Table Of ContentIntroduction 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).
I n t r od u c t i on
t o L o g i c
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 INTRODUCTION 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. I beefed up informal
fallacies, added inference to the best explanation, and corrected some typos. I
overhauled three difficult sections: on relational translations, belief-logic
proofs, and completeness. I did much tweaking of explanations (for example,
see the sections on the star test, Venn diagrams, and proofs). I tweaked some
exercises. I added an appendix on suggested further readings. I added a real
index (previously there was only an index of names); so now it’s easier to
research a topic. And I added a convenient list of rules to the inside covers. I cut
two parts that got little use: the appendix on how to download and use LogiCola
(the program is now so easy to download and use that this appendix isn’t
needed) and the glossary (which just repeated definitions from the text). I tried
