First-Order Logic: A Concise Introduction

303 Pages·2021·42.81 MB·English

by  John Heil

First-Order
Logic
A Concise
¬¬
Introduction
∧∧ S E C O N D E D I T I O N
∨∨ John Heil
⊃⊃
≡≡
DERIVATION RULES
Rules of Inference
Modus Ponens (MP) Modus Tollens (MT)
p q, p q p q, q p
Hypothetic⊃al Syllo⊦gism (HS) Construc⊃tiv e D¬ile⊦m ¬ma (CD)
p q, q r p r p q, p r, q s r s
Conjunc⊃ti on E⊃lim i⊦n at⊃io n ( E) Co∨njuncti⊃on Inse⊃rtio⊦n (∨I)
p q p p, q p q
∧ ∧
p q q
∧ ⊦ ⊦ ∧
Disjunction e∧lim⊦ination ( E) Disjunction Insertion ( I)
p q, p q p p q
∨ ∨
p q, q p
∨ ¬ ⊦ ⊦ ∨
Conditi∨onal¬ Pro⊦of (CP) Indirect Proof (IP)
p p

¬
q q q
⋮ ⋮
p q p
∧ ¬
⊃
Transformation Rules
Commutative Rule (Com) Associative Rule (Assoc)
p q q p p (q r) (p q) r
p q q p p (q r) (p q) r
∧ ⊣⊦ ∧ ∧ ∧ ⊣⊦ ∧ ∧
Principle ∨of T⊣au⊦tol∨ogy (Taut) D∨eMor∨gan’s⊣ L⊦aw∨ (DeM∨ )
p p p p q ( p q)
p p p p q ( p q)
⊣⊦ ∧ ∧ ⊣⊦¬ ¬ ∨¬
Distribut⊣iv⊦e R∨ule (Dist) Exp∨orta⊣t⊦i o¬n R¬ule∧ ( ¬Exp)
p (q r) (p q) (p r) (p q) r p (q r)
p (q r) (p q) (p r)
∧ ∨ ⊣⊦ ∧ ∨ ∧ ∧ ⊃ ⊣⊦ ⊃ ⊃
Co∨nditi∧onal ⊣E⊦quiv∨alen∧ce (C∨ond) Biconditional Equivalence (Bicond)
p q p q p q (p q) (q p)
⊃ ⊣⊦¬ ∨ Contraposition (Contra)≡ ⊣⊦ ⊃ ∧ ⊃
p q q p
⊃ ⊣⊦¬ ⊃¬
First-Order Logic
A Concise Introduction
Second Edition

First-Order Logic
A Concise Introduction
Second Edition
John Heil
Washington University in St. Louis
Durham University
Monash University
Hackett Publishing Company, Inc.
Indianapolis/Cambridge
For
Katherine Mark
John Jr. Gus
Henry Lilian
Lucy
Copyright © 2021 by Hackett Publishing Company, Inc.
All rights reserved
Printed in the United States of America
24 23 22 21 1 2 3 4 5 6 7
For further information, please address
Hackett Publishing Company, Inc.
P.O. Box 44937
Indianapolis, Indiana 46244-0937
www.hackettpublishing.com
Cover and text design and composition by E. L. Wilson
Illustration on p. ii from Philosophical Pictures © Charles B. Martin, 1990. Reproduced by permission.
Library of Congress Control Number: 2021933118
ISBN-13: 978-1-62466-992-7 (pbk.)
ISBN-13: 978-1-64792-010-4 (PDF ebook)
Contents
Preface to the First Edition viii
Preface to the Second Edition x
Acknowledgments xii
1 Introduction 1
1.00 Logic: What’s Not to Like? 1
1.01 Practice Makes Less Imperfect 4
1.02 Ls and Lp 5
L
2 The Language s 7
2.00 A Formal Language 7
2.01 Sentential Constants and Variables 7
2.02 Truth-Functional Connectives 10
2.03 Negation: 11
2.04 Conjunctio¬n: 12
2.05 Sentential Pun∧ctuation 14
2.06 Disjunction: 16
2.07 The Conditio∨nal: 19
2.08 Conditionals, Dep⊃endence, and Sentential Punctuation 25
2.09 The Biconditional: 28
2.10 Complex Truth Tab≡les 30
2.11 The Sheffer Stroke: 35
2.12 Translating English| into Ls 38
2.13 Conjunction 40
2.14 Disjunction 42
2.15 Conditionals and Biconditionals 44
2.16 Troublesome English Constructions 47
2.17 Truth Table Analyses of Ls Sentences 50
2.18 Contradictions and Logical Truths 53
v
Contents
2.19 Describing Ls 57
2.20 The Syntax of Ls 57
2.21 The Semantics of Ls 60
L
3 Derivations in s 63
3.00 Sentential Sequences 63
3.01 Object Language and Metalanguage 63
3.02 Derivations in Ls 66
3.03 The Principle of Form 70
3.04 Inference Rules: MP, MT 73
3.05 Sentence Valence 76
3.06 Hypothetical Syllogism: HS 77
3.07 Rules for Conjunction: I, E 79
3.08 Rules for Disjunction: ∧I, ∧E 82
3.09 Conditional Proof: CP ∨ ∨ 86
3.10 Indirect Proof: IP 89
3.11 Transformation Rules: Com, Assoc, Taut 93
3.12 Transformation Rules: DeM 98
3.13 Transformation Rules: Dist, Exp 101
3.14 Rules for Conditionals: Contra, Cond 103
3.15 Biconditional Sentences: Bicond 106
3.16 Constructive Dilemma: CD 108
3.17 Acquiring a Feel for Derivations 110
3.18 Proving Invalidity 113
3.19 Theorems 118
3.20 Soundness and Completeness of Ls 121
L
4 The Language p 127
4.00 Frege’s Legacy 127
4.01 Terms 127
4.02 Terms in Lp 131
4.03 Quantifiers and Variables 134
vi
Contents
4.04 Bound and Free Variables 140
4.05 Negation 142
4.06 Complex Terms 144
4.07 Mixed Quantification 147
4.08 Translational Odds and Ends 150
4.09 Identity 155
4.10 At Least, at Most, Exactly 159
4.11 Definite Descriptions 162
4.12 Comparatives, Superlatives, Exceptives 166
4.13 Times and Places 169
4.14 The Domain of Discourse 170
4.15 The Syntax of Lp 174
4.16 The Semantics of Lp 177
4.17 Logic and Ontology 181
L
5 Derivations in p 184
5.00 Preliminaries 184
5.01 Quantifier Transformation 187
5.02 Universal Instantiation: UI 190
5.03 Existential Generalization: EG 194
5.04 Existential Instantiation: EI 198
5.05 Universal Generalization: UG 202
5.06 Quantifier Rules Summary 208
5.07 Identity: ID 213
5.08 Theorems in Lp 218
5.09 Invalidity in Lp 220
5.10 Prenex Normal Form 231
5.11 Soundness and Completeness of Lp 232
Solutions to Even-Numbered Exercises 236
Index 283
vii
Preface to the First Edition
Why another logic textbook? Why indeed. The market is flooded with textbooks, each of which
fills, or purports to fill, a particular niche. Oddly, in spite of—or perhaps because of—the availabil-
ity of scores of textbooks, many teachers of logic spurn commercial texts and teach from notes and
handouts. This suggests that although there are many logic textbooks, there are not many good logic
textbooks.
Logic texts fall into two categories. Some, like S. C. Kleene’s Mathematical Logic (New York:
John Wiley and Sons, 1967) and Benson Mates’s Elementary Logic (New York: Oxford University
Press, 1972), emphasize logic as a distinctive subject matter to be explicated by articulating, as ele-
gantly as possible, the theory on which the subject matter rests. Others, too numerous to mention,
focus on applications of logic, treating logic as a skill to be mastered, refined, and applied to argu-
ments advanced by politicians, editorial writers, and talk show hosts. A few authors offer a middle
ground, notably E. J. Lemmon in Beginning Logic (originally published in 1965, reissued by Hackett
in 1978) and Paul Teller in his two-volume A Modern Formal Logic Primer (Englewood Cliffs, NJ:
Prentice-Hall, 1989). Lemmon and Teller embed discussions of theory within a context that encour-
ages the development of logical skills.
In what follows, I have elected to take this middle way. I focus on the construction of transla-
tions and derivations, but I locate these within a broader theoretical framework. The book assumes
no prior contact with, or enthusiasm for, formal logic. My aim has been to introduce the elements
of first-order logic gradually, in small steps, as clearly as possible. I have tried to write in a way that
is congenial to students (and instructors) who might feel uncomfortable in symbolic domains. My
approach to logic is not that of a card-carrying logician. This, I think, gives me something of an
advantage in understanding what nonlogicians and symbolphobes find difficult or unintuitive. As a
result, I spend more time explaining fundamental notions than other authors do. In my view, this
pays dividends in the long run.
The volume covers elementary first-order logic with identity. I have not attempted to offer proofs
for the soundness and completeness of the systems introduced. I have, however, offered sketches of
what such proofs involve. These are included, with materials on the syntax and semantics of the sys-
tems, in sections on metalogic at the end of chapters 2 through 5. These sections could be skipped
without loss of continuity. They are offered as springboards for more elaborate classroom discussions.
For my own part, I think it important to include a dose of metalogic in an introductory course. Met-
alogic brings order to materials that are apt to seem arbitrary and ad hoc otherwise. Less obviously,
an examination of the syntax, semantics, and metatheory of a formal system tells us something about
ourselves. In mastering a formal system we come to terms with a domain that can be given a precise
and elegant description. Any account of our psychology, then, must allow for our ability to under-
stand and deploy systems with these formal characteristics.
The book began life in the summer of 1972. I had received support from the National Endow-
ment for the Humanities to write a text that would combine logic with work in linguistic theory. My
thought was that this was a case in which learning two things together was easier, more efficient,
and more illuminating than learning either separately. The project culminated in a photocopied
text inflicted on successive generations of students. In the ensuing years, linguists progressed from
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