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Classical and Nonclassical Logics : An Introduction to the Mathematics of Propositions - Eric Schechter

Classical and Nonclassical Logics

An Introduction to the Mathematics of Propositions

Hardcover Published: 28th August 2005
ISBN: 9780691122793
Number Of Pages: 520

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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.

"We warmly welcome this book as an example of how the mathematical way of thinking can be made available and pleasant to a large group of students."--Solomon Marcus, Zentralblatt MATH

Preliminariesp. 1
Introduction for teachersp. 3
Purpose and intended audiencep. 3
Topics in the bookp. 6
Why pluralism?p. 13
Feedbackp. 18
Acknowledgmentsp. 19
Introduction for studentsp. 20
Who should study logic?p. 20
Formalism and certificationp. 25
Language and levelsp. 34
Semantics and syntacticsp. 39
Historical perspectivep. 49
Pluralismp. 57
Jarden's example (optional)p. 63
Informal set theoryp. 65
Sets and their membersp. 68
Russell's paradoxp. 77
Subsetsp. 79
Functionsp. 84
The Axiom of Choice (optional)p. 92
Operations on setsp. 94
Venn diagramsp. 102
Syllogisms (optional)p. 111
Infinite sets (postponable)p. 116
Topologies and interiors (postponable)p. 126
Topologiesp. 127
Interiorsp. 133
Generated topologies and finite topologies (optional)p. 139
English and informal classical logicp. 146
Language and biasp. 146
Parts of speechp. 150
Semantic valuesp. 151
Disjunction (or)p. 152
Conjunction (and)p. 155
Negation (not)p. 156
Material implicationp. 161
Cotenability, fusion, and constants (postponable)p. 170
Methods of proofp. 174
Working backwardsp. 177
Quantifiersp. 183
Inductionp. 195
Induction examples (optional)p. 199
Definition of a formal languagep. 206
The alphabetp. 206
The grammarp. 210
Removing parenthesesp. 215
Defined symbolsp. 219
Prefix notation (optional)p. 220
Variable sharingp. 221
Formula schemesp. 222
Order preserving or reversing subformulas (postponable)p. 228
Semanticsp. 233
Definitions for semanticsp. 235
Interpretationsp. 235
Functional interpretationsp. 237
Tautology and truth preservationp. 240
Numerically valued interpretationsp. 245
The two-valued interpretationp. 245
Fuzzy interpretationsp. 251
Two integer-valued interpretationsp. 258
More about comparative logicp. 262
More about Sugihara's interpretationp. 263
Set-valued interpretationsp. 269
Powerset interpretationsp. 269
Hexagon interpretation (optional)p. 272
The crystal interpretationp. 273
Church's diamond (optional)p. 277
Topological semantics (postponable)p. 281
Topological interpretationsp. 281
Examplesp. 282
Common tautologiesp. 285
Nonredundancy of symbolsp. 286
Variable sharingp. 289
Adequacy of finite topologies (optional)p. 290
Disjunction property (optional)p. 293
More advanced topics in semanticsp. 295
Common tautologiesp. 295
Images of interpretationsp. 301
Dugundji formulasp. 307
Basic syntacticsp. 311
Inference systemsp. 313
Basic implicationp. 318
Assumptions of basic implicationp. 319
A few easy derivationsp. 320
Lemmaless expansionsp. 326
Detachmental corollariesp. 330
Iterated implication (postponable)p. 332
Basic logicp. 336
Further assumptionsp. 336
Basic positive logicp. 339
Basic negationp. 341
Substitution principlesp. 343
One-formula extensionsp. 349
Contractionp. 351
Weak contractionp. 351
Contractionp. 355
Expansion and positive paradoxp. 357
Expansion and minglep. 357
Positive paradox (strong expansion)p. 359
Further consequences of positive paradoxp. 362
Explosionp. 365
Fusionp. 369
Not-eliminationp. 372
Not-elimination and contrapositivesp. 372
Interchangeability resultsp. 373
Miscellaneous consequences of noteliminationp. 375
Relativityp. 377
Soundness and major logicsp. 381
Soundnessp. 383
Constructive axioms: avoiding not-eliminationp. 385
Constructive implicationp. 386
Herbrand-Tarski Deduction Principlep. 387
Basic logic revisitedp. 393
Soundnessp. 397
Nonconstructive axioms and classical logicp. 399
Glivenko's Principlep. 402
Relevant axioms: avoiding expansionp. 405
Some syntactic resultsp. 405
Relevant deduction principle (optional)p. 407
Soundnessp. 408
Mingle: slightly irrelevantp. 411
Positive paradox and classical logicp. 415
Fuzzy axioms: avoiding contractionp. 417
Axiomsp. 417
Meredith's chain proofp. 419
Additional notationsp. 421
Wajsberg logicp. 422
Deduction principle for Wajsberg logicp. 426
Classical logicp. 430
Axiomsp. 430
Soundness resultsp. 431
Independence of axiomsp. 431
Abelian logicp. 437
Advanced resultsp. 441
Harrop's principle for constructive logicp. 443
Meyer's valuationp. 443
Harrop's principlep. 448
The disjunction propertyp. 451
Admissibilityp. 451
Results in other logicsp. 452
Multiple worlds for implicationsp. 454
Multiple worldsp. 454
Implication modelsp. 458
Soundnessp. 460
Canonical modelsp. 461
Completenessp. 464
Completeness via maximalityp. 466
Maximal unproving setsp. 466
Classical logicp. 470
Wajsberg logicp. 477
Constructive logicp. 479
Non-finitely-axiomatizable logicsp. 485
Referencesp. 487
Symbol listp. 493
Indexp. 495
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780691122793
ISBN-10: 0691122792
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 520
Published: 28th August 2005
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 4.45
Weight (kg): 0.88