© 2008 AGI-Information Management Consultants
May be used for personal purporses only or by
libraries associated to dandelon.com network.
Classical
Mathematical Logic
The Semantic Foundations of Logic
Richard L. Epstein
with contributions by
Leslaw W. Szczerba
Princeton University Press
Princeton and Oxford
Contents
Preface
Acknowledgments
Introduction
I
Classical Propositional Logic
A. Propositions
Other views of propositions
B. Types
• Exercises for Sections A and B
C. The Connectives of Propositional Logic
• Exercises for Section C
D. A Formal Language for Propositional Logic
1. Defining the formal language
A platonist definition of the formal language
2. The unique readability of wffs
3. Realizations
• Exercises for Section D
E. Classical Propositional Logic
1. The classical abstraction and truth-functions
2. Models
• Exercises for Sections E.I and E.2
3. Validity and semantic consequence
• Exercises for Section E.3
F. Formalizing Reasoning
• Exercises for Section F
Proof by induction
II
xvii
xix
xxi
1
2
3
4
5'
6
7
8
8
11
12
13
17
17
18
. 20
20
24
25
Abstracting and Axiomatizing
Classical Propositional Logic
A. The Fully General Abstraction
Platonists on the abstraction of models
B. A Mathematical Presentation of PC
1. Models and the semantic consequence relation
•Exercises for Sections A and B.I
2. The choice of language for PC
Normal forms
28
29
29
29
31
31
33
via Contents
3. The decidability of tautologies
•Exercises for Sections B.2 and B.3
C. Formalizing the Notion of Proof
1. Reasons for formalizing
2. Proof, syntactic consequence, and theories
3. Soundness and completeness
• Exercises for Section C
D. An Axiomatization of PC
1. The axiom system
•Exercises for Section D.I
2. A completeness proof
• Exercises for Section D.2
3. Independent axiom systems
4. Derived rules and substitution
5. An axiomatization of PC in L(~i, - » , A , V )
• Exercises for Sections D.3-D.5
A constructive proof of completeness for PC
III
36
37
39
39
39
42
42
45
45
46
\ Al
48
49
The Language of Predicate Logic
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
IV
33
35
Things, the World, and Propositions
Names and Predicates
Propositional Connectives
Variables and Quantifiers
Compound Predicates and Quantifiers
The Grammar of Predicate Logic
• Exercises for Sections A-F
A Formal Language for Predicate Logic
The Structure of the Formal Language
Free and Bound Variables
The Formal Language and Propositions
• Exercises for Sections G-J
53
55
56
57
59
60
60
61
63
65
66
67
The Semantics of Classical Predicate Logic
A. Names
B. Predicates
1. A predicate applies to an object
2. Predications involving relations
The platonist conception of predicates and predications
• Exercises for Sections A and B
C. The Universe of a Realization
D. The Self-Reference Exclusion Principle
• Exercises for Sections C and D
69
71
73
76
77
78
80
81
Contents
Models
1. The assumptions of the realization
2. Interpretations
3. The Fregean assumption and the division of form and content
4. The truth-value of a compound proposition: discussion
5. Truth in a model
6. The relation between V and 3
F. Validity and Semantic Consequence
• Exercises for Sections E and F
Summary: The definition of a model
ix
E.
V
Substitutions and Equivalences
A. Evaluating Quantifications
1. Superfluous quantifiers
2. Substitution of terms
3. The extensionality of predications
• Exercises for Section A
B. Propositional Logic within Predicate Logic
• Exercises for Section B
C. Distribution of Quantifiers
• Exercises for Section C
Prenex normal forms
D. Names and Quantifiers
E. The Partial Interpretation Theorem
• Exercises for Sections D and E
VI
99
100
102
102
104
105
106
107
107
109
110
112
Equality
A. The Equality Predicate
B. The Interpretation of'='in a Model
C. The Identity of Indiscernibles
D. Equivalence Relations
• Exercises for Chapter VI
VII
. . .
82
83
85
86
90
93
95
96
97
113
114
115
117
120
Examples of Formalization
A. Relative Quantification ;
B. Adverbs, Tenses, and Locations
C. Qualities, Collections, and Mass Terms
D. Finite Quantifiers
E. Examples from Mathematics
• Exercises for Chapter VII
121
125
128
130
135
137
x
Contents
VIII
Functions
A.
B.
C.
D.
Functions and Things
A Formal Language with Function Symbols and Equality
Realizations and Truth in a Model
Examples of Formalization
• Exercises for Sections A-D
E. Translating Functions into Predicates
• Exercises for Section E
IX
The Abstraction of Models
A. The Extension of a Predicate
• Exercises for Section A
B. Collections as Objects: Naive Set Theory
• Exercises for Section B
C. Classical Mathematical Models
• Exercises for Section C
X
153
157
157
162
164
165
Axiomatizing Classical Predicate Logic
A. An Axiomatization of Classical Predicate Logic
1. The axiom system
2. Some syntactic observations
3. Completeness of the axiomatization
4. Completeness for simpler languages
a. Languages with name symbols
b. Languages without name symbols
c. Languages without 3
5. Validity and mathematical validity
• Exercises for Section A
B. Axiomatizations for Richer Languages
1. Adding'='to the language
2. Adding function symbols to the language
• Exercises for Section B
Taking open wffs as true or false
XI
139
141
143
144
146
148
151
167
169
172
175
176
176
176
177
178
180
180
181
The Number of Objects in the Universe of a Model
Characterizing the Size of the Universe
• Exercises
Submodels and Skolem Functions
183
187
188
Contents xi
XII
Formalizing Group Theory
A. A Formal Theory of Groups
• Exercises for Section A
B. On Definitions
1. Eliminating 'e'
2. Eliminating'- 1 '
3. Extensions by definitions
• Exercises for Section B
Xin
199
203
205
206
Linear Orderings
A. Formal Theories of Orderings
• Exercises for Section A
B. Isomorphisms
• Exercises for Section B
C. Categoricity and Completeness
• Exercises for Section C
D. Set Theory as a Foundation of Mathematics?
Decidability by Elimination of Quantifiers
XIV
191
198
207
209
210
214
215
218
219
221
Second-Order Classical Predicate Logic
A. Quantifying over Predicates?
B. Predicate Variables and Their Interpretation: Avoiding Self-Reference
1. Predicate variables
2. The interpretation of predicate variables
Higher-order logics
C. A Formal Language for Second-Order Logic, L 2
• Exercises for Sections A-C
.
D. Realizations and Models
• Exercises for Section D
E. Examples of Formalization
• Exercises for Section E
F. Classical Mathematical Second-Order Predicate Logic
1. The abstraction of models
2. All things and all predicates
3. Examples of formalization
•Exercises for Sections F.1-F.3
4. The comprehension axioms
• Exercises for Section F.4
G. Quantifying over Functions
• Exercises for Section G
H. Other Kinds of Variables and Second-Order Logic
1. Many-sorted logic
2. General models for second-order logic
• Exercises for Section H
225
226
228
231
231
233
234
236
237
240
241
242
243
249
250
253
255
258
259
261
262
xii
Contents
XV
The Natural Numbers
A. The Theory of Successor
• Exercises for Section A
B. The Theory Q
1. Axiomatizing addition and multiplication
•Exercises for Section B.I
2. Proving is a computable procedure
3. The computable functions and Q
4. The undecidability of Q
• Exercises for Sections B.2-B.4
C. Theories of Arithmetic
1. Peano Arithmetic and Arithmetic
•Exercises for Sections C.I
2. The languages of arithmetic
• Exercises for Sections C.2
D. The Consistency of Theories of Arithmetic
• Exercises for Section D
E. Second-Order Arithmetic
• Exercises for Section E
F. Quantifying over Names
XVI
264
267
268
270
271
272
273
274
274
277
279
280
280
283
284
288
288
The Integers and Rationals
A. The Rational Numbers
1. A construction
2. A translation
• Exercises for Section A
B. Translations via Equivalence Relations
C. The Integers
,
• Exercises for Sections B and C
D. Relativizing Quantifiers and the Undecidability of
Z-Arithmetic and Q-Arithmetic
• Exercises for Section D
XVII
291
292
295
295
298
299
.
300
302
The Real Numbers
A. What Are the Real Numbers?
• Exercises for Section A
B. Divisible Groups
The decidability and completeness of the theory of divisible groups
• Exercises for Section B
C. Continuous Orderings
• Exercises for Section C
D. Ordered Divisible Groups
• Exercises for Section D
. . . .
303
305
306
308
309
309
311
312
315
Contents xiii
E. Real Closed Fields
1. Fields
•Exercises for Section E.I
2. Ordered
fields
• Exercises for Section E.2
3. Real closed fields
• Exercises for Section E.3
The theory of fields in the language of name quantification
Appendix: Real Numbers as Dedekind Cuts
316
317
317
319
320
323
324
326
XVm One-Dimensional Geometry
in collaboration with Leslaw Szczerba
A. What Are We Formalizing?
B. The One-Dimensional Theory of Betweenness
1. An axiom system for betweenness, Bl
• Exercises for Sections A and B.I
2. Some basic theorems of Bl
3. Vectors in the same direction
4. An ordering of points and Bl O i
5. Translating between Bl and the theory of dense linear orderings .
6. The second-order theory of betweenness
• Exercises for Section B
C. The One-Dimensional Theory of Congruence
1. An axiom system for congruence, Cl
2. Point symmetry
3. Addition of points
4. Congruence expressed in terms of addition
5. Translating between C l and the theory of 2-divisible groups . . .
6. Division axioms for C l and the theory of divisible groups . . . .
• Exercises for Section C
D. One-Dimensional Geometry
1. An axiom system for one-dimensional geometry, E l
2. Monotonicity of addition
3. Translating between E l and the theory of ordered divisible groups
4. Second-order one-dimensional geometry
• Exercises for Section D
E. Named Parameters
XIX
331
333
334
334
335
337
338
340
341
342
343
346
348
349
351
352
352
352
354
357
358
360
Two-Dimensional Euclidean Geometry
in collaboration with Leslaw Szczerba
A. The Axiom System E2
• Exercises for Section A
363
366
xiv
Contents
B. Deriving Geometric Notions
1. Basic properties of the primitive notions
2. Lines
3. One-dimensional geometry and point symmetry
4. Line symmetry
5. Perpendicular lines
6. Parallel lines
•Exercises for Sections B.l-B.6
7. Parallel projection
8. The Pappus-Pascal Theorem
9. Multiplication of points
C. Betweenness and Congruence Expressed Algebraically
D. Ordered Fields and Cartesian Planes
E. The Real Numbers
• Exercises for Sections C-E
Historical Remarks
XX
Translations within Classical Predicate Logic
A. What Is a Translation?
• Exercises for Section A
B. Examples
1. Translating between different languages of predicate logic . . . .
2. Converting functions into predicates
3. Translating predicates into formulas
4. Relativizing quantifiers
5. Establishing equivalence-relations
6. Adding and eliminating parameters
7. Composing translations
8. The general form of translations?
XXI
367
367
371
373
375
377
380
381
383
384
388
393
397
400
401
403
407
408
409
409
410
410
411
411
412
Classical Predicate Logic with Non-Referring Names
A. Logic for Nothing
B. Non-Referring Names in Classical Predicate Logic?
C. Semantics for Classical Predicate Logic with Non-Referring Names
1. Assignments of references and atomic predications
2. The quantifiers
3. Summary of the semantics for languages without equality . . . .
4. Equality
• Exercises for Sections A - C
D. An Axiomatization
• Exercises for Section D
E. Examples of Formalization
• Exercises for Section E
413
414
415
416
417
418
420
421
426
427
430
Contents
F. Classical Predicate Logic with Names for Partial Functions
1. Partial functions in mathematics
2. Semantics for partial functions
3. Examples
4. An axiomatization
• Exercises for Section F
G. A Mathematical Abstraction of the Semantics
XXII
xv
430
431
432
434
435
436
The Liar Paradox
A. The Self-Reference Exclusion Principle
B. Buridan's Resolution of the Liar Paradox
• Exercises for Sections A and B
C. A Formal Theory
• Exercises for Section C
D. Examples
• Exercises for Section D
E. One Language for Logic?
437
439
442
443
447
448
457
458
XXIII On Mathematical Logic and Mathematics
Concluding Remarks
461
Appendix: The Completeness of Classical Predicate Logic
Proved by Godel's Method
A. Description of the Method
B. Syntactic Derivations
C. The Completeness Theorem
.
465
466
468
Summary of Formal Systems
Propositional Logic
Classical Predicate Logic
Arithmetic
Linear Orderings
Groups
Fields
One-dimensional geometry
Two-dimensional Euclidean geometry
Classical Predicate Logic with Non-referring Names
Classical Predicate Logic with Name Quantification
Bibliography
Index of Notation
Index
475
476
477
478
479
481
482
484
485
486
487
495
499