Introduction to Logic “This new edition is a significant improvement on an already excellent text. The virtues of the original remain including clear expositions, an intuitive proof procedure that generalizes naturally from propositional logic to more advanced logics and a wealth of problems drawn from philosophical sources. There are new chapters on the history of logic, deviant logics and the philoso- phy of logic and the accompanying LogiCola program has been improved. This is a student friendly approach to logic.” Michael Bradie, Bowling Green State University Introduction to Logic combines likely the broadest scope of any logic textbook with clear, concise writing and interesting examples and arguments. Its key features, all retained in the Second Edition, include: • Simpler ways to test arguments, including the star test for syllogisms • A wide scope of materials, suiting it for introductory or intermediate courses • Engaging examples, from everyday life and the great philosophers • Useful for self-study and preparation for standardized tests, like the LSAT • A reasonable price (a third the cost of some competitors) • Exercises that correspond to the free LogiCola instructional program This Second Edition also: • Arranges chapters in a more natural way, going from easier to more difficult material • Adds new chapters on history of logic, deviant logic, and philosophy of logic • Refines many explanations and expands several sections (such as informal fallacies and relational translations) • Includes a fuller index and a new appendix on suggested further readings • Updates LogiCola, which is now more visually attractive and easier to down- load, update, and use (install from http://www.jcu.edu/philosophy/gensler/lc or http://www.routledge.com/textbooks/9780415996518) Harry J. Gensler, S.J., is Professor of Philosophy at John Carroll University in Cleveland. Some of his other books include Gödel’s Theorem Simplified (1984), Formal Ethics (1996), Anthology of Catholic Philosophy (2005), Historical Dictionary of Logic (2006), Historical Dictionary of Ethics (2008), and Ethics: A Contemporary Introduction, Second Edition (2011). Introduction to Logic Second Edition Harry J. Gensler First published 2002 by Routledge This edition published 2010 by Routledge 270 Madison Ave, New York, NY10016 Simultaneously published in the UK by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Routledge is an imprint of the Taylor & Francis Group This edition published in the Taylor & Francis e-Library, 2010. To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk. © 2010 Harry J. Gensler All rights reserved. No part of this book may be reprinted or reproduced or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Library of Congress Cataloging-in-Publication Data Gensler, Harry J., 1945— Introduction to logic / Harry J. Gensler. – 2nd ed. p. cm. Includes index. 1. Logic. I. Title. BC71.G37 2010 160–dc22 2009039539 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. ISBN 0-203-85500-0 Master e-book ISBN ISBN: 978–0–415–99650–1, ISBN 10: 0–415–99650–3 (hback) ISBN: 978–0–415–99651–8, ISBN 10: 0–415–99651–1 (pback) ISBN: 978–0–203–85500–3, ISBN 10: 0–203–85500–0 (ebook) Contents Preface ix Chapter 1: Introduction 1 1.1 Logic 1 1.3 Sound arguments 4 1.2 Valid arguments 2 1.4 The plan of this book 6 PART ONE SYLLOGISTIC, INFORMAL, AND INDUCTIVE LOGIC Chapter 2: Syllogistic Logic 7 2.1 Easier translations 7 2.5 Deriving conclusions 20 2.2 The star test 9 2.6 Venn diagrams 24 2.3 English arguments 13 2.7 Idiomatic arguments 28 2.4 Harder translations 17 2.8 The Aristotelian view 32 Chapter 3: Meaning and Definitions 34 3.1 Uses of language 34 3.5 Making distinctions 47 3.2 Lexical definitions 36 3.6 Analytic and synthetic 49 3.3 Stipulative definitions 41 3.7 A priori and a posteriori 51 3.4 Explaining meaning 43 Chapter 4: Fallacies and Argumentation 55 4.1 Good arguments 55 4.4 Constructing arguments 74 4.2 Informal fallacies 59 4.5 Analyzing arguments 77 4.3 Inconsistency 69 Chapter 5: Inductive Reasoning 80 5.1 The statistical syllogism 80 5.6 Analogy and other minds 98 5.2 Probability calculations 82 5.7 Mill’s methods 99 5.3 Philosophical questions 87 5.8 Scientific laws 104 5.4 Reasoning from a sample 92 5.9 Best-explanation reasoning 111 5.5 Analogical reasoning 95 5.10 Problems with induction 112 vi INTRODUCTION TO LOGIC PART TWO CLASSICAL SYMBOLIC LOGIC Chapter 6: Basic Propositional Logic 118 6.1 Easier translations 118 6.8 Harder translations 138 6.2 Simple truth tables 121 6.9 Idiomatic arguments 140 6.3 Truth evaluations 124 6.10 S-rules 143 6.4 Unknown evaluations 125 6.11 I-rules 146 6.5 Complex truth tables 126 6.12 Mixing S- and I-rules 149 6.6 The truth-table test 129 6.13 Extended inferences 150 6.7 The truth-assignment test 133 6.14 Logic and computers 152 Chapter 7: Propositional Proofs 153 7.1 Easier proofs 153 7.4 Harder refutations 175 7.2 Easier refutations 160 7.5 Other proof methods 178 7.3 Harder proofs 167 Chapter 8: Basic Quantificational Logic 182 8.1 Easier translations 182 8.4 Harder translations 197 8.2 Easier proofs 187 8.5 Harder proofs 199 8.3 Easier refutations 192 Chapter 9: Relations and Identity 205 9.1 Identity translations 205 9.4 Harder relations 214 9.2 Identity proofs 207 9.5 Relational proofs 218 9.3 Easier relations 212 9.6 Definite descriptions 226 CONTENTS vii PART THREE ADVANCED SYMBOLIC SYSTEMS Chapter 10: Basic Modal Logic 228 10.1 Translations 228 10.3 Refutations 240 10.2 Proofs 232 Chapter 11: Further Modal Systems 248 11.1 Galactic travel 248 11.3 Quantified proofs 256 11.2 Quantified translations 253 11.4 A sophisticated system 260 Chapter 12: Deontic and Imperative Logic 267 12.1 Imperative translations 267 12.3 Deontic translations 276 12.2 Imperative proofs 269 12.4 Deontic proofs 279 Chapter 13: Belief Logic 290 13.1 Belief translations 290 13.5 Rationality translations 304 13.2 Belief proofs 291 13.6 Rationality proofs 306 13.3 Believing and willing 299 13.7 A sophisticated system 310 13.4 Willing proofs 301 Chapter 14: A Formalized Ethical Theory 313 14.1 Practical rationality 313 14.4 Starting the GR proof 322 14.2 Consistency 315 14.5 GR logical machinery 326 14.3 The golden rule 317 14.6 The symbolic GR proof 333 viii INTRO DU CTION TO LOGIC PART FOUR FURTHER VISTAS Chapter 15: Metalogic 336 15.1 Metalogical questions 336 15.4 Completeness 340 15.2 Symbols 336 15.5 An axiomatic system 343 15.3 Soundness 338 15.6 Gödel’s theorem 344 Chapter 16: History of Logic 351 16.1 Ancient logic 351 16.4 Frege and Russell 358 16.2 Medieval logic 354 16.5 After Principia 360 16.3 Enlightenment logic 357 Chapter 17: Deviant Logics 363 17.1 Many-valued logic 363 17.3 Intuitionist logic 368 17.2 Paraconsistent logic 365 17.4 Relevance logic 369 Chapter 18: Philosophy of Logic 373 18.1 Abstract entities 373 18.4 Truth and paradoxes 379 18.2 Metaphysical structures 374 18.5 The scope of logic 382 18.3 The basis for logical laws 376 Appendix: For Further Reading 383 Answers to Selected Problems 384 Chapter 2 384 Chapter 07 393 Chapter 12 408 Chapter 3 385 Chapter 08 397 Chapter 13 410 Chapter 4 388 Chapter 09 400 Chapter 14 414 Chapter 5 389 Chapter 10 403 Chapter 6 391 Chapter 11 406 Index 415 Preface This is a comprehensive Introduction to Logic. It covers: • syllogisms; • informal aspects of reasoning (like meaning and fallacies); • inductive reasoning; • propositional and quantificational logic; • modal, deontic, and belief logic; • the formalization of an ethical theory about the golden rule; and • metalogic, history of logic, deviant logic, and philosophy of logic. Because of its broad scope, this book can be used for basic logic courses (where teachers can choose from a variety of topics) or more advanced ones (including graduate courses). The teacher manual and the end of Chapter 1 both talk about which chapters are suitable for which type of course. The first Routledge edition came out in 2002. Key features included: (a) clear, direct, concise writing; (b) interesting examples and arguments, often from everyday life or great philosophers; (c) simpler ways to test arguments, including the star test for syllogisms and an easier way to do proofs and refutations; (d) wide scope of materials (likely the widest of any logic text); (e) suitability for self-study and preparation for tests like the LSAT; (f) reasonable price (a third of the cost of some competitors); and (g) the companion LogiCola instructional program (which randomly generates problems, gives feedback on answers, provides help and explanations, and records progress). I’m happy with how the first edition has been received, often with lavish praise. I have made many improvements to this second edition. I have arranged the chapters in a more logical way; so they now go, roughly, from easier to harder material. I added new chapters on history of logic, deviant logic, and philosophy of logic; so the book is even broader in scope than before.