Table Of ContentN A T U R A L
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DEDUCTIONj
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ohn M. Anderson
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Henry W. Johnstone, Jr.
The Pennsylvania State University
WADSWORTH PUBLISHING COMPANY, INC.
Belmont, California
2
© 1962 by Wadsworth Publishing Company,
Inc., Belmont, California. All rights reserved.
No part of this book may be reproduced in any
form, by mimeograph or any other means, with
out permission in writing from the publisher.
L.C. Cat. Card No.: 62-11239
Printed in the United States of America
3
PREFACE
This book is based upon Gentzen's techniques of natural deduction. The
propositional and quantificational rules on which the exposition pivots are
stated essentially as Gentzen stated them, and even the names used for them
are free translations of his names. Gentzen's techniques constitute a very
natural approach to the study of the proofs occurring in axiom systems as well
as a sound basis for the analysis of the properties of formal systems as such.
The book is divided into two main parts. In Part One various rules and
sets of rules for deduction are presented. Part Two is largely concerned with
the metatheoretical analysis of deductions based upon these rules.
Part One begins with a chapter in which a simple set of axioms (those for
simple order) is presented and attention is called to the use of logical rules in
the deduction of theorems from these axioms. In Chapter Two all the stan
dard propositional rules are given. Here one of the great merits of Gentzen·s
formulation is obvious: it permits the analysis of proof in an axiom system
without presupposing another axiom system-namely, that of the logic
itself. The power of this formulation is revealed in the early use of conditional
proof, here introduced, of course, as a primitive rule. The analysis of negation
into four rules reveals something of its nature and facilitates transition in a
later chapter to the intuitionistic logic.
Chapter Three temporarily sets aside the basic exposition of the nature
of deduction in an axiom system in order to introduce truth tables and
normal forms. This material not only serves the science student interested in
computers, but also lays the ground for later metatheoretical discussions.
Chapter Four is intended to suggest that the problem of what constitutes a
deduction is by no means closed, for alternative logical systems such as those of
three-valued logic, intuitionistic logic, and modal logics are available and may
be used.
Chapter Five is occupied with the explicit development of monadic
predicate logic. The rigorous statement of the rules of predicate logic in a
form in which they lead, in Chapter Six, to a final justification for the steps
in deductions within first order axiom systems returns the reader to the
fundamental problem of axiomatic proof. Chapter Seven introduces the
concepts of jdentity and description, thus completing a standard first course
in modern logic.
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4
Preface
Part Two is concerned with the properties of deductions in axiom systems.
In Chapter Eight the problems of metatheory are introduced on the basis of a
distinction between the traditional axiom system of geometry regarded as a
description of space and modern abstract axiom systems. This distinction
makes clear the need for tests of consistency and completeness not dependent
on the literal truth of the theorems. The use of models in such tests is de
scribed, and it is pointed out that this use establishes consistency relative to the
model. The student is thus led to see the need for metatheoretical methods
capable of establishing absolute consistency.
In Chapter Nine the concept of a formal system is developed and meta
theoretical proofs of the absolute consistency and completeness of the propo
sitional rules are presented. Techniques used here are applied in Chapter
Ten to monadic predicate logic, for which several decision procedures are also
sketched. Chapter Eleven contains a further application of metatheoretical
techniques to general predicate logic. Henkin's completeness proof is given
in this chapter.
The difficult problems and apparent limitations of the axiomatic method
are introduced through the contradictions and paradoxes of set theory.
Chapter Twelve is an elementary account of the operations on sets and the
relation of set theory to logic and to mathematics. The reader already ac
quainted with the rudiments of set theory might skip this chapter. In Chapter
Thirteen some ofth e contradictions and paradoxes of set theory are presented,
and various attempts to deal with the problems raised are discussed: Zermelo's
axiomatization, Russell's theory of types, Brouwer·s intuitionism, and
Hilbert's formalism.
This book can be used in logic courses in many ways. Used in its entirety
it serves as the text of a year's course in intermediate logic, or of an intensive
semester's course for students with mathematical background. Part One
alone constitutes the basis for a course in the ~echniques of deduction.
Many teachers giving such a course, however, will probably prefer to omit
some sections or chapters in Part One and include some of the meta theoretical
material of Part Two. Perhaps the clearest way to indicate what selections
are possible is to say that Chapter Six presupposes Chapter Five, which in
turn presupposes Chapter Two; and that chapter Eleven presupposes
Chapter Ten, which in turn presupposes Chapter Nine, but all the other
chapters can be read more or less independently by a student having some
familiarity with logic since only fundamental concepts are presupposed in
them.
The authors wish to acknowledge with gratitude the help of a number of
anonymous publishers' readers whose criticism has been most constructive.
Among those critics that they can name they would like to thank Professor
Haskell B. Curry whose work on Gentzen 's systems is well known; Professor
Edward J. Cogan, who has read several drafts of the manuscript and has
5
Preface vii
made valuable suggestions; and Dr. Robert Price, who has also given us
considerable help, partly as a consequence of having used the manuscript as
a textbook in a course in logic. To Mr. William J. Coe and Mr. John Kozy,
Jr., we are indebted for much work on the exercises and for correcting a
number of drafts of the manuscript. The index was prepared by Miss
Margaret A. Mitchell. Messrs. Laurie W. Cameron, Gerald E. Rubin, James
C. Morrison, and Bernard M. Goldsmith have given substantial help with
proofreading. Miss Joan Lee has performed with distinction the difficult
and arduous task of typing the manuscript. For financial assistance at all
stages of our work on this book we are grateful to the Central Fund for
Research of The Pennsylvania State University.
J.M.A.
H. W. J., Jr.
University Park, Pennsylvania
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CONTENTS
Part One
Introduction to Logic
Chapter 1 Proof and Rules of Inference 3
1. Introduction 3
2. An Example of an Axiom System: Simple Order 6
3. Recapitulation 14
Suggested Readings 16
Chapter 2 The Propositional Rules 17
I. Introduction 17
2. The Rules for &, =>, v, and == 19
3. Negation Rules 32
4. A List of Important Propositional Rules 40
5. Names, Statements, and Schemata 46
6. Rule of Replacement 50
7. Rule of Substitution 55
8. A Brief Note on Translation 60
9. Some Applications of the Rules 63
10. Recapitulation: the Materials of Proof 73
Suggested Readings 77
Chapter 3 Matrices, Normal Forms, and Duality 79
I. Truth-Functional Tests of Deductions 79
2. Statements and Truth Values 89
3. Tautology, Contradiction, and Interdefinability 93
ix
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Contents X
4. Normal Forms 98
5. Duality 107
Suggested Readings 112
Chapter 4 Some Other Propositional Logics 113
I. Multi-valued Logics 113
2. lntuitionistic Logic 124
3. Modal Logic 129
Suggested Readings 137
Chapter 5 Introduction to Predicate Logic 139
I. Individual Variables 139
2. Quantification 141
3. Notes on Symbolizing Ordinary Language 145
4. Formulas, Scopes, Bound and Free Variables 153
5. Statement Functions and Propositional Rules of
Inference 157
6. Rules of Inference for Monadic Predicate Logic 159
7. Deductions in Monadic Predicate Logic 172
8. Derived Rules 180
Suggested Readings 185
Chapter 6 General Predicate Logic- First Order 187
l. Introduction 187
2. Relations 198
3. Derived Rules of GPL-1 211
4. General Predicate Logic-Higher Order 225
Suggested Readings 227
Chapter 7 Identity and Description 229
I. Identity 229
2. At Least and At Most 235
3. Descriptions 239
Suggested Readings 244
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xi Contents
Part Tivo
Introduction to Metatheory
Chapter 8 Axiom Systems and the Axiomatic Method 249
1. Euclidian Geometry 249
2. The Development of Axiom Systems 254
3. Boolean Algebra: an Axiom System 259
4. The Problem of Consistency 264
5. Independence and Completeness 272
Suggested Readings 276
Chapter 9 The Metatheory of Propositional Logic 279
1. Introduction 279
2. Consistency 281
3. Formal Systems 285
4. A Proof of the Consistency of PL 292
5. The Completeness of the Propositional Rules 301
6. Decidability 307
Suggested Readings 309
Chapter JO The Metatheory of Monadic Predicate Logic 311
1. The Formal System of MPL 311
2. Validity 313
3. The Consistency of the Rules of MPL 320
4. The Completeness of M PL 322
5. Decidability 333
6. Note on Decision Procedures 338
Suggested Readings 341
Chapter 11 Aspects of the Metatheory of General Predicate Logic 343
1. The Formal System of GPL-1 343
2. k-Validity and Consistency 344
3. Infinite Domains and the Metatheory of GPL-1 347
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Contents xii
4. Completeness of GPL-1: Preliminary Concepts 349
5. Completeness of G PL-I : Proof 355
6. Lowenheim-Skolem Theorem 360
7. Decidability 361
Suggested Readings 362
Chapter 12 Intuitive Set Theory 365
I. Introduction 365
2. Sets, Subsets, and Operations 366
3. Membership 368
4. Inclusion and Membership 372
5. Unit Sets 374
6. Sets and Predicates 375
7. Uses of Sets 379
Suggested Readings 385
Chapter 13 Logic and Set Theory 387
I. Contradictions and Paradoxes 387
2. The Importance of Sets 394
3. Axiomatic Set Theory 399
4. Logical Types 403
5. Intuitionism 405
6. Formalism 407
Suggested Readings 409
Index 411
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