, thus allow-that their self-help seminars can alter a person�s body ing only a small quantity of heat energy to flow to the chemistry so as to allow him or her to walk barefoot foot. As a result, the feet of fire walkers rarely sustain over a bed of glowing coals without being burned. injury (or at least serious injury). One of the leading gurus of this business was Tony For a third example of faked evidence, we need Robbins of the Robbins Research Institute. Robbins look no further than the thousands of fortune-tellers, used what he called �neurolinguistic programming� palm readers, and men-to cure all sorts of physical and psychological ail- Cold readers deceive talists who use the art of ments, from irrational fears and impotence to drug their clients. �cold reading� to divine addiction and tumors. As proof of the efficacy of this all sorts of amazing truths technique, he would invite those who had taken his about their clients� lives. Most people who engage the seminar to engage in a fire walk. By merely believing services of these �readers� do so because they have they would avoid burning their feet, he told them, problems concerning love, health, or finances. The they would survive the ordeal unharmed. reader knows this and often begins the reading with a

The truth is that anyone, whether or not he had flattering spiel that is tailored to fit practically every-

taken the seminar and regardless of what he believed, body. This recital is intended to put the client at ease could, under controlled conditions, walk across and condition him or her to open up to the reader.

Section 14.4 Integrity 553

All the while the reader is taking in every detail: the If the deceptive techniques of the magician who client�s age, sex, weight, posture, speech patterns, pretends to be a psychic, the neuro-linguistic pro-grammar, eye contact, build, hands, clothing (style, grammer, and the cold reader are accepted at face age, neatness, and cost), hairstyle, jewelry, and what- value, they appear to constitute evidence that really ever the client might be holding or carrying (books, supports the hypotheses underlying these activities. car keys, etc.). All of these provide clues to the per- But faking the evidence is not the only way in which sonality, intelligence, line of work, socioeconomic the practitioners of superstition lack integrity. The status, religion, education, and political affiliation of other way concerns the reaction of the community the client. of practitioners to problems that arise in connection

The reader uses this information to formulate with the adequacy, coherence, and external consis-

hypotheses that are then presented to the client in tency of those hypotheses. the form of subtle questions. Depending on the cli- Such problems arise in connection with scientific ent�s reactions�facial expression, eye motion, hypotheses no less often than they do with supersti-pupil dilation, gestures�the reader can often tell tious ones. When they arise if he or she is on the right track. Once the reader Scientists are puzzle in science, the community hits on something close to home, the client will solvers. of scientists shifts to what usually react in amazement and begin revealing more the philosopher Thomas personal details. After appropriate intervals, the reader Kuhn calls a puzzle-solving mode, and scientists will then rephrase this information in a different work on them with great persistence until the prob-sequence and feed it back to the client, to the client�s lems are solved. This puzzle-solving activity occupies ever-increasing amazement. The client then provides the attention of the vast majority of scientists for the even more details, which the reader weaves together greatest part of their careers, and it constitutes what with everything else learned to that point. The use of a Kuhn calls �normal science.� Furthermore, it is pre-crystal ball, satin cape, or tarot cards combined with a cisely this puzzle-solving character of normal sci-polished sense of confidence convey to the client that ence, Kuhn argues, that distinguishes science from the reader can literally read the client�s mind. pseudoscience.

The psychologist Ray Hyman, who as a teenager read palms to supplement his income, has studied the art of cold reading in some depth. �He relates a story about a young woman who visited a mind reader:

She was wearing expensive jewelry, a wedding band, and a black dress of cheap material. The observant reader noticed that she was wearing shoes that were currently being adver-tised for people with foot trouble. He assumed that this client came to see him, as did most of his female customers, because of a love or financial problem. The black dress and the wedding band led him to reason that her husband had died recently. The expensive jewelry suggested that she had been financially comfortable during marriage, but the cheap dress indicated that her husband�s death had left her penniless. The therapeutic shoes signified that she was now standing on her feet more than she used to, implying that she was working to support herself since her husband�s death.

The reader�s shrewdness led him to the following conclusion�which turned out to

be correct: The lady had met a man who had proposed to her. She wanted to marry the man to end her economic hardship. But she felt guilty about marrying so soon after her husband�s death. The reader told her what she had come to hear�that it was all right to marry without further delay.

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For example, after the Copernican hypothesis was need to show how infinitesimally small gravitational

introduced, a problem turned up in connection with fluctuations can affect people�s lives. On the other stellar parallax. If, as the hypothesis held, the earth hand, if some other causal influence is at work, the travels around the sun, then, in the course of its orbit, astrologers need to pin it down. What kind of laws the farthest stars should appear to shift in position govern it? Is it an inverse square law, like the law of with respect to the nearer ones. An analogous phe- gravity, or some other kind of law? Analogously, if nomenon can be observed as you change your posi- the lines on a person�s palms indicate something tion in a room. The distant lamp, which originally about the person�s life, then what form of causality is appeared to the left of the chair in the foreground, at work here? Do the lines influence the life, or is it now appears to the right of it. In the case of the stars, the other way around? And what laws does this form however, no parallax could be observed. The expla- of causality obey? nation given at the time was that the stars were too Any absence of a causal connection is a defect in far away for any parallax to be detectable. Neverthe- coherence, because it signals the lack of a connection less, stellar parallax constituted an adequacy prob- between ideas functioning in a hypothesis. However, lem that the community of astronomers regarded as such a lack of coherence need not be fatal to a hypoth-a puzzle, and they worked on it for 300 years. Even- esis. Physicians from the time of Hippocrates knew tually more-powerful telescopes were produced that that willow leaves, which contain the essential ingre-did indeed detect a change in position of the stars as dient of aspirin, had the power to relieve pain, but the earth orbited the sun. they failed to understand the causal connection until

In contrast, when an astrological prediction fails relatively recently. What distinguishes the biomedical

to materialize, the community of astrologers never community from the community of astrologers lies in sets to work to figure out what went wrong. Astrolo- their respective reactions to such problems. The gers never recheck the location and birth time of the members of the biomedical community recognized client or the exact position of the planets at the time the aspirin problem as a puzzle, and they worked on it of his birth. They merely charge forward and issue until they found the solution, but members of the more predictions. Similarly, when the bumps on a astrological community are unconcerned with identi-person�s head fail to indicate essential features of that fying the causal mechanism by which the planets person�s personality, or when the lines on her palm influence human lives. Similarly, members of the fail to reveal features of her life, the community of community of palm readers and members of the com-phrenologists and the community of palm readers munity of phrenologists care nothing about identify-never try to account for the failures. They just ignore ing the essential causal connections implied by their

a response reveals a lack of integrity on the part of An even more seri-Levitation is a puzzle these practitioners toward their respective hypoth-them and move on to the next batch of clients. Such respective hypotheses. no one wants to solve. ous problem is posed

eses. Something is clearly wrong with the hypotheses by hypotheses that are or with the measurements, but no one cares enough inconsistent with established theories or laws. A case to do anything about it. in point may be found in claims made by promoters

A similar response occurs in connection with of the Transcendental Meditation (TM) movement.

coherence problems. Most superstitions involve seri- The practice of TM was popularized in the 1960s ous incoherencies, many of which arise from the lack by the late Maharishi Mahesh Yogi, and since then of known causal connections. For example, if astrol- it has attracted thousands of adherents. It consists ogy claims that the planets influence our lives, then in the silent repetition of a mantra, which induces there must be some causal connection between the a mental state similar to self-hypnosis. For many planets and individual humans. But what could this who have tried it, the benefits are mental and phys-connection be? Is it gravity? If so, then astrologers ical relaxation leading to a sense of rejuvenation.

Section 14.4 Integrity 555

But with further instruction in TM (at consider- been written. This response is inconceivable for able cost to the student), longer and deeper trances any bona fide center of scientific research. can be induced that, the Maharishi claimed, allow On the preceding pages, we have examined some the meditator to levitate�to hover in the air with- features that are characteristic of scientific inquiry out any physical support. Thousands of disciples, and some contrasting features that are characteristic he claimed, have learned how to do this, and he of superstition. The purpose of this exposition has released photographs that purported to verify this not been to provide the sufficient and necessary con-claim. But of course if levitation actually occurs, it ditions for a bright demarcation line between science constitutes a violation of, or a suspension of, the and superstition. Rather, the purpose has been the law of gravity. more modest one of setting forth a group of family

The inconsistency of the Maharishi hypoth- resemblances that a fair-minded inquirer may use

esis with such a well-confirmed theory as the law in rendering a judgment that a set of beliefs is more of gravity is probably sufficient reason to assign it probably scientific or more probably superstitious. to the category of superstition. But the reaction of To the extent that a set of beliefs rests on hypoth-the community of TM practitioners to this incon- eses that are coherent, precisely tailored, narrowly sistency leaves little room for doubt. In 1971, the formulated, supported by genuine evidence, and Maharishi bought the grounds and buildings of productive of new insights, these beliefs can be con-what was formerly Parsons College in Fairfield, sidered scientifically grounded. This judgment is Iowa, and he converted the site into Maharishi reinforced by the conscientious response of the scien-International University. The university then tific community to problems that develop concerning became the home of the International Center for the adequacy, coherence, and external consistency of Scientific Research, which, one would think, would those hypotheses. But to the extent that a set of beliefs be the perfect forum for investigating levitation. rests on hypotheses that are incoherent, inconsistent Given the availability of scores of alleged levita- with well-established theories, vague, overly broad, tors, the �scientists� in residence could conduct motivated by emotional needs, and supported by evi-in-depth studies into this phenomenon. Their dence that fails to be trustworthy, as well as leading to findings could develop the basis for interplanetary no new insights, then those beliefs tend to be super-space travel, to say nothing of what they might do stitious. Such a judgment is reinforced by a reaction to achieve safer airplanes. However, from its incep- of oblivious unconcern on the part of the community tion, the International Center has conducted not of practitioners to problems that arise in connection a shred of research on levitation. No experiments with the adequacy, coherence, and external consis-have been performed and no scholarly papers have tency of those hypotheses.

14.5 Abusing Science

The foregoing observations about integrity should produce definitive results, and sometimes the �right� not be taken to imply that no scientist has ever results. This pressure has led a number of scientists, faked the evidence. Fraud has occurred and it results particularly in the biomedical and life sciences, it chiefly from changes that have taken place in the last seems, to fudge the data. Another motivation for fifty years or so. First, science today is a multibillion- fraud is the pressure in academia to �publish or dollar industry funded by grants from the govern- perish.� Without a solid publishing record, a gradu-ment, private foundations, and corporations. To get ate student in the sciences will never be awarded a funding, scientists are under constant pressure to PhD, never secure a postdoctoral appointment, and

556Chapter14 Science and Superstition

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). never hold down a position on a university faculty. no astrologer has ever been sent to prison for cast-Again, sometimes the pressure can be too much for ing fraudulent horoscopes and no palm reader has some individuals. ever been sanctioned for negligently interpreting the

A prime example of a scientist who has faked the lines on a client�s hands. Needless to say, no publi-

evidence is Dr. Scott Reuben, former Professor of cation of a psychic has ever been retracted for fake Anesthesiology at Tufts University and former chief predictions. In these endeavors, fraud is virtually of acute pain at Baystate Medical Center in Spring- meaningless. field, Massachusetts. Over a period of nearly thirteen Yet, there is a different, more insidious kind of years, Dr. Reuben wrote or co-wrote twenty-one dishonesty that is not measurable in terms of arti-articles that were published in scientific journals and cles retracted from scientific journals. This kind of that he claimed were supported by clinical trials, The dishonesty results from the corrupting influence so-called research had been funded by grants from of big money on science. Big corporate money can the pharmaceutical company Pfizer and it reported buy almost anything, including what people take to favorable results for Pfizer�s Bextra, Celebrex, and be scientific truth. The kind of �truth� that it buys Lyrica, as well as Merck�s Vioxx. Reuben�s work is whatever supports the sale of its products. Big led to the sale of billions of dollars of these drugs. money accomplishes this feat mainly through two After his work became suspect, Reuben admit- strategies: Whenever possible it blocks the publica-ted that every one of the clinical trials had been tion (or any dissemination) of scientific findings that fabricated, the names of many patients invented, tend to jeopardize the sale of its products, and when and the signatures of coauthors forged. In the end, negative scientific findings do manage to surface, it Dr. Reuben was sentenced to prison for fraud and his generates research programs leading to �facts� that practice of medicine was terminated. cast doubt on those findings.�

Granted that some scientific work is tainted The first strategy is achieved in part by issuing

with fraud, the next question is, how extensive grants to university scientists to conduct research is this abuse? One way of approaching this ques- aimed at promoting the sale of its products, but tion is to look at the number of articles in scien- these grants come with strings attached. The corpo-tific journals that are retracted each year�that is, ration issuing the grant reserves for itself the right to permanently withdrawn from the literature. The design the research program, to write up the find-highly respected scientific journal Nature pub- ings, and to veto the publication of any findings it lished such a study in its October 6, 2011, issue.* doesn�t like. The second strategy is accomplished by The author concluded that out of approximately engaging ethically challenged scientists to conduct 1,400,000 scientific articles published that year, research specifically designed to call into question perhaps 400 would be retracted, and half of those any scientific truth it doesn�t like. These scientists would involve some form of misconduct. That are either in-house people, members of university amounts to less than .02 percent. science departments, or mercenary scientists sup-

The fact that fraud is taken seriously by the scien- plied by specialty firms that engage in such business.

tific community distinguishes science from the prac- A large number of the world�s greatest corpora-tice of superstition. When fraud is discovered in sci- tions have used these tactics at one time or other to ence, the guilty scientist is often fired or even sent to avoid government regulation or to defend against prison. If fraudulent research serves as the basis for a litigation. As a result, if we want to examine the publication, the publication is retracted. In contrast, influence of big corporate money on science, there

*For another article on this topic, see �Misconduct Accounts for the

Majority of Retracted Scientific Publications.� In Proceedings of the �See David Michaels, The Triumph of Doubt: Dark Money and the National Academy of Sciences, vol. 110, issue 3, p. 1137 (2013). Science of Deception (New York: Oxford University Press, 2020).

Section 14.5 Abusing Science 557

Copyright 2024 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). are a lot of industries to pick from. For our purposes atmosphere. Instead, it responded by setting up some good candidates are the tobacco industry, the front groups having authoritative names includ-fossil fuel industry, the sugar industry, and the GMO ing the CO Coalition, the International Climate 2 (genetically modified organism) industry. Science Coalition, and the Global Climate Coali-

In the 1950s, the suspicion started to sink in that tion to attack the science of climate change. These

smoking cigarettes is linked to lung cancer and heart groups, which were staffed by some of the same disease. Big Tobacco�s initial response was to remind people who had attacked the science of smoking, the public that smokers are grown adults, and if engaged scientists to issue a stream of technical they choose to smoke, that is their right. The real reports�reports filled with charts, graphs, and trouble came with evidence that secondhand smoke citations to articles in scientific journals�that also causes these diseases, and the victims of second- raised doubts about climate change. The objec-hand smoke, who may be unsuspecting patrons of tive was to confuse the public with an avalanche restaurants, shopping malls, and workplaces, do not of false and misleading claims. choose to breathe the smoke they encounter. Coinci- These claims asserted that CO2, a major green-dent with these findings about secondhand smoke, a house gas, is not in fact increasing in the atmosphere

Japanese study came out showing that non-smoking (which is false), and even if it were, CO is essential 2 women who were married to smokers were more to life (true), so the more of it that we have in the likely to contract lung cancer than non-smoking atmosphere, the better off we are (false). Further, the women who were married to non-smokers. amount of CO that is naturally present in the atmo-2

Big Tobacco could have responded by attempt- sphere is very small (true) so even if it were doubled,

ing to make its product less hazardous. Instead, the effect would be minimal (false). There is no it responded by setting up a competing Japanese evidence of actual global warming (false), but even study showing that non-smoking women married if there were, it would result from natural causes to smokers were not more likely to contract lung (false). The temperature of the planet is not actu-cancer than non-smoking women married to non- ally increasing (false), but even if it were increasing smokers. The purpose of the second study was pre- the effect would be beneficial because it would open cisely to create doubt in the public mind about the shipping lanes in the north Atlantic (possibly true) findings of the first study, and company executives and extend the growing season in many regions hid the fact that they were the ones who funded, (true, but destroy it in others). Left unmentioned are designed, and rigged the second study. With this the facts that rising global temperatures will raise apparently �scientific� study, the tobacco com- sea levels worldwide, inundating coastal cities and panies could say that the science about second- towns, stronger storms will wreak havoc worldwide, hand smoke is ambiguous, so there is no reason and wildfires will become more frequent and more for anyone to be concerned about it or even about intense. firsthand smoke. Within the bona fide scientific community, agree-

The idea that burning large amounts of coal ment as to the causes and effects of climate change

would lead to catastrophic warming of the planet is so overwhelming that normally these contrary had been circulating since the early nineteenth voices would get little attention. But the climate century, but when it started to be taken seri- deniers raised so much noise with eye-catching ously in the middle of the twentieth century, the ads in newspapers and other media that people not fossil fuel industry came to realize that it posed versed in science responded by saying that both a serious threat to future profits. The responsible sides deserved to be heard. Of course this response and moral reaction would have been to conduct opened the door to the climate deniers to present research into how best to reduce the amount their �evidence,� and the overall effect has been to of greenhouse gasses being injected into the undermine the work of genuine science.

558Chapter14 Science and Superstition

In 2016, researchers at the University of Cali- Folta said, �I am grateful for this opportunity and

fornia, San Francisco, uncovered a disinformation promise a solid return on the investment.�* campaign by the sugar industry that had been kept In regard to how his company defended the secret for nearly fifty years. When evidence began to use of tobacco, an executive of the cigarette maker surface in the 1950s that sugar was a primary cause Brown & Williamson said famously, �Doubt is our of tooth decay, diabetes, obesity, and coronary heart product since it is the best means of competing disease, the Sugar Research Foundation, a front with the �body of fact� that exists in the mind of the group for Big Sugar, enlisted the services of Harvard general public.� In other words, instead of making University scientists to shift the blame from sugar to safer cigarettes, what the company did was to create dietary fats as the primary cause of coronary heart doubt in the public mind that smoking cigarettes is disease. The Foundation paid the Harvard scien- harmful at all. It did this by generating fake evidence tists $48,000 (in 2016 dollars) to initiate a research that conflicted with the solid findings of science program similar to the one Big Tobacco used to that cigarettes cause cancer and heart disease. And shift attention away from tobacco as a cause of lung this response by the cigarette executive typifies the cancer. The Foundation specified the objective of actions of countless big corporations in the face of the research, communicated with the scientists claims that their products are harmful or dangerous. throughout the course of their work, and approved The best response is to generate doubt in the public the final draft of the article before it was published mind about the harmfulness of the product. in the prestigious New England Journal of Medicine. The fossil fuel industry had known for a hundred In the end, the article accomplished what the Sugar years that burning fossil fuels causes global warming Foundation intended, and it made no mention of the but its executives lied about it. The sugar industry knew funding provided by the sugar industry. that there was something dishonest about their defense

Monsanto is the world�s largest seed supplier, of sugar, which is the reason they kept it secret for so

and it specializes in producing genetically modified long. And Monsanto knew that glyphosate was dan-seeds (GMOs). The main purpose of such modifica- gerous, which is the reason they wanted to keep their tion is to make the resulting plants immune to the grant to Kevin Folta a secret. University scientists tend effect of glyphosate, the active ingredient in their to be trusted by the public because they appear to be herbicide Roundup. The idea is that if farmers plant impartial and above the fray. If word gets out that they the genetically modified seeds, they can spray their have received money from an interested party, people fields with Roundup, causing the weeds to die but will be less likely to believe them. In connection with leaving the crop unscathed. Monsanto�s marketing the three criteria developed earlier in this chapter, the of Roundup together with Roundup-resistant seeds �science� produced by these corporations in defense was extremely profitable for years, and the future of their products lacks integrity. No science can claim looked bright until the World Health Organization integrity when it is based on secrets and lies. identified Roundup as a probable cause of cancer in Does the corrupting influence of big money on humans. science weaken the distinction between science

Monsanto reacted by giving grants to university and superstition? The answer is, for the most part,

scientists to write articles and give speeches arguing no. Science stands unshaken as the most successful that glyphosate is harmless to humans. One such sci- engine ever developed for disclosing the truth about entist was Kevin Folta of the University of Florida. the world around us. What big money has succeeded Folta was awarded an unrestricted grant of $25,000 in doing is to create an alternate practice, which I call together with expenses to pitch the safety of glypho-

was supposed to be kept secret, but somehow the ing War, Emails Show,� New York Times, September 5, 2015, www .nytimes.com/2015/09/06/us/food-industry-enlisted-academics-in word got out. In an email to a Monsanto executive, sate, which he did at multiple conferences. The grant *Eric Lipton, �Food Industry Enlisted Academics in GMO Lobby-

-gmo-lobbying-war-emails-show.html?_r=0.

Section 14.5 Abusing Science 559

�smoke-and-mirrors science,� that stands alongside same thing. Yet, beliefs about what science says are genuine science and competes with genuine science still highly important. Beliefs influence government for public attention. As a result of this competition, action, and if the day should come when big money, what people believe to be the findings of science has armed with smoke-and-mirrors science, dictates become vexed with confusion and uncertainty. But what most of us take to be true, then the future of beliefs about science and science itself are not the the human race will be bleak.

Key ideas introduced in this chapter

anecdotal evidence autokinetic effect replicability hypnagogic hallucinations falsifiable/disconfirmable hypnopompic hallucinations ad hoc modifications collective hallucinations Ockham�s razor confabulation placebo effect scientific progress pareidolia striking predictions perceptual set vague hypotheses

Exercise 14

I. Engage in the following discussion exercises.

1. Discuss the evidentiary problems raised by the following questions:

a. After drinking tea made from tarragon leaves every day for a year, your

grandfather reports that his bunions cleared up. Does this prove that tar-ragon tea has the power to cure bunions?

b. You go to a psychotherapist who diagnoses your emotional condition as re-

sulting from suppressed Oedipal tendencies. When these tendencies rise to the surface they can cause outbursts of emotion, but when they are held in check your behavior may be either normal or overly placid. Is the psycho-therapist�s hypothesis a good one?

c. Every time the star running back of your school�s football team makes a

touchdown, he kneels for a quick prayer in the end zone. Is God blessing this player�s performance on the field?

d. A few years ago, Rose Creek Health Products began marketing a product it

called �Vitamin O,� for which it charged $10 per ounce. In newspaper and In-ternet ads the company published several testimonials from users. One of them, from a lung cancer victim, read, �Three days after starting the Vitamin O, I threw my cane away. In November we went to Arizona and I bought myself a bicycle.� Another, about a man who had suffered severe headaches for twenty years stated, �The day he began taking Vitamin O his headaches disappeared.� Another said that a man received �very significant relief from his emphysema.�

560Chapter14 Science and Superstition

Yet another said, �After taking �Vitamin O� for several months, I find I have more energy and stamina and have been immune to colds and flu.� Do these testimonials prove that Vitamin O provides relief from disease and pain? Two years later the Federal Trade Commission obtained an injunction against Rose Creek after proving that Vitamin O is nothing more than ordinary salt water. Vitamin O is still sold online.

e. Shortly before drifting off to sleep, your roommate reports that she heard

someone call out her name in a loud, clear voice. Since no one else was in the room at that time, does this mean that the voice came from a ghost?

f. In 2002, the Hubble Space Science Institute released new high-resolution

photographs of the Cone Nebula that were taken by the recently refurbished Hubble Space Telescope. On viewing these images, hundreds of Christians thought they could see the face of Jesus in the nebula, which they renamed the Jesus Nebula (google on Jesus Nebula). Do these images prove that Jesus is alive in the cosmos?

2. Discuss the meaning of the following selections:

a. The universe is a macrocosm of creative energy and power, and every man,

woman, and child is the epitome of this totality of the cosmos. Within your individualized energy field, the microcosm called you, are twenty-two Causal Powers, or angels, that control your conscious behavior and govern the manifestation of all forms and experiences in your personal life.

( John Randolph Price, The Angels Within Us)

b. What is remarkable about the nervous system of the human species is that

it can command this infinite organizing power through conscious intent. Intent in the human species is not fixed or locked into a rigid network of energy and information. It has infinite flexibility. In other words, as long as you do not violate the other laws of nature, through your intent you can literally command the laws of nature to fulfill your dreams and desires.

( Deepak Chopra, The Seven Spiritual Laws of Success)

c. Every plant has an aura, the invisible charge of energy that exists around

all things. It is the refracted and reflected light from electrical impulses, heat, and vapor of the object. Or put another way, it is simply light dancing around, inside, and off all things.

(Laurie Cabot, The Power of the Witch)

d. The Pleiadians are back from the future to give us a broader perspective

on �the absurd times� we live in. They explain how we are fragmented and cut off from our connection to the whole of life. The crucial test for us will be to shift from lineal thinking to a multidimensional perspective and expand our understanding of the frequency of love�the gateway to freeing ourselves.

(Advertisement for Family of Light, by Barbara Marciniak)

e. When advice from your angel is incorrect, it ain�t your angel. Messages that

prove misleading or untrue are due to desire or fear that was not released

Section 14.5 Abusing Science 561

before you began talking with your heavenly helper. You�re aware, of course, that any situation has the likelihood of multiple outcomes.

(Alma Daniel, Timothy Wyllie, and Andrew Ramer, Ask Your Angels)

3. Discuss the following reports from residents of Hong Kong (taken from �Hong

Kong�s Feng Shui: Popular Magic in a Modern Urban Setting,� by Charles F. Emmons, Journal of Popular Culture, Summer 1992).

a. I have a little restaurant in Aberdeen, a seafood place facing the sea. I used

to be on the other side of the street and had very bad business. The feng shui man told me that it should face the sea. Now I have very good business.

b. I am a guard at a dockyard. The Westerners there believe in feng shui. They

changed the entrance to the place because the old one was unlucky. They also put out a [shiny] wok to deflect the bad luck, and a fok dzi [the Chinese Script character for good luck] upside down to bring luck in rather than out. All the workers there feel a lot better now and are in better health.

c. There were a lot of car accidents on the slope we faced, and even a murder

upstairs from there, so we put up a fork and mirror to eliminate the killing atmosphere. The mirror faced east and reflected the morning sun. A guy across from us came over and said he had been sick since we had put the mirror up, and asked us to take it down. We did, and then he said he had been well after that. But we moved to another place then.

d. Feng shui is what causes radios to work better facing one way than another.

4. One of the early creationist theories advanced in opposition to Darwin�s theory

of natural selection was put forward by the nineteenth-century naturalist Philip Gosse. According to this theory, God created the world in about 4000 b.c., and he did so in exactly six days, per the account in Genesis. To reconcile this story with indications that the world is much older, Gosse argued that God created the world with the appearance of a history. Adam was created with a navel, trees were created with rings, and fossils were implanted in geologic formations and in sea beds. Is Gosse�s theory plausible? Does his theory make God out to be a deceiver? Is a world with a fake history intrinsically better than one with no history? If we were to update Gosse�s theory, we would have to add that God created the world complete with light rays that appear to have been emitted by distant galaxies more than 10 billion years ago, as well as with fossils having ratios of carbon-12 to carbon-14 that indicate (by carbon-dating techniques) the fossils are millions of years old. Given these provisions, is there any way of disproving Gosse�s theory? In other words, is the theory disconfirm-able? If not, does this fact make the theory unscientific?

5. The modern creationist theory, called �scientific creationism,� propounded by

Duane Gish and Henry Morris, accounts for geologic formations and fossils by appealing to the great flood of Noah. This flood allegedly killed numer-ous mammals, birds, fish, and insects, and their carcasses settled in different strata amid the swirling waters, thus producing the fossils. But is it plausible that all the animals that populate the earth today were saved on Noah�s ark?

562Chapter14 Science and Superstition

These would include 25,000 species of birds, 15,000 species of mammals, 2,500 species of amphibians, 6,000 species of reptiles, including dinosaurs, and a mil-lion species of insects. How large must the ark have been, in comparison, say, with the Titanic, to house these animals together with sufficient food for a 300-day voyage? Based on the number of human hours it took to build the Titanic (do some research on this), how long do you think it would have taken Noah, his wife, and three sons to build the ark? Assuming the ark was made of wood, approximately how many board feet of lumber would have been needed to build it (include enough wood for walls and floors between the animals)? How long would it have taken Noah and his family to collect and load the animals on the ark? How much time would it have taken to provide one meal for the animals? Is scientific creationism a plausible theory?

6. Suppose you are a fortune-teller. You have a small shop in Milwaukee complete

with a crystal ball, beads hanging in the doorway, incense, and a shelf of books devoted to the dark arts. At about 4:00 in the afternoon in early March a couple in their mid-thirties comes into your shop. They are apparently healthy, and both are well dressed and wearing slightly worn gold wedding bands. They are obviously affectionate, and as they take a seat on your couch, the woman gives the man a quick kiss. You notice that their faces and arms are well tanned, and the man is holding a key ring with two keys. One is stamped with the word �Porsche.� When you glance through the front window, you notice a small, two-seat sports car of that make parked on the street. You introduce yourself as a fortune-teller famous throughout the world, and as you gaze into your crystal ball, you say, �I see . . .� (Do a cold reading of this couple.)

At about 10:00 the next morning a woman in her mid-forties comes

through the front door. She looks tired, her makeup is slightly smeared, and behind dark glasses you notice that her eyes look puffy. Her clothing is rum-pled, and you detect a slight scent of alcohol on her breath. As she takes a seat on your couch, you notice that she is agitated and nervous, and she fidgets with the wedding ring on her finger. In a subdued tone of voice she says that she wants her fortune told. You reply that she has come to the right place, and as you wave your hands over your crystal ball, you begin with the words �I see . . .� (Do a cold reading of this woman.)

7. Psychic phenomena include psychokinesis, telepathy, clairvoyance, and pre-

cognition. Psychokinesis is the ability to move objects through the direct power of the mind, telepathy is the power to perceive the thoughts of others without the use of the senses, clairvoyance is the ability to perceive distant objects with-out the use of the senses, and precognition is the ability to see the future with-out the use of the senses.

The psychologist Terence Hines argues that gambling casinos and govern-

ment-run lotteries provide a real-world test for the existence of psychic phe-nomena. Every roll of a pair of dice, every spin of a roulette wheel, every game of poker or blackjack, and every purchase of a lottery ticket is an opportunity for the operation of psychokinesis, telepathy, clairvoyance, or precognition. If these phenomena actually occurred, then one would expect that the earnings

Section 14.5 Abusing Science 563

of casinos and lotteries would be affected. But in fact these earnings are exactly as the laws of chance predict. Does this fact provide a fair test for the existence of psychic phenomena? Some defenders of psychic phenomena argue that psy-chic powers cannot be used for personal gain. Does this argument amount to an ad hoc modification of the hypothesis underlying psychic phenomena? Is the argument plausible?

8. During the past hundred years, thousands of experiments have been per-

formed to demonstrate unequivocally the existence of psychic phenomena, but thus far it appears that none has succeeded. To account for the negative results of some of these experiments, Gertrude Schmeidler invented the �sheep-goat� hypothesis, according to which such experiments are influenced by the atti-tudes of the experimenters. If the experimenters doubt the existence of psychic phenomena, then the experiment is doomed to failure; but if the experiment-ers are believers, then the experiment will succeed. Is this hypothesis believ-able? Does it qualify as an ad hoc modification? How might the attitude of the experimenters affect the outcome of the experiment? Could James Randi�s presence account for the inability of performers such as Uri Geller to demon-strate their alleged psychic powers?

9. Several years ago, the physicist David Simpson and his colleague Norman Fox-

well performed an experiment involving a group of UFO buffs in England. On a Saturday evening about thirty of these individuals had gathered on a remote hilltop where they thought there was a good chance of sighting a UFO. Simpson positioned himself on another hilltop, about three-quarters of a mile away, and he pointed a purple spotlight, connected to a 12-volt battery, in the direction of the group. At 11:00 p.m. he switched on the light for 5�seconds, and after a 5-second pause he switched it on again for 25 seconds. Mr. Foxwell, Simpson�s accomplice, was stationed among the observers, and to enhance the emotional effect, shortly after the purple light was sighted, Foxwell switched on a buzzer that was attached to a bogus magnetic-field sensor. According to UFO lore, UFOs are supposed to generate intense magnetic fields, so the buzzer signaled to the crowd that a UFO was operating nearby. To further enhance the effect, Foxwell had loaded a camera with a roll of film, two frames of which had been doctored with a fake latent image of a UFO. Foxwell proceeded to take two addi-tional snapshots in the general direction of the purple light (which by then had been turned off), and he then gave the film to one of the members of the group for processing.

When the film was developed, sure enough, a tiny image of what appeared

to be a UFO was visible on two frames. But those two frames (which were taken a year earlier) also contained geographic features that were inconsistent with the other two (genuine) frames. These inconsistencies were so glaring that anyone who examined all four frames closely would detect that at least two of them had been faked. However, when the photographs were examined in the laboratory of the Flying Saucer Review, an international publication devoted to UFO sightings, the experts announced that the negatives were �genuine be-yond all doubt.� Later, the director of research at the Astrophysical Institute of the French National Centre for Scientific Research published a �tentative

564Chapter14 Science and Superstition

interpretation� of the photographs, stating �In my opinion there is no question of the object photographed being in any possible way the result of faking.�

Four months after the sighting, a report of the incident was published in the

Flying Saucer Review. As you read the following selection from the report, re-member that the purple light was turned on for a total of 30 seconds, that it was absolutely stationary, and that it was located on the horizon, at zero degrees of elevation from the observers:

At 11:02 p.m. an object was seen at an elevation of approximately 20 degrees in the eastern sky. The object appeared very suddenly as if it came through the clouds, and appeared to the eye as a very bright ovoid light�purple in colour with a periphery of white. Two members of my group who observed the object through binoculars both remarked they could see a crimson light in the centre; this was also attested to by witnesses with good vision.

The object remained stationary for approximately 30 seconds, during which

time Mr. Foxwell was able to take the first of his photographs. The object then moved slowly to the right�towards the town�and lost a little altitude in the process. At one stage in the movement it dimmed considerably as though obscured by a low cloud. The object continued moving for approximately 20 to 30 seconds, and then stopped again. The light then increased considerably in intensity, though we could not be sure if the object was moving directly toward the observation point, or if it remained stationary. At this point the alarm of a detector sounded and a witness ran to switch it off. After 10 to 20 seconds the light dimmed and went out as though concealed by a cloud. However we were all certain that the object had not moved once more. The sighting had lasted for approximately one to one and a half minutes.

Why do you think the observers saw the purple light move? Was it because

they associated the light with the faked UFO images (the second of which ap-peared slightly below and to the right of the first)? Why did the observers see the light as elevated 20 degrees off the horizon? (The faked UFO images ap-peared approximately on the horizon.) Do you think that the expectations of the observers affected what they reportedly saw? Do you think that confabu-lation played a role in the report? How about the autokinetic effect? Or per-ceptual sets? Why do you think the specialists who examined the photographs failed to detect the inconsistencies? Could emotions have played a role in this oversight?

10. On the basis of a person�s natal chart (horoscope cast according to the per-

son�s date, time, and place of birth), astrologers claim to be able to analyze that person�s character and personality and make predictions about his or her life course. To test the claims of astrology about character analysis, UCLA physi-cist Shawn Carlson conducted a study involving thirty American and European astrologers considered by their peers to be among the best in the field. To com-plete the study Carlson selected 116 �clients,� who were then given the Califor-nia Personality Inventory (CPI), a widely used and scientifically accepted test that measures personality traits. The results of the CPI were then used to com-pile a personality profile for each of the clients. The astrologers were informed

Section 14.5 Abusing Science 565

about the nature of the CPI, and to prevent any anti-astrology bias, many of their suggestions were incorporated into the study.

The astrologers were then given each client�s natal chart together with three

personality profiles. One of the profiles matched the client in question, and the other two were chosen at random from a separate batch of personality profiles composed in the same way as the true one. No astrologer was able to confront any client face-to-face, and the study was done on a double-blind basis so that neither the astrologer nor the experimenters knew which profile corresponded with which natal chart. The astrologers predicted that they would be able to match a chart with the correct profile at least 50 percent of the time. How-ever, as it turned out, only one out of three matches was correct�exactly what would be expected by chance. The results of Carlson�s study were published in the widely respected scientific journal Nature.

Do you think that Carlson�s study tends to disprove the claim that astrology

can analyze personality features? Do you think that if the astrologers were al-lowed to meet the clients face-to-face, the results would have been different? Why?

Astrologers also claim the ability to make predictions. In 1978, R. N. Hunter

and J. S. Durr of the U.S. Geological Survey invited astrologers to submit pre-dictions about earthquakes. Hunter and Durr analyzed 240 predictions made by 27 astrologers and found that their accuracy was worse than would have been the case had they simply guessed. Also, in 1985, G. Chatillon analyzed 30 predictions for North America made by Huguette Hirsig, one of Montreal�s most famous astrologers. Only two were found to be correct. Do studies of this sort tend to disprove the claims that astrologers are able to make true predictions?

11. The first step in providing an astrological reading is to cast the chart. This is

followed by an analysis of interacting chart factors, which in turn is followed by the astrologer�s interpretation of these factors. In regard to the third step, D. Hamblin, former chairman of the United Kingdom Astrological Associa-tion, wrote:

If I find a very meek and unaggressive person with five planets in Aries, this does not cause me to doubt that Aries means aggression. I may be able to point to his Pisces Ascendant, or to his Sun conjunct Saturn, or to his ruler in the twelfth house; and if none of these alibis are available, I can simply say that he has not yet fulfilled his Aries potential. Or I can argue (as I have heard argued) that, if a person has an excess of planets in a particular sign, he will tend to suppress the characteristics of that sign because he is scared that, if he reveals them, he will carry them to excess. But if on the next day I meet a very aggressive person who also has five planets in Aries, I will change my tune: I will say that he had to be like that because of his planets in Aries.

Does this statement indicate that the step involving chart interpretation is

not subject to disconfirmation and is therefore unscientific? Is Hamblin saying here that the interpretation is not of the chart but of the face-to-face encounter with the client?

566Chapter14 Science and Superstition

12. In 1955, the psychologist N. D. Sundburg gave the Minnesota Multiphasic

Personality Inventory, a commonly used personality test, to forty-four stu-dents. Next he had two highly experienced psychologists draw up personality sketches for each student based on the results of the test. He then presented the individualized personality sketch to each student, together with a universal faked sketch�one provided for all the men and another for all the women. Fifty-nine percent of the students picked the faked sketch as describing their true personality. What do these findings imply about self-knowledge and the accuracy of personal validation? Are people prone to accept descriptions of themselves that are flattering? Do people tend to find some specific feature in their makeup that fits any generalization?

Specifically, Sundburg found that the following sketch was acceptable as de-

scribing the majority of college men:

You are a person who is very normal in his attitudes, behavior and relation-ships with people. You get along well without effort. People naturally like you, and you are not overly critical of them or yourself. You are neither overly con-ventional nor overly individualistic. Your prevailing mood is one of optimism and constructive effort, and you are not troubled by periods of depression, psychosomatic illness, or nervous symptoms.

The following sketch was found to be acceptable as describing the majority

of college women:

You appear to be a cheerful, well-balanced person. You may have some alter-nation of happy and unhappy moods, but they are not extreme now. You have few or no problems with your health. You are sociable and mix well with others. You are adaptable to social situations. You tend to be adventurous. Your interests are wide. You are fairly confident and usually think clearly.

Do you find that either of these descriptions fits you fairly well? If so, what

implications does this have for the astrologer, the tarot-card reader, or the fortune-teller?

13. On June 24 and 26, 1981, six children in the little village of Medjugorje in

Bosnia (former Yugoslavia) claimed to have had a vision of the Virgin Mary. In the days and weeks that followed, more alleged visions occurred, and reports began to attract visitors anxious for a vision of their own. From 1981 until 2010, approximately 30 million people, many from the United States, visited Medjugorje, and many reported that their hopes were fulfilled. By the time these visits began, claims of visions were popping up everywhere, to the obvi-ous glee of the charismatic pastor of the local church. However, not everyone accepted the authenticity of these visions. Among the detractors was the local bishop, Pavao Zanic, who, on July 25, 1987, issued a declaration that included the following:

The Madonna, they say, started to appear on the Podbrdo [a hamlet] of the Mountain Crnica, but when the militia forbade going there, she came into homes, into forests, fields, vineyards and tobacco fields; she appeared in the church, on the altar, in the sacristy, in the choir loft, on the roof, on the church

Section 14.5 Abusing Science 567

steeple, on the roads, on the way to Cerno, in a car, on buses, in classrooms, in several places in Mostar and Sarajevo, in monasteries in Zagreb, Verazdin, Switzerland, Italy, once again in the Podbrdo, atop Krizevac, in the parish, in the rectory, etc. It is certain that not even half of the alleged places where the apparitions have taken place have been mentioned. . . .

In 1991, after a lengthy inquiry, the Yugoslavian Bishops Conference issued

a statement that nothing miraculous had occurred at Medjugorje; nevertheless, thousands of pilgrims continue to stream into the village even to this day. Also, the original six visionaries continue to claim apparitions of the Virgin.

How do you think all these visions are best explained? Do you think they

are evidence of miracles? Or, could collective hallucination be responsible? Do people see what they want to see? What is your estimate of the average annual economic benefit that 30 million visitors would bring to a small village over a span of twenty-nine years? Do you think that this benefit has anything to do with the motives of the original six visionaries? Do you think that the pilgrims to Medjugorje are prone to superstition?

The philosopher Paul Kurtz suggests that religion, with its emphasis on mir-

acles and the supernatural, sensitizes people to believe in the superstitious and the paranormal. Do you think that Kurtz is correct in this assessment? Is there any essential difference between the miraculous and the superstitious? If so, is there any evidence to support this difference?

II. Design or perform the following experiments, as directed.

1. Psychic phenomena include psychokinesis, telepathy, clairvoyance, and pre-

cognition (see Exercise I, 7). Design an experiment to test for the existence of one or more forms of these phenomena. The experiment might use Zener cards, dice, remote-viewing tests, or Ganzfield (sensory deprivation) condi-tions. Be sure to build sufficient precautions into the experiment to prevent cheating.

2. Iridology is a diagnostic procedure that depends on the hypothesis that each

organ of the body is represented by a specific part of the iris. By examining the eyes of a patient, the iridologist is supposed to be able to detect disease. Design an experiment to test this hypothesis. Employ proper restrictions to prevent cheating.

3. Dowsing involves the use of a forked tree branch (�dowsing rod�) to find under-

ground water. Sometimes a pair of metal rods are used, and sometimes oil, minerals, and metal objects are claimed to have been found. When the dowser feels sudden jerks from the dowsing rod, he knows he is standing over a source of water (or oil, etc.). The practice depends on the hypothesis that water exerts some kind of magnetic influence over the rod or engages the psychic abilities of the dowser (or both). Design an experiment to test for the existence of this phenomenon.

4. Try your hand at cold reading. Have an assistant select some �clients� who are

willing to come in for a reading. They should be told in advance that you have genuine abilities in this area.

568Chapter14 Science and Superstition

5. Set up an experiment to test for the autokinetic effect. The experiment might

involve a small point of light in a large, dark room, or a flashlight some dis-tance away on a dark night. Select a small group of observers for this experi-ment, and include a test for the claim that one observer�s insistence that the light is moving influences what the others see.

6. Design an experiment to test whether people are prone to believing fantasti-

cal explanations of things they do not understand (including conjurer�s tricks.) Consider repeating the experiment (described in the text) that Singer and Benassi performed.

7. Design an experiment to test the claims of astrology. You might begin by select-

ing ten �clients�; obtain the exact time and location of their births and have an astrologer do a reading for each. Then see if the clients can recognize which reading (out of, say, a total of five) pertains to them. Or, alternately, for each reading you might prepare a �reverse reading� in which the character descrip-tion is the exact reverse of the �true� reading. Then give each client a copy of both readings (true and reverse) and see if he or she can distinguish the true one.

III. After researching one of the following topics, write an essay of eight to ten pages evalu-

ating the plausibility of the underlying hypothesis.

1. The existence of UFOs

2. Extraterrestrial abductions

3. Scientific creationism

4. Darwin�s theory of natural selection

5. Biorhythms

6. Phrenology

7. S�ances and channeling

Summary

Criteria that bear on the distinction between science and superstition:

■ Evidentiary support:

● Superstitious hypotheses tend to lack supporting evidence.

● Evidence is based on measurements expressed in mathematical terms.

● Anecdotal evidence is usually not acceptable.

● Experiments must be replicable at other times and places.

● Hypotheses must be framed in precise language.

Summary569

● Hypotheses must be disconfirmable.

● Ad hoc modifications must be avoided.

● Simple explanations are better than complex ones.

● The best hypotheses lead to predictions that turn out to be true.

● Superstitious hypotheses never lead to striking, new insights that turn out to be true.

■ Objectivity:

● Beliefs grounded in emotion are not well founded.

● Well-founded beliefs are not rooted in the preachings of a guru.

● Mental laziness and magical thinking must be avoided.

● The media are complicit in reinforcing magical thinking.

● The placebo effect is responsible for many apparent cures.

● Perceptions are affected by expectations.

● Hallucinations can distort the content of perception.

● Confabulation can distort the content of memory.

■ Integrity:

● Superstition often rests on faked evidence.

● Practitioners of superstition ignore bad predictions that arise in connection with

their practice.

● Practitioners of superstition ignore coherence problems in their hypotheses.

● Practitioners of superstition ignore conflicts between their hypotheses and other,

well-confirmed hypotheses.

● Scientists work on such problems until they are solved.

■ Abusing Science:

● Fraudulent work occurs occasionally in science, but it is not sufficient to erode the

distinction between science and superstition.

● To protect the sale of their products, big corporations may undermine genuine sci-

ence by promoting what is called smoke-and-mirrors science.

570Chapter14 Science and Superstition

Answers to Seleced Exercises

Exercise 1.116. P : The nations of planet earth have acquired nuclear 1 weapons with an explosive power equal to more than a million Hiroshima bombs. I. P : Studies suggest that explosion of only half these 2 weapons would produce enough soot, smoke, and 1. P: Carbon monoxide molecules happen to be just the dust to blanket the earth, block out the sun, and bring right size and shape, and happen to have just the on a nuclear winter that would threaten the survival right chemical properties, to fit neatly into cavities of the human race. within hemoglobin molecules in blood that are nor- C: Radioactive fallout isn�t the only concern in the mally reserved for oxygen molecules. aftermath of nuclear explosions. C: Carbon monoxide diminishes the oxygen-carrying 19. P : Antipoverty programs provide jobs for middle-class capacity of blood. 1 professionals in social work, penology, and public 4. P: When individuals voluntarily abandon property, they health. forfeit any expectation of privacy in it that they might P : Such workers� future advancement is tied to the con-have had. 2 tinued growth of bureaucracies dependent on the C: A warrantless search and seizure of abandoned existence of poverty. property is not unreasonable under the Fourth C: Poverty offers numerous benefits to the nonpoor. Amendment. 22. P: Take the nurse who alleges that physicians enrich 7. P 1 : We need sleep to think clearly, react quickly, and themselves in her hospital through unnecessary sur-create memories. gery; the engineer who discloses safety defects in P 2 : Studies show that people who are taught mentally the braking systems of a fleet of new rapid-transit challenging tasks do better after a good night�s sleep. vehicles; the Defense Department official who alerts P 3 : Other research suggests that sleep is needed for cre-Congress to military graft and overspending: All ative problem solving. know that they pose a threat to those whom they C: It really does matter if you get enough sleep. denounce and that their own careers may be at risk. 10. P 1 : Punishment, when speedy and specific, may suppress C: The stakes in whistle-blowing are high. undesirable behavior. 25. P : It is generally accepted that by constantly swimming P 2 : Punishment cannot teach or encourage desirable 1 with its mouth open, the shark is simply avoiding alternatives. suffocation. C: It is crucial to use positive techniques to model and P : This assures a continuous flow of oxygen-laden water reinforce appropriate behavior that the person can 2 into the shark�s mouth, over its gills, and out through use in place of the unacceptable response that has to the gill slits. be suppressed. C: Contrary to the tales of some scuba divers, the toothy, 13. P 1 : Private property helps people define themselves. gaping grin on the mouth of an approaching shark is P 2 : Private property frees people from mundane cares of not necessarily antici patory. daily subsistence. 28. P : Anyone familiar with our prison system knows that P 3 : Private property is finite. 1 there are some inmates who behave little better than C: No individual should accumulate so much property brute beasts. that others are prevented from accumulating the necessities of life.

571

P : If the death penalty had been truly effective as a deter-2 VI.

C: The very fact that these prisoners exist is a telling 1. Sufficient: If something is a tiger, then it is an animal. rent, such prisoners would long ago have vanished.

argument against the efficacy of capital punishment 4. Necessary: If a person has no racket, then he/she cannot as a deterrent. play tennis. Or, If a person plays tennis, then he/she has a

II. 7. Sufficient: If leaves burn, then smoke is produced. 10. Necessary: If a person does not intend to deceive someone, racket. 1. College sports are as much driven by money as are pro- then he or she does not tell a lie. Or, If a person tells a lie,

fessional sports. then he or she intends to deceive someone.

4. Business majors are robbing themselves of the true pur-

pose of collegiate academics, a sacrifice that outweighs Exercise 1.3 the future salary checks.

7. A person cannot reject free will and still insist on crimi-

10. Protecting the environment requires that we limit popu- 1. Deductive (argument based on mathematics; also, con-lation growth. clusion follows necessarily from the premises). nality and codes of moral behavior. I.

4. Deductive (categorical syllogism; also, conclusion fol-

Exercise 1.2 lows necessarily from the premises). 7. Inductive (causal inference; also, conclusion follows only

probably from the premise).

I.10. Inductive (argument from analogy; also, conclusion fol-

10. Nonargument; report. 19. Inductive (causal inference; also, conclusion follows only 13. Nonargument; illustration. probably from the premises). 16. Nonargument; piece of advice. 22. Deductive (conclusion follows necessarily from the premise; 19. Argument (conclusion: For organisms at the sea surface, this example might also be interpreted as an argument from sinking into deep water usually means death). definition�the definition of �refraction�). 22. Argument (conclusion: Atoms can combine to form mol-25. Inductive (causal inference: The dog�s familiarity with ecules whose properties generally are very different from the visitor caused the dog to be silent). those of the constituent atoms). 28. Inductive (causal inference; also, the word �may� sug-25. Nonargument; explanation. gests a probabilistic inference). 4. Nonargument; illustration. lows only probably from the premise). 7. Argument (conclusion: If stem-cell research is restricted, 16. Deductive (conclusion follows necessarily from the then people will die prematurely). premise). 1. Nonargument; explanation. 13. Inductive (argument from authority; also, conclusion fol-lows only probably from the premise). 28. Argument (conclusion: A person never becomes truly

31. This passage could be both an argument and an Exercise 1.4 self-reliant).

explanation (conclusion: In areas where rats are a problem,

34. Nonargument; loosely associated statements. it is very difficult to exterminate them with bait poison). I. 1. Valid, unsound; false premises, false conclusion. 4. Valid, sound; true premises, true conclusion. II. 7. Invalid, unsound; true premise, true conclusion. 1. Nonargument. 10. Valid, unsound; false premise, false conclusion. 4. Nonargument. 13. Invalid, unsound; true premises, true conclusion. 7. Argument (conclusion: Creating a third political party� II. the independent party�is a good idea). 10. Argument (conclusion: Strong anti-bullying programs 1. Strong, cogent; true premise, probably true conclusion. are needed to provide a means to report bullying anon- 4. Weak, uncogent; true premise, probably false conclusion. ymously, to train all school personnel to take reports of 7. Strong, uncogent; false premise, probably true conclusion. bullying seriously, and to offer workshops for children 10. Strong, cogent; true premise, probably true conclusion. on how to respond to being bullied). 13. Weak, uncogent; true premises, probably false conclusion.

572 Answers to Selected Exercises

III.Exercise 1.6

4.� Deductive, valid. I.1.� Deductive, valid.

7.� Inductive, weak. 1. 2 3 10.� Deductive, invalid.

13.� Inductive, weak.

16.� Deductive, invalid.

19.� Inductive, strong. 1

Exercise 1.5 4. 3

I. 2 4

1. All G are S. All cats are animals. (T)

All Q are S. All dogs are animals. (T)

All 1 G are Q . All cats are dogs. (F)

4. No U are P. No fish are mammals. (T) 7. 2 3 4 5 Some U are not F . Some fish are not cats. (T)

Some F are not P. Some cats are not mammals. (F) 7. No P are H. No dogs are fish. (T)

No 1 C are H . No mammals are fish. (T)

10. Some 10. 4 5 S No P are C. No dogs are mammals. (F) are not O. Some dogs are not fish. (T)

Some G are not O. Some animals are not fish. (T) Some S are not G. Some dogs are not animals. (F) 13. Some I are not E. Some animals are not 2 3

mammals. (T)

All C are E. All cats are mammals. (T) Some C are not I. Some cats are not animals. (F) 1

II.13. 1 2 3 1. If A then E.

Not A. 4 Not E .

George Washington is dead. (T) If George Washington was assassinated, then II.

George Washington was not assassinated. (T) 1. 3 4 5 6 George Washington is not dead. (F)

4. If E, then either D or C.

If D, then I. 2 If E, then I.

If Tom Cruise is a man, then he is either a mouse 4. 4 5

or a human. (T)

If Tom Cruise is a mouse, then he has a tail. (T)

If Tom Cruise is a man, then he has a tail. (F) 3 6 7

All cats are dogs. (F) 7. 5 8 9 10. All R that are F are either L or H . All R are H . 3 4 6 7 All F are L . All cats that are mammals are either dogs or animals. (T) All cats are animals. (T) All C are I. 1 2 All cats with fur are either mammals or dogs. (T) 7. All C with L are either S or I.

All mammals are dogs. (F) 1 2

Answers to Selected Exercises 573

10. 3 6 7 8 Liberalism has given more rights to the criminal than to

the ordinary citizen.

Liberalism has promoted sex and violence in the school

4 5 9 10 system.

Liberals have opposed prayer in the schools.

2 Therefore, liberalism is bad.

Note: Possible variations III.

exist for , , and . 5 6 7 1. Probably verbal; ambiguity. Does �sound� designate a

13. 7 8 air (or some other medium)? subjective perception or an objective disturbance of the

4. Probably verbal; ambiguity. By �violence� do we mean

6 intentional hostility exerted by one human against 11 12

another, or the operation of blind physical forces? Pos-

sibly a combination of verbal and f actual. Is human vio-

4 5 10 13 lence caused by the operation of physical forces just as

other physical events are?

7. Factual. Did Paul go to Knoxville or Nashville?

1 2 10. Factual. When was the Battle of Trafalgar fought, and

when did Nelson die?

13. Probably a combination of verbal (ambiguity) and factual.

Exercise 2.1 First, does �freedom� mean the absence of external con-

straint only, or the absence of both internal and external

constraint? Second, given the former, is it appropriate to

II. punish the perpetrator of evil acts even though those acts

1. In dog sled races the dogs are tortured. might be internally compelled? Torturing animals is morally wrong. 16. Verbal; vagueness. What is the meaning of �overpaid�? Therefore, dog sled races are morally wrong. 19. Verbal; vagueness. What is the meaning of �poverty�?

4. Free ownership of guns is as noble as belief in God and intestinal fortitude. Exercise 2.2 Belief in God and intestinal fortitude made our country

great and free. I.

Continued belief in God and intestinal fortitude are nec-

essary to keep our country the way it is. 4a. Plant, tree, conifer, spruce, Sitka spruce.

Therefore, gun control is wrong. and intestinal fortitude. Exercise 2.3 Free ownership of guns is no less important than God

7. The celebration of cultural diversity causes social I. fragmentation.

The celebration of cultural diversity is symptomatic of a 1. Precising definition. 16. Persuasive definition. split personality. 4. Lexical definition. 19. Lexical definition. The people who set this country up framed one nation, 7. Persuasive definition. 22. Precising definition. indivisible. 10. Theoretical definition. 25. Stipulative definition. The celebration of cultural diversity works against the 13. Stipulative definition. intention of these people. The celebration of cultural diversity erodes national identity. Exercise 2.4 Therefore, the celebration of cultural diversity is wrong. 10. Liberalism has excessively enlarged the welfare system. I. Liberalism has made welfare recipients indolent and irresponsible. 1. Definition by subclass. The liberals refuse to acknowledge or correct the defects 4. Enumerative definition. in this system. 7. Demonstrative definition. Liberalism has made the criminal justice system too sen-10. Operational definition. sitive to the criminal and too insensitive to the victim of 13. Definition by genus and difference. crime. 16. Etymological definition.

574 Answers to Selected Exercises

19. Enumerative definition. Exercise 3.222. Synonymous definition. 25. Definition by subclass. I.28. Definition by genus and difference.

II.1. Appeal to pity. 16.Ad hominem4. Accident. circumstantial. 7. Appeal to force. 1a. �Skyscraper� means the Empire State Building, Chrys-19. Straw man. ler Building, Willis Tower, and so on. Nonsynonymous 10.Tu quoque22. Appeal to the people, term: building. (you, too). indirect variety. 13. Red herring. 25. Missing the point. 3a. �Animal� means a horse, bear, lion, and so on. Nonsyn-onymous term: mammal. 5a. �Intersection� means crossing. 6a. A person is a �genius� if and only if that person can earn Exercise 3.3 a score of 140 on an IQ test. 7a. �Drake� means a male duck. I.8a. �Morphology� is derived from the Greek words morphe , 1. Hasty generalization 10. Appeal to unqualified meaning �form,� and logos , meaning reason, speech, (converse accident). authority. or account. The morphology of something (such as an 4. Slippery slope. 13. Weak analogy. animal or a plant) gives an account or explanation of the 7. Appeal to ignorance. form or structure of that thing.

Exercise 2.5III.

25. Rule 3: too broad (the definiens also describes violins, I. 1. Hasty generalization. 13. Red herring. 4. Ad hominem 16. Missing the point. 1. Rule 3: too narrow; the definiens excludes images made circumstantial. 19. Weak analogy. of bronze, wood, plaster, and so on. 7. False cause (gambler�s 22. No fallacy. 4. Rule 6: figurative language. fallacy). 25. Appeal to ignorance. 7. Rule 5: negative. 10. Straw man. 28. False cause. 10. Rule 7: affective terminology. 13. Rule 1: improper grammar. 16. Rule 4: circular. Exercise 3.419. Rule 6: vague. 22. Rule 1: improper grammar; Rule 6: vague; Rule 3: I. too broad (the definiens also includes ketches, sloops, and yawls). 1. False dichotomy. 16. Suppressed evidence. 4. Amphiboly. 19. Division. violas, and string basses). 7. Begging the question. 22. Complex question. 28. Rule 2: fails to convey the essential meaning; the defini-10. Equivocation. 25. Begging the question. tion says nothing about the purpose of a clock, which is 13. Composition. to tell the time; also too narrow: the definiens excludes twenty-four-hour clocks and clocks without twelve III. numerals on their face. 31. Rule 7: affective terminology and Rule 3: too narrow. 34. Rule 3: both too narrow and too broad; the defini- 1.Ad hominem25. Straw man. ens excludes instruments used for writing on canvas, circumstantial. 28. Accident. glass, metal, plastic, and so on, and it includes 4. Equivocation. 31. Red herring. pencils, crayons, and so on. 7. Begging the question. 34. Amphiboly. 10. Division. 37. False cause Exercise 3.113. False cause (over- (gambler�s fallacy). simplified cause). 40. Begging the question. 16. Appeal to unqualified 43. Missing the point or I. authority. suppressed evidence. 19. Composition. 1. Formal fallacy. 7. Informal fallacy. 46. Hasty generalization. 4. Informal fallacy. 10. Formal fallacy. 22. Weak analogy. 49. Composition.

Answers to Selected Exercises 575

Exercise 3.510. Quantifier: no; subject term: fast-food restaurants; copula: are; predicate term : establishments that promote public health. I.13. Quantifier : all; subject term : animal-rights activists; copula : are; predicate term : individuals motivated by 1. Missing the point or begging the question. empathy. 4. Composition. 16. Quantifier : some; subject term : presidential candi- 7. No fallacy? Weak analogy? dates; copula : are not; predicate term : threats to the 10. Appeal to unqualified authority. The statement �Only a establishment. fool . . .� involves an adhominem abusive. 13. False cause, suppressed evidence, begging the question. There is little or no evidence of any causal connection between malpractice suits and the decision of some obste-Exercise 4.2 tricians to leave the field. An unmentioned factor is the inconvenience of being on call twenty-four hours per day I. waiting for patients to deliver. There is also little or no evi-dence of any genuine �lawsuit crisis.� 1.E proposition, universal, negative, subject and predicate 16. Begging the question? 19. Appeal to pity. terms are distributed. (Strange argument!) 4.O proposition, particular, negative, subject term undis-22. False cause ( post hoc ergo propter hoc ). tributed, predicate term distributed. 25. Suppressed evidence? Weak analogy? Begging the ques-7.I proposition, particular, affirmative, subject and predi-tion? No fallacy? The Commerce Clause of the U.S. Con-cate terms undistributed. stitution and pertinent federal legislation prohibit unfair II. trade practices between states. No equivalent regulations exist for international trade. 1. No drunk drivers are threats to others on the highway. 28. Appeal to the people (direct variety). Also appeal 4. Some CIA operatives are champions of human rights. to pity? 7. Some stay-at-home parents are workers who deserve 31. Appeal to the people (direct variety)? Social Security. 34. False dichotomy? No fallacy? III.37. Appeal to unqualified authority, slippery slope. 40. Several cases of weak analogy. Also, a possible case of ad 1. Some owners of pit bull terriers are people who can hominem abusive. expect expensive lawsuits. 43. Begging the question; straw man. 4. No residents of Manhattan are people who can afford to 46. Appeal to unqualified authority. The last paragraph sug-live there. gests a hasty generalization. 7. Some prescription medicines are substances hazardous to 49. Argument against the person, circumstantial; also, beg-human health. ging the question. Are talk-show participants informed and unbiased spokespersons? IV.52. Argument against the person, abusive; tu quoque. Also, 1. Some oil spills are not events catastrophic to the begging the question. Does the possibility that the Japa-environment. nese would have used the atomic bomb against us justify 4. All corporate lawyers are people with a social conscience. our use of it against them? 7. Some economic forums are venues for airing financial 55. Weak analogy? No fallacy? fears.

Exercise 4.1Exercise 4.3

I.I.1.Quantifier: some; subject term: executive pay packages; 1. copula: are; predicate term: insults to ordinary workers. 4.Quantifier: some; subject term: preachers who are intoler-ant of others� beliefs; copula: are not; predicate term: tele- LH vision evangelists. 4. 7.Quantifier: no; subject term: sex education courses that are X taught competently; copula: are; predicate term: programs that are currently eroding public morals. FJ

576 Answers to Selected Exercises

7. Exercise 4.4 X

HCI.

1. No non-B are A. (true)

II. 4. All non-B are A. (false)

7. Contraposition. (undetermined)

1. Invalid. 10. Valid. 10. Obversion. (false)

7. Invalid. 4. Valid. 13. Invalid. II.

1a. All storms intensified by global warming are hurricanes.

III. (not logically equivalent)

2a. No radically egalitarian societies are societies that pre-

serve individual liberties. (logically equivalent)

1. No S are B. 3a. All physicians eligible to practice are physicians with

valid licenses. (logically equivalent)

SBIII.

All S are B. 1. Invalid (illicit conversion).

(invalid) 4. Invalid (illicit contraposition). S 7. Valid. B10. Valid. 4. All M are C. 13. Invalid (illicit conversion).

16. Invalid (illicit contraposition).

M19. Valid. C

False: Some Exercise 4.5M are not C . (valid) MCI.

7. No F are S. 1. (a) False, (b) True, (c) False.

4. (a) Undetermined, (b) True, (c) Undetermined.

FS 7. (a) False, (b) Undetermined, (c) Undetermined.

False: All II.F are S .

(invalid, existential fallacy) X 1. Valid.

FS 4. Invalid (existential fallacy).

7. Invalid (illicit contrary).

10. No V are A. 10. Invalid (illicit subcontrary).

13. Invalid (existential fallacy).

VAIII.

False: Some V are A. 1. Valid, unsound.

(valid) 4. Valid, sound.

VA 7. Invalid, unsound (existential fallacy).

10. Invalid, unsound (illicit subcontrary).

IV.

13. False: Some S are

not 1. All non-B are AO.. (true) S 4. Some non-A are B. (undetermined) O 7. No non-

Some 10. Some non-A are not non-B. (true) S are O . 13. Obversion. (false) A are B. (false)

existential fallacy) (invalid, X 16. Contradiction. (true)

SO19. Contrary. (undetermined)

Answers to Selected Exercises 577

V. Some A are not B. X (invalid, Boolean; conditionally 1. Valid. valid, Aristotelian) 4. Invalid (illicit contraposition). AB 7. Valid. II.10. Invalid (illicit contrary). 13. Invalid (illicit subcontrary). 1. No S are B . VI.SB

1. All I are C. False: Some S are B.

Some I are C. (subalternation) (valid, Boolean) Some C are I. (conversion)

4. All E are A. SB

False: No A are E. (conversion) X 4. False: Some D are A. False: All False: No E are A. (contrary)

A are non-E. (obversion)

7. Some P are not non-S. DA

Some P are S. (obversion) Some D are not A. Some S are P . (conversion) (invalid, Boolean; valid, Aristotelian; X False: No S are P . (contradiction) existential fallacy, Boolean) 10. False: Some F are not A .DA False: No F are A . (subalternation) False: No A are F . (conversion) False: All A are non-F . (obversion) X 7. All P are F.

Exercise 4.6 False: No P are F. X (invalid; existential fallacy, I. Boolean and Aristotelian) PF X 1. Some A are not B.

10. False: Some T are Q.

No A are B. TQ

(invalid)

AB All T are Q.

X (invalid)

4. All A are B. TQ

False: No 13. False: Some P are T. X A are B. X (invalid, Boolean; conditionally PT valid, Aristotelian) AB False: All P are T. (invalid, Boolean; valid, Aristotelian; X 7. False: Some A are B. existential fallacy, Boolean) PT

No A are B. Exercise 4.7

(valid, Boolean)

ABI.

10. No will fail. X A are B. 4. All substances identical to bromine are substances 1. All banks that make too many risky loans are banks that

AB extractable from seawater.

578 Answers to Selected Exercises

7. No halogens are chemically inert elements. 10. Some fake artworks are items sold on eBay. Therefore, 10. All ships that fly the Jolly Roger are pirate ships. some items sold on eBay are fake artworks. (valid, con-13. All bachelors are unmarried men. vert either the premise or the conclusion) 16. Some organic silicones are things used as lubricants.

19. Some giant stars are things in the Tarantula Nebula. Or All things identical to the Tarantula Nebula are things Exercise 5.1

that contain a giant star.

22. All people who require insulin treatments are diabetics. I.25. All cities identical to Berlin are cities that were the set-

ting for the 1936 Olympic Games. Or All events identical 1. Major term: things that produce intense gravity. to the 1936 Olympic Games are events that took place in Minor term: extremely dense objects. Berlin. Middle term: neutron stars.

28. All places there is smoke are places there is fire. Mood, figure:AAA-3; invalid. 31. All ores identical to pitchblende are radioactive ores. 4. Major term: good witnesses. 34. All novels written by John Grisham are novels about Minor term: hypnotized people.

lawyers. Middle term: people who mix fact with fantasy.

37. All times a rainbow occurs are times the sun is shining. Mood, figure:EIO-1; valid, Boolean. 40. Some corporate raiders are people known for their integ-

rity, and some corporate raiders are not people known II.

43. All people identical to me are people who like strawber- 1. All B are D. ries. for their integrity.

like. Or All things identical to strawberries are things I No R are D.

46. All places Talisha wants to travel are places Talisha AEE-2 No R are B.

travels. valid, Boolean

49. No physicists are people who understand the oper ation 4. No M are F.

of superconductors. All M are I.

52. All measures that increase efficiency are measures that Some I are not F.

improve profitability. EAO-3

55. Some picnics are events entirely free of ants, and some invalid, Boolean;

picnics are not events entirely free of ants. valid, Aristotelian

58. Some contestants are people who won prizes. 7. All P are E.

All L are P.

II. Some L are E.

AAI-1

1. Some figure skating finalists are performers who may invalid

win medals. 10. Some O are not C.

4. No downhill skiers who suffer from altitude sickness are All S are O.

effective competitors. Some S are C.

7. No matadors are performers who succumb easily to fear. OAI-1 10. All hungry crocodiles are dangerous animals. invalid

III.

1. Some M are not P.

1. No flu vaccines are completely effective medicines. All M are S.

Therefore, some flu vaccines are not completely effective No S are P. medicines. (valid) 4. Some M are P.

4. All dates who spend every minute texting are dates who All S are M.

are bad dinner companions. Therefore, all dates who No S are P. are bad dinner companions are dates who spend every 7. All M are P. minute texting. (invalid, illicit conversion; contrapose All S are M. premise and conclusion) All S are P.

7. No tax cuts are measures that please everyone.Therefore, 10. Some P are not M.

all tax cuts are measures that please everyone. (invalid, No M are S. illicit contrary) All S are P.

Answers to Selected Exercises 579

IV.19. No T are R. T

1. No dogmatists are scholars who encourage free thinking. Some G are not R. X EAO-3 All T are G. Some theologians are scholars who encourage free thinking. invalid, Boolean; G Some theologians are not dogmatists. valid, Aristotelian R

4. Some viruses are not things capable of replicating by II.

themselves.

All viruses are structures that invade cells. 1. M Some structures that invade cells are not things capable

of replicating by themselves.

Exercise 5.2SP

Conclusion: No S are P.

I. 4. M

1. All C are U. U X Some U are I . Some I are C . X AII-4 SP invalid ICConclusion: Some S are not P.

Some D are not P. X Some P are not H . AOO-4 4. All H are D. D 7. M

invalid PHSP

7. No P are I. IConclusion: All S are P. All F are I. 10. M No F are P.

EAE-2

valid, Boolean FP

10. No SPC are O .

Some OConclusion: None. D are not O .

EOO-2 X Exercise 5.3 Some D are not C.

invalid DC

All 1. All M are P. � D13. No P are W. PM are P.

No All M are S. D are W .

EAE-1 All S are P. valid, Boolean invalid; DW illicit minor SP

16. All C are G. G 4. All P are M. M All S are M . All G are E. Some S are P . Some E are C . X invalid; AAI-4 undistributed middle invalid ECSP

580 Answers to Selected Exercises

7. No M are P. 10. All S are M. MM All S are M . All M are P . All S are P . Some P are S . X �invalid; invalid, Boolean; drawing affirmative valid, Aristotelian; PS conclusion from SP existential fallacy, negative premise Boolean

10. Some M are P. MExercise 5.4

Some S are not P. X I. All M are S.

invalid; illicit major; SP 1. Some non-T are M. (convert, obvert) drawing negative

conclusion from Some All non-I are non-M. (contrapose)

affirmative premises Some I are T.

16. No no rules broken I are C. SP All C are non-P. Some non-M are P . MI are not P. (contrapose) No S are M . Some I are C . No S are P . All C are non-P . invalid; Some non-P are not I . exclusive premises invalid; illicit major SP 7. All non-M are non-E . (contrapose) 19. All P are M . M Some P are not M . Some S are not M . Some P are non-E . (obvert) Some No I are T. M are S . invalid; drawing affirmative conclusion No S are P . from negative premise valid, Boolean; 4. Some 13. All All M are I. P are M . M Some M are not T.

valid, Boolean; S are not P. All E are M. X Some P are not M. no rules broken SP Some P are not E . valid

II.10. Some F are non-D. (obvert)

Some F are not D. C are O . No D are V . Some O are N . X X Some F are not V . invalid; invalid; exclusive premises 1. Some N are C. C Some non-V are F. (convert, obvert) Some No D are V.

undistributed middle ON

4. Some C are not M. CExercise 5.5 No C are I . X Some I are not M . I. invalid; exclusive premises IM 1. All scientists who theorized about the nature of time are physicists. 7. No S are V . V All people identical to Stephen Hawking are scientists Some W are V . who theorized about the nature of time. Some W are not S. X All people identical to Stephen Hawking are physicists. valid, Boolean; valid no rules broken WS 4. All people who wrote the Declaration of Inde pendence are people who had a big impact on civilization.

Answers to Selected Exercises 581

All people identical to Thomas Jefferson are people who 7. All messages from friends are welcome messages.

had a big impact on civilization. Some text messages are not welcome messages.

All people identical to Thomas Jefferson are people who Some text messages are not messages from friends.

wrote the Declaration of Independence. valid

invalid, undistributed middle 10. All times administrative salaries have exploded are times 7. Some songs Miranda Lambert sings are country songs. attempts to reduce tuition have failed. All songs Miranda Lambert wants to sing are songs All recent years are times administrative salaries have

Miranda Lambert sings. exploded.

Some songs Miranda Lambert wants to sing are country All recent years are times attempts to reduce tuition have

songs. failed.

invalid, undistributed middle valid 10. All hikers who can finish the trek are endurance hikers 13. No people who think that everything is governed by

who can tolerate hardship. deterministic laws are people who believe in free will.

All hikers who hike Death Valley are endurance hikers All mechanistic materialists are people who think

who can tolerate hardship. everything is governed by deterministic laws.

All hikers who hike Death Valley are hikers who can No mechanistic materialists are people who believe in

finish the trek. free will.

Invalid, undistributed middle valid 13. All diseases carried by recessive genes are diseases that

All diseases identical to cystic fibrosis are diseases car- III. can be inherited by offspring of two carriers.

ried by recessive genes.

All diseases identical to cystic fibrosis are diseases that 1. No organizations that make alcohol readily available and

can be inherited by offspring of two carriers. acceptable are organizations that are serious about fight-

valid ing alcohol abuse.

All organizations identical to the Defense Depart ment

Exercise 5.6 are organizations that make alcohol readily available and

acceptable.

I. No organizations identical to the Defense Depart ment

are organizations that are serious about fighting alcohol

1. Premise missing: Some police chiefs fix parking tickets. abuse. 4. Conclusion missing: A few fraternities have no 4. All policies that help price petroleum products at a level

legitimate role in campus life. that reflects their true cost to society are policies that

7. Conclusion missing: Some text messages are not from place a fee on carbon.

friends. All effective policies to combat climate change are poli-

10. Premise missing: Whenever administrative salaries have cies that help price petroleum products at a level that

exploded, attempts to reduce tuition have failed. reflects their true cost to society.

13. Premise missing: No one who thinks that everything is All effective policies to combat climate change are policies

governed by deterministic laws believes in free will. that place a fee on carbon.

and

All policies that place a fee on carbon are policies that make

II. renewable energy more attractive.

1. All people who fix parking tickets are people who that place a fee on carbon. All effective policies to combat climate change are policies All effective policies to combat climate change are policies

undermine the evenhanded enforcement of the law.

Some police chiefs are people who fix parking tickets. that make renewable energy more attractive. Some police chiefs are people who undermine the even- 7. All policies that promote more college graduates

handed enforcement of the law. tomorrow are policies that result in higher tax revenues

valid tomorrow. 4. No groups that have dangerous initiation rites are groups All policies that offer financial aid to college students

that have a legitimate role in campus life. today are policies that promote more college graduates

Some fraternities are groups that have dangerous initia- tomorrow.

tion rites. All policies that offer financial aid to college students

Some fraternities are not groups that have a legitimate today are policies that result in higher tax revenues

role in campus life. tomorrow.

valid and

582 Answers to Selected Exercises

All policies that offer financial aid to college students T are K Some C . Some are not N. All All policies that result in higher tax revenues tomorrow 4. No K are N. are good investments in the future. T are not N. Some T are C.

tomorrow. Some C are Q. today are policies that result in higher tax revenues

All policies that offer financial aid to college students Some Q are not N.

today are good investments in the future. invalid KTC

All responsible people who want a great pet are people animals are people who adopt a pet from a shelter. X X X X 10. All people who want to benefit both themselves and

who want to benefit both themselves and animals.

All responsible people who want a great pet are people TNCNQN

All people who will save money over what breed-who adopt a pet from a shelter. 7. After contraposing the first premise, obverting the and second premise and the conclusion, and rearranging the premises, we have ers charge are people who will search online for rescue No X are W . groups that are breed-specific. No U are X . No V are X . All people who want a purebred animal and can�t find All U are W . one at a shelter are people who will save money over what All V are U . breeders charge. All V are Y All people who want a purebred animal and can�t find No Y are X . one at a shelter are people who will search online for invalid rescue groups that are breed-specific. WUVand All people who adopt an abandoned animal are people who benefit both themselves and animals. All responsible people who want a great pet and find an abandoned animal are people who adopt an abandoned UXVXYX animal. 10. After converting and obverting the second and fourth All responsible people who want a great pet and find an premises, obverting the third and fifth premises and the abandoned animal are people who benefit both them-conclusion, and rearranging the premises, we have selves and animals. All P are Q . All P are R . V. All Q are R . All P are S . 1. All couples who experience higher levels of emotional All R are S . No P are T . well-being are happier couples. No T are S . All average childless couples are couples who experience All T are V . higher levels of emotional well-being. Therefore, all average childless couples are happier No V are P . couples. invalid

QRST

Exercise 5.7

I.PRPSPTVP

1. All A are B. C are A. No B are C . No Some D are C . II.

Some D are not A. 1. Valid�no rules broken. valid B 4. Invalid�breaks Rule 1: C is undistributed in both C premises.

X 7. Invalid�breaks Rule 2: Y is distributed in the conclusion

but not in the premise.

10. Invalid�breaks Rule 2: V is distributed in the conclusion

CADA but not in the premise.

Answers to Selected Exercises 583

III.

1. All things that produce oxygen are things that support human life. All O are S. All rain forests are things that produce oxygen. All R are O. No things that support human life are things that should be destroyed. No S are D. No rain forests are things that should be destroyed. No R are D. After rearranging the premises, we have

All O are D. No S are D.

O are S. No

All R are O.

No R are D. SO no rules broken

ODRD

4. No brittle things are ductile things. No B are D. All superconductors are ceramics. All S are C. All things that can be pulled into wires are ductile things. All P are D. All ceramics are brittle things. All C are B. No superconductors are things that can be pulled into wires. No S are P.

All P are D. No B are P. No After rearranging the premises, we have

All B are D. No C are P. C are B .

All S are C.

No S are P. DBC

no rules broken

BPCPSP

7. All theories that attract morons are theories that contaminate

the public consciousness. All A are P. Some conspiracy theories are theories based on lies. Some C are B. All theories that contaminate the public consciousness are

theories that can harm innocent victims. All P are H . No conspiracy theories based on lies are theories that No B are non-A. fail to attract morons.

Therefore, some conspiracy theories are theories that can Some C are H. harm innocent victims.

After obverting the fourth premise and rearranging the premises, we have

All P are H.

All A are H.

All A are P.

All B are H.

All B are A.

Some C are B.

Some C are H. PAB no rules broken

AHBHCH

584 Answers to Selected Exercises

10. All things that promote skin cancer are things that cause death. All S are C. All things that preserve the ozone layer are things that prevent the release of CFCs. All O are non-R. No things that resist skin cancer are things that increase UV radiation. No non-S are U. All things that do not preserve the ozone layer are things that increase UV radiation. All non-O are U. Some packaging materials are things that release CFCs. Some M are R. No things that cause death are things that should be legal. No C are L. Some packaging materials are things that should not be legal. Some M are non-L. After contraposing the second premise, converting and obverting the third

premise, and obverting the conclusion, we have

No C are L. No S are L. All S are C . No U are L. All U are S . No non-O are L. All non-O are U . No R are L. All R are non-O .

Some M are R.

Some M are not L.

no rules broken

CSUnon-OR

SLULnon-O LRLML

IV.

1. No ducks are waltzers. No D are W. No officers are non-waltzers. No O are non-W. All poultry of mine are ducks. All P are D. No poultry of mine are officers. No P are O. After obverting the second premise and rearranging the premises, we have

All O are W.

No D are O.

No D are W.

All P are D.

No P are O.

no rules broken

DOPO

4. All hummingbirds are richly colored birds. All H are R. No large birds are birds that live on honey. No L are O. All birds that do not live on honey are birds that are dull in color. All non-O are non-R. All hummingbirds are small birds. All H are non-L.

Answers to Selected Exercises 585

After contraposing the third premise, obverting the conclusion, and

rearranging the premises, we have

No L are O.

No L are R.

All R are O.

All H are R.

No H are L.

no rules broken

LRHL

7. All books in this library that I do not recommend are books

that are unhealthy in tone. All non-R are non-H.

All the bound books are well-written books. All B are W. All the romances are books that are healthy in tone. All O are H. All the unbound books are books in this library that I do not recommend. All non-B are non-R. All the romances are well-written books. All O are W. After contraposing the first and fourth premises, we have

All B are W.

All R are B. All All R are W.

H are W.

All H are R.

All O are H.

All O are W.

no rules broken BRH

RWHWOW

10. All animals that belong to me are animals I trust. All A are T. All dogs are animals that gnaw bones. All D are G. All animals I admit to my study are animals that beg when told to do so. All S are B. All the animals in the yard are animals that belong to me. All Y are A. All animals I trust are animals I admit into my study. All T are S. All animals that are willing to beg when told to do so are dogs. All B are D. All the animals in the yard are animals that gnaw bones. All Y are G. After rearranging the premises, we have

All D are G.

All B are D. All B are G.

All S are B. All S are G.

All T are S. All T are G. All A are T. All A are G. All Y are A.

All Y are G.

no rules broken DBSTA

BGSGTGAGYG

586 Answers to Selected Exercises

Exercise 6.1III.

28. ( I. 1. A ● X T ○ F F 4. ∼ ∼ C �� Z 1. ∼ ∼ C 4. H ● F T ○ F F B 7. ∼ ∼ X ⊃ Z 7. T K P T F ○ F F 10. P ⊃ A10. ∼ ∼ ( A ● ∼ ∼ Z) 13. A ● ( K �� D ) ○ F T T T F 16. ∼ ∼ T ● ∼ ∼ P13. ( A ● Y ) �� ( ∼ ∼ Z19. ∼ ∼ ( F �� M ) or ∼ ∼ F ● ● C) ∼ ∼ M T F F 22. ( T ⊃ D ) ⊃ J ○ T T F T T 16. ( C K ∼ ∼ A ) �� ( Y K Z) 25. ∼ ∼ R �� ( M ⊃ P ) T F F T ○ T F T F ∼ ∼ P ● ∼ ∼ C ) ⊃ M31. ( T ● B ) �� ( ∼ ∼ Z ● 19. ∼ ∼ [ ∼ ∼ ( X ⊃ C ) K ∼ ∼ ( B ⊃ Z)] ∼ ∼ K ) ○ T F F T T F T T F F 34. S ● [ N ⊃ ( C �� J )] 37. M ⊃ [ N ⊃ ( S ● 22. ∼ ∼ [( A K X ) �� ( Z K Y ) ] �� [( ∼ ∼ Y ⊃ B) ● (Z ⊃ C)] T )] F T F F T F T F ○ T T F T T T F T T 40. ∼ ∼ [( M �� K ) ● ( A �� M )] 25. ( Z ⊃ C ) ⊃ {[( ∼ ∼ X ⊃ B ) ⊃ ( C ⊃ Y )] K 43. I K P46. ∼ ∼ [( W ⊃ T ) ● F T T ○ T T F T T F T F F T ( C ⊃ K )] 49. [( C ● [( Z ⊃ X ) ⊃ ( ∼ ∼ Y ⊃ Z )]} J ) K K ] ⊃ ∼ ∼ ( B �� S ) F T F F T F F F

II.IV.

4. 1. �� F T R 1. A �� P ○ T ? ∼ ∼ ( H �� S ) 4. Q ● A 7. S ⊃ G ? ○ ? T 10. A ⊃ ( ∼ ∼ N ⊃ M ) 7. A ⊃ P13. ( C ● K ) K P T ○ ? ? 16. [( T ● L ) ⊃ H ] ● ( H ⊃ I ) 10. ( P ⊃ A) K (Q ⊃ B) 19. ( C ⊃ M ) ● [( M ● F ) ⊃ ( N ● P )] ? T T ○ T ? T T 13. ∼ ∼ ( Q ● Y ) K ∼ ∼ ( Q �� A) Exercise 6.2 T ? F F ○ F F ? T T

I.Exercise 6.3

1. Dot. I. 4. Triple bar.

7. Horseshoe. 1. N ⊃ (N ⊃ N) 10. Wedge. T T T T T

II. tautologous 4. [( F T F T F

4. ○ F T H ● F T T T T F F ∼ ∼ N F T F F F T F T ○ 1. T T T T T T T ∼ ∼ H T F F T F T T E ⊃ F ) ⊃ F ] ⊃ E

10. 7. F F T contingent W ⊃ E 7. [( Z ⊃ X ) ● (X �� Z)] ⊃ X F ○ T F T T T T T T T T T H ⊃ ( C �� E ) T F F F F T T T F T ○ T T T F 13. ∼ ∼ ( E �� F ) ⊃ ( H ● F T T T T T F T T L ) F T F F F F F T F T F F F ○ T T T T tautologous

Answers to Selected Exercises 587

10. [ 13. H ● (K �� J ) ( JG ⊃ ( N ⊃ ∼ ∼ G )] ● ● H ) �� (H [( N K G ) ● ● K ) ( N �� G )]

T F T F F T F T T T T T T T T T T T T T T T T T T T

T T F T F T F F F T F F T T T T T T F F F T T T T T

F T T T T F F T F F F T T F T T F T T T T T T T F F

F T F T T F F F T F F F F F T F F F F F F T F T F F

self-contradictory F F T T T T F F F F F T

F F T T F F F F F F F T

13. [U ● (T �� S)] K [(∼ ∼ T �� ∼ ∼ U ) ● (∼ ∼ S �� ∼ ∼ U )] F F F T T T F F F F F F

T T T T T F F T F F T F F T F F T F F F F F F F F F F F F

T T T T F F F T F F T F T F T F T logically equivalent

T T F T T F T F T F T F F T F F T

T F F F F F T F T F T T T F T F T III.

F F T T T F F T T T F T F T T T F

F F T T F F F T T T F T T F T T F 1. Carlson�s prediction is false (self-contradictory).

F F F T T F T F T T F T F T T T F 4. It is possible that both astronomers are correct. If they

F F F F F F T F T T F T T F T T F are, a supernova will not occur within 10 light years of

self-contradictory the earth.

7. It is possible that both stockbrokers are correct. If they

II. are, then Datapro will cut back its workforce. We can

conclude nothing about Netmark and Compucel.

10. It is possible that Nicole�s philosophy makes sense. If it

1. ∼ ∼ D �� B ∼ ∼ (D ● ∼ ∼ B) does, then the mind is not identical to the brain, personal

F T T T T T F F T freedom exists, and humans are responsible for their

F T F F F T T T F actions.

T F T T T F F F T

T F T F T F F T F Exercise 6.4 logically equivalent

4. R �� ∼ ∼ SS ● ∼ ∼ RI. T T F T T F F T

T T T F F F F T 1. N ⊃ S / / ∼ ∼ N ⊃ ∼ ∼ S F F F T T T T F T T T F T T F T F T T F F F T F T F F F T T T F contradictory F ○ T T T F ○ F F T F T F T F T T F 7. ( E ⊃ C ) ⊃ LE ⊃ ( C ⊃ L ) invalid T T T T T T T T T T T T T F F T F T F F 4. D ⊃ W / D / / W T F F T T T T F T T T T T T T T F F T F T T F T F T F F T F F T T T T F T T T T F T T F T F T T F F F T T F F F T F F F F T F T T F T F T T valid F T F F F F T F T F 7. Invalid (fails on first line). consistent

10. K (B ● T) W ● (T ⊃ ∼ ∼ B)

W10. Valid.

T F T F F T T T T T T F T F F T II. T T F T F T

F 7. Invalid (fails on fourth line). F T T T F F T F F T 10. Invalid (fails on fourth and sixth lines). F T T F F F F F T F T 13. Valid. F T F F F F 1. Valid. 4. Valid. T T F T T F T F F F T T T T T T F F T F F F 16. Invalid (fails on third line). 19. Valid. F F F T T F T F F T F F T T T F

inconsistent 22. Invalid (fails on eighth line). 25. Valid.

588 Answers to Selected Exercises

Exercise 6.5

1. D = True

4. N = Unknown

7. C = True

10. J = Unknown

13. C = True, P = False

16. G = False, K = False

19. L = True

22. H = False, B = False

25. C = Unknown, R = Unknown

II.

Note: Truth values may vary depending on how the problem is done. 1. B ⊃ C / ∼ ∼ C / / ∼ ∼ B

T T F T F F T

valid

4. ∼ ∼ (I K J ) / / ∼ ∼ (I ⊃ J )

T T F F F T F F

T F F T F F T T

invalid

7. G ⊃ H / H ⊃ I / ∼ ∼ J ⊃ G / ∼ ∼ I / / J

T T T T T T T F T T T T F

valid

10. (M �� N) ⊃ O / O ⊃ (N �� P) / M ⊃ (∼ ∼ Q ⊃ N) / (Q ⊃ M) ⊃ ∼ ∼ P / / N K O

T T T F T T T T F F

T T T T F T T F T T F F T F T F F T

invalid

13. (A �� B) ⊃ (C ● D) / ( X �� ∼ ∼Y) ⊃ (∼ ∼ C ● ∼ ∼ W) / ( X �� Z) ⊃ (A ● E) / / ∼ ∼ X

T T T F F T T T T F T T F T T T T T T F T valid

III.

1. K K (R �� M) / K ● ∼ ∼ R / M ⊃ ∼ ∼ K

T T F T T T T T F T T F T

inconsistent

4. (N �� C) K E / N ⊃ ∼ ∼ (C �� H) / H ⊃ E / C ⊃ H

F T T T T F T F T T T T T T T T T

F T T

F T F

consistent

7. S ⊃ (R K A) / A ⊃ (W ● ∼ ∼ R) / R K (W �� T ) / S ● U / U ⊃ T T T T T F F T F F T T T T T T T T T T T inconsistent

10. A �� Z / A ⊃ (T ● F ) / Z ⊃ (M ● Q) / Q ⊃ ∼ ∼ F / T ⊃ ∼ ∼ M / M ⊃ A T T T T T F F F T T T T T T T T F F T F T T T T T T F T T T T T F T F F F F T F T T T T F F T T

F T F

consistent

Answers to Selected Exercises 589

Exercise 6.6 constructing a counterdilemma. If Melinda adequately prepares for the test before the party, then she does not spend the party night studying and she does not fail the I. test. This would falsify the right-hand conjunct of the first premise, thus falsifying the entire premise. Here is 1. MT�valid. a counterdilemma: 4. CD�valid. If Melinda spends the night studying, she will pass 7. DD�valid. the test tomorrow; and if she doesn�t spend the night 10. DA�invalid. studying, she will go to the party. She will either 13. DS�valid. spend the night studying or not studying. Therefore, 16. AC�invalid. she will either pass the test or go to the party. 19. Invalid. 4. ( C ⊃ ∼ ∼ S ) ● ( E ⊃ S ) Numbers 7 and 13 must be rewritten as follows: S �� ∼ ∼ S 7. ( E ⊃ N ) ● ( ∼ ∼ L ⊃ ∼ ∼ K ) ∼ ∼ C �� ∼ ∼ E ∼ ∼ N �� ∼∼ ∼∼ K rewritten: ( C ⊃ ∼ ∼ S ) ● ( E ⊃ S ) ∼ ∼ E �� ∼∼ ∼∼ L ∼∼ ∼∼ S �� ∼ ∼ S13. ∼ ∼ S �� P ∼ ∼ C �� ∼ ∼ E DD ∼∼ ∼∼ S The second premise is a tautology, so it is impossi-P ble to escape between the horns. One could grasp the dilemma by the horns by arguing that corporations II. could share the cost of neutralizing toxic waste, thus preserving the competitive edge. Here is a constructive 1. B ⊃ G counterdilemma: ∼ ∼ G If corporations spend money to neutralize their ∼ ∼ B MT�valid toxic waste, then the environment will be pre- 4. W �� ∼ ∼ M served; but if corporations do not spend money to ∼ ∼ W neutralize their toxic waste, then they will remain ∼ ∼ M DS�valid competitive. Corporations will do one or the other. 7. N Therefore, either the environment will be preserved N ⊃ B or corporations will remain competitive. B MP�valid 7. ( C ⊃ L ) ● ( J ⊃ B ) 10. ( L ⊃ ∼ ∼ A ) ● ( C ⊃ F ) ∼ ∼ L �� ∼ ∼ B ∼ ∼ L ● ∼ ∼ C ∼ ∼ C �� ∼ ∼ J DD A ● ∼ ∼ F invalid Here the second premise is not a tautology, so it is pos-13. ( A ⊃ E ) ● ( P ⊃ ∼ ∼ E ) sible to escape between the horns. Perhaps students E �� ∼ ∼ E could take a double major in liberal arts and business. ∼ ∼ A �� ∼ ∼ P One could also grasp the dilemma by the horns by argu- rewritten: ing that students could major in a liberal arts field where ( A ⊃ E ) ● ( P ⊃ ∼ ∼ E ) a job would be available upon graduation. Here is a con-∼ ∼ E �� ∼∼ ∼∼ E structive counterdilemma: ∼ ∼ A �� ∼ ∼ P DD�valid If students major in liberal arts, then they will take 16. C ⊃ M courses that are interesting and rewarding; but if M they major in business, then they will have a job C AC�invalid when they graduate. Students will either major in 19. S ⊃ C liberal arts or business. Therefore, either they will I ⊃ S take courses that are interesting and rewarding or I ⊃ C HS�valid they will have a job when they graduate. 10. ( P ⊃ R ) ● ( T ⊃ E ) III.P �� TR �� E CD 1. ( S ⊃ M ) ● ( ∼ ∼ S ⊃ F ) The second premise is not a tautology, so it is at least pos-S �� ∼ ∼ S sible to escape between the horns. If we instructed coun-M �� F CD terterrorist squads to execute terrorists on the spot, we Since the second premise is a tautology, it is impos-would neither prosecute them nor release them. Can you sible to escape between the horns. The two available think of a way to grasp the dilemma by the horns? Here is strategies are therefore grasping by the horns and a counterdilemma:

590 Answers to Selected Exercises

If we prosecute suspected terrorists, then we dis- II. courage terrorism; but if we release them, then we

must either prosecute or release suspected terror- 1, 2, DS avoid the risk of retaliation by other terrorists. We 1. ∼ ∼B

4. R ⊃ C

ists. Therefore, either we will discourage terror- 1, 2, HS

terrorists. 2, 3, MP ism or we will avoid the risk of retaliation by other 7. Q

10. ∼ ∼A

1, 4, MT

V. 3, 4, MT 16. 13. ∼∼ ∼∼S ∼ ∼Z

1. If human organs are given first to registered donors, 3, 4, MP

19. H �� G

then more people will register as donors. If more 2, 4, MP people register as donors, then the supply of organs III.

will increase. Therefore, if human organs are given first (1) 1. ∼ ∼C ⊃ (A ⊃ C) to registered donors, then the supply of organs will 2. ∼ ∼C / ∼ ∼ A

4. If group problem solving is important, then we should 4. increase. (HS) 3. A ⊃ C 1, 2, MP ∼ ∼A 2, 3, MT

not emphasize individual testing. Group problem solving (4) 1. P ⊃ (G ⊃ T) is important. Therefore, we should not emphasize indi- 2. Q ⊃ (T ⊃ E) vidual testing. (MP) 3. P If we should not emphasize individual testing, then the 4. Q / G ⊃ E national math test is a mistake. We should not emphasize 5. G ⊃ T 1, 3, MP individual testing. Therefore, the national math test is a mis- 6. T ⊃ E 2, 4, MP

7. If we close the library at Central Juvenile Hall, then delin- (7) 1. take. (MP) 7. G ⊃ E 5, 6, HS ∼ ∼S ⊃ D

quents will be deprived of an opportunity to read. If delin- 2. ∼ ∼S �� (∼ ∼D ⊃ K) quents are deprived of an opportunity to read, then they 3. ∼ ∼D / K will not have access to ideas, dreams, and alternative ways of 4. ∼∼ ∼∼S 1, 3, MT living. Therefore, if we close the library at Central Juvenile 5. ∼ ∼D ⊃ K 2, 4, DS Hall, then delinquents will not have access to ideas, dreams, 6. K 3, 5, MP and alternative ways of living. (HS) (10) 1. N ⊃ (J ⊃ P)

If we close the library at Central Juvenile Hall, then 2. (J ⊃ P) ⊃ (N ⊃ J)

delinquents will not have access to ideas, dreams, and 3. N / P alternative ways of living. Delinquents must have access 4. J ⊃ P 1, 3, MP to ideas, dreams, and alternative ways of living. There- 5. N ⊃ J 2, 4, MP fore, we must not close the library at Central Juvenile 6. N ⊃ P 4, 5, HS

10. If viewing adult videos led to violent sex crimes, then (13) 1. Hall. (MT) 7. P 3, 6, MP R ⊃ (G �� ∼ ∼A)

there would be over a million violent sex crimes per 2. (G �� ∼ ∼A) ⊃ ∼ ∼S week. It is not the case that there are over a million vio- 3. G ⊃ S lent sex crimes per week. Therefore, viewing adult videos 4. R / ∼ ∼ A does not lead to violent sex crimes. (MT) 5. G �� ∼ ∼A 1, 4, MP

6. ∼ ∼S 2, 5, MP

Exercise 7.17. ∼ ∼G 3, 6, MT 8. ∼ ∼A 5, 7, DS

I.2. B ⊃ K3. (16) 1. (B ⊃ ∼ ∼M) ⊃ (T ⊃ ∼ ∼S) K ⊃ ∼ ∼M

1. ∼ ∼G 1, 2, MT 4. ∼ ∼S ⊃ N / T ⊃ N 4. C 1, 2, DS 5. B ⊃ ∼ ∼M 2, 3, HS 7. F ⊃ D 1, 3, HS 6. T ⊃ ∼ ∼S 1, 5, MP 10. G ⊃ A 1, 4, HS 7. T ⊃ N 4, 6, HS 13. ∼∼ ∼∼C 1, 3, MT (19) 1. ∼ ∼G ⊃ [G �� (S ⊃ G)] 16. ∼ ∼P 1, 2, MP 2. (S �� L) ⊃ ∼ ∼G19. ∼ ∼(S �� C) 1, 3, MT 3. S �� L / L

Answers to Selected Exercises 591

(22) 1. ( 4. ∼ ∼G 2, 3, MP Exercise 7.25. G �� ( S ⊃ G ) 1, 4, MP 6. S ⊃ G 4, 5, DS I.7. ∼ ∼ S 4, 6, MT 8. L 3, 7, DS 1. B 2 C ⊃ M ) ⊃ ( N ⊃ P ) 4. H �� F 1 2. ( C ⊃ N ) ⊃ ( N ⊃ M ) 7. Q �� K 1 3. ( C ⊃ P ) ⊃ ∼ ∼ M 10. ∼ ∼ L �� M 1, 2 4. C ⊃ N / ∼ ∼ C5. N ⊃ M 2, 4, MP II.6. C ⊃ M 4, 5, HS 7. N ⊃ P 1, 6, MP 1. G 2, Simp 8. C ⊃ P 4, 7, HS 3, Add 9. ∼ ∼ M 3, 8, MP 4. T �� U 1, Add 10. ∼ ∼ C 6, 9, MT 3, 4, MP (25) 1. ∼ ∼ N ⊃ [( B ⊃ D ) ⊃ ( N �� ∼ ∼ E )] 2. ( B ⊃ E ) ⊃ ∼ ∼ N 7. ∼ ∼ F 2, 3, MT 3. B ⊃ D 1, 4, Conj 4. D ⊃ E / ∼ ∼ D10. M ● E 1, 3, Conj 5. B ⊃ E 3, 4, HS 2, 4, MP 6. ∼ ∼ N 2, 5, MP 7. ( B ⊃ D ) ⊃ ( N �� ∼ ∼ E ) 1, 6, MP III.8. N �� ∼ ∼ E 3, 7, MP 9. ∼ ∼ (1) 1. ∼ ∼ M ⊃ E 6, 8, DS Q10. ∼ ∼ 2. R ⊃ ∼ ∼ D 4, 9, MT T3. ∼ ∼ M �� R / Q �� ∼ ∼ T4. ( ∼ ∼ M ⊃ Q ) ● ( R ⊃ ∼ ∼ T ) 1, 2, Conj IV.5. Q �� ∼ ∼ T 3, 4, CD (4) 1. ( H �� ∼ ∼ B ) ⊃ R (1) 1. O ⊃ ( P �� T ) 2. ( H �� ∼ ∼ M ) ⊃ P2. ∼ ∼ P3. H / R ● P3. O / T4. H �� ∼ ∼ B 3, Add 4. P �� T 1, 3, MP 5. R 1, 4, MP 5. T 2, 4, DS 6. H �� ∼ ∼ M 3, Add (4) 1. ( R ⊃ L ) ⊃ ( L ⊃ ∼ ∼ F ) 7. P 2, 6, MP 2. ∼ ∼ F �� ( R ⊃ L ) 8. R ● P 5, 7, Conj 3. ∼∼ ∼∼ F / ∼ ∼ R (7) 1. ( ∼ ∼ F �� X ) ⊃ ( P �� T ) 4. R ⊃ L 2, 3, DS 2. F ⊃ P5. L ⊃ ∼ ∼ F 1, 4, MP 3. ∼ ∼ P / T6. ∼ ∼ L 3, 5, MT 4. ∼ ∼ F 2, 3, MT 7. ∼ ∼ R 4, 6, MT 5. ∼ ∼ F �� X 4, Add (7) 1. H ⊃ ( D K A ) 6. P �� T 1, 5, MP 2. V �� ( R ⊃ V ) 7. T 3, 6, DS 3. R �� H (10) 1. ( D �� E ) ⊃ ( G ● H ) 4. ∼ ∼ V / D K A2. G ⊃ ∼ ∼ D5. R ⊃ V 2, 4, DS 3. D ● F / M6. ∼ ∼ R 4, 5, MT 4. D 3, Simp 7. H 3, 6, DS 5. D �� E 4, Add 8. D K A 1, 7, MP 6. G ● H 1, 5, MP (10) 1. ∼ ∼ C ⊃ [ C �� ( J ⊃ D )] 2. C ⊃ ( C ● 7. G 6, Simp U ) 3. ∼ ∼ ( C ● 8. ∼ ∼ D 2, 7, MP U ) 9. D �� M 4, Add 4. ∼ ∼ D / ∼ ∼ J10. M 8, 9, DS 5. ∼ ∼ C 2, 3, MT (13) 1. ( C ⊃ N ) ● E6. C �� ( J ⊃ D ) 1, 5, MP 2. D �� ( N ⊃ D ) 7. J ⊃ D 5, 6, DS 3. ∼ ∼ D / ∼ ∼ C �� P8. ∼ ∼ J 4, 7, MT

592 Answers to Selected Exercises

4. N ⊃ D 2, 3, DS IV.

5. ∼ ∼N 3, 4, MT

7. 2. T ● C / Q �� O ∼ ∼ C 5, 6, MT 3. 6. (1) 1. T ⊃ (Q ● F) C ⊃ N 1, Simp

8. T 2, Simp ∼ ∼ C �� P 7, Add

(16) 1. ( 4. QC �� ∼ ∼ G ) ⊃ ( ∼ ∼ P ● ● F 1, 3, MP L ) 2. (∼ ∼P ● 5. Q 4, Simp C ) ⊃ ( C ⊃ D ) 6.

5. 2. (P �� S) (R ● D) C �� ∼ ∼ G 4, Add 3. 6. ∼ ∼ P ● ∼ ∼ M / RL 1, 5, MP 4. P 1, 3, DS 7. ∼ ∼ P 6, Simp 8. 4. (4) 1. M �� PC 3, Simp ⊃ 3. C ● Q �� O 5, Add ∼ ∼ R / D �� R

(25) 1. ( 10. 5. 2. V ● IK ⊃ N ∼ 4. ∼ E ⊃ (P �� J) / J3. ∼ ∼ N ● ● ∼ ∼ EB5. V 3, Simp 4. ∼ ∼ P �� ∼ ∼ R / G6. ∼ ∼ E 2, 5, MP ∼ ∼ N 3, Simp 7. V ● ∼ ∼ E 5, 6, Conj 6. ∼ ∼ K 2, 5, MT 8. P ⊃ E 1, 7, MP 7. ∼ ∼ K ● ∼ ∼ N 5, 6, Conj 8. ( ∼ ∼ P ⊃ K ) ● 9. ∼ ∼ P 6, 8, MT ( ∼ ∼ R ⊃ G ) 1, 7, MP 10. P �� J 4, 6, MP 9. K �� G 4, 8, CD 11. J 9, 10, DS G 6, 9, DS 12. J ● ∼ ∼ E 6, 11, Conj ∼ ∼ M ● N ) ⊃ [( ∼ ∼ M �� H ) ⊃ ( K ● L )] 2. ∼ ∼ M ● ( C ⊃ D ) 3. ∼ ∼ N ● ( F K G ) / K ● ∼ ∼ NExercise 7.34. ∼ ∼ M 2, Simp 5. ∼ ∼ N 3, Simp 6. ∼ ∼ M ● ∼ ∼ N 4, 5, Conj I.7. ( ∼ ∼ M �� H ) ⊃ ( K ● L ) 1, 6, MP 1. ∼ ∼ N ● ∼ ∼ G 2 8. ∼ ∼ M �� H 4, Add 4. A ● S 3 9. K ● L 7, 8, MP 7. ∼ ∼ G �� ∼∼ ∼∼ Q 1 10. K 9, Simp 10. ∼ ∼ ( R ● P ) 1 11. K ● ∼ ∼ N 5, 10, Conj 13. H ⊃ ∼ ∼ ( L �� D ) 2 (28) 1. ( D ⊃ B ) ● ( C ⊃ D ) 2. ( B ⊃ D ) ● ( E ⊃ C ) II.3. B �� E / D �� B 1. C �� K 1, Com 4. D �� C 2, 3, CD 2, 3, DS 5. B �� D 1, 4, CD 6. B ⊃ D 2, Simp 4. L ● ( S ● F ) 1, Assoc 7. D ⊃ B 1, Simp 2, Simp 8. ( (22) 1. ( 6. 2. ∼ ∼C / B ∼ ∼ O ⊃ U ∼ 4. ∼ C �� ∼ ∼ M 3, Add 3. ∼ ∼ P ⊃ O4. ∼ ∼ O ● 5. ∼ ∼ C ⊃ T 1, 4, MP T / Q6. T 3, 5, MP 5. ∼ ∼ O 4, Simp 7. T �� B 6, Add U 2, 5, MP 8. ∼ ∼ T 2, 3, DS 7. ∼∼ ∼∼ P 3, 5, MT 8. U ● 9. B 7, 8, DS ∼∼ ∼∼ P 6, 7, Conj (10) 1. ( V ● ∼ ∼ E ) ⊃ ( P ⊃ E ) 9. Q 1, 8, MP ∼ ∼ K ● ∼ ∼ N ) ⊃ [( ∼ ∼ P ⊃ K ) ● 2. V ⊃ ∼ ∼ E ( ∼ ∼ R ⊃ G )] 3. (19) 1. ( 10. R 6, Simp D 4, 9, MP ∼ (7) 1. ( ∼ C �� ∼ ∼ M ) ⊃ ( ∼ ∼ C ⊃ T ) 11. D �� R 10, Add U ● 2. C �� ∼ ∼ T ∼∼ ∼∼ P ) ⊃ Q3. 9. 6. R ● D 2, 5, MP C ⊃ D 2, 8 MP 7. ∼ ∼P ● 5. P �� S 4, Add C 4, 7, Conj

9. B ⊃ D) ● (D ⊃ B) 6, 7, Conj ● 7. D (M �� N) 1, Dist D �� B 5, 8, CD 2, Simp

Answers to Selected Exercises 593

10. (D �� N) ● (D �� H) 1, Dist 5. (I �� M) �� G 4, Assoc

III. 2, 3, DS M 6, 7, DS (22) 1. S �� ( I ● ∼ ∼ J ) 2. S ⊃ ∼ ∼ R3. 13. 7. G �� M 2, Com M �� ( G �� T ) 1, Assoc 8. 2, Simp 6. ∼ ∼G 1, 5, MP

(1) 1. (∼ ∼M ⊃ P) ● (∼ ∼N ⊃ Q) 4. (S �� I) ● (S �� ∼ ∼J) 1, Dist ∼ 2. ∼ ( M ● N ) / P �� Q5. ( ∼ ∼J ⊃ ∼ ∼Q / ∼ ∼ ( R ● Q )

3. ∼ ∼M �� ∼ ∼N 2, DM 6. S �� ∼ ∼J) ● (S �� I) 4, Com

4. P �� Q 1, 3, CD 7. ( S �� ∼ ∼J 5, Simp

(4) 1. ∼ ∼(N ● T) 8. ∼ ∼R �� ∼ ∼Q 6, 7, CD ∼ 2. T / ∼ N9. S ⊃ ∼ ∼R) ● (∼ ∼J ⊃ ∼ ∼Q) 2, 3, Conj

3. ∼ ∼N �� ∼ ∼T 1, DM (25) 1. ∼ ∼(R ● Q) 8, DM

4. ∼ ∼T �� ∼ ∼N 3, Com 2. ( E �� ∼ ∼(D �� C)

5. ∼∼ ∼∼T 2, DN 3. E �� ∼ ∼D) ⊃ C / E

6. ∼ ∼N 4, 5, DS 4. ( E �� (∼ ∼D ● ∼ ∼C) 1, DM

(7) 1. T ⊃ (B �� E) 5. E �� ∼ ∼D 4, Simp ∼ 2. ∼ E ● T / B6. E �� ∼ ∼D) ● (E �� ∼ ∼C) 3, Dist

3. T ● ∼ ∼E 2, Com 7. ( C 2, 5, MP

4. T 3, Simp 8. E �� ∼ ∼C) ● (E �� ∼ ∼D) 4, Com

5. B �� E 1, 4 MP 9. E �� ∼ ∼C 7, Simp

6. E �� B 5, Com 10. ∼ ∼C �� E 8, Com

7. ∼ ∼E 2, Simp 11. ∼∼ ∼∼C 6, DN

8. B 6, 7, DS (28) 1. E 9, 10, DS

(10) 1. (K ● H) �� (K ● L) 2. (P �� I) ⊃ ∼ ∼(L �� C) ∼ 2. ∼ L / H3. ( P �� (I ● L)

3. K ● (H �� L) 1, Dist 4. (P ● ∼ ∼C) ⊃ (E ● F) / F �� D

4. (H �� L) ● K 3, Com 5. P �� I) ● (P �� L) 1, Dist

5. H �� L 4, Simp 6. P �� I 4, Simp

6. L �� H 5, Com 7. ∼ ∼(L �� C) 2, 5, MP

7. H 2, 6, DS 8. ∼ ∼L ● ∼ ∼C 6, DM

(13) 1. (E ● I) �� (M ● U) 9. (P �� L) ● (P �� I) 4, Com ∼ 2. ∼ E / ∼ ∼ ( E �� ∼ ∼ M ) 10. ∼ ∼L 7, Simp

3. ∼ ∼E �� ∼ ∼I 2, Add 11. P �� L 9, Simp

4. ∼ ∼(E ● I) 3, DM 12. L �� P 10, Com

5. M ● U 1, 4, DS 13. P 8, 11, DS

6. M 5, Simp 14. ∼ ∼C ● ∼ ∼L 7, Com

7. ∼∼ ∼∼M 6, DN 15. ∼ ∼C 13, Simp

8. ∼ ∼E ● ∼∼ ∼∼M 2, 7, Conj 16. P ● ∼ ∼C 12, 14, Conj

9. ∼ ∼(E �� ∼ ∼M) 8, DM 17. E ● F 3, 15, MP

(16) 1. (Q ● N) �� (N ● T) 18. F 17, Simp ⊃ 2. ( Q �� C ) ∼ ∼ N / T19. F ● E 16, Com

3. (N ● Q) �� (N ● T) 1, Com (31) 1. ( F �� D 18, Add

4. N ● (Q �� T) 3, Dist 2. (∼ ∼ R �� D) ⊃ ∼ ∼(F ● G)

5. N 4, Simp 3. F ● R) ⊃ S6. ∼∼ ∼∼N 5, DN 4. F ● ∼ ∼S / ∼ ∼ ( S �� G )

7. ∼ ∼(Q �� C) 2, 6, MT 5. ∼ ∼S ● F 3, Com

8. ∼ ∼Q ● ∼ ∼C 7, DM 6. ∼ ∼S 4, Simp

9. ∼ ∼Q 8, Simp 7. ∼ ∼(F ● R) 2, 5, MT

10. (Q �� T) ● N 4, Com 8. ∼ ∼F �� ∼ ∼R 6, DM

11. Q �� T 10, Simp 9. F 3, Simp

12. T 9, 11, DS 10. ∼∼ ∼∼F 8, DN ∼ ∼R 7, 9, DS

(19) 1. [(I �� M) �� G] ⊃ ∼ ∼G11. ∼ ∼R �� D 10, Add 2. M �� G / M12.

3. (M �� G) �� I 2, Add 13. ∼ ∼(F ● G) 1, 11, MP

4. I �� (M �� G) 3, Com 14. ∼ ∼F �� ∼ ∼G 12, DM ∼ ∼G 9, 13, DS

594 Answers to Selected Exercises

15. ∼ ∼S ● ∼ ∼G 5, 14, Conj 7. (∼ ∼E �� G) ● (∼ ∼E �� W) 2, Dist

16. ∼ ∼(S �� G) 15, DM 8. (∼ ∼E �� W) ● (∼ ∼E �� G) 7, Com (34) 1. (M ● N) �� (O ● P) 9. ∼ ∼E �� W 8, Simp

2. (N �� O) ⊃ ∼ ∼P / N10. (∼ ∼E �� P) ● (∼ ∼E �� W) 6, 9, Conj

3. [(M ● N ) �� O] ● [(M ● N) �� P] 1, Dist 11. ∼ ∼E �� (P ● W) 10, Dist

4. (M ● N) �� O 3, Simp 12. (P ● W) �� ∼ ∼E 11, Com

5. O �� (M ● N) 4, Com 13. ∼ ∼(P ● W) 3, DM

6. (O �� M) ● (O �� N) 5, Dist 14. ∼ ∼E 12, 13, DS

7. (O �� N) ● (O �� M) 6, Com

9. N �� O 8, Com Exercise 7.48. O �� N 7, Simp

10. ∼ ∼P 2, 9, MP

IV.15. M ● 4. B K N 1 N 10, 13, DS 7. N ● ∼∼ ∼∼ C �� ∼ ∼ F 1 M 14, Com 10. S ⊃ G 3 16. N 15, Simp 13. W K ∼ ∼ T 2 II.12. (M ● N) �� P 11, Simp 1. 13. P �� ( M ●G ⊃ Q 3 N ) 12, Com 14. 11. [(M ● N) �� P] ● [(M ● N) �� O] 3, Com I. (1) 1. (S ● D) �� (S ● H)

2. S ⊃ (I ● R) / S ● R1. J ⊃ M 1, Impl

3. S ● (D �� H) 1, Dist 2, 3, HS

4. S 3, Simp 4. K �� K 1, 2, CD

5. I ● R 2, 4, MP 3, Taut

6. R ● I 5, Com 7. H ⊃ (C ⊃ R) 1, Impl

7. R 6, Simp 2, Exp

8. S ● R 4, 7, Conj 10. ∼ ∼H �� ∼ ∼H 1, Impl (4) 1. G �� (R ● E) 2, Taut

2. (G �� E) ⊃ ∼ ∼R / G �� M13. (N ⊃ A) ● (A ⊃ N) 1, Trans

3. (G �� R) ● (G �� E) 1, Dist 2, Equiv

4. (G �� E) ● (G �� R) 3, Com

5. G �� E 4, Simp III.

(7) 1. 10. 6. ∼ ∼R 2, 5, MP (1) 1. (S ● K) ⊃ R7. G �� R 3, Simp 2. K / S ⊃ R8. R �� G 7, Com 3. ( K ● S ) ⊃ R 1, Com 9. G 6, 8, DS 4. K ⊃ ( S ⊃ R ) 3, Exp G �� M 9, Add 5. S ⊃ R 2, 4, MP R ⊃ ( C �� M ) (4) 1. S K Q2. ∼ ∼ ( I �� C ) 2. ∼ ∼ S/ ∼ ∼ Q3. ∼ ∼ ( A �� M ) / ∼ ∼ R4. ∼ ∼ I ● 3. ( S ⊃ Q ) ● ( Q ⊃ S ) 1, Equiv ∼ ∼ C 2, DM 4. ( Q ⊃ S ) ● ( S ⊃ 5. ∼ ∼ A ● Q ) 3, Com ∼ ∼ M 3, DM 5. Q ⊃ 6. ∼ ∼ C ● S 4, Simp ∼ ∼ I 4, Com 6. ∼ ∼ Q 2, 5, MT 7. ∼ ∼ C 6, Simp (7) 1. ( B ⊃ M ) ● ( D ⊃ 8. ∼ ∼ M ● M ) ∼ ∼ A 5, Com 2. B �� D / M9. ∼ ∼ M 8, Simp 10. ∼ ∼ C ● 3. M �� M 1, 2, CD ∼ ∼ M 7, 9, Conj 4. M 3, Taut 11. ∼ ∼ ( C �� M ) 10, DM (10) 1. ( B ⊃ G ) ● ( F ⊃ N ) 12. ∼ ∼ R 1, 11, MT 2. ∼ ∼ ( G ● N ) / ∼ ∼ ( B ● (10) 1. ∼ ∼ E �� ( B ● F ) P ) 3. ∼ ∼ G �� ∼ ∼ 2. ∼ ∼ E �� ( G ● N 2, DM W ) 4. ( ∼ ∼ G ⊃ ∼ ∼ B ) ● ( F ⊃ N ) 1, Trans 3. ∼ ∼ P �� ∼ ∼ W / ∼ ∼ E5. ( ∼ ∼ G ⊃ ∼ ∼ B ) ● ( ∼ ∼ N ⊃ ∼ ∼ 4. ( ∼ ∼ E �� B ) ● F ) 4, Trans ( ∼ ∼ E �� P ) 1, Dist 6. ∼ ∼ B �� ∼ ∼ 5. ( ∼ ∼ E �� P ) ● F 3, 5, CD ( ∼ ∼ E �� B ) 4, Com 7. ∼ ∼ ( B ● F ) 6, DM 6. ∼ ∼ E �� P 5, Simp

Answers to Selected Exercises 595

(13) 1. K ⊃ (B ⊃ ∼ ∼M) 6. F ⊃ N 3, 5, MP

2. D ⊃ (K ● M) / D ⊃ ∼ ∼ B7. ∼ ∼N ⊃ ∼ ∼F 6, Trans

3. K ⊃ (∼∼ ∼∼M ⊃ ∼ ∼B) 1, Trans 8. ∼∼ ∼∼F �� S 2, DN

4. K ⊃ (M ⊃ ∼ ∼B) 3, DN 9. ∼ ∼F ⊃ S 8, Impl

5. (K ● M) ⊃ ∼ ∼B 4, Exp 10. ∼ ∼N ⊃ S 7, 9, HS

6. D ⊃ ∼ ∼B 2, 5, HS 11. ∼∼ ∼∼N �� S 10, Impl (16) 1. T ⊃ R12. N �� S 11, DN

2. T ⊃ ∼ ∼R / ∼ ∼ T (37) 1. (D ⊃ E) ⊃ (E ⊃ D)

3. ∼∼ ∼∼R ⊃ ∼ ∼T 2, Trans 2. (D K E) ⊃ ∼ ∼(G ● ∼ ∼H) 4. R ⊃ ∼ ∼T 3, DN 3. E ● G / G ● H

5. T ⊃ ∼ ∼T 1, 4, HS 4. E 3, Simp

6. ∼ ∼T �� ∼ ∼T 5, Impl 5. E �� ∼ ∼D 4, Add

7. ∼ ∼T 6, Taut 6. ∼ ∼D �� E 5, Com (19) 1. ∼ ∼R �� P7. D ⊃ E 6, Impl

2. R �� ∼ ∼P / R K P8. E ⊃ D 1, 7, MP

3. R ⊃ P 1, Impl 9. (D ⊃ E) ● (E ⊃ D) 7, 8, Conj

4. ∼ ∼P �� R 2, Com 10. D K E 9, Equiv

5. P ⊃ R 4, Impl 11. ∼ ∼(G ● ∼ ∼H) 2, 10, MP

6. (R ⊃ P) ● (P ⊃ R) 3, 5, Conj 12. ∼ ∼G �� ∼∼ ∼∼H 11, DM

7. R K P 6, Equiv 13. ∼ ∼G �� H 12, DN (22) 1. S ⊃ (L ● M) 14. G ● E 3, Com

2. M ⊃ (L ⊃ R) / S ⊃ R15. G 14, Simp

3. (M ● L) ⊃ R 2, Exp 16. ∼∼ ∼∼G 15, DN

4. (L ● M) ⊃ R 3, Com 17. H 13, 16, DS

5. S ⊃ R 1, 4, HS 18. G ● H 15, 17, Conj (25) 1. T ⊃ G (40) 1. A K W

2. S ⊃ G / (T �� S) ⊃ G2. ∼ ∼A �� ∼ ∼W

3. ∼ ∼T �� G 1, Impl 3. R ⊃ A / ∼ ∼ ( W �� R )

4. ∼ ∼S �� G 2, Impl 4. (A ● W) �� (∼ ∼A ● ∼ ∼W) 1, Equiv

5. G �� ∼ ∼T 3, Com 5. ∼ ∼(A ● W) 2, DM

6. G �� ∼ ∼S 4, Com 6. ∼ ∼A ● ∼ ∼W 4, 5, DS

7. (G �� ∼ ∼T) ● (G �� ∼ ∼S) 5, 6, Conj 7. ∼ ∼A 6, Simp

8. G �� (∼ ∼T ● ∼ ∼S) 7, Dist 8. ∼ ∼R 3, 7, MT

9. (∼ ∼T ● ∼ ∼S) �� G 8, Com 9. ∼ ∼W ● ∼ ∼A 6, Com

10. ∼ ∼(T �� S) �� G 9, DM 10. ∼ ∼W 9, Simp

11. (T �� S) ⊃ G 10, Impl 11. ∼ ∼W ● ∼ ∼R 8, 10, Conj (28) 1. P ⊃ (∼ ∼E ⊃ B) 12. ∼ ∼(W �� R) 11, DM

2. ∼ ∼(B �� E) / ∼ ∼ P (43) 1. O ⊃ (Q ● N)

3. ∼ ∼(E �� B) 2, Com 2. (N �� E) ⊃ S / O ⊃ S

4. ∼ ∼(∼∼ ∼∼E �� B) 3, DN 3. ∼ ∼O �� (Q ● N) 1, Impl

5. ∼ ∼(∼ ∼E ⊃ B) 4, Impl 4. (∼ ∼O �� Q) ● (∼ ∼O �� N) 3, Dist

6. ∼ ∼P 1, 5, MT 5. (∼ ∼O �� N) ● (∼ ∼O �� Q) 4, Com (31) 1. K K R6. ∼ ∼O �� N 5, Simp

2. K ⊃ (R ⊃ P) 7. O ⊃ N 6, Impl 3. ∼ ∼P / ∼ ∼ R8. ∼ ∼(N �� E) �� S 2, Impl

4. (K ● R) �� (∼ ∼K ● ∼ ∼R) 1, Equiv 9. (∼ ∼N ● ∼ ∼E) �� S 8, DM

5. (K ● R) ⊃ P 2, Exp 10. S �� (∼ ∼N ● ∼ ∼E) 9, Com

6. ∼ ∼(K ● R) 3, 5, MT 11. (S �� ∼ ∼N) ● (S �� ∼ ∼E) 10, Dist

7. ∼ ∼K ● ∼ ∼R 4, 6, DS 12. S �� ∼ ∼N 11, Simp

8. ∼ ∼R ● ∼ ∼K 7, Com 13. ∼ ∼N �� S 12, Com

9. ∼ ∼R 8, Simp 14. N ⊃ S 13, Impl (34) 1. (F ● H) ⊃ N15. O ⊃ S 7, 14, HS

2. F �� S (45) 1. P ⊃ A3. H / N �� S2. Q ⊃ B / (P �� Q) ⊃ (A �� B)

4. (H ● F) ⊃ N 1, Com 3. ∼ ∼P �� A 1, Impl

5. H ⊃ (F ⊃ N) 4, Exp 4. ∼ ∼Q �� B 2, Impl

596 Answers to Selected Exercises

5. (∼ ∼P �� A) �� B 3, Add (4) 1. (G �� H) ⊃ (S ● T )

6. (∼ ∼Q �� B) �� A 4, Add 2. (T �� U ) ⊃ (C ● D) / G ⊃ C

7. ∼ ∼P �� (A �� B) 5, Assoc 3. G ACP

8. (A �� B) �� ∼ ∼P 7, Com 4. G �� H 3, Add

9. ∼ ∼Q �� (B �� A) 6, Assoc 5. S ● T 1, 4, MP

10. ∼ ∼Q �� (A �� B) 9, Com 6. T ● S 5, Com

11. (A �� B) �� ∼ ∼Q 10, Com 7. T 6, Simp

12. [(A �� B) �� ∼ ∼P] ● [(A �� B) �� ∼ ∼Q] 8, 11, Conj 8. T �� U 7, Add

13. (A �� B) �� (∼ ∼P ● ∼ ∼Q) 12, Dist 9. C ● D 2, 8, MP

14. (∼ ∼P ● ∼ ∼Q) �� (A �� B) 13, Com 10. C 9, Simp

15. ∼ ∼(P �� Q) �� (A �� B) 14, DM 11. G ⊃ C 3�10, CP

IV.2. ∼ ∼M ACP 3. 16. (P �� Q) ⊃ (A �� B) 15, Impl (7) 1. M �� (N ● O) / ∼ ∼ N ⊃ MN ● O 1, 2, DS (1) 1. 4. N 3, Simp F ⊃ D

2. ∼ ∼(D ● 5. ∼ ∼M ⊃ N 2�4, CP ∼ ∼ H ) / F ⊃ H

3. 6. ∼ ∼N ⊃ ∼∼ ∼∼M 5, Trans ∼ ∼ D �� ∼∼ ∼∼ H 2, DM

4. 7. ∼ ∼N ⊃ M 6, DN D ⊃ ∼∼ ∼∼ H 3, Impl

5. (10) 1. C ⊃ (A ● D) D ⊃ H 4, DN

6. 2. B ⊃ (A ● E) / (C �� B) ⊃ AF ⊃ H 1, 5, HS

(4) 1. D ⊃ P / (I ● 3. C�� B ACP D ) ⊃ P

2. ( 4. [ C ⊃ (A ● D)] ● [B ⊃ (A ● E )] D ⊃ P ) �� ∼ ∼ I 1, Add

3. 1, 2, Conj ∼ ∼ I �� (D ⊃ P) 2, Com

4. 5. ( A ● D) �� (A ● E) 3, 4, CD I ⊃ (D ⊃ P) 3, Impl

5. ( 6. AI ● ● (D �� E) 5, Dist D ) ⊃ P 4, Exp

(7) 1. 7. A 6, Simp G ⊃ A

2. G ⊃ L / G ⊃ ( A ● 8. (C �� B) ⊃ A 3�7, CP L )

3. (13) 1. R ⊃ B ∼ ∼ G �� A 1, Impl

4. 2. R ⊃ (B ⊃ F) ∼ ∼ G �� L 2, Impl

5. (∼ ∼G �� A) ● 3. B ⊃ (F ⊃ H) / R ⊃ H ( ∼ ∼ G �� L ) 3, 4, Conj

6. 4. R ACP ∼ ∼ G �� ( A ● L ) 5, Dist

7. 5. B 1, 4, MP G ⊃ ( A ● L ) 6, Impl

(10) 1. ( 6. B ⊃ F 2, 4, MP A ● U ) K ∼ ∼ R

2. 7. F 5, 6, MP ∼ ∼ ( ∼ ∼ R �� ∼ ∼ A ) / ∼ ∼ U

3. [(A ● U ) ⊃ ∼ ∼R] ● [∼ ∼R ⊃ (A ● 8. F ⊃ H 3, 5, MP U )] 1, Equiv

4. (A ● 9. H 7, 8, MP U ) ⊃ ∼ ∼ R 3, Simp

5. ∼∼ ∼∼R ● 10. R ⊃ H 4�9, CP ∼∼ ∼∼ A 2, DM

6. (16) 1. Q ⊃ (R ⊃ S) ∼∼ ∼∼ R 5, Simp

7. 2. Q ⊃ (T ⊃ ∼ ∼U) ∼ ∼ ( A ● U ) 4, 6, MT

8. 3. U ⊃ (R �� T) / Q ⊃ ( U ⊃ S ) ∼ ∼ A �� ∼ ∼ U 7, DM

9. ∼∼ ∼∼A ● 4. Q ACP ∼∼ ∼∼ R 5, Com

10. 5. U ACP ∼∼ ∼∼ A 9, Simp

11. 6. R ⊃ S 1, 4, MP ∼ ∼ U 8, 10, DS

7. T ⊃ ∼ ∼U 2, 4, MP

Exercise 7.58. ∼∼ ∼∼U 5, DN 9. ∼ ∼T 7, 8, MT

I.11. T �� R 10, Com 12. 10. R �� T 3, 5, MP R 9, 11, DS

(1) 1. N ⊃ O13. S 6, 12, MP

2. N ⊃ P / N ⊃ ( O ● P ) 14. U ⊃ S 5�13, CP

3. N ACP 15. Q ⊃ (U ⊃ S) 4�14, CP 4. O 1, 3, MP (19) 1. P ⊃ [(L �� M) ⊃ (N ● O)] 5. P 2, 3, MP 2. (O �� T ) ⊃ W / P ⊃ (M ⊃ W) 6. O ● P 4, 5, Conj 3. P ACP

7. N ⊃ (O ● P) 3�6, CP 4. M ACP

Answers to Selected Exercises 597

5. ( L �� M) ⊃ (N ● O) 1, 3, MP 12. ∼ ∼(H ● L) 3�11, IP

6. M �� L 4, Add 13. ∼ ∼H �� ∼ ∼L 12, DM

7. L�� M 6, Com (7) 1. (E �� F) ⊃ (C ● D)

8. N ● O 5, 7, MP 2. (D �� G) ⊃ H

9. O ● N 8, Com 3. E �� G / H

10. O 9, Simp 4. ∼ ∼H AIP

11. O �� T 10, Add 5. ∼ ∼(D �� G) 2, 4, MT

12. W 2, 11, MP 6. ∼ ∼D ● ∼ ∼G 5, DM

13. M ⊃ W 4�12, CP 7. ∼ ∼D 6, Simp

14. P ⊃ (M ⊃ W) 3�13, CP 8. ∼ ∼D �� ∼ ∼C 7, Add

II. 9. ∼ ∼C �� ∼ ∼D 8, Com

Exercise 7.612. 11. 4. T ⊃ S 4, Impl J ACP 6. S ⊃ ( T ⊃ S ) 2�5, CP 5. N ACP (13) 1. [ C ⊃ ( D ⊃ C )] ⊃ E / E 6. D 1, 4, MP 2. C ACP 7. J ● D 4, 6, Conj 3. C �� ∼ ∼ D 2, Add 8. C 2, 7, MP 4. ∼ ∼ D �� C 3, Com 9. N ● C 5, 8, Conj 5. D ⊃ C 4, Impl 10. I 3, 9, MP 6. C ⊃ ( D ⊃ C ) 2�5, CP N ⊃ I 5�10, CP 7. E 1, 6, MP J ⊃ ( N ⊃ I ) 4�11, CP (16) 1. ( N �� O ) ⊃ ( C ● D ) 2. ( D �� K ) ⊃ ( P �� ∼ ∼ C ) 3. ( P �� G ) ⊃ ∼ ∼ ( N ● D ) / ∼ ∼ N 4. N AIP I. 5. N �� O 4, Add 6. C ● D 1, 5, MP (1) 1. ( S �� T ) ⊃ ∼ ∼ S / ∼ ∼ S 7. D ● C 6, Com 2. S AIP 8. D 7, Simp 3. S �� T 2, Add 9. D �� K 8, Add 4. ∼ ∼ S 1, 3, MP 10. P �� ∼ ∼ C 2, 9, MP 5. S ● ∼ ∼ S 2, 4, Conj 11. C 6, Simp 6. ∼ ∼ S 2�5, IP 12. ∼ ∼ C �� P 10, Com (4) 1. H ⊃ ( L ⊃ K ) 13. ∼∼ ∼∼ C 11, DN 2. L ⊃ ( K ⊃ ∼ ∼ L ) / ∼ ∼ H �� ∼ ∼ L14. P 12, 13, DS 3. H ● L AIP 15. P �� G 14, Add 4. H 3, Simp 16. ∼ ∼ ( N ● D ) 3, 15, MP 5. L ⊃ K 1, 4, MP 17. ∼ ∼ N �� ∼ ∼ D 16, DM 6. L ● H 3, Com 18. ∼∼ ∼∼ N 4, DN 7. L 6, Simp 19. (4) 1. 10. ( 5. ∼ ∼GD 1, 4, MP ● ∼ ∼D 6, Com 16. ∼ ∼ G 15, Simp 6. U ● H 3, Com 17. G7. U 6, Simp ● ∼ ∼ G 14, 16, Conj 18. ∼∼ ∼∼ H 4�17, IP 8. S 2, 7, MP 19. H 18, DN 9. S ● D 5, 8, Conj (10) 1. K / S ⊃ ( T ⊃ S ) H ● U ) ⊃ ( S ● D ) 3�9, CP 2. S ACP J ⊃ D3. S �� ∼ ∼ T 2, Add 2. ( J ● D ) ⊃ C4. ∼ ∼ T �� S 3, Com 3. ( N ● C ) ⊃ I / J ⊃ ( N ⊃ I ) 5. (1) 1. 11. ∼ ∼(E �� F) 1, 10, MT H ⊃ D ∼ 10. ∼ ∼(C ● D) 9, DM 2. 12. ∼EU ⊃ S / ( H ● ● ∼ ∼F 11, DM U ) ⊃ ( S ● D ) 13. ∼ ∼ E 12, Simp 3. H ● U ACP 14. G 3, 13, DS 4. H 3, Simp 15. ∼ ∼D 17, 18, DS

8. K ⊃ ∼ ∼L 2, 7, MP 20. D ● ∼ ∼D 8, 19, Conj 9. K 5, 7, MP 21. ∼ ∼N 4�20, IP 10. ∼ ∼L 8, 9, MP (19) 1. A ⊃ [(N �� ∼ ∼N) ⊃ (S �� T)] 11. L ● ∼ ∼L 7, 10, Conj 2. T ⊃ ∼ ∼(F �� ∼ ∼F) / A ⊃ S

598 Answers to Selected Exercises

3. A ● ∼ ∼S AIP 3. Q 1, 2, MP

4. A 3, Simp 4. (P ⊃ Q) ⊃ Q 2�3, CP

5. (N �� ∼ ∼N) ⊃ (S �� T) 1, 4, MP 5. P ⊃ [(P ⊃ Q) ⊃ Q] 1�4, CP

6. N ACP (4) / (P ⊃ Q) ⊃ [(P ● R) ⊃ (Q ● R)] 7. N �� N 6, Add 1.P ⊃ Q ACP 8. N 7, Taut 2. P ● R ACP

9. N ⊃ N 6�8, CP 3. P 2, Simp 10. ∼ ∼N �� N 9, Impl 4. Q 1, 3, MP 11. N �� ∼ ∼N 10, Com 5. R ● P 2, Com 12. S �� T 5, 11, MP 6. R 5, Simp 13. ∼ ∼S ● A 3, Com 7. Q ● R 4, 6, Conj 14. ∼ ∼S 13, Simp 8. ( P ● R) ⊃ (Q ● R) 2�7, CP 15. T 12, 14, DS 9. ( P⊃ Q) ⊃ [(P ● R) ⊃ (Q ● R)] 1�8, CP 16. ∼ ∼(F �� ∼ ∼F) 2, 15, MP (7) / (P ⊃ Q) �� (∼ ∼Q ⊃ P) 17. ∼ ∼F ● ∼∼ ∼∼F 16, DM 1. ∼ ∼[(P ⊃ Q) �� (∼ ∼Q ⊃ P)] AIP

18. ∼ ∼(A ● ∼ ∼S) 3�17, IP 2. ∼ ∼(P ⊃ Q) ● ∼ ∼(∼ ∼Q ⊃ P) 1, DM

19. ∼ ∼A �� ∼∼ ∼∼S 18, DM 3. ∼ ∼(P ⊃ Q) 2, Simp

20. ∼ ∼A �� S 19, DN 4. ∼ ∼(∼ ∼P �� Q) 3, Impl

II.6. P ● ∼ ∼Q 5, DN 7. 21. A ⊃ S 20, Impl 5. ∼∼ ∼∼P ● ∼ ∼Q 4, DM P 6, Simp

(1) 1. (C ● R) ⊃ (I ● D) 8. ∼ ∼(∼ ∼Q ⊃ P) ● ∼ ∼(P ⊃ Q) 2, Com

2. R ⊃ ∼ ∼D / ∼ ∼ C �� ∼ ∼ R9. ∼ ∼(∼ ∼Q ⊃ P) 8, Simp

3. C ● R AIP 10. ∼ ∼(∼∼ ∼∼Q �� P) 9, Impl 4. I ● D 1, 3, MP 11. ∼ ∼(Q �� P) 10, DN 5. D ● I 4, Com 12. ∼ ∼Q ● ∼ ∼P 11, DM 6. D 5, Simp 13. ∼ ∼P ● ∼ ∼Q 12, Com 7. R ● C 3, Com 14. ∼ ∼P 13, Simp 8. R 7, Simp 15. P ● ∼ ∼P 7, 14, Conj 9. ∼ ∼D 2, 8, MP 16. ∼∼ ∼∼[(P ⊃ Q) �� (∼ ∼Q ⊃ P)] 1�15, IP 10. D ● ∼ ∼D 6, 9, Conj 17. ( P ⊃ Q) �� (∼ ∼Q ⊃ P) 16, DN

11. ∼ ∼(C ● R) 3�10, IP (10) / [∼ ∼(P ● ∼ ∼Q) ● ∼ ∼Q] ⊃ ∼ ∼P

12. ∼ ∼C �� ∼ ∼R 11, DM 1. ∼ ∼(P ● ∼ ∼Q) ● ∼ ∼Q ACP (4) 1. (Z ⊃ C) ⊃ B2. ∼ ∼(P ● ∼ ∼Q) 1, Simp

2. (V ⊃ Z ) ⊃ B / B3. ∼ ∼P �� ∼∼ ∼∼Q 2, DM

3. ∼ ∼B AIP 4. ∼ ∼P �� Q 3, DN 4. ∼ ∼(Z ⊃ C) 1, 3, MT 5. ∼ ∼Q ● ∼ ∼(P ● ∼ ∼Q) 1, Com 5. ∼ ∼(∼ ∼Z �� C) 4, Impl 6. ∼ ∼Q 5, Simp 6. ∼∼ ∼∼Z ● ∼ ∼C 5, DM 7. Q �� ∼ ∼P 4, Com 7. ∼∼ ∼∼Z 6, Simp 8. ∼ ∼P 6, 7, DS 8. ∼ ∼(V ⊃ Z) 2, 3, MT 9. [ ∼ ∼(P ● ∼ ∼Q) ● ∼ ∼Q] ⊃ ∼ ∼P 1�8, CP 9. ∼ ∼(∼ ∼V �� Z) 8, Impl (13) / (P ⊃ Q) ⊃ [(P ⊃ ∼ ∼Q) ⊃ ∼ ∼P] 10. ∼∼ ∼∼V ● ∼ ∼Z 9, DM 1. P ⊃ Q ACP 11. ∼ ∼Z ● ∼∼ ∼∼V 10, Com 2. P ⊃ ∼ ∼Q ACP 12. ∼ ∼Z 11, Simp 3. ∼∼ ∼∼Q ⊃ ∼ ∼P 2, Trans 13. ∼ ∼Z ● ∼∼ ∼∼Z 7, 12, Conj 4. Q ⊃ ∼ ∼P 3, DN

14. ∼∼ ∼∼B 3�13, IP 5. P ⊃ ∼ ∼P 1, 4, HS 15. B 14, DN 6. ∼ ∼P �� ∼ ∼P 5, Impl

Exercise 7.78. ( P ⊃ ∼ ∼Q) ⊃ ∼ ∼P 2�7, CP 9. 7. ∼ ∼P 6, Taut (P ⊃ Q) ⊃ [(P ⊃ ∼ ∼Q) ⊃ ∼ ∼P] 1�8, CP

I. (16) / ∼ ∼[(P ⊃ ∼ ∼P) ● (∼ ∼P ⊃ P)]

1. (P ⊃ ∼ ∼P) ● (∼ ∼P ⊃ P) AIP (1) / P ⊃ [( P ⊃ Q ) ⊃ Q ] 2. ( ∼ ∼P �� ∼ ∼P) ● (∼ ∼P ⊃ P) 1, Impl

1.P ACP 3. ∼ ∼P ● (∼ ∼P ⊃ P) 2, Taut

2. P ⊃ Q ACP 4. ∼ ∼P ● (∼∼ ∼∼P �� P) 3, Impl

Answers to Selected Exercises 599

5. ∼ ∼P ● (P �� P) 4, DN 5. ∼ ∼Bx �� Ax 4, Com

6. ∼ ∼P ● P 5, Taut 6. Bx ⊃ Ax 5, Impl

7. P ● ∼ ∼P 6, Com 7. (Ax ⊃ Bx) ● (Bx ⊃ Ax) 3, 6, Conj 8. ∼ ∼ [(P ⊃ ∼ ∼P) ● (∼ ∼P ⊃ P)] 1�7, IP 8. Ax K Bx 7, Equiv (19) / P K [P �� (Q ● ∼ ∼Q)] 9. (x)(Ax K Bx) 8, UG

1. P ACP (7) 1. (x)[ Jx ⊃ (Kx ● Lx)]

2. P �� (Q ● ∼ ∼Q) 1, Add 2. (∃y) ∼ ∼Ky / (∃z) ∼ ∼Jz

3. P ⊃ [P �� (Q ● ∼ ∼Q)] 1�2, CP 3. ∼ ∼Km 2, EI

4. P �� (Q ● ∼ ∼Q) ACP 4. Jm ⊃ (Km ● Lm) 1, UI

5. ∼ ∼P AIP 5. ∼ ∼Km �� ∼ ∼Lm 3, Add

6. Q ● ∼ ∼Q 4, 5, DS 6. ∼ ∼(Km ● Lm) 5, DM

7. ∼∼ ∼∼P 5�6, IP 7. ∼ ∼Jm 4, 6, MT

8. P 7, DN 8. (∃z) ∼ ∼Jz 7, EG

9. [P �� (Q ● ∼ ∼Q)] ⊃ P 4�8, CP (10) 1. (x)(Ax ⊃ Bx) 10. {line 3} ● {line 9} 3, 9, Conj 2. Am �� An / Bm �� Bn 11. PK [P �� (Q ● ∼ ∼Q)] 10, Equiv 3. Am ⊃ Bm 1, UI

Exercise 8.15. (Am ⊃ Bm) ● (An ⊃ Bn) 3, 4, Conj 6. 4. An ⊃ Bn 1, UI Bm �� Bn 2, 5, CD

I. (13) 1. (x)[(Ax ● Bx) ⊃ Cx]

2. (∃x)(Bx ● ∼ ∼Cx) / (∃x)∼ ∼Ax 1. Ce3. Bm ● ∼ ∼Cm 2, EI 4. Jr �� Nr4. (Am ● Bm) ⊃ Cm 1, UI 7. (x)(Mx ⊃ Tx) 5. ∼ ∼Cm ● Bm 3, Com 10. (∃x)(Hx ● ∼ ∼Rx) 6. ∼ ∼Cm 5, Simp 13. (∃x)Tx7. ∼ ∼(Am ● Bm) 4, 6, MT 16. (∃x)(Sx ● ∼ ∼Gx) 8. ∼ ∼Am �� ∼ ∼Bm 7, DM 19. (x)(Sx ⊃ Vx) 9. Bm 3, Simp 22. (x)(Cx ⊃ ∼ ∼Hx) 10. ∼∼ ∼∼Bm 9, DN 25. (x)(Tx ⊃ Hx) 11. ∼ ∼Bm �� ∼ ∼Am 8, Com 28. (x)(Hx ⊃ ∼ ∼Ex) 12. ∼ ∼Am 10, 11, DS 31. (∃x)[Cx ● ∼ ∼(Sx �� Bx)] 13. (∃x)∼ ∼Ax 12, EG 34. (∃x)[Dx ● (Bx (16) 1. (∃x)Ax ⊃ (x)Bx K Tx )]

37. (∃x)[Cx ● (Ax ⊃ Tx)] 2. (∃x)Cx ⊃ (∃x)Dx40. (x)[(Wx ● Cx) ⊃ Rx] 3. An ● Cn / (∃x)(Bx ● Dx) 43. (x)[(Vx �� Cx) ⊃ (Sx ● Ix)] 4. An 2, Simp 46. (∃x)[(Fx ● Rx) ● Ex] 5. (∃x)Ax 4, EG 49. Gt K (x)(Wx ⊃ Cx) 6. (x)Bx 1, 5, MP 52. (∃x)(Ix ● Mx) ⊃ Ir7. Cn ● An 3, Com 55. (x)[(Bx ● Mx) ⊃ Sx] ⊃ Sc8. Cn 7, Simp 58. (∃x)(Ex ● Rx) K (∃x)(Mx ● Ox) 9. (∃x)Cx 8, EG

Exercise 8.211. Dm 10, EI 12. 10. (∃x)Dx 2, 9, MP

I.13. Bm ● Dm 11, 12, Conj 14. ( Bm 6, UI ∃x)(Bx ● Dx) 13, EG

(1) 1. (x)(Ax ⊃ Bx) (19) 1. (∃x)Ax ⊃ (x)(Cx ⊃ Bx)

2. (x)(Bx ⊃ Cx) / (x)(Ax ⊃ Cx) 2. (∃x)(Ax �� Bx)

3. Ax ⊃ Bx 1, UI 3. (x)(Bx ⊃ Ax) / (x)(Cx ⊃ Ax)

4. Bx ⊃ Cx 2, UI 4. Am �� Bm 2, EI

5. Ax ⊃ Cx 3, 4, HS 5. Bm ⊃ Am 3, UI

6. (x)(Ax ⊃ Cx) 5, UG 6. ∼∼ ∼∼Am �� Bm 4, DN (4) 1. (x)(Ax ⊃ Bx) 7. ∼ ∼Am ⊃ Bm 6, Impl

2. (y)(Ay �� ∼ ∼By) / (x)(Ax K Bx) 8. ∼ ∼Am ⊃ Am 5, 7, HS

3. Ax ⊃ Bx 1, UI 9. ∼∼ ∼∼Am �� Am 8, Impl

4. Ax �� ∼ ∼Bx 2, UI 10. Am �� Am 9, DN

600 Answers to Selected Exercises

11. Am 10, Taut (4) 1. (x)(Cx ⊃ Vx) ● (x)(Px ⊃ Fx)

12. (∃x)Ax 11, EG 2. (∃x)(Cx ● Gx) ● / ( ∃ x )( Vx ● Gx ) ●

13. (x)(Cx ⊃ Bx) 1, 12, MP (∃x)(Px ● Gx) (∃x)(Fx ● Gx)

14. Cx ⊃ Bx 13, UI 3. (∃x)(Cx ● Gx) 2, Simp

15. Bx ⊃ Ax 3, UI 4. Cm ● Gm 3, EI

16. Cx ⊃ Ax 14, 15, HS 5. (∃x)(Px ● Gx) ● (∃x)(Cx ● Gx) 2, Com

17. (x)(Cx ⊃ Ax) 16, UG 6. (∃x)(Px ● Gx) 5, Simp (22) 1. (x)(Ax ● ∼ ∼Bx) 7. Pn ● Gn 6, EI

2. An ⊃ [∼ ∼(∃x)Cx ⊃ Bc] 8. (x)(Cx ⊃ Vx) 1, Simp

3. Am ⊃ (x)(Cx ⊃ Dx) / ( ∃ x ) Dx9. Cm ⊃ Vm 8, UI

4. An ● ∼ ∼Bn 1, UI 10. Cm 4, Simp

5. An 4, Simp 11. Vm 9, 10, MP

6. 12. Gm ● Cm ∼ ∼ ( ∃ x ) Cx ⊃ Bc 2, 5, MP 4, Com

7. Am ● ∼ ∼Bm 1, UI 13. Gm 12, Simp

8. 14. Vm ● Gm 11, 13, Conj Am 7, Simp

9. (x)(Cx ⊃ Dx) 3, 8, MP 15. (∃x)(Vx ● Gx) 14, EG

10. 16. (x)(Px ⊃ Fx) Ac ● ∼ ∼ Bc 1, UI● 1, Com

11. (x)(Cx ⊃ Vx) ∼ ∼ Bc ● Ac 10 , Com

12. ∼ ∼Bc 11, Simp 17. (x)(Px ⊃ Fx) 16, Simp

13. 18. Pn ⊃ Fn 17, UI ∼∼ ∼∼ ( ∃ x ) Cx 6, 12, MT

14. (∃x)Cx 13, DN 19. Pn 7, Simp

15. 20. Fn 18, 19, MP Cs 14 , EI

16. Cs ⊃ Ds 9, UI 21. Gn ● Pn 7, Com

18. ( 23. Fn ∃ x ) Dx 17 , EG ● Gn 20, 22, Conj 24. ( 17. 22. Gn 21, Simp Ds 15 , 16 , MP

(25) 1. ( ∃x)(Fx ∃ x ) Ax ⊃ ( x )( Ax ⊃ Bx ) ● Gx) 23, EG 2. ( 25. (∃x)(Vx ● Gx) ∃ x ) Cx ⊃ ( x ) ∼ ∼ ( Dx ● Bx )● 15, 24, Conj ( 3. ( ∃x)(Fx ● Gx) ∃ x )( Ax ● Cx ) / ( ∃ x ) ∼ ∼ Dx ⊃ (7) 1. ( x )[ Gx ( Ix ● Px)] 4. An ● Cn 3, EI 2. ( x )[( Ix5. An 4, Simp ● Px ) ⊃ Rx ] 3. 10. ( 8. ( 6. GaCn ● ● Gc / Ra An 4, Com ● Rc4. Gx ⊃ ( Ix7. Cn 6, Simp ● Px ) 1, UI 5. ( Ix ∃ x ) Ax 5, EG ● Px ) ⊃ Rx 2, UI 6. Gx ⊃ Rx 4, 5, HS 9. ( ∃ x ) Cx 7, EG 7. ( x )( Gx ⊃ Rx ) 6, UG x )( Ax ⊃ Bx ) 1, 8, MP 8. Ga ⊃ Ra 7, UI 11. An ⊃ Bn 10 , UI 9. Ga 3, Simp 12. Bn 5, 11 , MP 10. Ra 8, 9, MP 13. ( x ) ∼ ∼ ( Dx ● Bx ) 2, 9, MP 11. Gc ⊃ Rc 7, UI 14. ∼ ∼ ( Dn ● Bn ) 13 , UI 12.

II.19. ( 16. 15. Gc ● Ga 3, Com ∼ ∼ Dn �� ∼ ∼ Bn 14 , DM 13. Gc 12, Simp ∼ ∼ Bn �� ∼ ∼ Dn 15 , Com 14. Rc 11, 13, MP 17. ∼∼ ∼∼ Bn 12 , DN 15. Ra ● Rc 10, 14, Conj 18. ∼ ∼ Dn 16 , 17 , DS (10) 1. ( x )[( Ax ● Kx ) ⊃ Rx ] ⊃ ( x )( Gx ⊃ Sx ) ∃ x ) ∼ ∼ Dx 18 , EG 2. ( x )[( Ax ● Kx ) ⊃ Fx ] ⊃ ( x )( Gx ⊃ Px ) 3. ( x )[( Ax ● Kx ) ⊃ ( Rx ● Fx )] / ( x )[ Gx ⊃ ( Sx ● Px)] 4. ( Ax ● Kx ) ⊃ ( Rx ● Fx ) 3, UI (1) 1. ( x )( Ox ⊃ Sx ) 5. ∼ ∼ ( Ax ● Kx ) �� ( Rx ● Fx ) 4, Impl 2. ( x )( Ox ⊃ Fx ) / ( x )[ Ox ⊃ ( Sx ● Fx )] 6. [ ∼ ∼ ( Ax ● Kx ) �� Rx ] ● 3. Ox ⊃ Sx 1, UI [ ∼ ∼ ( Ax ● Kx ) �� Fx ] 5, Dist 4. Ox ⊃ Fx 2, UI 7. ∼ ∼ ( Ax ● Kx ) �� Rx 6, Simp 5. ∼ ∼ Ox �� Sx 3, Impl 8. [ ∼ ∼ ( Ax ● Kx ) �� Fx ] ● 6. ∼ ∼ Ox �� Fx 4, Impl [ ∼ ∼ ( Ax ● Kx ) �� Rx ] 6, Com 7. ( ∼ ∼ Ox �� Sx ) ● ( ∼ ∼ Ox �� Fx ) 5, 6, Conj 9. ∼ ∼ ( Ax ● Kx ) �� Fx 8, Simp 8. ∼ ∼ Ox �� ( Sx ● Fx ) 7, Dist 10. ( Ax ● Kx ) ⊃ Rx 7, Impl 9. Ox ⊃ ( Sx ● Fx ) 8, Impl 11. ( Ax ● Kx ) ⊃ Fx 9, Impl 10. ( x )[ Ox ⊃ ( Sx ● Fx )] 9, UG 12. ( x )[( Ax ● Kx ) ⊃ Rx ] 10, UG

Answers to Selected Exercises 601

13. (x)[(Ax ● Kx) ⊃ Fx] 11, UG (13) 1. (x)(Ax ● ∼ ∼Bx) ⊃ (∃x)Cx

14. (x)(Gx ⊃ Sx) 1, 12, MP 2. ∼ ∼(∃x)(Cx �� Bx) / ∼ ∼ ( x ) Ax

15. (x)(Gx ⊃ Px) 2, 13, MP 3. (x)∼ ∼(Cx �� Bx) 2, QN

16. Gx ⊃ Sx 14, UI 4. ∼ ∼(Cx �� Bx) 3, UI

17. Gx ⊃ Px 15, UI 5. ∼ ∼Cx ● ∼ ∼Bx 4, DM

18. ∼ ∼Gx �� Sx 16, Impl 6. ∼ ∼Cx 5, Simp

19. ∼ ∼Gx �� Px 17, Impl 7. (x)∼ ∼Cx 6, UG

20. (∼ ∼Gx �� Sx) ● (∼ ∼Gx �� Px) 18, 19, Conj 8. ∼ ∼(∃x)Cx 7, QN

21. 9. ∼ ∼ ∼ ∼ Gx �� ( Sx ● Px ) 20, Dist(x)(Ax ● ∼ ∼Bx) 1, 8, MT

22. Gx ⊃ (Sx ● Px) 21, Impl 10. (∃x)∼ ∼(Ax ● ∼ ∼Bx) 9, QN

23. (x)[Gx ⊃ (Sx ● Px)] 22, UG 11. ∼ ∼(Am ● ∼ ∼Bm) 10, EI

12. ∼ ∼Am �� ∼∼ ∼∼Bm 11, DM

Exercise 8.313. ∼ ∼Am �� Bm 12, DN

14. ∼ ∼Bx ● ∼ ∼Cx 5, Com

I.16. ( 15. ∼ ∼Bx 14, Simp

(7) 1. ( 10. ( 9. ( ∃x)∼ ∼(Cx ∃ x )( Bx ● Cx ) �� ( ∃ x ) Ax 1, Com● Dx) 8, EG 10. ∼ ∼ ( x )( Cx ● Dx ) 9, QN ∃ x ) Ax 8, 9, DS 11. ∼ ∼ ( ∃ x )( Ax x )( Ax ⊃ Bx ) ● Bx ) 1, 10 , MT 12. ( x ) ∼ ∼ ( Ax ● Bx ) 11 , QN 2. ∼ ∼ ( x ) Cx �� ( x ) Ax13. ∼ ∼ ( Ax ● Bx ) 12 , UI 3. ∼ ∼ ( x ) Bx / ( ∃ x ) ∼ ∼ Cx14. ∼ ∼ Ax �� ∼ ∼ Bx 13 , DM 4. ( ∃ x ) ∼ ∼ Bx 3, QN 15. ( Ax �� Ex ) ● ( Bx �� Fx ) 2, UI 5. ∼ ∼ Bm 4, EI 16. ( ∼∼ ∼∼ Ax �� Ex ) ● ( Bx �� Fx ) 15 , DN 6. Am ⊃ Bm 1, UI 7. ∼ ∼ Am17. ( ∼∼ ∼∼ Ax �� Ex ) ● ( ∼∼ ∼∼ Bx �� Fx ) 16 , DN 5, 6, MT 18. ( ∼ ∼ Ax ⊃ Ex ) ● ( ∼∼ ∼∼ Bx �� Fx ) 17 , Impl 8. ( ∃ x ) ∼ ∼ Ax 7, EG 19. ( ∼ ∼ Ax ⊃ Ex ) ● ( ∼ ∼ Bx ⊃ Fx ) 18 , Impl 9. ∼ ∼ ( x ) Ax 8, QN 20. Ex 10. ( �� Fx 14 , 19 , CD x ) Ax �� ∼ ∼ ( x ) Cx 2, Com 11. ∼ ∼ ( x ) Cx 9, 10, DS 21. ( x )( Ex �� Fx ) 20 , UG (19) 1. ( ∃ x )( Ax ● Bx ) ⊃ ( x )( Bx ⊃ Cx ) 12. ( ∃ x ) ∼ ∼ Cx 11, QN (10) 1. ∼ ∼ ( ∃ x )( Ax ● ∼ ∼ Bx ) 2. Bn ● ∼ ∼ Cn / ∼ ∼ ( x ) Ax2. ∼ ∼ ( ∃ x )( Bx ● ∼ ∼ Cx ) / ( x )( Ax ⊃ Cx ) 3. ( ∃ x )( Bx ● ∼ ∼ Cx ) 2, EG 4. ( ∃ x )( ∼∼ ∼∼ Bx ● ∼ ∼ 3. ( x ) ∼ ∼ ( Ax ● ∼ ∼ Bx ) 1, QN Cx ) 3, DN 4. ( x ) ∼ ∼ ( Bx ● ∼ ∼ Cx ) 2, QN 5. ( ∃ x ) ∼ ∼ ( ∼ ∼ Bx �� Cx) 4, DM 5. ∼ ∼ ( Ax ● ∼ ∼ Bx ) 3, UI 6. ( ∃ x ) ∼ ∼ ( Bx ⊃ Cx ) 5, Impl 6. ∼ ∼ ( Bx ● ∼ ∼ Cx ) 4, UI 7. ∼ ∼ ( x )( Bx ⊃ Cx ) 6, QN 7. ∼ ∼ Ax �� ∼∼ ∼∼ Bx 5, DM 8. ∼ ∼ ( ∃ x )( Ax ● Bx ) 1, 7, MT 8. ∼ ∼ Ax �� Bx 7, DN 9. ( x ) ∼ ∼ ( Ax ● Bx ) 8, QN 9. ∼ ∼ Bx �� ∼∼ ∼∼ Cx 6, DM 10. ∼ ∼ ( An ● Bn ) 9, UI 10. ∼ ∼ Bx �� Cx 9, DN 11. ∼ ∼ An �� ∼ ∼ Bn 10 , DM 11. Ax ⊃ Bx 8, Impl 12. ∼ ∼ Bn �� ∼ ∼ An 11 , Com 12. Bx ⊃ Cx 10, Impl 13. Bn 2, Simp 13. Ax ⊃ Cx 11, 12, HS 14. ∼∼ ∼∼ Bn 13 , DN 14. ( (4) 1. ( 5. ( ∼ ∼(x)Ax 20, QN ∃ x ) ∼ ∼ Ax 4, QN (16) 1. ( ∃ x )( Ax ● Bx ) ⊃ ( x )( Cx ● Dx ) ∃ x ) Ax �� ( ∃ x )( Bx ● Cx ) 2. ( x )[( Ax �� Ex ) ● ( Bx �� Fx )] 2. ∼ ∼ ( ∃ x ) Bx / ( ∃ x ) Ax3. ∼ ∼ ( x ) Dx / ( x )( Ex �� Fx) 3. ( x ) ∼ ∼ Bx 2, QN 4. ( ∃ x ) ∼ ∼ Dx 3, QN 4. ∼ ∼ Bx 3, UI 5. ∼ ∼ Dn 4, EI 5. ∼ ∼ Bx �� ∼ ∼ Cx 4, Add 6. ∼ ∼ Dn �� ∼ ∼ Cn 5, Add 6. ∼ ∼ ( Bx ● Cx ) 5, DM 7. ( x ) ∼ ∼ ( Bx ● 7. ∼ ∼ Cn �� ∼ ∼ Dn 6, Com Cx ) 6, UG 8. ∼ ∼ ( Cn ● Dn ) 7, DM 8. ∼ ∼ ( ∃ x )( Bx ● Cx ) 7, QN 9. ( (1)1. ( 17. ∼ ∼Bm 16, UI x ) Ax ⊃ ( ∃ x ) Bx18. x)∼ ∼Bx 15, UG 2. ( Bm �� ∼ ∼Am 13, Com x ) ∼ ∼ Bx / ( ∃ x ) ∼ ∼ Ax19. ∼ ∼ Am 17, 18, DS 3. ∼ ∼ ( ∃ x ) Bx 2, QN 20. ( ∃ x ) ∼ ∼ Ax 19, EG 4. ∼ ∼ ( x ) Ax 1, 3, MT 21. x)(Ax ⊃ Cx) 13, UG

602 Answers to Selected Exercises

15. ∼ ∼An 12, 14, DS 3. (∃x)(Px ● Cx) �� (∃x)(Px ● Nx) / (∃x)(Cx ● ∼ ∼Gx)

16. (∃x)∼ ∼Ax 15, EG 4. (x)∼ ∼[Px ● (Gx �� Hx)] 1, QN

II.6. ∼ ∼Px �� ∼ ∼(Gx �� Hx) 5, DM 7. 17. ∼ ∼(x)Ax 16, QN 5. ∼ ∼[Px ● (Gx �� Hx)] 4, UI

(4) 1. ( 17. ( 15. ∼ ∼Nm ● 21. (∃x)(Px ● Cx) 19, 20, DS ( Pm ● ∼ ∼ Hm ) 13, Com 22. Pm ● Cm 21, EI ∼ ∼ Nm 14, Simp 16. Pm ● 23. ( x )( ∼ ∼ Px �� ∼ ∼ Gx ) 9, UG ∼ ∼ Nm 12, 15, Conj 24. ∼ ∼ Pm �� ∼ ∼ Gm 23, UI ∃ x )( Px ● ∼ ∼ Nx ) 16, EG ∃ x )( Gx ● Px ) �� ( ∃ x )( Sx ● 25. Pm 22, Simp Ex ) 26. ∼∼ ∼∼ Pm 25, DN 2. ∼ ∼ ( ∃ x ) Ex / ( ∃ x ) Px27. ∼ ∼ Gm 24, 26, DS 3. ( x ) ∼ ∼ Ex 2, QN 28. Cm ● Pm 22, Com 4. ∼ ∼ Ex 3, UI 29. Cm 28, Simp 5. ∼ ∼ Ex �� ∼ ∼ Sx 4, Add 30. Cm ● ∼ ∼ Gm 27, 29, Conj 6. ∼ ∼ Sx �� ∼ ∼ Ex 5, Com 31. ( ∃ x )( Cx ● ∼ ∼ Gx ) 30, EG 7. ∼ ∼ ( Sx ● Ex ) 6, DM 8. ( x ) ∼ ∼ ( Sx ● Ex ) 7, UG 9. ∼ ∼ ( ∃ x )( Sx ● Ex ) 8, QN Exercise 8.410. ( ∃ x )( Sx ● Ex ) �� ( ∃ x )( Gx ● Px ) 1, Com 11. ( ∃ x )( Gx ● Px ) 9, 10, DS I.12. Gm ● Pm 11, EI 13. Pm ● Gm 12, Com (1) 1. ( x )( Ax ⊃ Bx ) 14. Pm 13, Simp 2. ( x )( Ax ⊃ Cx ) / ( x )[ Ax ⊃ ( Bx● Cx)] 15. ( ∃ x ) Px 14, EG 3. Ax ACP (7) 1. ( x )( Px ⊃ Sx ) ● ( x )( Ix ⊃ Gx ) 4. Ax ⊃ Bx 1, UI 2. ∼ ∼ ( ∃ x )( Sx ● Gx ) / ∼ ∼ ( ∃ x )( Px ● Ix ) 5. Ax ⊃ Cx 2, UI 3. ( x ) ∼ ∼ ( Sx ● Gx ) 2, QN 6. Bx 3, 4, MP 4. ∼ ∼ ( Sx ● Gx ) 3, UI 7. Cx 3, 5, MP 5. ∼ ∼ Sx �� ∼ ∼ Gx 4, DM 8. Bx ● Cx 6, 7, Conj 6. ( x )( Px ⊃ Sx ) 1, Simp 9. Ax ⊃ ( Bx ● Cx ) 3�8, CP 7. ( x )( Ix ⊃ Gx ) ● ( x )( Px ⊃ Sx ) 1, Com 10. ( x )[ Ax ⊃ ( Bx ● Cx )] 9, UG 8. ( x )( Ix ⊃ Gx ) 7, Simp (4) 1. ( x )( Ax ⊃ Cx ) 9. (1) 1. ( 8. (∼ ∼Px �� ∼ ∼Gx) ● (∼ ∼Px �� ∼ ∼Hx) 7, Dist x )[ Px ⊃ ( Hx �� Nx )] ⊃ ∼ ∼ ( ∃ x ) Cx9. ∼ ∼Px �� (∼ ∼Gx ● ∼ ∼Hx) 6, DM 2. Cf / (∃x)(Px ● ∼ ∼Px �� ∼ ∼Gx 8, Simp ∼ ∼ Nx ) 10. ( ∼ ∼ Px �� ∼ ∼ Hx ) ● ( ∼ ∼ Px �� ∼ ∼ Gx ) 8, Com 3. ( ∃ x ) Cx 2, EG 11. ∼ ∼ Px �� ∼ ∼ Hx 10, Simp 4. ∼∼ ∼∼ ( ∃ x ) Cx 3, DN 12. 13. ( 12. 10. 6. ( 5. ∼ ∼(Px ● Hx) 11, DM ∼ ∼ ( x )[ Px ⊃ ( Hx �� Nx )] 1, 4, MT 13. Nx ⊃ ( Px ● Hx ) 2, UI ∃ x ) ∼ ∼ [ Px ⊃ ( Hx �� Nx )] 5, QN 14. ∼ ∼ Nx 12, 13, MT 7. ∼ ∼ [ Pm ⊃ ( Hm �� Nm )] 6, EI 15. ∼ ∼ Nx �� ∼ ∼ Px 14, Add 8. ∼ ∼ [ ∼ ∼ Pm �� ( Hm �� Nm )] 7, Impl 16. ∼ ∼ Px �� ∼ ∼ Nx 15, Com 9. ∼∼ ∼∼ Pm ● ∼ ∼ ( Hm �� Nm ) 8, DM Pm ● 17. ∼ ∼ ( Px ● Nx ) 16, DM ∼ ∼ ( Hm �� Nm ) 9, DN 18. ( x ) ∼ ∼ ( Px ● Nx ) 17, UG 11. Pm ● ( ∼ ∼ Hm ● ∼ ∼ Nm ) 10, DM 19. ∼ ∼ ( ∃ x )( Px ● Nx ) 18, QN Pm 11, Simp 20. ( ∃ x )( Px ● Nx ) �� ( ∃ x )( Px ● Cx ) 3, Com Pm ● ∼ ∼ Hm ) ● ∼ ∼ Nm 11, Assoc 14. Px ⊃ Sx 6, UI 2. (∃x)Cx ⊃ (∃x)(Bx ● Dx) / (∃x)Ax ⊃ (∃x)Bx

10. Ix ⊃ Gx 8, UI 3. (∃x)Ax ACP

11. ∼ ∼Sx ⊃ ∼ ∼Px 9, Trans 4. Am 3, EI

12. ∼ ∼Gx ⊃ ∼ ∼Ix 10, Trans 5. Am ⊃ Cm 1, UI

13. (∼ ∼Sx ⊃ ∼ ∼Px) ● (∼ ∼Gx ⊃ ∼ ∼Ix) 11, 12, Conj 6. Cm 4, 5, MP

14. ∼ ∼Px �� ∼ ∼Ix 5, 13, CD 7. (∃x)Cx 6, EG

15. ∼ ∼(Px ● Ix) 14, DM 8. (∃x)(Bx ● Dx) 2, 7, MP

16. (x)∼ ∼(Px ● Ix) 15, UG 9. Bn ● Dn 8, EI

17. ∼ ∼(∃x)(Px ● Ix) 16, QN 10. Bn 9, Simp (10) 1. ∼ ∼(∃x)[Px ● (Gx �� Hx)] 11. ( ∃x)Bx 10, EG

2. (x)[Nx ⊃ (Px ● Hx)] 12. (∃x)Ax ⊃ (∃x)Bx 3�11, CP

Answers to Selected Exercises 603

(7) 1. (x)[(Ax �� Bx) ⊃ Cx] 15. ∼ ∼(x)Ex ● ∼ ∼(x)Cx 4, Com

2. (x)[(Cx �� Dx) ⊃ Ex] / (x)(Ax ⊃ Ex) 16. ∼ ∼(x)Ex 15, Simp

3. Ax ACP 17. ( x)Ex ● ∼ ∼(x)Ex 14, 16, Conj

4. (Ax �� Bx) ⊃ Cx 1, UI 18. ∼∼ ∼∼[(x)Cx �� (x)Ex] 3�17, IP

5. (Cx �� Dx) ⊃ Ex 2, UI 19. (x)Cx �� (x)Ex 18, DN

6. Ax �� Bx 3, Add (19) 1. (x)[Bx ⊃ (Cx ● Dx)] / (x)(Ax ⊃ Bx) ⊃

7. Cx 4, 6, MP (x)(Ax ⊃ Dx)

8. Cx �� Dx 7, Add 2. (x)(Ax ⊃ Bx) ACP

9. Ex 5, 8, MP 3. Ax ACP

10. Ax ⊃ Ex 3�9, CP 4. Ax ⊃ Bx 2, UI

11. (x)(Ax ⊃ Ex) 10, UG 5. Bx 3, 4, MP (10) 1. (x)(Ax ⊃ Bx) 6. Bx ⊃ (C x ● Dx) 1, UI

2.Am �� An / (∃x)Bx7. Cx ● Dx 5, 6, MP

3. ∼ ∼(∃x)Bx AIP 8. Dx ● Cx 7, Com

4. (x)∼ ∼Bx 3, QN 9. Dx 8, Simp

5. Am ⊃ Bm 1, UI 10. Ax ⊃ Dx 3�9, CP

6. An ⊃ Bn 1, UI 11. ( x )(Ax ⊃ Dx) 10, UG

7. (Am ⊃ Bm) ● (An ⊃ Bn) 5, 6, Conj 12. (x)(Ax ⊃ Bx) ⊃ (x)(Ax ⊃ Dx) 2�11, CP

8. Bm �� Bn 2, 7, CD (22) 1. (x)Ax �� (x)Bx / (x)(Ax �� Bx)

9. ∼ ∼Bm 4, UI 2. ∼ ∼(x)(Ax �� Bx) AIP

10. Bn 8, 9, DS 3. (∃x)∼ ∼(Ax �� Bx) 2, QN

11. ∼ ∼Bn 4, UI 4. (∃x)(∼ ∼Ax ● ∼ ∼Bx) 3, DM

12. Bn ● ∼ ∼Bn 10, 11, Conj 5. ∼ ∼An ● ∼ ∼Bn 4, EI

13. ∼∼ ∼∼(∃x)Bx 3�12, IP 6. ∼ ∼An 5, Simp

14. (∃x)Bx 13, DN 7. (∃x)∼ ∼Ax 6, EG (13) 1. (∃x)Ax ⊃ (x)(Bx ⊃ Cx) 8. ∼ ∼(x)Ax 7, QN

2. (∃x)Dx ⊃ (∃x)Bx / (∃x)(Ax ● Dx) ⊃ 9. (x)Bx 1, 8, DS

(∃x)Cx 10. Bn 9, UI

3. (∃x)(Ax ● Dx) ACP 11. ∼ ∼Bn ● ∼ ∼An 5, Com

4. Am ● Dm 3, EI 12. ∼ ∼Bn 11, Simp

5. Am 4, Simp 13. Bn ● ∼ ∼Bn 10, 12, Conj

6. (∃x)Ax 5, EG 14. ∼∼ ∼∼(x)(Ax �� Bx) 2�13, IP

7. (x)(Bx ⊃ Cx) 1, 6, MP 15. (x)(Ax �� Bx) 14, DN

8. Dm ● Am 4, Com (25) 1. (∃x)Ax ⊃ (x)(Bx ⊃ Cx)

9. Dm 8, Simp 2. (∃x)Dx ⊃ (x)(Ex ⊃ ∼ ∼Bx)

10. ( ∃x)Dx 9, EG 3. ∼ ∼(∃x)(Cx ● ∼ ∼Ex) / (∃x)(Ax ● Dx) ⊃

11. ( ∃x)Bx 2, 10, MP ∼ ∼(∃x)Bx

12. Bn 11, EI 4. (∃x)(Ax ● Dx) ACP

13. Bn ⊃ Cn 7, UI 5. An ● Dn 4, EI

14. Cn 12, 13, MP 6. An 5, Simp

15. ( ∃x)Cx 14, EG 7. (∃x)Ax 6, EG

16. (∃x)(Ax ● Dx) ⊃ (∃x)Cx 3�15, CP 8. (x)(Bx ⊃ Cx) 1, 7, MP (16) 1. (x)[(Ax �� Bx) ⊃ Cx] 9. Dn ● An 5, Com

2. (∃x)(∼ ∼Ax �� Dx) ⊃ (x)Ex / (x)Cx �� (x)Ex 10. Dn 9, Simp

3. ∼ ∼[(x)Cx �� (x)Ex] AIP 11. ( ∃x)Dx 10, EG

4. ∼ ∼(x)Cx ● ∼ ∼(x)Ex 3, DM 12. ( x)(Ex ⊃ ∼ ∼Bx) 2, 11, MP

5. ∼ ∼(x)Cx 4, Simp 13. ( x)∼ ∼(Cx ● ∼ ∼Ex) 3, QN

6. (∃x)∼ ∼Cx 5, QN 14. ∼ ∼(Cx ● ∼ ∼Ex) 13, UI

7. ∼ ∼Cm 6, EI 15. ∼ ∼Cx �� ∼∼ ∼∼Ex) 14, DM

8. (Am �� Bm) ⊃ Cm 1, UI 16. ∼ ∼Cx �� Ex 15, DN

9. ∼ ∼(Am �� Bm) 7, 8, MT 17. Cx ⊃ Ex 16, Impl

10. ∼ ∼Am ● ∼ ∼Bm 9, DM 18. Bx ⊃ Cx 8, UI

11. ∼ ∼Am 10, Simp 19. Bx ⊃ Ex 17, 18, HS

12. ∼ ∼Am �� Dm 11, Add 20. Ex ⊃ ∼ ∼Bx 12, UI

13. ( ∃x)(∼ ∼Ax �� Dx) 12, EG 21. Bx ⊃ ∼ ∼Bx 19, 20, HS

14. ( x)Ex 2, 13, MP 22. ∼ ∼Bx �� ∼ ∼Bx 21, Impl

604 Answers to Selected Exercises

23. ∼ ∼Bx 22, Taut 21. ∼∼ ∼∼(∃x)(Dx ● Sx) 3�20, IP

24. ( x )∼ ∼Bx 23, UG 22. (∃x)(Dx ● Sx) 21, DN

25. ∼ ∼(∃x)Bx 24, QN (10) 1. (∃x)(Gx ● Px) �� (∃x)(Ax ● Px)

26. (∃x)(Ax ● Dx) ⊃ ∼ ∼(∃x)Bx 4�25, CP 2. (∃x)Px ⊃ (∃x) [Ax ● (Cx ● Dx)] / (∃x)(Dx ● Cx)

II.3. ∼ ∼(∃x)Px AIP

4. (x)∼ ∼Px 3, QN

(1) 1. (x)(Ax ⊃ Wx) 6. ∼ ∼Px �� ∼ ∼Gx 5, Add ⊃ 5. ∼ ∼Px 4, UI 2. (x)(RxCx) / (x)[(Rx ● Ax) ⊃ 7. ∼ ∼Gx �� ∼ ∼Px 6, Com ( Cx ● Wx )] 8. ∼ ∼ ( Gx ● Px ) 7, DM 3. Rx ● Ax ACP 9. ( x)∼ ∼(Gx ● Px) 8, UG

4. Rx 3, Simp 10. ∼ ∼(∃x)(Gx ● Px) 9, QN

10. 7. AxWx 1, UI 13. Pm ● Am 12, Com ⊃ 8. RxCx 2, UI 14. Pm 13, Simp 9. Cx 4, 8, MP 15. ∼ ∼ Pm 4, UI 6. Ax 5, Simp 12. Am ● Pm 11, EI ⊃ 5. Ax ● Rx 3, Com 11. ( ∃x)(Ax ● Px) 1, 10, DS

(4) 1. (x)[(Rx ● Ax) ⊃ (Cx ● Wx)] 12, UG 19. ( ∃x)[Ax ● (Cx ● Dx)] 2, 18, MP ⊃ x )[( Sx �� Ux ) ( Ix ● Cx )] 20. An ● ( Cn ● Dn ) 19, EI 2. ( x )[( Cx �� Vx ) ⊃ ( Rx ● Ax )] / ( x )( Sx ⊃ Ax ) 21. ( Cn ● Dn ) ● An 20, Com 3. Sx ACP 22. Cn ● Dn 21, Simp 4. Sx �� Ux 3, Add 23. Dn ● Cn 22, Com 5. ( Sx �� Ux ) ⊃ ( Ix ● Cx ) 1, UI 24. ( ∃ x )( Dx ● Cx ) 23, EG 6. 13. ( 12. ( 11. Cx ● Wx 9, 10, Conj 17. ∼∼ ∼∼(∃x)Px 3�16, IP ⊃ Rx ● Ax ) ( Cx ● Wx ) 3�11, CP 18. ( ∃ x ) Px 17, DN Wx 6, 7, MP 16. Pm ● ∼ ∼Pm 14, 15, Conj Ix ● Cx 4, 5, MP

7. Cx ● Ix 6, Com

8. Cx 7, Simp Exercise 8.5

9. Cx �� Vx 8, Add

10. ( Cx �� Vx) ⊃ (Rx ● Ax) 2, UI

11. Rx ● Ax 9, 10, MP I.

12. Ax ● Rx 11, Com

13. Ax 12, Simp 1. All cats are animals.

14. Sx ⊃ Ax 3�13, CP No cats are dogs.

15. (x)(Sx ⊃ Ax) 14, UG No dogs are animals.

(7) 1. (∃x)Cx ⊃ (x)[Ax ⊃ (Sx ● Dx)] 4. Some mammals are dogs. ⊃ 2. ( x )( Cx ∼ ∼ Ax ) ⊃ ( ∃ x )( Dx ● Sx ) / ( ∃ x )( Dx ● Sx ) Some mammals write books. 3. ∼ ∼ ( ∃ x )( Dx ● Sx ) AIP Some mammals are dogs that write books. 4. ∼ ∼ ( x )( Cx ⊃ ∼ ∼ Ax ) 2, 3, MT 5. ( ∃ x ) ∼ ∼ ( Cx ⊃ ∼ ∼ Ax ) 4, QN 7. There are flowers. 6. ∼ ∼ ( Cm ⊃ ∼ ∼ Am ) 5, EI There are dogs. 7. ∼ ∼ ( ∼ ∼ Cm �� ∼ ∼ Am ) 6, Impl No flowers are animals. 8. ∼∼ ∼∼ Cm ● ∼∼ ∼∼ Am 7, DM Some dogs are not animals. 9. Cm ● ∼∼ ∼∼ Am 8, DN 10. Cm ● 10. Some mammals are felines. Am 9, DN Some animals are not felines. 11. Cm 10, Simp All mammals are animals. 12. ( ∃ x ) Cx 11, EG Some feline animals are not mammals. 13. ( x )[ Ax ⊃ ( Sx ● Dx )] 1, 12, MP 14. Am ⊃ ( Sm ● Dm ) 13, UI II.15. Am ● Cm 10, Com 16. Am 15, Simp (1) 1. ( x )( Ax ⊃ Bx ) 17. Sm ● Dm 14, 16, MP 18. Dm ● 2. ( x )( Ax ⊃ Cx ) / ( x )( Bx ⊃ Cx) Sm 17, Com 19. ( ∃ x )( Dx ● Sx ) 18, EG For a universe consisting of one member, we have 20. ( ∃ x )( Dx ● Sx ) ● 3, 19, Conj Aa ⊃ Ba / Aa ⊃ Ca // Ba ⊃ Ca ∼ ∼ ( ∃ x )( Dx ● Sx ) F T T F T F T F F

Answers to Selected Exercises 605

(4) 1. (x)(Ax ⊃ Bx)

2. (∃x)Ax / (x)Bx

For a universe consisting of two members, we have

(Aa ⊃ Ba) ● (Ab ⊃ Bb) / Aa �� Ab // Ba ● Bb

T T T T F T F T T F T F F

(7) 1. (x)(Ax ⊃ Bx)

2. (∃x)Bx ⊃ (∃x)Cx / (x)(Ax ⊃ Cx)

For a universe consisting of two members, we have

(Aa ⊃ Ba) ● (Ab ⊃ Bb) / (Ba �� Bb) ⊃ (Ca �� Cb) // (Aa ⊃ Ca) ● (Ab ⊃ Cb)

T T T T T T T T T T T T T F T T T F T F F (10) 1. (∃x)(Ax ● Bx)

2. (∃x)(∼ ∼Ax ● ∼ ∼Bx) / (x)(Ax K Bx)

For a universe consisting of one member, we have

Aa ● Ba / ∼ ∼ Aa ● ∼ ∼ Ba // Aa K Ba

T T F T T F

For a universe consisting of two members, we have

(Aa ● Ba) �� (Ab ● Bb) / (∼ ∼ Aa ● ∼ ∼ Ba) �� (∼ ∼ Ab ● ∼ ∼ Bb) // (Aa K Ba) ● (Ab K Bb)

T T T T T F F F T F F T T F T F T F T T T F T F F

For a universe consisting of three members, we have

(Aa ● Ba) �� [(Ab ● Bb) �� (Ac ● Bc)] / (∼ ∼ Aa ● ∼ ∼ Ba) �� [(∼ ∼ Ab ● ∼ ∼ Bb) �� (∼ ∼ Ac ● ∼ ∼ Bc)]

T T T T T F F T F F F F T F F T T F T F T F T T F T T F

// (Aa K Ba) ● [(Ab K Bb) ● (Ac K Bc)]

T T T F T F F F F T F

III.

(1) 1. (x)[(Vx ● Px) ⊃ (Ax ● Mx)]

2. (∃x)(Vx ● Ox) / (∃x)(Mx ● Ax)

For a universe consisting of one member, we have

(Va ● Pa) ⊃ (Aa ● Ma) / Va ● Oa // Ma ● Aa

T F F T F F T T T T T F F

(4) 1. (x)(Tx ⊃ Hx)

2. (∃x) (Tx ● Hx) ⊃ (∃x)(Px ● Ox) / ( x)(Tx ⊃ Ox)

For a universe consisting of two members, we have

(Ta ⊃ Ha) ● (Tb ⊃ Hb) / [(Ta ● Ha) �� (Tb ● Hb)] ⊃ [(Pa ● Oa) �� (Pb ● Ob)]

T T T T T T T T T T T T T T T F F F T T T T

// (Ta ⊃ Oa) ● (Tb ⊃ Ob)

T F F F T T T

Exercise 8.6 3. Am ⊃ (y)Bmy 1, UI 4. ( y ) Bmy 2, 3, MP (4) 1. ( x )( ∃ y )( Ax ⊃ By ) / ( x ) Ax⊃ (∃y)ByI.2. ( x ) Ax ACP 3. Ax 2, UI 1. Rcp4. ( ∃ y )( Ax ⊃ By ) 1, UI 4. ( ∃ x ) Fxv ⊃ Fmv5. Ax ⊃ Bm 4, EI 7. ( x )[ Px ⊃ ( ∃ y ) Sxy ] 6. Bm 3, 5, MP 10. ( ∃ x )[ Px ● ( y ) Sxy ] 7. ( ∃ y ) By 6, EG 13. ( ∃ x )[( Cx ● Lx ) ● Dsx ] 8. ( x ) Ax ⊃ ( ∃ y ) By 2�7, CP 16. ( ∃ x )( Fxc ● Icx ) (7) 1. ( ∃ x )[ Ax ● ( y )( Ay ⊃ Bxy )] / ( ∃ x ) Bxx19. ( x ){ Px ⊃ ( ∃ y )[ Py ● ( Mxy ⊃ Axy )]} 2. Am22. ( ∃ x ){ Px ● ( y )[( Ty ● Sxy ) ⊃ Axy ]} ● ( y )( Ay ⊃ Bmy ) 1, EI 3. Am 2, Simp 25. ( x ){ Lx ⊃ ( y )[( Wy ● Cy ) ⊃ Rxy ]} 28. ( x ){( Cx ● Fx ) ⊃ ( y )[( By ● Ly ) ⊃ Rxy4. ( y )( Ay ⊃ Bmy ) ● Am 2, Com ]} 5. ( y )( Ay ⊃ Bmy ) 4, Simp II.6. Am ⊃ Bmm 5, UI 7. Bmm 3, 6, MP (1) 1. ( x )[ Ax ⊃ ( y ) Bxy ] 8. ( ∃ x ) Bxx 7, EG 2. Am / ( y ) Bmy

606 Answers to Selected Exercises

(10) 1. (x)(∃y)Axy ⊃ (x)(∃y)Bxy3. Pg ⊃ (y)(Ay ⊃ Ogy) 1, UI

2. (∃x)(y)∼ ∼Bxy / (∃x)(y)∼ ∼Axy4. Pg 2, Simp

3. (∃x)∼ ∼(∃y)Bxy 2, QN 5. (y)(Ay ⊃ Ogy) 3, 4, MP

4. ∼ ∼(x)(∃y)Bxy 3, QN 6. Am ⊃ Ogm 5, UI

5. ∼ ∼(x)(∃y)Axy 1, 4, MT 7. ∼ ∼Ogm ● Pg 2, Com

6. (∃x)∼ ∼(∃y)Axy 5, QN 8. ∼ ∼Ogm 7, Simp

7. (∃x)(y)∼ ∼Axy 6, QN 9. ∼ ∼Am 6, 8, MT (13) 1. (∃x){Ax ● (y)[(By �� Cy) ⊃ Dxy]} (4) 1. Po

2. (∃x)Ax ⊃ (∃y)By / (∃x)(∃y)Dxy2. (x)[(Px ● Cx) ⊃ Sox]

3. Am ● (y)[(By �� Cy) ⊃ Dmy] 1, EI 3. (x)(Px ⊃ ∼ ∼Sxx) / ∼ ∼ Co

4. Am 3, Simp 4. Co AIP

5. (∃x)Ax 4, EG 5. ( Po ● Co) ⊃ Soo 2, UI

6. (∃y)By 2, 5, MP 6. Po ● Co 1, 4, Conj

7. Bn 6, EI 7. Soo 5, 6, MP

8. (y)[(By �� Cy) ⊃ Dmy] ● Am 3, Com 8. Po ⊃ ∼ ∼Soo 3, UI

9. (y)[(By �� Cy) ⊃ Dmy] 8, Simp 9. ∼ ∼Soo 1, 8, MP

10. (Bn �� Cn) ⊃ Dmn 9, UI 10. Soo ● ∼ ∼Soo 7, 9, Conj 11. Bn �� Cn 7, Add 11. ∼ ∼Co 4�10, IP 12. Dmn 10, 11, MP (7) 1. (∃x){Px ● (y)[(Py ● Kxy) ⊃ Fxy]} 13. (∃y)Dmy 12, EG 2. (x)[Px ⊃ (∃y)(Py ● Kxy)] / (∃x)(∃y) 14. (∃x)(∃y)Dxy 13, EG [(Px● Py) ● Fxy ]

(16) 1. (x)(Ax ⊃ Bx) 3. Pm ● (y)[(Py ● Kmy) ⊃ Fmy] 1, EI

2. ( ∃x)Bx ⊃ ∼ ∼(∃x)(∃y)Cxy / (∃x)Ax ⊃ ∼ ∼Cmn4. Pm ⊃ (∃y)(Py ● Kmy) 2, UI

3. ( ∃ x )Ax ACP 5. Pm 3, Simp

4. Ae 3, EI 6. (∃y)(Py ● Kmy) 4, 5, MP

5. Ae ⊃ Be 1, UI 7. Pn ● Kmn 6, EI

6. Be 4, 5, MP 8. (y)[(Py ● Kmy) ⊃ Fmy] ● Pm 3, Com

7. ( ∃x)Bx 6, EG 9. (y)[(Py ● Kmy) ⊃ Fmy] 8, Simp

8. ∼ ∼(∃x)(∃y)Cxy 2, 7, MP 10. ( Pn ● Kmn) ⊃ Fmn 9, UI

9. ( x)∼ ∼(∃y)Cxy 8, QN 11. Fmn 7, 10, MP

10. ( x)(y)∼ ∼Cxy 9, QN 12. Pn 7, Simp

11. ( y)∼ ∼Cmy 10, UI 13. Pm ● Pn 5, 12, Conj

12. ∼ ∼Cmn 11, UI 14. ( Pm ● Pn) ● Fmn 11, 13, Conj 13. ( ∃x)Ax ⊃ ∼ ∼Cmn 3�12, CP 15. ( ∃y)[(Pm ● Py) ● Fmy] 14, EG (19) 1. (∃x)(y)Ayx �� (x)(y)Bxy 16. ( ∃x)(∃y)[(Px ● Py) ● Fxy] 15, EG

2. (∃x)(y)(Cy ⊃ ∼ ∼Byx) / (x)(∃y) (10) 1. (x){Ix ⊃ [(∃y)(Cy ● Ay) ⊃ Ex]}

(Cx ⊃ Axy) 2. [(∃x)Tx �� (∃x)Wx] ⊃ [(∃x)Ix ● (∃x)Cx]

3. Cx ACP 3. (x)(Cx ⊃ Ax) / (∃x)Tx ⊃ (∃x)(Ix ● Ex)

4. ( y)(Cy ⊃ ∼ ∼Bym) 2, EI 4. ( ∃x)Tx ACP

5. Cx ⊃ ∼ ∼Bxm 4, UI 5. ( ∃x)Tx �� (∃x)Wx 4, Add

6. ∼ ∼Bxm 3, 5, MP 6. ( ∃x)Ix ● (∃x)Cx 2, 5, MP

7. ( ∃y)∼ ∼Bxy 6, EG 7. ( ∃x)Ix 6, Simp

8. ( ∃x)(∃y)∼ ∼Bxy 7, EG 8. Im 7, EI

9. ( ∃x)∼ ∼(y)Bxy 8, QN 9. Im ⊃ [(∃y)(Cy ● Ay) ⊃ Em] 1, UI

10. ∼ ∼(x)(y)Bxy 9, QN 10. ( ∃y)(Cy ● Ay) ⊃ Em 8, 9, MP

11. ( x)(y)Bxy �� (∃x)(y)Ayx 1, Com 11. ( ∃x)Cx ● (∃x)Ix 6, Com

12. (∃x)(y)Ayx 10, 11, DS 12. ( ∃x)Cx 11, Simp

13. (y)Ayn 12, EI 13. Cn 12, EI

14. Axn 13, UI 14. Cn ⊃ An 3, UI 15. Cx ⊃ Axn 3�14, CP 15. An 13, 14, MP 16. ( ∃y)(Cx ⊃ Axy) 15, EG 16. Cn ● An 13, 15, Conj 17. ( x)(∃y)(Cx ⊃ Axy) 16, UG 17. ( ∃y)(Cy ● Ay) 16, EG

III.18. Em 10, 17, MP

19. Im ● Em 8, 18, Conj

20. ( ∃x)(Ix ● Ex) 19, EG

(1) 1. (x)[Px ⊃ (y)(Ay ⊃ Oxy)] 21. ( ∃x)Tx ⊃ (∃x)(Ix ● Ex) 4�20, CP ∼ 2. Pg ● ∼ Ogm / ∼ ∼ Am

Answers to Selected Exercises 607

Exercise 8.74. Px ACP 5. Px ⊃ x = a 1, UI 6. x = a 4, 5, MP I.7. x = c 3, 6, Id 8. x = c ⊃ Qx 2, UI 1. s = g9. Qx 7, 8, MP 4. h ≠ g10. Px ⊃ Qx 4�9, CP 5. Wp ● ( x )( Wx ⊃ x = p ) 11. ( x )( Px ⊃ Qx ) 10, UG 8. Nc ● Mc ● ( x )[( Nx ● Mx ) ⊃ x = c ] (13) 1. ( x )( Ba ⊃ x ≠ a ) 11. Sh ● Ph ● ( x )[( Sx ● Px ) ⊃ x = h ] 2. Bc / a ≠ c12. Eh ● ( x )[( Ex ● x ≠ h ) ⊃ Lhx ] 15. Rd ● Nd ● 3. a = c AIP ( x )[( Rx ● Nx ● x ≠ d ) ⊃ Ldx ] 4. c = a 3, Id 16. Ci ● ∼ ∼ Si ● ( x )[( Cx ● x ≠ i ) ⊃ Sx ] 19. Pc ● ∼ ∼ Wc ● 5. Ba 2, 4, Id ( x )[( Px ● x ≠ c ) ⊃ Wx ] 6. Ba ⊃ c ≠ a 1, UI 20. ( x )( y )[( Cx ● Bx ● Cy ● By ) ⊃ x = y ] 7. c ≠ a 5, 6, MP 23. ( x )( y )( z )[( Cx ● Mx ● Cy ● My ● Cz ● Mz ) ⊃ 8. c = a ● c ≠ a 4, 7, Conj ( x = y �� x = z �� y = z )] 9. a ≠ c 3�8, IP 26. ( ∃ x )( ∃ y )( ∃ z )( Cx ● Cy ● Cz ● x ≠ y ● x ≠ z ● y ≠ z ) 29. ( ∃ x )( ∃ y ){ Sx ● Bx ● Gx ● Sy ● By ● Gy ● x ≠ y ● (16) 1. ( x )[ Nx ⊃ ( Px ● x = m )] 2. ∼ ∼ Pm / ∼ ∼ Ne ( z )[( Sz ● Bz ● Gz ) ⊃ ( z = x �� z = y )]} 30. ( ∃ x )[ Wxv ● ( y )( Wyv ⊃ y = x ) ● 3. Ne AIP Bx ] 33. ( ∃ x ){ Ax ● Pxa ● ( y )[( Ay ● 4. Ne ⊃ ( PePya ) ⊃ y = x ] ● ● e = m ) 1, UI x = b } 5. Pe35. Sr ● ● e = m 3, 4, MP ( x )[( Sx ● x ≠ r ) ⊃ Srx ] 6. Pe 5, Simp 38. Ar ● Er ● ( x )[( Ax ● Ex ) ⊃ x = r ] 7. e = m ● Pe 5, Com 41. ( x )( y )[( Sxh ● Syh ) ⊃ x = y ] 8. e = m 7, Simp 44. m ≠ b47. ( ∃ x )( ∃ y ){ Tx ● Cx ● Ty ● Cy ● x ≠ y ● 9. Pm 6, 8, Id ( z )[( Tz ● Cz ) ⊃ 10. Pm ● ∼ ∼ Pm 2, 9, Conj ( z = x �� z = y )]} 50. ( ∃ x )( ∃ y )( ∃ z )( Sx ● Ox ● Sy ● Oy ● Sz ● Oz ● x ≠ y ● 11. ∼ ∼ Ne 3�10, IP x ≠ z ● (19) 1. ( x )( ∃ y )( Cxy ⊃ x = y ) y ≠ z ) 2. ( ∃ x )( y )( Cxy ● x = a ) / CaaII.3. ( y )( Cny ● n = a ) 2, EI 4. ( ∃ y )( Cay ⊃ a = y ) 1, UI (1) 1. ( x )( x = a ) 5. Cam ⊃ a = m 4, EI 2. ( ∃ x ) Rx / Ra6. Cnm ● n = a 3, UI 3. Ri 2, EI 7. Cnm 6, Simp 4. i = a 1, UI 8. n = a ● Cnm 6, Com 5. Ra 3, 4, Id 9. n = a 8, Simp (4) 1. ( ∃ x )( x = g ) 10. Cam 7, 9, Id 2. ( x )( x = i ) / g = i11. a = m 5, 10, MP 3. n = g 1, EI 12. m = a 11, Id 4. n = i 2, UI 13. Caa 10, 12, Id 5. g = n 3, Id III.6. g = i 4, 5, Id

(7) 1. ( (1) 1. (∃x)(Nx ● Wjx ● Ix) x )( x = a )

2. 2. Nc ● Wjc ● (x)[(NxFa / Fm ● ● Wjx) ⊃ Fn

3. x = c] / Icm = a 1, UI

4. 3. Na ● Wja ● Ia 1, EI a = m 3, Id

5. 4. (x)[(Nx ● Wjx) ⊃ x = c] Fm 2, 4, Id●

6. Nc ● Wjc 2, Com n = a 1, UI

7. 5. (x)[(Nx ● Wjx) ⊃ x = c] 4, Simp a = n 6, Id

8. 6. ( Na ● Wja) ⊃ a = c 5, UI Fn 2, 7, Id

(10) 1. (x)(Pxx = a) 9. Ia ● Na ● Wja 3, Com = 2. ( x )( xc ⊃ Qx ) 10. ⊃ 8. a = c 6, 7, MP 9. 7. NaFm ● ● Wja 3, Simp Fn 5, 8, Conj Ia 9, Simp

3. a = c / (x)(Px ⊃ Qx) 11. Ic 8, 10, Id

608 Answers to Selected Exercises

(4) 1. (∃x){Nx ● Tx ● 12. (y)[(By ● y ≠ a) ⊃

(y)[(Ny ● Ty) ⊃ y = x] ● Wmx} Tay] ● Ca ● Ba 4, Com

2. Ng ● Wmg ● / (∃x){Nx ● Tx ● 13. ( y)[(By ● y ≠ a) ⊃ Tay] 12, Simp

(x) [(Nx ● Wmx) ⊃ x = g] (y)[(Ny ● Ty) ⊃ y = x] ● 14. ( Bo ● o ≠ a) ⊃ Tao 13, UI

x = g} 15. Bo 1, Simp

3. Na ● Ta ● 16. o ≠ a 8, Id

(y)[(Ny ● Ty) ⊃ y = a] ● 17. Bo ● o ≠ a 15, 16, Conj

Wma 1, EI 18. Tao 14, 17, MP

4. (x)[(Nx ● Wmx) ⊃ x = g] ● 19. (y)(Tay ⊃ ∼ ∼Tya) 3, UI

Ng ● Wmg 2, Com 20. Tao ⊃ ∼ ∼Toa 19, UI

5. (x)[(Nx ● Wmx) ⊃ x = g] 4, Simp 21. ∼ ∼Toa 18, 20, MP

6. (Na ● Wma) ⊃ a = g 5, UI 22. Toa ● ∼ ∼Toa 11, 21, Conj

7. Na 3, Simp 23. ∼ ∼(a ≠ o) 8�22, IP

8. Wma ● Na ● Ta ● 24. a= o 23, DN

(y)[(Ny ● Ty) ⊃ y = a] 3, Com 25. Ca ● Ba ● (y)[(By ● y ≠ a) ⊃ Tay] 4, Com

9. Wma 8, Simp 26. Ca 25, Simp

10. Na ● Wma 7, 9, Conj 27. Co 24, 26, Id 11. a = g 6, 10, MP (13) 1. (∃x)(∃y)(Ax ● Ox ● Ay ● Oy ● x ≠ y) 12. Na ● Ta ● 2. (x)(Ax ⊃ Px)

(y)[(Ny ● Ty) ⊃ y = a] 3, Simp 3. (x)(y)(z)[(Px ● Ox ● Py ● Oy ● Pz ●

13. Na ● Ta ● (y)[(Ny ● Ty) ⊃ Oz) ⊃ (x = y �� x = z �� y = z)]

y = a] ● a = g 11, 12, Conj / (∃x)(∃y){Px ● Ox ● Py ● Oy ● x ≠ y ●

14. (∃x){Nx ● Tx ● (y)[(Ny ● Ty) ⊃ (z)[(Pz ● Oz) ⊃ (z = x �� z = y)]}

y = x] ● x = g} 13, EG 4. (∃y)(Aa ● Oa ● Ay ● Oy ● a ≠ y) 1, EI

(7) 1. Me ● ∼ ∼Se ● 5. Aa ● Oa ● Ab ● Ob ● a ≠ b 4, EI

(x)[(Mx ● x ≠ e) ⊃ Sx] 6. Aa ⊃ Pa 2, UI

2. Mn ● ∼ ∼Gn ● (x)[(Mx ● x ≠ n) ⊃ Gx] 7. Aa 5, Simp 3. e ≠ n / Ge ● Sn8. Pa 6, 7, MP

4. (x)[(Mx ● x ≠ e) ⊃ Sx] ● 9. Ab ⊃ Pb 2, UI

Me ● ∼ ∼Se 1, Com 10. Ab ● Ob ● a ≠ b ● Aa ● Oa 5, Com

5. (x)[(Mx ● x ≠ e) ⊃ Sx] 4, Simp 11. Ab 10, Simp

6. (Mn ● n ≠ e) ⊃ Sn 5, UI 12. Pb 9, 11, MP

7. Mn 2, Simp 13. Oa ● Ab ● Ob ● a ≠ b ● Aa 10, Com

8. n ≠ e 3, Id 14. Oa 13, Simp

9. Mn ● n ≠ e 7, 8, Conj 15. Ob ● a ≠ b ● Aa ● Oa ● Ab 5, Com

10. Sn 6, 9, MP 16. Ob 15, Simp 11. (x)[(Mx ● x ≠ n) ⊃ Gx] ● 17. a ≠ b ● Aa ● Oa ● Ab ● Ob 5, Com

Mn ● ∼ ∼Gn 2, Com 18. a ≠ b 17, Simp

12. (x)[(Mx ● x ≠ n) ⊃ Gx] 11, Simp 19. Pa ● Oa ● Pb ● Ob ● a ≠ b 8, 14, 12, 16, 13. (Me ● e ≠ n) ⊃ Ge 12, UI 18, Conj 14. Me 1, Simp 20. ∼ ∼ (z)[(Pz ● Oz) ⊃ 15. Me ● e ≠ n 3, 14, Conj (z = a �� z = b)] AIP 16. Ge 13, 15, MP 21. ( ∃z)∼ ∼[(Pz ● Oz) ⊃ 17. Ge ● Sn 10, 16, Conj (z = a �� z = b)] 20, QN

(10) 1. Bo ● (x)[(Bx ● x ≠ o) ⊃ Tox] 22. ∼ ∼[(Pc ● Oc) ⊃ (c = a �� c = b)] 21, EI

2. (∃x){Bx ● (y)[(By ● y ≠ x) ⊃ Txy] ● Cx} 23. ∼ ∼[∼ ∼(Pc ● Oc) �� (c = a �� c = b)] 22, Impl 3. (x)(y)(Txy ⊃ ∼ ∼Tyx) / Co24. ∼∼ ∼∼(Pc ● Oc) ● ∼ ∼(c= a �� c = b) 23, DM 4. Ba ● (y)[(By ● y ≠ a) ⊃ 25. Pc ● Oc ● ∼ ∼(c = a �� c = b) 24, DN

Tay] ● Ca 2, EI 26. ( y)(z)[(Pa ● Oa ● Py ● Oy ● Pz ● Oz) ⊃

5. (x)[(Bx ● x ≠ o) ⊃ Tox] ● Bo 1, Com (a = y �� a = z �� y = z)] 3, UI 6. (x)[(Bx ● x ≠ o) ⊃ Tox] 5, Simp 27. ( z)[(Pa ● Oa ● Pb ● Ob ● Pz ● Oz) ⊃ 7. (Ba ● a ≠ o) ⊃ Toa 6, UI (a = b �� a = z �� b = z)] 26, UI

8. a ≠ o AIP 28. ( Pa ● Oa ● Pb ● Ob ● Pc ● Oc) ⊃

9. Ba 4, Simp (a = b �� a = c �� b = c) 27, UI

10. Ba ● a ≠ o 8, 9, Conj 29. Pc ● Oc 25, Simp

11. Toa 7, 10, MP 30. Pa ● Oa ● Pb ● Ob 19, Simp

Answers to Selected Exercises 609

31. Pa ● Oa ● Pb ● Ob● Pc ● Oc 29, 30, Conj 36. ∼ ∼Da ● ∼ ∼Pa 33, 35, MP

32. a= b �� a = c �� b = c 28, 31, MP 37. ( x)[(Sx ● x ≠ c ● x ≠ n) ⊃ Px] ●

33. a= c �� b = c 18, 32, DS Sc ● ∼ ∼Pc ● Sn ● ∼ ∼Pn 1, Com

34. ∼ ∼(c = a �� c = b) ● Pc ● Oc 25, Com 38. ( x)[(Sx ● x ≠ c ● x ≠ n) ⊃ Px] 37, Simp

35. ∼ ∼(c = a �� c = b) 34, Simp 39. ( Sa ● a ≠ c ● a ≠ n) ⊃ Pa 38, UI

36. ∼ ∼(a = c �� c = b) 35, Id 40. ( Sa ● a ≠ c) ⊃ (a ≠ n ⊃ Pa) 39, Exp

37. ∼ ∼(a = c �� b = c) 36, Id 41. a≠ c ● Sa ● Ra 28, Com

38. (a = c�� b = c) ● ∼ ∼(a = c �� b = c) 33, 37, Conj 42. a≠ c 41, Simp 39. ∼∼ ∼∼(z)[(Pz ● Oz) ⊃ z = a �� z = b)] 20�38, IP 43. Sa ● a ≠ c 30, 42, Conj 40. ( z)[(Pz ● Oz) ⊃ (z = a �� z = b)] 39, DN 44. a≠ n ⊃ Pa 40, 43, MP 41. Pa ● Oa ● Pb ● Ob ● a ≠ b ● 45. ∼ ∼Pa ● ∼ ∼Da 36, Com

(z)[(Pz ● Oz) ⊃ (z = a �� z = b)] 19, 40, Conj 46. ∼ ∼Pa 45, Simp 42. ( ∃y){Pa ● Oa ● Py ● Oy ● a ≠ y ● 47. ∼ ∼(a ≠ n) 44, 46, MT

(z)[(Pz ● Oz) ⊃ (z = a �� z = y)]} 41, EG 48. a= n 47, DN 43. ( ∃x)(∃y){Px ● Ox ● Py ● Oy ● x ≠ y ● 49. n= a 48, Id

(z)[(Pz ● Oz) ⊃ (z = x �� z = y)]} 42, EG 50. Dn ● (x)[(Sx ● Dx) ⊃ x = n] ● (16) 1. Sc ● ∼ ∼Pc ● Sn ● ∼ ∼Pn ● Sn 2, Com

(x)[(Sx ● x ≠ c ● x ≠ n) ⊃ Px] 51. Dn 50, Simp

2. Sn ● Dn ● (x)[(Sx ● Dx) ⊃ x = n] 52. Da 49, 51, Id

3. (x){Sx ⊃ [Rx K (∼ ∼Dx ● ∼ ∼Px)]} 53. ∼ ∼Da 36, Simp

4. c ≠ n/ (∃x){Sx ● Rx ● (y)[(Sy ● Ry) ⊃ y = x]} 54. Da ● ∼ ∼Da 52, 53, Conj

5. Sc 1, Simp 55. ∼∼ ∼∼(y)[(Sy ● Ry) ⊃ y = c] 23�54, IP

6. Sc ⊃ [Rc K (∼ ∼Dc ● ∼ ∼Pc)] 3, UI 56. ( y)[(Sy ● Ry) ⊃ y = c] 55, DN

7. Rc K (∼ ∼Dc ● ∼ ∼Pc) 5, 6, MP 57. Sc ● Rc ● (y)[(Sy ● Ry) ⊃ y = c] 22, 56, Conj

8. [Rc ⊃ (∼ ∼Dc ● ∼ ∼Pc)] ● 58. ( ∃x){Sx ● Rx ●

[(∼ ∼Dc ● ∼ ∼Pc) ⊃ Rc] 7, Equiv (y)[(Sy ● Ry) ⊃ y = x]} 57, EG

9. [(∼ ∼Dc ● ∼ ∼Pc) ⊃ Rc] ●

10. (∼ ∼Dc ● ∼ ∼Pc) ⊃ Rc 9, Simp Exercise 9 [Rc ⊃ (∼ ∼Dc ● ∼ ∼Pc)] 8, Com

11. (x)[(Sx ● Dx) ⊃ x = n] ● Sn ● Dn 2, Com

13. (Sc ● Dc) ⊃ c = n 12, UI 1. a. Has no effect. 14. ∼ ∼ ( Sc ● f. Strengthens. Dc ) 4, 13, MT b. Strengthens. 12. (x)[(Sx ● Dx) ⊃ x = n] 11, Simp II.15. g. Weakens. ∼ ∼ Sc �� ∼ ∼ Dc 14, DM c. Weakens. h. Strengthens. 16. ∼∼ ∼∼ Sc 5, DN d. Weakens. i. Strengthens. 17. ∼ ∼ Dc 15, 16, DS e. Strengthens. j. Weakens. 18. ∼ ∼ Pc ● Sn ● ∼ ∼ Pn ● ( 19. x)[(Sx ● x ≠ c ● x ≠ n) ⊃ Px] ● 4. a. Has no effect. f. Has no effect. Sc 1, Com b. Weakens. g. Strengthens. ∼ ∼ Pc 18, Simp 20. ∼ ∼ Dc ● c. Strengthens. h. Weakens. ∼ ∼ Pc 17, 19, Conj d. Weakens. i. Weakens. 21. Rc 10, 20, MP 22. Sc ● e. Strengthens. j. Strengthens. Rc 5, 21, Conj 7. a. Weakens. f. Weakens. 23. ∼ ∼ ( y )[( Sy ● Ry ) ⊃ y = c ] AIP b. Strengthens. g. Weakens. 24. ( ∃ y ) ∼ ∼ [( Sy ● Ry ) ⊃ y = c ] 23, QN 25. ∼ ∼ [( Sa ● c. Has no effect. h. Strengthens. Ra ) ⊃ a = c ] 24, EI d. Strengthens. i. Strengthens. 26. ∼ ∼ [ ∼ ∼ ( Sa ● Ra ) �� a = c ] 25, Impl 27. ∼∼ ∼∼ ( Sa ● Ra ) ● e. Strengthens. j. Weakens. a ≠ c 26, DM 10. Hint: For Maxie�s argument, concentrate on the similari-28. Sa ● Ra ● a ≠ c 27, DN 29. Sa ⊃ [ Ra K ( ∼ ∼ Da ● ties between a home and a car (a car is an extended living ∼ ∼ Pa )] 3, UI space with heating, air conditioning, stereo, comfortable 30.

32. [ Ra K (∼ ∼Da ● ∼ ∼Pa) 29, 30, MP and a car (greater difficulty in controlling, ease of cross-31. seating, and so on), the dissimilarities between a plane Sa 28, Simp

[(∼ ∼Da ● ing international borders, greater danger in operating, and Ra ⊃ (∼ ∼Da ● ∼ ∼Pa)] ●

33. so on). Also, people outside U.S. borders are not accorded ∼ ∼Pa) ⊃ Ra] 31, Equiv

34. Ra ● a ≠ c ● the same constitutional protections as people inside, and Ra ⊃ (∼ ∼Da ● ∼ ∼Pa) 32, Simp

35. a phone message normally suggests greater privacy than a Sa 28, Com

Ra 34, Simp

radio message. In addition, the teenagers parked in the lot

610 Answers to Selected Exercises

were acting in plain view, and a telescope only enhances drop a brick, the fact that the brick would strike Hunt, ordinary sense perception (so there was no search). Maxie, etc. The facts are dissimilar to those in Sacco v. Lane, on the other hand, was talking while speeding down the where Lane was in direct control over the flames in the freeway and was thus not acting in plain view (in the same barbecue, and it was those very same flames that spread sense), and a radio receiver (used by the agents) does not to the houses. enhance ordinary sense perception. For Mary�s argument note that Liz initiated a chain

For the agents� argument, concentrate on the similarities of events that flowed naturally from the car accident to

between a car and a plane (both are means of transporta- the amputation. Once the car accident occurred, it was tion, both can cross state lines, both are relatively hard to foreseeable that Mary would be taken to the hospital; keep track of, and so on) and the dissimilarities between once Mary was in the hospital it was foreseeable that a car or plane and a house (houses do not move, so it will mix-ups would occur (after all, mix-ups occur in stay in place while a search warrant is obtained). Also, cell hospitals every day); given the nature of these mix-ups, phones use radio transmitters just like planes, both the it was foreseeable that Mary�s leg would be amputated. cell phone message and the radio message from the plane Granted, Liz might not have been able to foresee were received inside U.S. borders, and the cell phone mes- each event in the chain, but once an event occurred, sage was received inadvertently, just as the radio message someone familiar with it could have foreseen the next from the plane and the image through the telescope were event. Therefore, given that each event was foreseeable (thus making it impossible to plan ahead for a search war- by someone�at least some hypothetical person�Liz rant). Also, in a sense, Maxie was acting in plain view: A should be held liable. The events are similar to those lip reader traveling in an adjacent car might interpret his in Sacco v. Lane. When the flames were leaping from message. Lastly, controlling illicit drugs is a high priority the barbecue, it was foreseeable that they would ignite for the government. the trees; once the trees were aflame, it was foreseeable

13. Hint: Couch your arguments in terms of foreseeability. that a house would be ignited, then another house,

For Liz�s argument note that there were several etc. Lane would not have been able to foresee the

events that intervened between the car accident and the whole chain of events at the time he was tending the amputation: Mary�s being taken to the hospital, Mary�s barbecue, but once one event occurred, the next was being treated by doctors for bumps and bruises, Mary�s foreseeable. Also, even though the wind constituted an apparent mix-up with some other patient scheduled intervening event, Lane was still held liable. Finally, the for leg amputation, the doctors� failure to check Mary�s facts are dissimilar to those in Gomez v. Hunt. When proper identity before operating, etc. Liz did not directly Hunt was walking home, he was in complete control control any of these events, and therefore she could not over his own actions. He freely chose to walk past the have foreseen them. Liz could not have foreseen what building under construction, and he should have been hospital Mary would be taken to, the fact that Mary on the lookout for falling objects. The fact that he was would be mixed up with another patient, etc. Because Liz struck by a falling brick was partly the result of his own should not be held liable for an event utterly unforeseen failure to observe. On the contrary, from the time of the to her, she should not be liable for Mary�s amputated leg. accident until the amputation, Mary was in the hands of The facts are similar to those in Gomez v. Hunt, where others: the person who took her to the hospital, nurses Gomez could not foresee the exact route that Hunt would in the hospital, etc. In no sense was the amputation the take when walking home, the fact that a worker would result of Mary�s free choices.

Exercise 10

1. Sufficient condition. The window can also be broken by throwing a stone or baseball through it. 4. Necessary condition. For an image to appear on the screen there must be sufficient light and the battery must be

charged.

7. Sufficient condition. The fire will also be extinguished if it is smothered.

10. Necessary condition. Electricity must also be supplied from the main lines.

13. Necessary condition. Without exercise a person cannot be healthy, but other things are needed such as proper nutrition. 16. Necessary condition. The check must also be dated and the payee must be named. 19. Necessary condition. If the batter does not pick up the bat, he/she cannot hit a home run. But the batter must also hit the

ball.

Answers to Selected Exercises 611

II.

1. A = a certain make, B = a certain year, C = a certain model, D = driven 30,000 miles, E = a certain gasoline,

F = a certain oil, G = a certain driver, H = certain road conditions, I = the additive.

Possible ConditionsPhenomenon

Occurrence(less wear)ABCDEFGHI

Engine 1 * * * * * * * * * * Engine 2 * * * * * * * * � �

Method of difference, sufficient condition.

4. A = lecture hall, B = gym, C = library, D = activities center, E = book store, F = cafeteria.

Possible ConditionsPhenomenon

OccurrenceABCDEF(stolen article)

(Vivian) * * � * * � * (Jamar) � * * * � * *

(Martin) * * * � * * *

(Elena) � * � * * * *

(Shandra) * * � * * � *

(Hank) * * * � * * *

(Daniela) * * * * � * *

Access to lockers in the gym is the cause in the sense of a necessary condition. Method of agreement.

7. A = camomile tea, B = late dinner, C = hot bath, D = read book, E = walk, F = wine, G = massage.

Possible ConditionsPhenomenon

Occurrence(slept well)ABCDEFG

Mon * * * * * � � *

Tu � * � * � � � �

Wed � � � � * * * �

Th � * * * � * * * Fri * * � * * � * * Sat * � * � * * * * Sun * � * * � � � �

None of the possible conditions is a cause of the phenomenon.

Joint method of agreement and difference.

10. A = corporal punishment, B = siblings, C = adopted, D = male parent figure, E = sexual abuse, F = domineering mother,

G = uprooted often, H = day-care center.

Possible ConditionsPhenomenon

Occurrence(blurred bound)ABCDEFGH

Meg * * * � * * � � * Sue * * � * * � � * * Dot * � * * * * * * * Jane * * * * * * * � * Lynn � * * * * � � * *

Flo * * � * * * * � *

612 Answers to Selected Exercises

Sexual abuse is the cause of the phenomenon in the sense of a necessary condition. Method of agreement.

13. A = soapy water, B = American Beauty, C = five years old, D = a certain location, E = a certain amount of water,

F = a certain amount of sun, G = a certain kind of soil, H = a certain amount of cultivation, I = a certain amount of bone meal rose food.

Possible ConditionsPhenomenon

Occurrence(no aphids)ABCDEFGH

1 * * * * * * * * * 2 � * * * * * * * �

Method of difference, sufficient condition.

III. b. P(Y and G) = P(Y1 and G2) + P(G1 and Y2)

4. By an unnamed method, of a necessary condition. 10/33 = .303 = c. P ( R or G ) 1 − P ( Y 1 and Y2) D is the cause in the sense of a = 1 − (5/12 � 4/11) sufficient condition. = 1. By the method of agreement, = 20/132 + 20/132 D is the cause in the sense = = (5/12 � 4/11) + (4/12 � 5/11)

1 − 20/132

Exercise 11 d. P(G 1 or G2) = 1 − P(not G) = = 28/33 = .848

1 − [P(R 1 and R2) + P(R1 and Y2)

I. + P(Y1 and R2) + P(Y1 and Y2)]

b. 13. 4/9 19/33 = .57 e. P (same color) = P ( R 1 and R2) + P(G1 and G2) + P(Y1 and Y 2 ) II. = (3/12 � 2/11) + (4/12 � 3/11) + �(5/12 � 4/11) 1. P (6 or 1) = P (6) + P (1) = 1/6 + 1/6 = 2/6 = 1/3 = 6/132 + 12/132 + 20/132 4. a. P ( A 1 and A 2 ) = P ( A 1 ) � P ( A 2 ) = = 19/66 = .288 4/52 � 4/52 = 1/169 = .0059 13. a. P ( M or W ) = P ( M ) + P ( W ) − P ( M and W ) P ( A 1 and A 2 ) = P ( A 1 ) � P ( A 2 given A 1 ) = .74 + .82 − (.74 � .82) = 4. .853 1 − [6/132 + 15/132 + 15/132 + �20/132] 7. 1/4 or .25 = 1 − 56/132 10. Approximately $17 = 1. 1/6; 1:5 (5/12 � 3/11) + (5/12 � 4/11)] = = 1 − [(3/12 � 2/11) + (3/12 � 5/11) +

7. First compute the probability of getting no sixes: = 4/52 � 3/51 = .95 1/221 = .0045 b. P ( M and W and S) = P(M) � P(W) � P(S) = .74 � .82 � 8/9 P (no sixes) = 5/6 � 5/6 � 5/6 = .54 = 125/216 Then use the negation rule: P (at least one six) = 1 − P (no sixes) 1 = − 125/216 91/216 = = .4213 10. a. P ( R 1 and R 2 ) = P ( R 1 ) � P ( R 2 given R 1 ) = 3/12 � 2/11 = 6/132 = .045

Answers to Selected Exercises 613

P(N) 3 P(R given N)

16. PNP( given )] [(P ( N given R ) = [() 3 RN 1 PO) 3 P(R given O)]

5 3/5 3 7/15 21/75 5

[3/5 3 7/15] 1 [2/5 3 5/15] 21/75 1 10/75

5 21/75 5 21/31 5 .68

31/75

Answer: new urn.

19. (R given N) =

PP(R) 3 P(N given R)

[P(R) 3 P(N given R)] 1 [P(T) 3 P(N given T)]

5 .2 3 .7 .14 5 5 .14/.30 5 .47

(.2 3 .7) 1 (.8 3 .2) .14 1 .16

Exercise 12 For 2%, n = (1.29/.02)2 = 64.52 = 4,160 For 3%, n = (1/29/.03) 2 = 43 2 = 1,849 For 4%, n = (1.29/.04) 2 = 32.25 2 = 1,040 I. For 5%, n = (1.29/.05) 2 = 25.8 2 = 666 For 6%, n = (1.29/.06) 2 = 21.5 2 = 462 1. Since the water in the lake might not be circulating, the For 7%, n = (1.29/.07) 2 = 18.43 2 = 340 algae content of the water at one end might not be rep- For 10%, n = (1.29/.1) 2 = 12.9 2 = 166 resentative of the whole lake. Thus, the sample might be 4. biased. 4. According to Table 12.1, the margin of error for a 3 random sample of 600 is � 5 percent. Since the sample 2 taken indicates a difference of only 2� percent, the results 1 of the sample are inconclusive. Frequency 7. Since no mention is made of the size of the sample or 1 2 3 4 5 6 of the expected sampling error, the sample might be Age biased. The manufacturer might have taken 25 separate samples consisting of ten dentists per sample and reported Mean age = 3, variance = 3, standard deviation = 1.73. the results of only the most favorable one. 10. The problem concerns the meaning of �average.� If the III. average is a mean, most of the toys could be over $15, and a few very cheap toys could bring the average down 1. False. to $15. If the average is a mode, there might be a few toys 4. False. priced at $15, and all the other toys might have varying 7. True. prices exceeding $15. Only if the average is a median can 10. False. one be assured that half the toys are $15 or less. 13. True. 13. Since no mention is made of the dispersion, the 16. False. argument is weak. The rock pile might consist of several 19. False. pieces weighing 500 pounds and enough weighing only 4 or 5 pounds to bring the average down to 50 pounds. If the range were only 10�pounds or so, the conclusion Exercise 13 would follow. 16. If the scale on the vertical axis does not begin at zero, the V. conclusion does not follow. 19. Since there were many more cars on the road in 2016 than 1. True. there were in 1995, the comparison is faulty. 4. False. 7. True. II.10. True. 1. Margin of error = 1.29/√ n13. True. Solving for n , n = (1.29/margin of error) 2 16. False. For 1%, n = (1.29/.01) 2 = 129 2 = 16,641 19. False.

614 Answers to Selected Exercises

Glossary/Index

A proposition: A categorical proposition having the form Amyntas II, King, 13

�All S are P,� 4, 179�180 Analogical reasoning, 447�450

Abelard, Peter, 5, 87 Analogue, 448�449 Absorption, 347n Analogy, 447�457; argument from, 32, 448�453 Accident: An informal fallacy that occurs when a general rule Anecdotal evidence, 545

is wrongly applied to an atypical specific case, 116�117 Antecedent: (1) The component of a conditional statement

Ackermann, Wilhelm, 421n immediately following the word �if,� 19; (2) the component Ad hoc modifications, 546 of a conditional statement to the left of the horseshoe, 277 Ad hominem. See Argument against the person Appeal to fear: A variety of appeal to the people that occurs Ad hominemabusive: A variety of the argument-against-the- when an arguer trumps up a fear of something and then

person fallacy that occurs when an arguer verbally abuses a uses that fear as the premise for some conclusion, 112 second arguer for the purpose of discrediting that person�s Appeal to force: An informal fallacy that occurs when an argument, 115, 160, 161, 173 arguer threatens a reader or listener for the purpose of get-

Ad hominemcircumstantial: A variety of the argument- ting him or her to accept a conclusion, 110�111, 161

against-the-person fallacy that occurs when an arguer cites Appeal to ignorance: An informal fallacy that occurs when an circumstances that affect a second arguer, for the purpose arguer uses the fact that nothing has been proved about some-of discrediting that person�s argument, 114 thing, as evidence in support of some conclusion about that

Adams, John Couch, 529, 530, 534, 537 thing, 126�127 Addition: A valid rule of inference: �p // p or q,� 347�350, Appeal to pity: An informal fallacy that occurs when an

377; with relational predicates, 428 arguer attempts to evoke pity from a reader or listener for

Adequacy: The extent to which a hypothesis fits the facts it is the purpose of getting him or her to accept a conclusion,

intended to explain, 536, 542, 544, 554, 556 111, 161�162

Advice. fallacy that occurs when the arguer plays on the reader�s or See Piece of advice listener�s need to feel superior, 113 Adverbs, translation of, 220�221 Appeal to snobbery: A variety of the appeal-to-the-people

Affirming the consequent: An invalid argument form: �If p 111�114, 161, 549 then q / q // p, � 321, 323, 335 Appeal to tradition: A fallacy that occurs when an arguer Alexander the Great, 13 cites the fact that something has become a tradition as �All except,� �all but,� 223, 433�434 grounds for a conclusion, 114 Ambiguity, fallacies of. Affirmative proposition/statement: A proposition/state-Affective terminology and definitions, 103 Appeal to the people: An informal fallacy that occurs when ment that asserts class membership, 179�181 an arguer plays on certain psychological needs for the pur-pose of getting the reader or listener to accept a conclusion,

See Fallacies of ambiguity Appeal to unqualified authority: An informal fallacy that

Ambiguous definitions, 103 occurs when an arguer cites the testimony of an unqualified Ambiguous expression: An expression that can be inter- authority in support of a conclusion, 125�126, 162

preted as having more than one distinct meaning in a given Appeal to vanity: A variety of the appeal-to-the-people fal-context, 73�74, 80, 141 lacy that occurs when an arguer plays on the vanity of the

Ambrose, Alice, 177 reader or listener, 113 Amphiboly: An informal fallacy that occurs when the con- Argument: A group of statements, one or more of which (the

clusion of an argument depends on the misinterpretation premises) are claimed to provide support for, or reasons to of a statement that is ambiguous owing to some structural believe, one of the others (the conclusion), 2; cogent, 44; defect, 147�149

615

conditional statements and, 19�21; definitions and, 58�61; Autokinetic effect, 551 explanation and, 18�19; extended, 58�61; form of, 52�56, Average, meaning of, 511�512 318�326, 337; in science, 34; recognition of, 12�21; sound, Avoiding fallacies, 160�164

Argument against the person: An informal fallacy that 39�45; strong, 42�45; valid, 39�42, 52�53. See also Deduc- Axiom of replacement: An axiom that states that logically tive argument; Inductive argument equivalent expressions may replace one another in a proof sequence, 355 occurs when an arguer verbally attacks the person of a Babbage, Charles, 306 second arguer for the purpose of discrediting his or her Baliani, Giovanni, 531 argument, 114�116 Bandwagon argument: A variety of the appeal-to-the-people Argument based on mathematics: A deductive argument in fallacy that occurs when the arguer plays on the reader�s or which the conclusion depends on some purely arithmetic listener�s need to feel part of a group, 113 or geometric computation or measurement, 31 �Barbara,� syllogism, 5, 234, 242, 259 Argument based on signs: An inductive argument that pro-Barcan Formula, 30 ceeds from the knowledge of a sign to a claim about the Barometer, invention of, 531�532 thing or situation that the sign symbolizes, 33 Barrow, Harriet, 471 Argument form: (1) An arrangement of words and let-Bayes, Thomas, 498 ters such that the uniform substitution of terms or state-ments in place of the letters results in an argument, 52�56; Bayes�s theorem: In probability theory, a rule for evaluating (2) an arrangement of statement variables and operators the conditional probability of two or more mutually exclu-such that the uniform substitution of statements in place sive and jointly exhaustive events, 498�500 of the variables results in an argument, 318�326, 336� �Because,� 14, 18 337; invalid, 52�56, 323�326, 335; valid, 52�53, 318�321, Becquerel, Henri, 529 336�337 Begging the question: An informal fallacy that occurs when Argument from analogy: An inductive argument that the arguer creates the illusion that inadequate premises pro-depends on the existence of a similarity between two things vide adequate support for the conclusion�by leaving out a or states of affairs, 32�33, 448 key premise, by restating the conclusion as a premise, or by Argument from authority: An inductive argument in which reasoning in a circle, 141�143, 160, 162 the conclusion rests on a statement made by some pre-Benassi, Victor A., 549 sumed authority or witness, 33 Bentham, Jeremy, 471 Argument from compassion, 111 Berti, Gasparo, 531 Argument from definition: A deductive argument in which Biased sample: A sample that is not representative of the the conclusion is claimed to depend merely on the defini-population from which it was selected, 507 tion of some word or phrase used in the premise or conclu-Biconditional statement (biconditional): A statement sion, 31 having a triple bar as its main operator, 277, 280; compari-Argument from example: An argument that purports to son of with ordinary language, 291�293; relating logically prove something by giving one or more examples of it, 17 equivalent statements, 300; subjunctive, 293; truth-func-Argumentum ad bacculum.See Appeal to force tional definition of, 288 Argumentum ad hominem.See Argument against the person Boethius, 5 Argumentum ad ignorantiam.See Appeal to ignorance Bolzano, Bernard, 5 Argumentum ad miserecordiam.See Appeal to pity Boole, George, 5, 184, 185, 191�193, 240 Argumentum ad populum. See Appeal to the people Boolean standpoint/interpretation (modern standpoint), Argumentum ad verecundiam. See Appeal to unqualified 184, 186, 188, 191, 205�207, 213, 215�216, 220, 233�234,

authority 239�243, 248�251; in predicate logic, 390�396

Aristotelian sorites, 267n �Both . . . not,� 280�281 Aristotelian standpoint/interpretation (traditional stand- Bound variable: A variable that is bound by a quantifier, 393

point), 184, 186, 203�206, 213, 214n, 215�217, 220, 234, Bouvard, Alexis, 530 239, 243�245, 251�252; in predicate logic, 394; Venn dia- Boyle, Robert, 481 grams for, 213�217 Boyle�s law, 34

Associativity: A valid rule of inference that allows for the 282, 527, 531, 532 Brackets, 277�278, 280 Aristotle, 4�5, 10, 13, 25, 29, 34, 47, 108, 161, 177, 184, 248, Braces, 277, 280, 281

relocation of parentheses in conjunctions and disjunctions, Broad hypotheses, 546, 556 Broad definitions, 101�102

Atmospheric pressure, discovery of, 531�532, 534, 536, 538 Bruner, Jerome S., 551 355�357

616 Glossary/Index

Bryan, William Jennings, 112 Compound statement: A statement that contains at least Byron, Ada, 306 one simple statement as a component, 276; truth values of, Campbell, Stephen K., 519n 287�293 Carroll, Lewis, 271 Conclusion: The statement in an argument that the premises Categorical proposition: A proposition that relates two are claimed to support or imply, 2�6; tautologous, 305

classes (or categories), 175�224; letter names of, 179�182; Conclusion indicator: A word that provides a clue to identi-standard form of, 176�178 fying a conclusion, 2

Categorical syllogism: A syllogism in which all three state- Conditional proof: A method of proof that consists of

ments are categorical propositions, 32, 178, 231�274; assuming the antecedent of a required conditional state-exceptive propositions in, 259; figure of, 231�235; form of, ment on the first line of an indented sequence, deriving the 231�235; in ordinary language, 258�260; mood of, 231� consequent on a subsequent line, and then discharging the 235; proving the rules for, 252�253; reconstruction of, from indented sequence in a conditional statement that exactly mood and figure, 231�235; reducing the number of terms replicates the one to be obtained, 377�380; indirect proof in, 255�257; rules for, 248�252; standard form of, 231�235; and, 382�384; in predicate logic, 413�415; to prove logical

Causal inference: An inductive inference that proceeds from Conditional statement (conditional): (1) An �if . . . then,� statement, 19�21; (2) a statement having a horseshoe as its Venn diagrams for, 238�245 truths, 387

knowledge of a cause to a claim about an effect, or from

Causality, 467�468 language, 291�293; in propositional logic, 277�279, 289; subjunctive, 293; translating into categorical propositions, knowledge of an effect to a claim about a cause, 33 main operator, 276�279, 281; comparison of with ordinary

Chrysippus, 4�5, 47 231�232; truth-functional definition of, 292. See also Cor-Charles, Jacques Alexandre, 481

responding conditional

Cicero, 47 Conditionally valid: Valid from the Aristotelian standpoint Circular definitions, 102 on condition that the subject term of the premise (or prem-Circular reasoning. See Begging the question ises) denotes actually existing things ; conditionally valid Class complement, 196 inferences, 206�207, 217; conditionally valid syllogisms, Class statement, 150�151 244�245; conditionally valid syllogistic forms, 233�234 Classical theory of probability: The theory according to Confabulation, 552

which probabilities are computed a priori by dividing the Confidence level, 509 number of favorable outcomes by the number of possible Conjoint premises, 69 outcomes, 491�494, 500 Conjunct: The component in a conjunctive statement on

Cogent argument: An inductive argument that is strong, Conjunction: (1) A statement having a dot as its main operator, has all true premises, and meets the total evidence require-Cleanthes, 47 either side of the main operator, 277

276�278; (2) a valid rule of inference: � p / q // p and q,� 347�

ment, 45 350; comparison of with ordinary language, 291�293; truth-

Cognitive meaning: The meaning by which terminology functional definition of, 288

conveys information, 71�73 Conjunctive statement (conjunction): A statement having a

Coherence. See Internal coherence dot as its main operator, 277�278 Cold reading, 553�554 Connectives: Symbols used to connect or negate propositions Collective hallucination, 551 in propositional logic, 275 Collective predication: An attribute is predicated collectively Connotation: Intensional meaning or intension, 81; conven-

when it is assigned to a class as a whole, 148�150 tional, 81

Common names, 80 Connotative definition. See Intensional definition Commutativity: A valid rule of inference that provides for Consequent: (1) The component of a conditional statement

the rearrangement of conjunctions and disjunctions, 325, immediately following the word �then,�; the component 355�356 of a conditional statement that is not the antecedent, 19;

Complex question: An informal fallacy that occurs when (2) the component of a conditional statement to the right of

a single question that is really two or more questions is the horseshoe, 277 asked, and a single answer is applied to both questions, 141, Consistency, indirect truth tables for, 312�314; truth tables 143�144, 161 for, 300�301. See also External consistency

Composition: An informal fallacy that occurs when the con- Consistent statements: Statements for which there is at least

clusion of an argument depends on the erroneous transfer- one line on their truth tables in which all of them are true, ence of an attribute from the parts of something onto the 300, 314�315 whole, 141, 149�150 Constant. See Individual constant

Glossary/Index 617

Constructive dilemma: A valid argument form/rule of infer- D�niken, Erich von, 547

ence: �If p then q, and if r then s / p or r // q or s,� 320, 322, Darwin, Charles, 112, 527 347; refuting, 321�322 Decreasing extension: The order of decreasing class size, 82

Context, definitions and, 103 Decreasing intension: The order of decreasing specificity or Contingent statement: A statement that is neither necessar- increasing generality, 82

ily true nor necessarily false, 299 Deduction. See Deductive argument; Natural deduction

Contradictory relation: The relation that exists between Deductive argument: An argument incorporating the claim

statements that necessarily have opposite truth values, 188, that it is impossible for the conclusion to be false given that 204�206 the premises are true, 29, 31�32, 40�42; invalid, 40�41, 45;

Contradictory statements: Statements that necessarily have sound, 45; unsound, 42; valid, 40�41, 45

opposite truth values, 195, 299�300 Definiendum: In a definition, the word or group of words that

Contraposition: An operation that consists in switching the are proposed to be defined, 84

subject and predicate terms in a standard-form categorical Definiens: In a definition, the word or group of words that do proposition and replacing each with its term complement, the defining, 84 194�200, 206�207; to reduce number of terms in a syllo- Definite description, 441

Contrapositive, 198�200 word or group of words, 84; argument and, 58�74; by genus gism, 255�257 Definition: A group of words that assigns a meaning to a Contrary relation: The relation that exists between two state- and difference, 95�97, 101�102; by subclass, 93�94, 97; cir-

ments that are necessarily not both true, 204�207, 214 cular, 102; criteria for, 101�103; demonstrative (ostensive),

Controlled experiments, 477�478, 481 92, 97; enumerative, 92�93, 97; etymological, 94�95, 97; Conventional connotation: The intensional meaning con- extensional (denotative), 93�94, 101; figurative, 102�103;

ventionally agreed on by the members of the community intensional, 94�96; lexical, 86, 93, 96, 97, 101�103; negative, who speak the language in question, 81 102; obscure, 102; operational, 95, 101; persuasive, 88�89,

Converse, 195�196, 200 93, 96, 97; precising, 86�88, 93, 97; purposes of, 84�89; stip-Converse accident. ulative, 85�86, 89, 93, 96, 97; synonymous, 94, 96, 97, 101; See Hasty generalization

Conversion: An operation that consists in switching the theoretical, 88, 93, 96, 97; vague, 102�103

subject and predicate terms in a standard-form categorical Definition by genus and difference: A definition that assigns proposition, 194�196, 199�200, 207; to reduce number of a meaning to a term by identifying a genus term and one terms in a syllogism, 255�257 or more difference words that, when combined, convey the

Conversion by limitation, 191 same meaning as the term being defined, 95�96, 101�102

Copula: In standard-form categorical propositions, the words to a term by naming subclasses of the class that the term denotes, 93�94 Copernicus, Nicholaus, 534, 536, 544 Definition by subclass: A definition that assigns a meaning

Counteranalogy, 449 having the conjunction of an argument�s premises as its class that the word denotes, 92, 97 De Morgan, Augustus, 5, 240, 306, 315, 355, 535 antecedent and the conclusion as its consequent, 305 De Morgan, George, 315 De Morgan�s rule: A valid rule of inference that allows tildes Counterdilemma, 322 to be moved inside and outside of parentheses, 47, 129, 280, Counterexample method: A method for proving invalidity; 281, 349, 355�356, 359, 367, 396 consists in constructing a substitution instance having true Denotation: Extensional meaning or extension, 81 premises and false conclusion, 54�56; in predicate logic, Denotative definition. See Extensional (denotative) definition 418�419 Denying the antecedent: An invalid argument form: �If Correlational method, 480 Definitions of the logical operators, 287�290 Demonstrative (ostensive) definition: A definition that Correlations, 132, 480�481 assigns a meaning to a word by pointing to members of the Corresponding conditional: The conditional statement Correlation coefficient, 480 Definitional techniques, 92�97 �are,� and �are not,� 176�177

Critical term: The term in a categorical syllogism which, p then q / not p // not q, � 321, 326, 335, 418 when it denotes at least one existing thing, guarantees that Dependent events, 496, 498 the subject of the conclusion denotes at least one existing thing, 234, 244, 251�252; of a sorites, 268 Descartes, Ren�, 532, 537 Descriptive phrases, 80 Curie, Marie, 529�530, 537 Destructive dilemma: A valid argument form/rule of infer-Curie, Pierre, 529�530 ence: �If p then q , and if r then s / not q or not s // not p or Daisy commercial, 112 not r, � 320�321, 347; refuting, 321�322 Dalton, John, 527

618 Glossary/Index

Detecting fallacies, 159�160 Emotional disposition and avoiding fallacies, 161, 163, 164 Difference, 95�96 Emotive meaning: The meaning by which terminology Dilemma. See Constructive dilemma; Destructive dilemma expresses or evokes feelings, 71�73 Disanalogy, 448 Empirical hypotheses: Hypotheses that concern the produc-Disconfirmability, 546 tion of some thing or the occurrence of some event that can Disjunct: The component in a disjunctive statement on either be observed, 534

Disjunctive statement (disjunction): A statement having a something that does not exist; the null class, 81 Empty intension, 81 side of the main operator, 276�278 Empty extension: The extension of a term that denotes

wedge as its main operator, 277�278; comparison of with

ordinary language, 291�293; exclusive, 278, 288�289, 292; Enthymeme: A categorical syllogism that is missing a prem-inclusive, 278, 288, 292, 320; truth-functional definition of, ise or conclusion, 261�263; an argument that is missing a 288�290 premise or conclusion, 20n, 261�263

Disjunctive syllogism: (1) A syllogism having a disjunctive Enumerative definition: A definition that assigns a meaning

statement for one or both of its premises, 32; (2) a valid to a word by naming the members of the class that the word argument form/rule of inference: �p or q / not p // q,� 320, denotes, 92�93, 96 337�341 Equivalence. See Biconditional statement; Logically equiva-

Dispersion: In statistics, an indicator of how spread out the lent statements; Material equivalence

data are in regard to numerical value, 512�516 Equivocation: An informal fallacy that occurs because some

Disputes, 74 word or group of words is used either implicitly or explic-Distribution: (1) An attribute possessed by a term in a cat- itly in two different senses, 141, 147, 174, 519; division and,

egorical proposition if and only if the proposition makes 150�151 a claim about all the members of the class denoted by the Escaping between the horns of a dilemma, 322 term, 179�182, 248�249; (2) a valid rule of inference that Essential meaning, definitions and, 101 allows a conjunct/disjunct to be distributed through a dis- Etymological definition: A definition that assigns a meaning junction/conjunction, 355�357, 359 to a word by disclosing the word�s ancestry in both its own

Distributive predication: An attribute is predicated distributively language and other languages, 94�97

when it is assigned to each and every member of a class, 150 Everest, Mary, 185

Division: An informal fallacy that occurs when the conclusion Evidentiary support, 544�548

of an argument depends on the erroneous transference of an Exceptive propositions, 223�224; in predicate logic, 433; syl-

Dot, 276�278; comparison of with ordinary language, 291� 141, 150�151 Exclusive disjunction, 288, 292 attribute from a whole (or class) onto its parts (or members), logisms containing, 258�259

293; truth-functional definition of, 288; use in predicate Exclusive premises: A formal fallacy that occurs when both

logic, 393 Exclusive propositions, 222�223; in predicate logic, 394, 433 premises of a categorical syllogism are negative, 250

Double colon: The metalogical symbol that designates logical ment is invalid merely because the premises lack existential Double blind studies, 478, 510 Existential fallacy: A fallacy that occurs whenever an argu-

equivalence, 355 import, 191, 206�207, 216�217, 250�251

Double colon with super-scribed �c,� 419 Existential generalization: A rule of inference that intro-Double negation: A valid rule of inference that allows the duces existential quantifiers, 398, 400�401, 403; improper

introduction or deletion of pairs of negation signs, 325, or invalid applications of, 404 356�357 Existential import: An attribute of a categorical proposition

Drawing an affirmative/negative conclusion from nega- by which it implies that one or more things denoted by the

tive/affirmative premises: A formal fallacy that occurs in subject term actually exist, 183�184, 191�193 a categorical syllogism when an affirmative conclusion is Existential instantiation: A rule of inference that removes drawn from a negative premise or a negative conclusion is existential quantifiers, 401�403; invalid applications of, 400, drawn from affirmative premises, 250 404; restrictions on, 401�403

E proposition: A categorical proposition having the form Existential quantifier: The quantifier used to translate par-�No S are P,� 179�180 Duns Scotus, 129 Existential names, 401�402

ticular statements in predicate logic, 393�395

Edmonstone, Susanna Carnegie, 240 Explanandum: The component of an explanation that Einstein, Albert, 29, 435, 501, 527, 536�538, 548 describes the event or phenomenon to be explained, 18 �Either,� 278 Explanans: The component of an explanation that explains Eminent logicians, See Logicians the event or phenomenon indicated by the explanandum, 18

Glossary/Index 619

Explanation: An expression that purports to shed light on Figure: An attribute of a categorical syllogism that specifies

some event or phenomenon, 18�19 the location of the middle term, 231�235

Exportation: A valid rule of inference that allows conditional Finite universe method: A method for proving invalidity in

statements having conjunctive antecedents to be replaced predicate logic that consists in reducing the universe to a with conditional statements having conditional conse- single object and then sequentially increasing it until one is quents, and vice versa, 366�368 found in which the premises of an argument turn out true

Expository passage: A kind of discourse that begins with and the conclusion false, 418�421

a topic sentence followed by one or more sentences that Folta, Kevin, 559 develop the topic sentence, 16�17 �For the reason that,� 3

Extended arguments, 58�61 �For this reason,� 3 Extensional (denotative) definition: A definition that Form of a categorical syllogism, 231�235

assigns a meaning to a term by indicating the members of Form of an argument, 52�55, 149, 320; invalid, 321, 323, 337, the class that the term denotes, 92�94 340�341

Extensional meaning (extension): The members of the class Formal fallacy: A fallacy that can be identified by merely

that a term denotes, 80�81; empty, 81 examining the form or structure of an argument, 107�108.

External consistency: The extent to which a hypothesis See also Fallacies (of various kinds)

agrees with other, well-confirmed hypotheses, 536�537, Free variable: A variable that is not bound by a quantifier, 544, 552, 554, 556 393

Factual disputes, 74 Frege, Gottlob, 5, 13, 378 Fallacies of ambiguity: A group of informal fallacies that Fruitfulness: The extent to which a hypothesis suggests new

occur because of an ambiguity in the premises or conclu- ideas for future analysis and confirmation, 537�538, 544 sion, 141, Fulbert, 87

Fallacies of illicit transference: A group of informal fallacies Galilei, Galileo, 9, 37 that involves the incorrect transference of an attribute from Fallacies of categorical syllogisms, 248�253 Galen, 5

the parts of something onto the whole, or from the whole Gambler�s fallacy, 131 Galle, Johann, 529, 530, 534

Fallacies of presumption: A group of informal fallacies that Gay-Lussac, Joseph Luis, 546 onto the parts, 141

occur when the premises of an argument presume what Geller, Uri, 552

Fallacies of relevance: A group of informal fallacies that computing the probability of two events occurring together whether or not they are independent, 495�496, 498, 499 they purport to prove, 141�151 General conjunction rule: In probability theory, a rule for

occur because the premises of an argument are irrelevant to

Fallacies of weak induction: A group of informal fallacies computing the probability of either of two events whether or not they are mutually exclusive, 497, 498, 500 the conclusion, 110�120 General disjunction rule: In probability theory, a rule for

that occur because the connection between the premises

and conclusion is not strong enough to support the conclu- General statement: A statement that makes a claim about all

Fallacy: A defect in an argument that arises from a mistake Generalization: An inductive argument that proceeds from the knowledge of a selected sample to some claim about the sion, 125�134 the members of a class, 34, 150�151

in reasoning or the creation of an illusion that makes a bad

argument appear good, 107. whole group, 33; in predicate logic, 399. See also Existential See also Fallacies (of various kinds); Formal fallacy; Informal fallacy generalization; Universal generalization

False cause: An informal fallacy that occurs when the con- Genus, 95�96

clusion of an argument depends on some imagined causal Gestalt, 550 connection that probably does not exist, 130�132, 162�163, Goclenian sorites, 267n 174, 545, 549; and appeal to the people, 113 G�del, Kurt, 6, 435

False dichotomy: An informal fallacy that is committed when Goldwater, Barry, 112

as if they were jointly exhaustive and then eliminates one, Grammar, definitions and, 101 leaving the other as the conclusion, 141, 144�145, 161�162 an arguer presents two nonjointly exhaustive alternatives Gordon, George, 306

Feng shui, 545 Hallucination, 551 Fermat, Pierre de, 491 Hallucinations distort perception, 551 �Few,� �a few,� 221�222 Hampshire, Stuart, 358 Falsifiability criterion, 546 Grasping a dilemma by the horns, 322 Graphs, 516�518 Figurative definitions, 102�103

620 Glossary/Index

Hasty generalization: An informal fallacy that occurs when Illicit transference, See Fallacies of illicit transference

a general conclusion is drawn from atypical specific cases, Illustration: An expression involving one or more examples 127�128, 149�151, 162, 549 that is intended to show what something means or how it

Helmont, Jan Baptista Van, 532 is done, 17 Heloise, 87 Immediate inference: An argument having a single premise, Herschel, William, 530 188�191, 205�207, 215�217 Hippocrates, 555 Implication, rules of, 336�341, 347�350. See also Material Histogram, 513�515 implication History of logic, 4�6 Inclusive disjunction, 288, 291, 320 Horizontal pattern, 59 Inconsistent premises, 305, 349 Horns of a dilemma, 321 Inconsistent statements: Statements such that there is no line Horseshoe, 276�279; comparison of with ordinary language, on their truth tables in which all of them are true, 300

291�293; truth-functional definition of, 289; use in predi- Increasing extension: The order of increasing class size, 82 cate logic, 392 Increasing intension: The order of increasing specificity, 82

Huff, Darrell, 513n Independent events, 496�498 Hyman, Ray, 554 Indirect proof: A method of proof that consists of assum-Hypnagogic hallucination, 551 ing the negation of a required statement on the first line of

Hypotheses: Conjectures offered as possible explanations for sequent line, and then discharging the indented sequence by asserting the negation of the assumed statement, 382� Hypnopompic hallucination, 551 an indented sequence, deriving a contradiction on a sub-

Hypothetical reasoning: The reasoning process used to pro-534; vague, 545 truths, 387 Indirect truth tables, 310�314; for arguments, 311�313; for series of statements, 313�314; in predicate logic, 423�428 of, 534�536; tentative acceptance of, 536�538; theoretical, 384; in predicate logic, 413�415, 427�428; to prove logical a phenomenon, 526�529; broad, 546; empirical, 534; proof

I proposition: A categorical proposition having the form statement for one or both of its premises, 32. w, x, y, z) used to rep-See also Pure resent anything at random in the universe, 392, 394, 402 hypothetical syllogism Inductive argument: An argument incorporating the claim that it is improbable that the conclusion is false given Hypothetical syllogism: A syllogism having a conditional used to name individuals, 391, 402 Individual variable: A lowercase letter ( tions, 526�538 duce hypotheses, draw implications, and test the implica- Individual constant: A lowercase letter (a, b, c . . . t, u, v)

Ignoratio elenchi. See strong, 42�44 Missing the point Inference: The reasoning process expressed by an argument, 4; Illicit contradictory, 205 conditional statements and, 19�21; rules of, 336�341, 377, 379, Illicit contraposition: A formal fallacy that occurs when the Identity, 432�439; rules of, 450�451 that the premises are true, 32�33, 42�45; cogency of, 45; �Some S are P,� 179�180, 187

398�404

Illicit contrary: A formal fallacy that occurs when the con-conclusion of an argument depends on the contraposition Informal fallacy: A fallacy that can be detected only through of an E or I statement, 199�200 analysis of the content of an argument, 108; avoiding, 160� 164; detecting in ordinary language, 159�161; generally, clusion of an argument depends on an incorrect applica-248. See also Fallacies (of various kinds) tion of the contrary relation, 205 Instantial letter: The letter (a variable or constant) intro-Illicit conversion: A formal fallacy that occurs when the con-duced by universal instantiation or existential instantiation, clusion of an argument depends on the conversion of an A 399�400 or O statement, 196, 200 Instantiation, 400. See also Existential instantiation; Universal Illicit major: A formal fallacy that occurs when the major instantiation term in a categorical syllogism is distributed in the conclu-Integrity, 544, 552�556 sion but not in the premise, 249 Intensional definition: A definition that assigns a meaning Illicit minor: A formal fallacy that occurs when the minor to a word by indicating the qualities or attributes that the term in a categorical syllogism is distributed in the conclu-word connotes, 94�96 sion but not in the premise, 249 Intensional meaning (intension): The qualities or attributes Illicit subalternation: A formal fallacy that occurs when the that a term connotes, 80�81, 94 conclusion of an argument depends on an incorrect appli-Intent, role of, in fallacies, 161 cation of the subalternation relation, 205 Internal coherence: The extent to which the ideas or terms Illicit subcontrary: A formal fallacy that occurs when the in a hypothesis are rationally interconnected, 536�537, conclusion of an argument depends on an incorrect appli-544, 549 cation of the subcontrary relation, 205

Glossary/Index 621

Invalid argument forms, 53, 321, 323, 418 Logicians (biographies): Abelard, 87; Ambrose, 177; Aristo-Invalid deductive argument: A deductive argument in which tle, 13; Boole, 185; Byron, 306; Chrysippus, 47; De Morgan,

it is possible for the conclusion to be false given that the 315; Frege, 378; G�del, 435; Kripke, 256; Leibniz, 282; premises are true, 40, 46 Marcus, 30; Mill, 471; Ockham, 129; Peirce, 535; Quine,

Invalidity, proving, 52�56; proving in predicate logic, 418�421 358; Russell, 409; Venn, 240; Whitehead, 409

Johnson, Lyndon, 112 same general subject and that lack an inferential relation-ship, 15 Jevons, William Stanley, 5 Loosely associated statements: Statements that are about the

Joint method of agreement and difference: A method for identifying a causal connection between an effect and a Ludwig of Bavaria, 129

single factor that is present in two or more occurrences in Lyons, Eugene, 519n which the effect is present and absent from two or more McCarthy, Joseph, 113 occurrences in which the effect is absent, but never present Maharishi Mahesh Yogi, 555 when the effect is absent nor absent when the effect is pres- Main operator: The operator (connective) in a compound ent, 469, 473�475 statement that has as its scope everything else in the state-

Kant, Immanuel, 13, 527 ment, 277, 290, 299�300 Kepler, Johannes, 536 Major premise: In a categorical syllogism, the premise that King, William, 306 contains the major term, 232 Kripke, Saul, 256 Major term: In a standard-form categorical syllogism, the Kuhn, Thomas, 554 predicate of the conclusion, 232 Lakatos, Imre, 548 Marcus, Jules Alexander, 30 Lalande, J. J., 534 Marcus, Ruth Barcan, 30 Laplace, Pierre, 530 Margin of Error, 509

Leading question, 144 functional biconditional, 277, 293; comparison with ordi-nary language, 291; truth-functional definition of, 290; (2) a Lazerowitz, Morris, 177 Material equivalence: (1) The relation expressed by a truth-

Legal reasoning, 450�453 valid rule of inference that allows an equivalence statement Leavitt, Henrietta Swan, 481

Leibniz, Gottfried Wilhelm, 5, 185, 282 statement, 366 to be replaced by a conjunctive statement or a disjunctive Leikind, Bernard J., 553 Material implication: (1) The relation expressed by a truth-Letter names of categorical propositions, 179�182 functional conditional, 277, 279; (2) a valid rule of inference Level of confidence, 509 that allows an implication sign to be replaced by a disjunc-Leverrier, U. J. J., 529, 530, 534, 537 tion sign if and only if the antecedent is negated, 366; com-Levitation, 555 parison with ordinary language, 292�293; truth-functional Lexical definition: A definition intended to report the way a definition of, 289

word is actually used in a language, 86, 89, 93, 96; criteria Maxwell, James Clerk, 537�538 for, 101�103 Mean: The arithmetical average, 511�516

Liesburg, Margaret, 378 Meaning, 70�73, 80�81; cognitive, 71�73; emotive, 71�73; Literary Digest poll, 508 extensional, 80�81, 92�94; intensional, 80�81 93; varieties Logic: The science that evaluates arguments, 1; history of, of, 70�74

4�6 Median: The middle point when data are arranged in ascend-

Logical operators, definitions of, 287�290 ing order, 511�512, 516 Logically equivalent statements: (1) Statements that neces- Mendeleev, Dmitri, 537

sarily have the same truth value, 195, 300, 355, 356; consis- Mental carelessness, 161, 163 tency and, 301; (2) statements having the same truth value Mention of a word, 80 on each line under their main operators, 300�301 Method of agreement: A method for identifying a causal

Logically false statement: A statement that is necessarily connection between an effect and a single factor that is

false, a self-contradictory statement, 299�300 present in several occurrences in which the effect is present,

Logically true statement: A statement that is necessarily 469�470, 479

true; a tautology, 299�300; proving, 300, 387 Method of concomitant variation: A method for identifying

Logically undetermined truth value: A condition that exists a causal connection between two conditions by matching

when a certain statement is not necessarily either true or variations in one condition with variations in another, 469, false, given the truth value of some related statement, 188, 476�477, 479�481 204�205, 214�215

622 Glossary/Index

Method of difference: A method for identifying a causal con- inference, 336�337; in predicate logic, 398; in propositional

nection between an effect and a single factor that is present logic, 336�388 in an occurrence in which the effect is present and absent Naturalistic explanations, 547 from an occurrence in which the effect is absent, 470�473 Necessary and sufficient condition, 252, 279; causality and,

Method of residues: A method of identifying a causal con- 467�468

nection by subtracting already-known strands of causal Necessary condition: The condition represented by the con-connection from a compound causal connection, 475�476 sequent in a conditional statement, 20�21, 279; causality

Michael of Cesena, 129 and, 467�470, 474 Middle term: In a standard-form categorical syllogism, the Needham, John, 533�534, 536

term that occurs only in the premises, 232; of a sorites, 268 Negation: A statement having a tilde as its main operator,

Milbanke, Annabella, 306 276�277; truth-functional definition of, 288 Mill, James, 471 Negation rule: A rule for computing the probability of an Mill, John Stuart, 5, 88, 469, 471 event from the probability of the event not happening, Mill�s methods and science, 477�481 498�501 Mill�s methods of induction, 240 Negative definitions, 102 Minor premise: In a categorical syllogism, the premise that Negative proposition/statement: A proposition/statement

contains the minor term, 232 that denies class membership, 179

Minor term: In a standard-form categorical syllogism, the Neptune, discovery of, 530�531, 534, 536�537

subject of the conclusion, 232 Neurolinguistic programming, 553

Missing the point: An informal fallacy that occurs when the New England Journal of Medicine, 559

premise of an argument entails one particular conclusion Newton, Isaac, 532, 536�538, 546 but a completely different conclusion is actually drawn, �No . . . except,� 222�224, 432�433

Mob mentality, 112�114 �None but,� 222, 223; in predicate logic, 394 Modal logic: A kind of logic that deals with concepts such as �None except,� 222�224 possibility, necessity, belief, and doubt, 4�6, 256 Nonstandard quantifiers, 221�222 Mode: The value that occurs with the greatest frequency in a Nonstandard verbs, translation of, 219 set of data, 511�512, 516 Normal probability distribution: A distribution of random Modern square of opposition: A diagram that illustrates the Mnemonic device, for distribution, 181; for sufficient condi- Non sequitur, 107 tions, necessary conditions, 279 Nonarguments, typical kinds of, 14 118�119, 162, 174, 519 Non causa pro causa, 131

Modus ponens: necessary relations that prevail between the four kinds of phenomena that has the shape of a bell, 479, 515 �Not both,� 280�281 standard-form categorical propositions as interpreted from �Not either,� 281 the Boolean standpoint, 183�193 Numerical statement, 434�436 A valid argument form/rule of inference: �If p then q / p // q, � 32, 47, 319�321, 323, 324, 326, 337�341, O proposition: A categorical proposition having the form 347�348; in predicate logic, 399�401 �Some S are not P,� 179, 181, 184 Objectivity, 544, 548�552 Modus tollens: A valid argument form/rule of inference: �If p then q / not q // not p, � 326, 337�339, 341, 349, 368, 528 Obscure definitions, 102�103 Monadic predicate: A predicate used to assign an attribute to Obverse, 196�197, 200

individual things, 421, 423�425 Obversion: An operation that consists of changing the qual-

Monsanto, 559 ity of a standard-form categorical proposition and replacing Mood: An attribute of a categorical syllogism that specifies the predicate term with its term complement, 194�200, 207,

the kind of statements (A, E, I, O) that make it up, 231�235 215; to reduce the number of terms in a syllogism, 255�257

Moore, G. E., 409 Ockham, William of, 5, 129, 547 Moral reasoning, 447�452 Ockham�s razor, 129, 547, 560 Multiple conclusion, 59�60 Odds of an event happening, 491 Mutually exclusive events, 496, 497 Oerstead, Hans Christian, 481 Names, 80, 82; existential, 402, 427 �Only,� 222, 223; in predicate logic, 432, 433 Narrow definitions, 101�102 �Only if,� 222, 224, 278�279 Natural deduction: A method for deriving the conclusion of Operational definition: A definition that assigns a mean-

a valid argument in which each step is justified by a rule of ing to a word by specifying experimental procedures that

Glossary/Index 623

determine whether or not the word applies to a certain Predicate symbol: An uppercase letter used to translate a thing, 95�97, 101 predicate, 391, 493

Operators: Symbols used to connect simple propositions in Predicate term: In a standard-form categorical proposition,

propositional logic, 275�277; truth-functional definitions the term that comes immediately after the copula, 175�177 of, 288�290 Predicate variable, 402

Opinion: A kind of nonargument composed of statements Predication. See Collective predication; Distributive

that express the personal conviction of a speaker or writer predication without giving any evidence in support of that conviction, 15 Prediction: An inductive argument that proceeds from

Ostensive definition. See Demonstrative (ostensive) definition knowledge of some event in the relative past to a claim Overlapping quantifiers: Quantifiers that lie within the about some other event in the relative future, 32

scope of one another, 424, 426�427 Premise: A statement in an argument that sets forth evidence,

Oversimplified cause, 130�131 1�6; exclusive, 250; inconsistent, 300�301, 306, 349 Parameter: A phrase that, when introduced into a statement, Premise indicator: A word that provides a clue to identifying

affects the form but not the meaning, 219 a premise, 3

Pareidolia, 550, 560 Presumption, fallacies of. See Fallacies of presumption Parentheses, 280�281 Presuppositions, 162�163, 174 Particular proposition/statement: A proposition/statement Primary analogue, 448�449, 451

that makes a claim about one or more (but not all) mem- Principle of indifference: In the classical theory of prob-bers of a class, 34, 179, 184, 189, 206; in a restricted uni- ability, the principle that the various possible outcomes are verse, 426�427; in predicate logic, 391, 393�396 equally probable, 492

Pascal, Blaise, 491, 532 Probability, 490�501; classical theory of, 491�492, 494, 500; Pasteur, Louis, 529, 533�534, 536, 537 relative frequency theory of, 491�494, 500; subjectivist Peirce, Benjamin, 535 theory of, 491, 493�494, 500 Peirce, Charles Sanders, 5, 185, 535 Probability calculus: A set of rules for computing the prob-

Perceptual set, 550, 560 events, 494�501 Probability of a necessary event, 493 Percentages, 518�520 ability of compound events from the probabilities of simple Perier, F., 532

Persuasive definition: A definition intended to engender a Probability of an impossible event, 492

favorable or unfavorable attitude toward what is denoted by Progress in science, 548 the definiendum, 88�89, 93, 96 Pronouns, translation of, 223

Peter of Spain, 5 Proper names, 80, 82; intension of, 80�82 Petitio principii. See Begging the question Proposition: The information content of a statement, 4; Philip of Macedonia, 13 exceptive, 223, 224, 259; exclusive, 222�223, 250, 394. See

Piece of advice: A form of expression that makes a recommen- 275 Prospective study, 480 dation about some future decision or course of conduct, 15 Proving invalidity, 52�56; in predicate logic, 418�421 Placebo effect, 550 Proving logical truths, 387 Plato, 13, 84, 435, 527, 543, 544, 550 Proving the traditional square of opposition, 203�207 Pope John XXII, 129 Psychological factors affecting a sample, 510 Pictogram: A diagram that compares two situations through Propositional logic: A kind of logic in which the funda-mental components are whole statements or propositions, drawings that differ in either size or number, 516�518 Philo of Megara, 47 also Categorical proposition Post, Emil, 5

Post hoc ergo propter hoc, Ptolemy, 534, 536 130, 545

Precising definition: A definition intended to reduce the Putnam, Hillary, 177 vagueness of a word, 86�88, 93 Quality: The attribute of a categorical proposition by which it Predicate: An expression of the form �is a bird,� �is a house,� is either affirmative or negative, 179�182 and �are fish,� 391; distinguished from predicate term, 176� Quantifier: In standard-form categorical propositions, the 177; monadic, 427�428; relational, 427�428 Popper, Karl, 546 inference: �If p then q / If q then r // If p then r,� 319�320, 337, 339�341 Postman, Leo J., 551 Pure hypothetical syllogism: A valid argument form/rule of

Predicate logic: A kind of logic that combines the symbolism words �all,� �no,� and �some,� 176�177; existential, 393�396; negated, 404; nonstandard 221�222; overlapping, 423�428; of propositional logic with symbols used to translate predi-unexpressed, 221; universal, 392�396 cates, 390�439

624 Glossary/Index

Quantifier negation rule: A rule of inference that allows one 347�348; for relational predicates and overlapping quanti-

kind of quantifier to be replaced by another, provided that fiers, 423�428; in predicate logic, 398�404 certain negation signs are deleted or introduced, 408�410, Rule of replacement: A rule consisting of two logically 414; with overlapping quantifiers, 426�427 equivalent statement forms by means of which the conclu-

Quantity: The attribute of a categorical proposition by which sion of an argument is derived from the premises, 355�359,

it is either universal or particular, 179�182 366�369

Quine, Willard Van Orman, 358, 409 Rules for categorical syllogisms, 248�253 Radium, discovery of, 529�530, 536 Rules for sorites, 266�269 Randi, James, 552 Russell, Bertrand, 5, 13, 177, 315, 358, 378, 409, 535 Random sample: A sample in which every member of the Russell�s paradox, 378

population has an equal chance of being selected, 507�509 Samples, 507�510

Range: In statistics, the difference between the largest and Sampling error: The difference between the relative fre-

smallest values in a set of data, 512�514 quency with which some characteristic occurs in a sample

Rawls, John, 177 and the relative frequency with which the same characteris-Red herring: A fallacy that occurs when the arguer diverts tic occurs in the population, 508�510

the attention of the reader or listener by addressing extrane- Scatter diagram, 480 ous issues and finishes by presuming that some conclusion Science and superstition, 543�560 has been established, 120, 160 Scientific arguments, 33

Redi, Francesco, 532 Scientific hypotheses, 546 Reducing the number of terms in a categorical syllogism, Scientific progress, 548�549

255�257 Scientific reasoning, 526�538

Reference, 81 Secondary analogue, 448�449, 457 Regression line, 480 Self-contradictory statement: A statement that is necessar-Relation, 423�428 ily false, a logically false statement, 299�300; and inconsis-Relational predicate: A predicate that expresses a connection tency, 301

between or among two or more individuals, 423�428 Seneca, 47

Relative frequency theory of probability: The theory Sense, 81 according to which probabilities are computed by dividing Shannon, Claude, 185 the number of observed favorable events by the number of

Relevance, fallacies of. See Fallacies of relevance Sidgwick, Henry, 240 Replacement, axiom of, 355; rules of, 355�359, 366�369 Simple identity statements, 432�433 Replicability, 545, 560 observed events, 492�493 Shirt collar model, 233 Sherwood, William of, 5 Report: A kind of nonargument consisting of one or more Simple noninferential passages, 15�17

Restricted conjunction rule: In probability theory, a rule p and q // p,� 347�350, 358 for computing the probability of two independent events �Since,� 14 occurring together, 495�498, 500 Singer, Barry F., 549 event, 15 other statement as a component, 276 Simplification: A valid rule of inference, � statements that convey information about some topic or Simple statement: A statement that does not contain any Restricted disjunction rule: In probability theory, a rule for

computing the probability of either of two mutually exclu- Singleton, Donald, 553 sive events, 496�499 Singular proposition/statement: A proposition/statement

Reuben, Scott, 557 place, thing, or time, 219�220; in predicate logic, 391 Size of sample, 507, 509 Retrospective study, 479 that makes an assertion about a specifically named person, Rice, N. L., 133

Rule of implication: A rule consisting of a basic argument chain reaction will actually take place, 132�133, 162 form by means of which the conclusion of an argument is Smoke-and-mirrors science, 560 derived from the premises, 336�341, 347�350 �Some,� 54�55, 176�177, 184n Rule of inference: A rule by means of which the conclusion Sorites: A chain of categorical syllogisms in which the of an argument is derived from the premises, 336, 338, Root, Elihu, 113 conclusion of an argument rests on an alleged chain reac-tion, and there is not sufficient reason to think that the Robbins, Tony, 553 Slippery slope: An informal fallacy that occurs when the

intermediate conclusions have been left out, 266�269;

Glossary/Index 625

Aristotelian, 267n; Goclenian, 267n; rules for, 267�268; Straw man: A fallacy that occurs when the arguer misinterprets standard form of, 275; Venn diagrams for, 268 an opponent�s position for the purpose of more easily attack-

Sound argument: A deductive argument that is valid and has ing it, demolishes the misinterpreted argument, and then

all true premises, 42, 207 proceeds to conclude that the original argument has been

Spallanzani, Lazzaro, 533 demolished, 117�118, 120, 160�162 Species, 95 Striking predictions, 560

Spinoza, Benedict, 537 which it is improbable that the conclusion be false given that the premises are true 42�43 Specific difference, 95 Strong inductive argument: An inductive argument in

Square of opposition. Study, 471 Spontaneous generation, 532�534

Standard-form categorical syllogism: A categorical syllo-�Some S are P,� �Some S are not P,� 176 Subject, 175�177 Subject term: In a standard-form categorical proposition, the term that comes immediately after the quantifier, 175�177 gism in which all three statements are standard-form cat-Subjectivist theory of probability: The theory according to egorical propositions, the two occurrences of each term are which probabilities are computed from the odds that people identical, each term is used in the same sense throughout would accept on a bet, 493, 535 the argument, and the major premise is listed first, the Subjunctive biconditionals, 293 minor premise second, and the conclusion last, 232 Subjunctive conditionals, 293 Standard-form sorites: A sorites in which each of the com-ponent propositions is in standard form, each term occurs Substitution instance: An argument or statement that has twice, the predicate of the conclusion is in the first premise, the same form as a given argument form or statement and each successive premise has a term in common with form; of an argument form, 52�54, 318�326; of a statement the preceding one, 267 form, 287 Sufficient and necessary condition, 279; causality and, 468� Statement: (1) A sentence that is either true or false, 2, 44; (2) in predicate logic, an expression involving bound vari-Standard deviation: In statistics, a measure of how far the sarily implies a false A or E statement, respectively, 204, data vary or deviate from the mean value the square root of 206, 215 the variance, 513�514 Subcontrary relation: The relation that exists between two Standard-form categorical proposition: A proposition that statements that are necessarily not both false, 204, 205, 214 has one of the following forms: �All S are P,� �No S are P,� Stalin, Joseph, 519 E statement necessarily implies a true I or O statement, respectively, and by which a false I or O statement neces-ditional square of opposition See Modern square of opposition; Tra- Subalternation relation: The relation by which a true A or 469, 473�475

ables or constants throughout, 391�396, 402. See also Com- Sufficient condition: The condition represented by the ante-pound statement; General statement; Numerical statement; cedent in a conditional statement, 21, 279; causality and, Particular statement; Simple statement; Singular proposi- 468, 470�477 tion; Superlative statement; Universal statement Superfluous distribution rule, 251, 268

Statement form: An arrangement of statement variables and Superlative statement, 434

operators such that the uniform substitution of statements Superstition, 543�560 in place of the variables results in a statement, 287 Suppressed evidence: A fallacy that occurs when the arguer

Statement function: In predicate logic, the expression that ignores relevant evidence that outweighs the presented evi-

remains when a quantifier is removed from a statement, dence and entails a very different conclusion, 141, 145�147, 393, 400, 402 161�162, 519

Statement of belief: A kind of nonargument composed of Syllogism: A deductive argument consisting of two premises

statements that express the personal conviction of a speaker and one conclusion, 32, 231. See also Categorical syllogism; or writer without giving any evidence in support of that con- Disjunctive syllogism; Hypothetical syllogism; Pure hypo-

Statement variable: A lowercase letter, such as p or q, that Syllogistic logic: The logic that deals with categorical propo-can represent any statement, 287 viction, 15 thetical syllogism

sitions and categorical syllogisms, 4; predicate logic and,

Statistical reasoning, 506�520 391, 394 Stellar parallax, 554�555 Synonymous definition: A definition in which the definiens Stipulative definition: A definition that assigns a meaning to is a single word that connotes the same attributes as the

a word for the first time, 85�86, 88, 93, 96 definiendum, 94, 96, 101

Stipulative use of a word, 85�86 Tautologous conclusion, 305 Stoicism, 47 Tautologous statement: A statement that is necessarily true a

logically true statement, 299, 300, 305

626 Glossary/Index

Tautology: (1) A tautologous statement, 299, 305, 387; (2) A Tu quoque: A variety of the argument-against-the-person fal-

rule of inference that eliminates redundancy in conjunc- lacy that occurs when an arguer shifts the burden of guilt tions and disjunctions, 366�369, 387; with relational predi- onto a second arguer for the purpose of discrediting his or cates, 427 her argument, 113�115

Taylor, Harriet, 471 Uncogent argument: An inductive argument that is weak, Term: A word or group of words that can serve as the subject has one or more false premises, fails to meet the total evi-

of a statement, 80�82. See also Predicate term; Subject term dence requirement, or any combination of these, 45

Term complement: The word or group of words that denotes Unconditionally valid: Valid from the Boolean standpoint;

the class complement, 196�198 for immediate inferences, 188, 207; for syllogistic forms,

�The only,� 233, 432�433 See Logically undetermined truth value Theophrastus, 47 Undistributed middle: A formal fallacy that occurs when the Theoretical definition: A definition that assigns a meaning middle term in a categorical syllogism is undistributed in Thagard, Paul, 548 ally existing things, 188 Undetermined truth value. Terms without nouns, translation of, 219 233�234; valid regardless of whether the terms denote actu-

acterization to the entities that the term denotes, 88, 93, both premises, 249 to a word by suggesting a theory that gives a certain char-

Theoretical hypotheses: Hypotheses that concern how Uniformity of nature, 45�43 Universal generalization: A rule of inference that introduces 96, 97 Unexpressed quantifiers, 221

�Thus,� 14, 17 universal quantifiers, 398�403; invalid applications of, 404, 414�415, 418�421; restrictions on, 400�403, 414�415, something should be conceptualized, 534, 536

Torricelli, Evangelista, 529, 531�532, 536�538 426�427 Tilde, 276�277, 280; truth-functional definition of, 288 Total evidence requirement, 44�45 Universal instantiation: A valid rule of inference that

Traditional square of opposition: A diagram that illus- invalid applications of, 401, 402, 427 removes universal quantifiers, 398�402, 414, 427�428;

Traditional standpoint. trates the necessary relations that prevail between the four Universal quantifier: In predicate logic, the quantifier used kinds of standard-form categorical propositions as inter-to translate universal statements, 392, 394�396 preted from the Aristotelian standpoint, 203�207; proof of, Universal proposition/statement: A proposition/statement 214�215 that makes an assertion about every member of its subject See Aristotelian standpoint class, 179, 183�184, 186, 191�192; in a restricted universe, Translating ordinary language statements into categorical 419�421; in predicate logic, 390�396 form, 219�224 �Unless,� 222, 278, 291�292, 395 Translating relational statements, 424�426 Unsound argument: A deductive argument that is invalid, Translating statements in predicate logic, 394�395 has one or more false premises, or both, 42 Translating statements in propositional logic, 275�281 Use of a word, 80 Translating statements into categorical form, 219�224 Vacuously false: Falsity that results merely from the fact that Transposition: A valid rule of inference that allows the ante-the subject class is empty, 188, 206n cedent and consequent of a conditional statement to switch Vacuously true: Truth that results merely from the fact that places if and only if both are negated, 366 the subject class is empty, 188, 206n Triple bar, 276, 279�281; comparison of with ordinary lan-Vague definitions, 102�103 guage, 291�293; truth-functional definition of, 289�291 Vague expression: An expression that allows for border-Truth, and strength, 42�45; and validity, 40�42; vacuous, 188. line cases in which it is impossible to tell if the expression See also Logically true statement applies or does not apply, 73, 86, 93 Truth function: A compound proposition whose truth value Vague hypotheses, 560 is completely determined by the truth values of its compo-Valid argument forms, 52, 318�319, 321. See also Rules of nents, 287�293 inference; Valid syllogistic forms Truth table: An arrangement of truth values that shows in Valid deductive argument: An argument in which it is every possible case how the truth value of a compound impossible for the conclusion to be false given that the proposition is determined by the truth values of its simple premises are true, 40, 43 components, 288�289; for arguments, 304�305; for propo-Valid syllogistic forms, 233 sitions, 297�301. See also Indirect truth tables Validity, 40�42; form of an argument and, 52, 318�326 Truth value: The attribute by which a statement is either Value claim: A claim that something is good, bad, right, or true or false, 2; of compound statements, 287�293; logically wrong, 71 undetermined, 188, 196, 199, 204�205, 214�215

Glossary/Index 627

Variable, bound, 393; free, 392; individual, 392; predicate, Weak analogy: An informal fallacy that occurs when the

402; statement, 287 conclusion of an argument depends on an analogy (or simi-

Variance: In statistics, a measure of how far the data vary larity) that is not strong enough to support the conclusion,

from the mean value, 513�515 133�134, 162�163, 174

Venn, John, 5, 184, 188, 195�193, 231, 240 Weak induction, fallacies of. See Fallacies of weak induction Venn, John Archibald, 240 Weak inductive argument: An inductive argument in which Venn diagram: A diagram consisting of two or more circles the conclusion does not follow probably from the premises

used to represent the information content of categorical even though it is claimed to do so, 42 propositions, 186�187, 189�190; and the Aristotelian (tra- Wedge, 276�278, 281; comparison with ordinary language, ditional) standpoint, 213�217; for categorical syllogisms, 291�293; truth-functional definition of, 288�290 238�245; for particular statements in predicate logic, 391; Well-formed formula (WFF): A syntactically correct for proving the traditional square of opposition, 213�217; arrangement of symbols, 281 for sorites, 267; for testing immediate inferences, 188� Whitehead, Alfred North, 5, 13, 177, 315, 358, 409, 535 191, 215�217; for universal statements in predicate logic, William of Champeaux, 87 391�396 William of Ockham, 5, 129, 547

Verbal disputes, 74 Wittgenstein, Ludwig, 5, 70, 177, 409

Viviani, Vincenzo, 531 World Health Organization, 559 Wade, Evelyn, 409 Worldview, 161�163 Vertical pattern, 58�59 Wordsworth, William, 433, 471

Warning: A form of expression intended to put someone on Zeno of Citium, 47 guard against a dangerous or detrimental situation, 15

628 Glossary/Index

A Concise Introduction to Logic, Fourteenth Edition

Patrick J. Hurley

Guide to Important Rules & Argument Forms

Traditional Square of Opposition Logically Equivalent Statement Forms

A Contrary Given statementE Conversion Converse

T T E: No S are P. No P are S.

Subalternation Con y I: Some S are P. Some P are S. tor tra dic dic Obversion Given statementObverse Subalternation tra tor Con y A: All S are P . No S are non-P.

F F SE: No are P. All S are non-P.

I: Some S are P. Some S are not non-P.

IO Subcontrary SO: Some are not P. Some S are non-P.

Contraposition Given statementContrapositive

A: All S are P. All non-P are non-S.

Valid Syllogistic Forms O: Some S are not P. Some non-P are not non-S.

Unconditionally Valid Forms

Figure 1Figure 2Figure 3Figure 4

AAAEAEIAIAEEEAEAEEAIIIAI

AIIEIOOAOEIOEIOAOOEIO

Conditionally Valid Forms

Figure 1Figure 2Figure 3Figure 4Required condition

AAIAEOAEOSexistEAOEAO

AAIEAOMexistEAO

AAI . Pexist

Rules for Categorical Syllogisms

, by Patrick Hurley

Rule 1:The middle term must be distributed at least once.

Fallacy: Undistributed middle

Rule 2:If a term is distributed in the conclusion, then it must be distributed in a premise. Fallacy: Illicit major; illicit minor

Rule 3:Two negative premises are not allowed.

Fallacy: Exclusive premises A Concise Introduction to LogicRule 4:A negative premise requires a negative conclusion, and a negative conclusion requires a negative premise. Fallacy: Drawing an affirmative conclusion from a negative premise; drawing a negative conclusion from affirmative

premises

Rule 5:If both premises are universal, the conclusion cannot be particular. Fallacy: Existential fallacy

NOTE: If only Rule 5 is broken, the syllogism is valid from the Aristotelian standpoint if the

Rules of Implication Rules of Replacement 1. Modus ponens (MP) 2. Modus tollens (MT) 9. De Morgan�s rule (DM) ∼ ∼( p ● q) :: (∼ ∼p �� ∼ ∼q)

p ⊃ qp ⊃ q ∼ ∼( p �� q) :: (∼ ∼p ● ∼ ∼q)

q 10. Commutativity (Com) ( p �� q) :: (q �� p) p ∼ ∼q

∼ ∼p

( p ● q) :: (q ● p)

3. Pure hypothetical syllogism (HS) 4. Disjunctive syllogism (DS)

p ⊃ qp �� q 11. Associativity (Assoc) [ p �� (q �� r)] :: [( p �� q) �� r] q ⊃ r ∼ ∼ p [ p ● (q ● r)] :: [( p ● q) ● r]

p ⊃ rq 12. Distribution (Dist) [ p ● (q �� r)] :: [( p ● q) �� ( p ● r)] 5. Constructive dilemma (CD) 6. Simplification (Simp) [ p �� (q ● r)] :: [( p �� q) ● ( p �� r)]

(p ⊃ q) ● (r ⊃ s) p ● qp 13. Double negation (DN) p:: ∼∼ ∼∼p �� r p

q �� s 14. Transposition (Trans) ( p ⊃ q) :: (∼ ∼q ⊃ ∼ ∼p)

7. Conjunction (Conj) 8. Addition (Add) 15. Material implication (Impl) ( p ⊃ q) :: (∼ ∼p �� q) pp

q p �� q 16. Material equivalence (Equiv) ( p ≡ q) :: [( p ⊃ q) ● (q ⊃ p)] p ● q ( p ≡ q) :: [( p ● q) �� (∼ ∼p ● ∼ ∼q)]

17. Exportation (Exp) [( p ● q) ⊃ r] :: [ p ⊃ (q ⊃ r)] 18. Tautology (Taut) p:: ( p �� p)

p:: ( p ● p)

Truth Tables for the Propositional Conditional Proof Operators �

�

pq ∼ ∼pp ● qp �� qp ⊃ qp ≡ q � /__ TTFTTTTp ACP

TFFFTFF � � FTTFTTF �

FqFTFFTT CP p ⊃ q

Quantifiers Indirect Proof Rules for Removing and Introducing

�

(a, b, c, . . . u, v,w are individual constants; x, y, z are individual variables) �

� /__

1. Universal instantiation (UI) (x)ℱx (x)ℱxp AIP

. ℱy ℱa �

2. Universal generalization (UG) � ℱ y ℱ a � ( x ) ℱ xnot allowed: ( x ) ℱ xq ● ∼ ∼q

, by Patrick Hurley Restrictions: (conditional (1) UG must not be used within the scope of an ∼ ∼p IP

and indirect proof) indented sequence if the instantial variable y

is free in the first line of that sequence.

(overlapping quantifiers) (2) UG must not be used if the instantial variable y Rules for the Probability Calculus

is free in any preceding line obtained by EI.

A Concise Introduction to Logic ℱ P(A and B) = P(A) � P(B) (when A and B are independent) anot allowed: ℱ y4. P ( A and B ) = P ( A ) � P ( B given A ) Restriction: The existential name a must be a new name that does not 5. P ( A or B ) = P ( A ) + P ( B ) (when A and B are mutually exclusive) appear in any previous line (including the conclusion line). 6. 3. Existential instantiation (EI) 2. P(A and not A) = 0 ( ∃ x ) ℱ x ( ∃ x ) ℱ x3. 1. P(A or not A) = 1 P(A or B) = P(A) + P(B) 2 P(A and B)

4. Existential generalization (EG) ℱa ℱy7. P(A) = 1 2 P(not A)

(∃x)ℱx (∃x)ℱx

Quantifier Negation Rule (QN) Identity Rules (Id)

Rules and Argument Forms Card for ( 1. Prem. 2.x = y :: y = x3. ℱxx ) ℱ x :: ∼ ( ∃ x ) ∼ ℱ x ( ∃ x ) ℱ x :: ∼ ( x ) ∼ ℱ x = ∼ ( x ) ℱ xxx = y Copyright � Cengage Learning. All Rights Reserved. x :: ( ∃ x ) ∼ ℱ x ∼ ( ∃ x ) ℱ x :: ( x ) ∼ ℱ x ℱ y