Classical Mathematical Logic

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 xvii xix xxi I II Classical Propositional Logic A. Propositions 1 Other views of propositions 2 B. Types 3 Exercises for Sections A and B 4 C. The Connectives of Propositional Logic 5' Exercises for Section C 6 D. A Formal Language for Propositional Logic 1. Defining the formal language 7 A platonist definition of the formal language 8 2. The unique readability of wffs 8 3. Realizations 11 Exercises for Section D 12 E. Classical Propositional Logic 1. The classical abstraction and truth-functions 13 2. Models 17 Exercises for Sections E.I and E.2 17 3. Validity and semantic consequence 18 Exercises for Section E.3. 20 F. Formalizing Reasoning 20 Exercises for Section F 24 Proof by induction 25 Abstracting and Axiomatizing Classical Propositional Logic A. The Fully General Abstraction 28 Platonists on the abstraction of models 29 B. A Mathematical Presentation of PC 29 1. Models and the semantic consequence relation 29 Exercises for Sections A and B.I 31 2. The choice of language for PC 31 Normal forms 33

via Contents 3. The decidability of tautologies 33 Exercises for Sections B.2 and B.3 35 C. Formalizing the Notion of Proof 1. Reasons for formalizing 36 2. Proof, syntactic consequence, and theories 37 3. Soundness and completeness 39 Exercises for Section C 39 D. An Axiomatization of PC 1. The axiom system 39 Exercises for Section D.I 42 2. A completeness proof 42 Exercises for Section D.2 45 3. Independent axiom systems 45 4. Derived rules and substitution 46 5. An axiomatization of PC in L(~i, -», A, V) \ Al Exercises for Sections D.3-D.5 48 A constructive proof of completeness for PC 49 III IV The Language of Predicate Logic A. Things, the World, and Propositions 53 B. Names and Predicates 55 C. Propositional Connectives 56 D. Variables and Quantifiers 57 E. Compound Predicates and Quantifiers 59 F. The Grammar of Predicate Logic 60 Exercises for Sections A-F 60 G. A Formal Language for Predicate Logic 61 H. The Structure of the Formal Language 63 I. Free and Bound Variables 65 J. The Formal Language and Propositions 66 Exercises for Sections G-J 67 The Semantics of Classical Predicate Logic A. Names 69 B. Predicates 1. A predicate applies to an object 71 2. Predications involving relations 73 The platonist conception of predicates and predications 76 Exercises for Sections A and B 77 C. The Universe of a Realization 78 D. The Self-Reference Exclusion Principle 80 Exercises for Sections C and D 81

Contents ix E. Models 1. The assumptions of the realization 82 2. Interpretations 83 3. The Fregean assumption and the division of form and content... 85 4. The truth-value of a compound proposition: discussion 86 5. Truth in a model 90 6. The relation between V and 3 93 F. Validity and Semantic Consequence 95 Exercises for Sections E and F 96 Summary: The definition of a model 97 V VI VII Substitutions and Equivalences A. Evaluating Quantifications 1. Superfluous quantifiers 99 2. Substitution of terms 100 3. The extensionality of predications 102 Exercises for Section A 102 B. Propositional Logic within Predicate Logic 104 Exercises for Section B 105 C. Distribution of Quantifiers 106 Exercises for Section C 107 Prenex normal forms 107 D. Names and Quantifiers 109 E. The Partial Interpretation Theorem 110 Exercises for Sections D and E 112 Equality A. The Equality Predicate 113 B. The Interpretation of'='in a Model 114 C. The Identity of Indiscernibles 115 D. Equivalence Relations 117 Exercises for Chapter VI 120 Examples of Formalization A. Relative Quantification ; 121 B. Adverbs, Tenses, and Locations 125 C. Qualities, Collections, and Mass Terms 128 D. Finite Quantifiers 130 E. Examples from Mathematics 135 Exercises for Chapter VII 137

x Contents VIII IX X XI Functions A. Functions and Things 139 B. A Formal Language with Function Symbols and Equality 141 C. Realizations and Truth in a Model 143 D. Examples of Formalization 144 Exercises for Sections A-D 146 E. Translating Functions into Predicates 148 Exercises for Section E 151 The Abstraction of Models A. The Extension of a Predicate 153 Exercises for Section A 157 B. Collections as Objects: Naive Set Theory 157 Exercises for Section B 162 C. Classical Mathematical Models 164 Exercises for Section C 165 Axiomatizing Classical Predicate Logic A. An Axiomatization of Classical Predicate Logic 1. The axiom system 167 2. Some syntactic observations 169 3. Completeness of the axiomatization 172 4. Completeness for simpler languages a. Languages with name symbols 175 b. Languages without name symbols 176 c. Languages without 3 176 5. Validity and mathematical validity 176 Exercises for Section A 177 B. Axiomatizations for Richer Languages 1. Adding'='to the language 178 2. Adding function symbols to the language 180 Exercises for Section B 180 Taking open wffs as true or false 181 The Number of Objects in the Universe of a Model Characterizing the Size of the Universe 183 Exercises 187 Submodels and Skolem Functions 188

Contents xi XII Xin XIV Formalizing Group Theory A. A Formal Theory of Groups 191 Exercises for Section A 198 B. On Definitions 1. Eliminating 'e' 199 2. Eliminating'- 1 ' 203 3. Extensions by definitions 205 Exercises for Section B 206 Linear Orderings A. Formal Theories of Orderings 207 Exercises for Section A 209 B. Isomorphisms 210 Exercises for Section B 214 C. Categoricity and Completeness 215 Exercises for Section C 218 D. Set Theory as a Foundation of Mathematics? 219 Decidability by Elimination of Quantifiers 221 Second-Order Classical Predicate Logic A. Quantifying over Predicates? 225 B. Predicate Variables and Their Interpretation: Avoiding Self-Reference 1. Predicate variables 226 2. The interpretation of predicate variables 228 Higher-order logics 231 C. A Formal Language for Second-Order Logic, L 2 231 Exercises for Sections A-C. 233 D. Realizations and Models 234 Exercises for Section D 236 E. Examples of Formalization 237 Exercises for Section E 240 F. Classical Mathematical Second-Order Predicate Logic 1. The abstraction of models 241 2. All things and all predicates 242 3. Examples of formalization 243 Exercises for Sections F.1-F.3 249 4. The comprehension axioms 250 Exercises for Section F.4 253 G. Quantifying over Functions 255 Exercises for Section G 258 H. Other Kinds of Variables and Second-Order Logic 1. Many-sorted logic 259 2. General models for second-order logic 261 Exercises for Section H 262

xii Contents XV XVI XVII The Natural Numbers A. The Theory of Successor 264 Exercises for Section A 267 B. The Theory Q 1. Axiomatizing addition and multiplication 268 Exercises for Section B.I 270 2. Proving is a computable procedure 271 3. The computable functions and Q 272 4. The undecidability of Q 273 Exercises for Sections B.2-B.4 274 C. Theories of Arithmetic 1. Peano Arithmetic and Arithmetic 274 Exercises for Sections C.I 277 2. The languages of arithmetic 279 Exercises for Sections C.2 280 D. The Consistency of Theories of Arithmetic 280 Exercises for Section D 283 E. Second-Order Arithmetic 284 Exercises for Section E 288 F. Quantifying over Names 288 The Integers and Rationals A. The Rational Numbers 1. A construction 291 2. A translation 292 Exercises for Section A 295 B. Translations via Equivalence Relations 295 C. The Integers, 298 Exercises for Sections B and C 299 D. Relativizing Quantifiers and the Undecidability of Z-Arithmetic and Q-Arithmetic 300 Exercises for Section D. 302 The Real Numbers A. What Are the Real Numbers? 303 Exercises for Section A 305 B. Divisible Groups 306 The decidability and completeness of the theory of divisible groups.... 308 Exercises for Section B 309 C. Continuous Orderings 309 Exercises for Section C 311 D. Ordered Divisible Groups 312 Exercises for Section D 315

Contents xiii E. Real Closed Fields 1. Fields 316 Exercises for Section E.I 317 2. Ordered fields 317 Exercises for Section E.2 319 3. Real closed fields 320 Exercises for Section E.3 323 The theory of fields in the language of name quantification 324 Appendix: Real Numbers as Dedekind Cuts 326 XVm One-Dimensional Geometry in collaboration with Leslaw Szczerba A. What Are We Formalizing? 331 B. The One-Dimensional Theory of Betweenness 1. An axiom system for betweenness, Bl 333 Exercises for Sections A and B.I 334 2. Some basic theorems of Bl 334 3. Vectors in the same direction 335 4. An ordering of points and Bl O i 337 5. Translating between Bl and the theory of dense linear orderings. 338 6. The second-order theory of betweenness 340 Exercises for Section B 341 C. The One-Dimensional Theory of Congruence 1. An axiom system for congruence, Cl 342 2. Point symmetry 343 3. Addition of points 346 4. Congruence expressed in terms of addition 348 5. Translating between Cl and the theory of 2-divisible groups... 349 6. Division axioms for Cl and the theory of divisible groups... 351 Exercises for Section C 352 D. One-Dimensional Geometry 1. An axiom system for one-dimensional geometry, El 352 2. Monotonicity of addition 352 3. Translating between El and the theory of ordered divisible groups 354 4. Second-order one-dimensional geometry 357 Exercises for Section D 358 E. Named Parameters 360 XIX Two-Dimensional Euclidean Geometry in collaboration with Leslaw Szczerba A. The Axiom System E2 363 Exercises for Section A 366

xiv Contents B. Deriving Geometric Notions 1. Basic properties of the primitive notions 367 2. Lines 367 3. One-dimensional geometry and point symmetry 371 4. Line symmetry 373 5. Perpendicular lines 375 6. Parallel lines 377 Exercises for Sections B.l-B.6 380 7. Parallel projection 381 8. The Pappus-Pascal Theorem 383 9. Multiplication of points 384 C. Betweenness and Congruence Expressed Algebraically 388 D. Ordered Fields and Cartesian Planes 393 E. The Real Numbers 397 Exercises for Sections C-E 400 Historical Remarks 401 XX XXI Translations within Classical Predicate Logic A. What Is a Translation? 403 Exercises for Section A 407 B. Examples 1. Translating between different languages of predicate logic.... 408 2. Converting functions into predicates 409 3. Translating predicates into formulas 409 4. Relativizing quantifiers 410 5. Establishing equivalence-relations 410 6. Adding and eliminating parameters 411 7. Composing translations 411 8. The general form of translations? 412 Classical Predicate Logic with Non-Referring Names A. Logic for Nothing 413 B. Non-Referring Names in Classical Predicate Logic? 414 C. Semantics for Classical Predicate Logic with Non-Referring Names 1. Assignments of references and atomic predications 415 2. The quantifiers 416 3. Summary of the semantics for languages without equality.... 417 4. Equality 418 Exercises for Sections A-C 420 D. An Axiomatization 421 Exercises for Section D 426 E. Examples of Formalization 427 Exercises for Section E 430

Contents xv F. Classical Predicate Logic with Names for Partial Functions 1. Partial functions in mathematics 430 2. Semantics for partial functions 431 3. Examples 432 4. An axiomatization 434 Exercises for Section F 435 G. A Mathematical Abstraction of the Semantics 436 XXII The Liar Paradox A. The Self-Reference Exclusion Principle 437 B. Buridan's Resolution of the Liar Paradox 439 Exercises for Sections A and B 442 C. A Formal Theory 443 Exercises for Section C 447 D. Examples 448 Exercises for Section D 457 E. One Language for Logic? 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 465 B. Syntactic Derivations 466 C. The Completeness Theorem. 468 Summary of Formal Systems Propositional Logic 475 Classical Predicate Logic 476 Arithmetic 477 Linear Orderings 478 Groups 479 Fields 481 One-dimensional geometry 482 Two-dimensional Euclidean geometry 484 Classical Predicate Logic with Non-referring Names 485 Classical Predicate Logic with Name Quantification 486 Bibliography 487 Index of Notation 495 Index 499