Buch, Englisch, 536 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 936 g
An Introduction to the Mathematics of Propositions
Buch, Englisch, 536 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 936 g
ISBN: 978-0-691-12279-3
Verlag: Princeton University Press
So-called classical logic--the logic developed in the early twentieth century by Gottlob Frege, Bertrand Russell, and others--is computationally the simplest of the major logics, and it is adequate for the needs of most mathematicians. But it is just one of the many kinds of reasoning in everyday thought. Consequently, when presented by itself--as in most introductory texts on logic--it seems arbitrary and unnatural to students new to the subject. In Classical and Nonclassical Logics, Eric Schechter introduces classical logic alongside constructive, relevant, comparative, and other nonclassical logics. Such logics have been investigated for decades in research journals and advanced books, but this is the first textbook to make this subject accessible to beginners. While presenting an assortment of logics separately, it also conveys the deeper ideas (such as derivations and soundness) that apply to all logics. The book leads up to proofs of the Disjunction Property of constructive logic and completeness for several logics. The book begins with brief introductions to informal set theory and general topology, and avoids advanced algebra; thus it is self-contained and suitable for readers with little background in mathematics. It is intended primarily for undergraduate students with no previous experience of formal logic, but advanced students as well as researchers will also profit from this book.
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Fachgebiete
Weitere Infos & Material
A Preliminaries 1
Chapter 1: Introduction for teachers 3
Purpose and intended audience, 3
Topics in the book, 6
Why pluralism?, 13
Feedback, 18
Acknowledgments, 19
Chapter 2: Introduction for students 20
Who should study logic?, 20
Formalism and certication, 25
Language and levels, 34
Semantics and syntactics, 39
Historical perspective, 49
Pluralism, 57
Jarden's example (optional), 63
Chapter 3: Informal set theory 65
Sets and their members, 68
Russell's paradox, 77
Subsets, 79
Functions, 84
The Axiom of Choice (optional), 92
Operations on sets, 94
Venn diagrams, 102
Syllogisms (optional), 111
Infinite sets (postponable), 116
Chapter 4: Topologies and interiors (postponable) 126
Topologies, 127
Interiors, 133
Generated topologies and finite topologies (optional), 139
Chapter 5: English and informal classical logic 146
Language and bias, 146
Parts of speech, 150
Semantic values, 151
Disjunction (or), 152
Conjunction (and), 155
Negation (not), 156
Material implication, 161
Cotenability, fusion, and constants (postponable), 170
Methods of proof, 174
Working backwards, 177
Quantifiers, 183
Induction, 195
Induction examples (optional), 199
Chapter 6: Definition of a formal language 206
The alphabet, 206
The grammar, 210
Removing parentheses, 215
Defined symbols, 219
Prefix notation (optional), 220
Variable sharing, 221
Formula schemes, 222
Order preserving or reversing subformulas (postponable), 228
B Semantics 233
Chapter 7: Definitions for semantics 235
Interpretations, 235
Functional interpretations, 237
Tautology and truth preservation, 240
Chapter 8: Numerically valued interpretations 245
The two-valued interpretation, 245
Fuzzy interpretations, 251
Two integer-valued interpretations, 258
More about comparative logic, 262
More about Sugihara's interpretation, 263
Chapter 9: Set-valued interpretations 269
Powerset interpretations, 269
Hexagon interpretation (optional), 272
The crystal interpretation, 273
Church's diamond (optional), 277
Chapter 10: Topological semantics (postponable) 281
Topological interpretations, 281
Examples, 282
Common tautologies, 285
Nonredundancy of symbols, 286
Variable sharing, 289
Adequacy of finite topologies (optional), 290
Disjunction property (optional), 293
Chapter 11: More advanced topics in semantics 295
Common tautologies, 295
Images of interpretations, 301
Dugundji formulas, 307
C Basic syntactics 311
Chapter 12: Inference systems 313
Chapter 13: Basic implication 318
Assumptions of basic implication, 319
A few easy derivations, 320
Lemmaless expansions, 326
Detachmental corollaries, 330
Iterated implication (postponable), 332
Chapter 14: Basic logic 336
Further assumptions, 336
Basic positive logic, 339
Basic negation, 341
Substitution principles, 343
D One-formula extensions 349
Chapter 15: Contraction 351
Weak contraction, 351
Contraction, 355
Chapter 16: Expansion and positive paradox 357
Expansion and mingle, 357
Positive paradox (strong expansion), 359
Further consequences of positive paradox, 362
Chapter 17: Explosion 365
Chapter 18: Fusion 369
Chapter 19: Not-elimination 372
Not-elimination and contrapositives, 372
Interchangeability results, 373
Miscellaneous consequences of not-elimination, 375
Chapter 20: Relativity 377
E Soundness and major logics 381
Chapter 21: Soundness 383
Chapter 22: Constructive axioms: avoiding not-elimination 385
Constructive implication, 386
Herbrand-Tarski Deduction Principle, 387
Basic logic revisited, 393
Soundness, 397
Nonconstructive axioms and classical logic, 399
Glivenko's Principle, 402
Chapter 23: Relevant axioms: avoiding expansion 405
Some syntactic results, 405
Relevant deduction principle (optional), 407
Soundness, 408
Mingle: slightly irrelevant, 411
