Classical mathematical logic: the semantic foundations of logic
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Princeton, NJ [u.a.]
Princeton University Press
2006
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| Schlagworte: | |
| Online-Zugang: | Publisher description Contributor biographical information Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references (S. 487 - 493) and indexes |
| Beschreibung: | XXII, 522 S. |
| ISBN: | 0691123004 9780691123004 |
Internformat
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| 100 | 1 | |a Epstein, Richard L. |d 1947- |e Verfasser |0 (DE-588)109298241 |4 aut | |
| 245 | 1 | 0 | |a Classical mathematical logic |b the semantic foundations of logic |c Richard L. Epstein, with contributions by Lesław W. Szczerba |
| 264 | 1 | |a Princeton, NJ [u.a.] |b Princeton University Press |c 2006 | |
| 300 | |a XXII, 522 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references (S. 487 - 493) and indexes | ||
| 650 | 4 | |a Logique symbolique et mathématique | |
| 650 | 7 | |a Mathematische Logik |2 swd | |
| 650 | 7 | |a Philosophische Semantik |2 swd | |
| 650 | 7 | |a Semantiek |2 gtt | |
| 650 | 4 | |a Sémantique (Philosophie) | |
| 650 | 7 | |a Wiskundige logica |2 gtt | |
| 650 | 4 | |a Semantik | |
| 650 | 4 | |a Logic, Symbolic and mathematical | |
| 650 | 4 | |a Semantics (Philosophy) | |
| 856 | 4 | |u http://www.loc.gov/catdir/enhancements/fy0654/2005055239-d.html |3 Publisher description | |
| 856 | 4 | |u http://www.loc.gov/catdir/enhancements/fy0708/2005055239-b.html |3 Contributor biographical information | |
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Datensatz im Suchindex
| _version_ | 1804137205107523584 |
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| adam_text | 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 XVII ACKNOWLEDGMENTS XIX
INTRODUCTION XXI I 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 II 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 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 IV 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
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 VI
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 VII 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 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
IX 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 X 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 XI 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
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 XIN 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 XIV 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 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 XVI 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 XVII 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 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 XXI 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
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| adam_txt |
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 XVII ACKNOWLEDGMENTS XIX
INTRODUCTION XXI I 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 II 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 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 IV 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
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 VI
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 VII 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 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
IX 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 X 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 XI 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
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 XIN 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 XIV 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 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 XVI 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 XVII 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 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 XXI 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 |
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| author | Epstein, Richard L. 1947- |
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| dewey-full | 511.3 |
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| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.3 |
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| dewey-sort | 3511.3 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik Philosophie |
| discipline_str_mv | Mathematik Philosophie |
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| id | DE-604.BV022959210 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T19:04:03Z |
| indexdate | 2024-07-09T21:08:35Z |
| institution | BVB |
| isbn | 0691123004 9780691123004 |
| language | English |
| lccn | 2005055239 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016163597 |
| oclc_num | 62290616 |
| open_access_boolean | |
| owner | DE-19 DE-BY-UBM DE-11 |
| owner_facet | DE-19 DE-BY-UBM DE-11 |
| physical | XXII, 522 S. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Princeton University Press |
| record_format | marc |
| spelling | Epstein, Richard L. 1947- Verfasser (DE-588)109298241 aut Classical mathematical logic the semantic foundations of logic Richard L. Epstein, with contributions by Lesław W. Szczerba Princeton, NJ [u.a.] Princeton University Press 2006 XXII, 522 S. txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references (S. 487 - 493) and indexes Logique symbolique et mathématique Mathematische Logik swd Philosophische Semantik swd Semantiek gtt Sémantique (Philosophie) Wiskundige logica gtt Semantik Logic, Symbolic and mathematical Semantics (Philosophy) http://www.loc.gov/catdir/enhancements/fy0654/2005055239-d.html Publisher description http://www.loc.gov/catdir/enhancements/fy0708/2005055239-b.html Contributor biographical information GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016163597&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Epstein, Richard L. 1947- Classical mathematical logic the semantic foundations of logic Logique symbolique et mathématique Mathematische Logik swd Philosophische Semantik swd Semantiek gtt Sémantique (Philosophie) Wiskundige logica gtt Semantik Logic, Symbolic and mathematical Semantics (Philosophy) |
| title | Classical mathematical logic the semantic foundations of logic |
| title_auth | Classical mathematical logic the semantic foundations of logic |
| title_exact_search | Classical mathematical logic the semantic foundations of logic |
| title_exact_search_txtP | Classical mathematical logic the semantic foundations of logic |
| title_full | Classical mathematical logic the semantic foundations of logic Richard L. Epstein, with contributions by Lesław W. Szczerba |
| title_fullStr | Classical mathematical logic the semantic foundations of logic Richard L. Epstein, with contributions by Lesław W. Szczerba |
| title_full_unstemmed | Classical mathematical logic the semantic foundations of logic Richard L. Epstein, with contributions by Lesław W. Szczerba |
| title_short | Classical mathematical logic |
| title_sort | classical mathematical logic the semantic foundations of logic |
| title_sub | the semantic foundations of logic |
| topic | Logique symbolique et mathématique Mathematische Logik swd Philosophische Semantik swd Semantiek gtt Sémantique (Philosophie) Wiskundige logica gtt Semantik Logic, Symbolic and mathematical Semantics (Philosophy) |
| topic_facet | Logique symbolique et mathématique Mathematische Logik Philosophische Semantik Semantiek Sémantique (Philosophie) Wiskundige logica Semantik Logic, Symbolic and mathematical Semantics (Philosophy) |
| url | http://www.loc.gov/catdir/enhancements/fy0654/2005055239-d.html http://www.loc.gov/catdir/enhancements/fy0708/2005055239-b.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016163597&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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