The Logic Book

696 Pages·2008·9.461 MB·English

by  Merrie Bergmann, James Moor, Jack Nelson

THE
LOGIC
BOOK
Fifth Edition
MERRIE BERGMANN
Smith College, Emerita
JAMES MOOR
Dartmouth College
JACK NELSON
Arizona State University
Published by McGraw-Hill, an imprint of The McGraw-Hill Companies, Inc.,
1221 Avenue of the Americas, New York, NY 10020. Copyright © 2009, 2004, 1998,
1990, and 1980 by Merrie Bergmann, James Moor, and Jack Nelson. All rights reserved.
No part of this publication may be reproduced or distributed in any form or by any
means, or stored in a database or retrieval system, without the prior written consent
of The McGraw-Hill Companies, Inc., including, but not limited to, in any network or
other electronic storage or transmission, or broadcast for distance learning.
This book is printed on acid-free paper.
1 2 3 4 5 6 7 8 9 0 DOC/DOC 0 9 8
ISBN 978-0-07-353563-0
MHID 0-07-353563-X
Editor in Chief: Michael Ryan
Publisher: Beth Mejia
Sponsoring Editor: Mark Georgiev
Marketing Manager: Pamela S.Cooper
Production Editor: Leslie LaDow
Designer: Margarite Reynolds
Production Supervisor: Tandra Jorgensen
Media Project Manager: Thomas Brierly
Composition: 10/12 New Baskerville by Aptara, Inc.
Printing: 45# New Era Matte Plus by R.R. Donnelley & Sons
Cover: Jacopo de'Barbari, (c.1460/70–c.1516). Portrait of the mathematician Luca
Pacioli, the “father of accounting,” and an unknown young man. Museo Nazionale di
Capodimonte, Naples, Italy. © Scala/Art Resource, NY
Library of Congress Cataloging-in-Publication Data
Bergmann, Merrie.
The logic book / Merrie Bergmann, James Moor, Jack Nelson.—5th ed.
p. cm.
Includes bibliographical references and index.
ISBN–13: 978-0-07-353563-0 (alk. paper)
ISBN–10: 0-07-353563-X (alk. paper)
1. Logic, Symbolic and mathematical. 2. Predicate (Logic) I. Moor,
James, 1942-II. Nelson, Jack, 1944-III. Title.
BC135.B435 2009
160—dc22
2008020421
The Internet addresses listed in the text were accurate at the time of publication.
The inclusion of a Web site does not indicate an endorsement by the authors or
McGraw-Hill, and McGraw-Hill does not guarantee the accuracy of the information
presented at these sites.
www.mhhe.com
ABOUT THE AUTHORS
MERRIE BERGMANN received her Ph.D. in philosophy from the University of
Toronto. She is an emerita professor of computer science at Smith College. She
has published articles in formal semantics and logic, philosophy of logic, phi-
losophy of language, philosophy of humor, and computational linguistics and
is author of An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras,
and Derivation Systems. She and her husband are currently circumnavigating the
earth on their 44 sailboat.
JAMES MOOR received his Ph.D. in history and philosophy of science from
Indiana University. He is currently a professor of philosophy at Dartmouth Col-
lege. He has published articles in philosophy of science, philosophy of mind,
logic, philosophy of artificial intelligence, and computer ethics. He is author
of The Turing Test: The Elusive Standard of Artificial Intelligence, and co-editor, with
Terrell Bynum, of Cyberphilosophy: The Intersection of Computing and Philosophy and
The Digital Phoenix: How Computers Are Changing Philosophy. Moor is also editor
of Minds and Machines.
JACK NELSON received his Ph.D. in philosophy from the University of
Chicago. He is currently Interim Chair of the Philosophy Department of Ari-
zona State University and Associate Dean for Student and Academic Programs
in the College of Liberal Arts and Sciences. He has published articles in epis-
temology, identity, and the philosophy of science and is co-author, with Lynn
Hankinson Nelson, of On Quine.
ABOUT THE AUTHORS iii

CONTENTS
Preface xi
CHAPTER 1 BASIC NOTIONS OF LOGIC 1
1.1 Background 1
1.2 Why Study Logic? 6
1.3 Sentences, Truth-Values, and Arguments 7
1.4 Deductive Validity and Soundness 12
1.5 Inductive Arguments 17
1.6 Logical Consistency, Truth, Falsity, and Equivalence 19
1.7 Special Cases of Validity 24
CHAPTER 2 SENTENTIAL LOGIC: SYMBOLIZATION AND SYNTAX 28
2.1 Symbolization and Truth-Functional Connectives 28
2.2 Complex Symbolizations 50
2.3 Non-Truth-Functional Connectives 60
2.4 The Syntax of SL 67
CHAPTER 3 SENTENTIAL LOGIC: SEMANTICS 75
3.1 Truth-Value Assignments and Truth-Tables for Sentences 75
3.2 Truth-Functional Truth, Falsity, and Indeterminacy 83
CONTENTS v
3.3 Truth-Functional Equivalence 93
3.4 Truth-Functional Consistency 98
3.5 Truth-Functional Entailment and Truth-Functional
Validity 101
3.6 Truth-Functional Properties and Truth-Functional
Consistency 110
CHAPTER 4 SENTENTIAL LOGIC: TRUTH-TREES 115
4.1 The Truth-Tree Method 115
4.2 Truth-Tree Rules for Sentences Containing ‘∼’, ‘∨’,
and ‘&’ 116
4.3 Rules for Sentences Containing ‘⊃’ and ‘ ’ 130
4.4 More Complex Truth-Trees 137
4.5 Using Truth-Trees to Test for Truth-Functional Truth, Falsity,
and Indeterminacy 144
4.6 Truth-Functional Equivalence 150
4.7 Truth-Functional Entailment and Truth-Functional
Validity 153
CHAPTER 5 SENTENTIAL LOGIC: DERIVATIONS 160
5.1 The Derivation System SD 160
5.2 Basic Concepts of SD 189
5.3 Strategies for Constructing Derivations in SD 193
5.4 The Derivation System SD 228
CHAPTER 6 SENTENTIAL LOGIC: METATHEORY 240
6.1 Mathematical Induction 240
6.2 Truth-Functional Completeness 248
6.3 The Soundness of SD and SD 258
6.4 The Completeness of SD and SD 266
CHAPTER 7 PREDICATE LOGIC: SYMBOLIZATION AND SYNTAX 276
7.1 The Limitations of SL 276
7.2 Predicates, Individual Constants, and Quantity Terms of
English 279
7.3 Introduction to PL 284
7.4 Quantifiers Introduced 290
7.5 The Formal Syntax of PL 297
7.6 A-, E-, I-, and O-Sentences 308
vi CONTENTS
7.7 Symbolization Techniques 322
7.8 Multiple Quantifiers with Overlapping Scope 342
7.9 Identity, Definite Descriptions, Properties of Relations, and
Functions 359
CHAPTER 8 PREDICATE LOGIC: SEMANTICS 378
8.1 Informal Semantics for PL 378
8.2 Quantificational Truth, Falsehood, and Indeterminacy 391
8.3 Quantificational Equivalence and Consistency 398
8.4 Quantificational Entailment and Validity 403
8.5 Truth-Functional Expansions 409
8.6 Semantics for Predicate Logic with Identity and Functors 424
8.7 Formal Semantics of PL and PLE 443
CHAPTER 9 PREDICATE LOGIC: TRUTH-TREES 458
9.1 Expanding the Rules for Truth-Trees 458
9.2 Truth-Trees and Quantificational Consistency 467
9.3 Truth-Trees and Other Semantic Properties 474
9.4 Fine-Tuning the Tree Method 482
9.5 Trees for PLE 498
9.6 Fine-Tuning the Tree Method for PLE 514
CHAPTER 10 PREDICATE LOGIC: DERIVATIONS 532
10.1 The Derivation System PD 532
10.2 Using Derivations to Establish Syntactic Properties of PD 551
10.3 The Derivation System PD 583
10.4 The Derivation System PDE 588
CHAPTER 11 PREDICATE LOGIC: METATHEORY 608
11.1 Semantic Preliminaries for PL 608
11.2 Semantic Preliminaries for PLE 623
11.3 The Soundness of PD, PD , and PDE 627
11.4 The Completeness of PD, PD , and PDE 633
11.5 The Soundness of the Tree Method 650
11.6 The Completeness of the Tree Method 660
Selected Bibliography B-1
Index I-1
Index of Symbols I-7
CONTENTS vii

PREFACE
In the fifth edition of The Logic Book we retain our overall goal: to present sym-
bolic logic in an accessible yet formally rigorous manner. This involved a major
overhaul of several chapters, along with some terminological and notational
changes.
Most of the material in Chapters 5 and 10 is new or extensively rewrit-
ten. We have tried to present the derivation systems we develop in ways that make
them more transparent. We emphasize the need for and use of specific strategies
in constructing derivations, and in each chapter we explicitly list those strategies.
We have also introduced new annotations for the assumptions that begin sub-
derivations, annotations that specify the reason these assumptions are being made.
We use the notation “A / I”, for example, to indicate that an auxiliary assump-
tion has been made to introduce a Conditional Introduction subderivation. We
believe that these annotations underscore the point that auxiliary assumptions
should always be made with clear strategies in mind. We have significantly
expanded the number and variety of exercises in these chapters as well.
There are also significant changes to Chapters 4 and 9. The latter sections
of Chapter 9 have been reorganized so that systematic trees for PL are presented
prior to, and independently of, trees for PLE, and the sections motivating the rules
for PLE are less circuitous. In Chapter 4 we dispense with talk of fragments of
truth-value assignments and instead adopt the convention that a display such as
A B C D
T F T T
specifies the infinitely many truth-value assignments that each assign the spec-
ified values to ‘A’, ‘B’, ‘C’, and ‘D’. In both chapters we have also introduced
PREFACE ix
a new annotation for trees to indicate completed open branches (‘o’) as well
as closed ones (‘x’).
Chapter 8 now introduces the concept of a model, to be used there and
in subsequent chapters.
As always, we have corrected known errors and typos from previous
editions.
The Logic Book presupposes no previous training in logic, and because
it covers sentential logic through the metatheory of first-order predicate logic,
it is suitable for both introductory and intermediate courses in symbolic logic.
There is sufficient material in the text for a one- or two-semester course. There
are several sequences of chapters that would form good syllabi for courses in
symbolic logic. If an instructor has two semesters (or one semester with
advanced students), it is possible to work through the entire book.
The instructor who does not want to emphasize metatheory can sim-
ply omit Chapters 6 and 11. The chapters on truth-trees and the chapters on
derivations are independent, so it is possible to cover truth-trees but not deri-
vations, and vice versa. The chapters on truth-trees do depend on the chapters
presenting semantics; that is, Chapter 4 depends on Chapter 3, and Chapter 9
depends on Chapter 8. And although most instructors will want to cover seman-
tics before derivations, the opposite order is possible. Finally, in covering pred-
icate logic, the instructor has the option in each chapter of skipping material
on identity and functions, as well as the option of including the material on
identity but omitting that on functions.
The Logic Book includes large numbers of exercises in all chapters.
Answers to the starred exercises appear in the Instructor’s Manual; answers to
the unstarred exercises appear in the Student Solutions Manual.
SOFTWARE
Two software packages, BERTIE and TWOOTIE, are available for use with The
Logic Book. BERTIE is a program that allows students to construct derivations
online and checks those derivations for accuracy. TWOOTIE allows students to
construct truth-trees online (and checks those trees for accuracy) and also pro-
duces trees for specified sets of sentences. Both programs were written by
Austen Clarke; BERTIE is based on an earlier program by James Moor and Jack
Nelson. Both programs run in a DOS environment. Information on the soft-
ware can be found at
http://selfpace.uconn.edu/BertieTwootie/software.htm
Both software packages can also be downloaded from this site.
x PREFACE