A Treatise on Many-Valued Logics

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A growing interest in many-valued logics has developed over recent years, which to a large extent is based on applications, intended as well as already realised ones. These applications range from the field of computer science, e.g. in the areas of automated theorem proving, approximate reasoning, multi-agent systems, switching theory, and program verification, through the field of pure mathematics, e.g. in independence of consistency proofs, in generalised set theories, or in the theory of particular algebraic structures.

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Editorial Reviews

For advanced undergraduate students of logic or computer science with a knowledge of elementary notions from classical logic and set theory, and lattices and other algebraic structures, Gottwald (logic and philosophy of science, U. of Leipzig, Germany) explains the theory underling many-valued logic, and surveys a broad class of applications. It is the growing applications that have driven recent interest in the logic, especially in computer science for automated theorem proving, approximate reasoning, multi-agent systems, switching theory, program verification, and other tricks. Annotation c. Book News, Inc., Portland, OR (booknews.com)
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Product Details

  • ISBN-13: 9780863802621
  • Publisher: Research Studies Press Limited
  • Publication date: 11/1/2000
  • Series: Studies in Logic and Computation
  • Pages: 616
  • Product dimensions: 6.38 (w) x 9.35 (h) x 1.68 (d)

Table of Contents

Part I. Basic Notions
1. General Background 3
1.1 Classical and Many-Valued Logic 3
1.2 Preliminary Notions 6
2. The Formalized Language and its Interpretations 15
2.1 Propositional Syntax 15
2.2 Propositional Semantics 17
2.3 First-Order Syntax 21
2.4 Many-Valued Predicates 24
2.5 First-Order Semantics 26
3. Logical Validity and Entailment 29
3.1 Designated Truth Degrees 29
3.2 The Propositional Situation 31
3.3 The First-Order Situation 38
3.4 Elementary Model Theory 40
4. Outline of the History of Many-Valued Logic 55
Part II. General Theory
5. Particular Connectives and Truth Degree Sets 63
5.1 Conjunction Connectives 65
5.2 Negation Connectives 84
5.3 Disjunction Connectives 88
5.4 Implication Connectives 91
5.5 The J-Connectives 104
6. Axiomatizability 107
6.1 The Axiomatizability Problem 107
6.2 Axiomatizing Propositional Systems 108
6.3 Axiomatizing First-Order Systems 120
6.4 Axiomatizing the Entailment Relation 128
7. Sequent and Tableau Calculi 137
7.1 Tableau Calculi for Many-Valued Logic 138
7.2 Sequent Calculi for Many-Valued Logic 149
8. Some Further Topics 161
8.1 Functional Completeness 161
8.2 Decidability of Propositional Systems 171
8.3 Product Systems 173
Part III. Particular Systems of Many-Valued Logic
9. The Lukasiewicz Systems 179
9.1 The Propositional Systems 179
9.1.1 Important tautologies of the Lukasiewicz systems 181
9.1.2 Characterizing the number of truth degrees 185
9.1.3 Axiomatizability 193
9.1.4 Decidability of the system L[subscript infinity] 199
9.1.5 Representability of truth degree functions 201
9.2 Algebraic Structures for Lukasiewicz Systems 214
9.2.1 MV-algebras 215
9.2.2 MV-algebras and axiomatizations of the L-systems 234
9.2.3 Wajsberg algebras 242
9.2.4 Lukasiewicz algebras 247
9.3 The First-Order Systems 249
9.3.1 Important logically valid formulas 250
9.3.2 Theoretical results for the L-systems 253
9.3.3 The infinitely many-valued L-system 259
10. The Godel Systems 267
10.1 The Propositional Systems 267
10.2 The First-Order Systems 284
11. Product Logic 291
11.1 The Propositional System 291
11.2 The First-Order System 308
12. The Post Systems 313
12.1 The Original Presentation 313
12.2 The Present Form 318
13. t-Norm Based Systems 327
13.1 The Propositional Systems 327
13.2 The First-Order Systems 338
14. Axiomatizing t-Norm Based Logics 345
14.1 The Propositional Systems 345
14.1.1 Some particular cases 345
14.1.2 A global approach 346
14.1.3 Monoidal logic 352
14.1.4 Monoidal t-norm logic 362
14.1.5 Basic t-norm logic 367
14.1.6 Completeness under continuous t-norms 370
14.2 The First-Order Systems 374
15. Some Three- and Four-Valued Systems 385
15.1 Three-Valued Systems 385
15.2 Four-Valued Systems 393
16. Systems with Graded Identity 401
16.1 Graded Identity Relations 401
16.2 Identity: the Absolute Point of View 403
16.3 Identity: the Liberal Point of View 406
16.4 Identity and Extent of Existence 413
Part IV. Applications of Many-Valued Logic
17. The Problem of Applications 419
18. Fuzzy Sets, Vague Notions, and Many-Valued Logic 423
18.1 Vagueness of Notions and Fuzzy Sets 423
18.2 Basic Theory of Fuzzy Sets 425
18.2.1 Elementary set algebraic operations 426
18.2.2 Graded inclusion of fuzzy sets 429
18.2.3 Particular fuzzy sets 431
18.2.4 Generalized set algebraic operations 433
18.2.5 Fuzzy cartesian products 435
18.2.6 The extension principle 437
18.3 Fuzzy Relations 438
18.4 The Full Image Under a Relation 442
18.5 Special Types of Fuzzy Relations 445
18.5.1 Fuzzy equivalence relations 446
18.5.2 Fuzzy partitions of fuzzy sets 448
18.5.3 Transitive hulls 452
18.5.4 Fuzzy ordering relations 454
18.6 Graded Properties of Fuzzy Relations 460
19. Fuzzy Logic 471
19.1 Many-Valued Logic with Graded Consequences 472
19.2 The Semantic Approach 473
19.3 The Syntactic Approach 475
19.4 Axiomatizing Fuzzy Logic 477
19.5 Partial Soundness of Inference Rules 480
19.5.1 Formalizing the problem 480
19.5.2 Partially sound rules in many-valued and fuzzy logics 482
19.6 Some Theoretical Results 484
19.7 The Algebraic Approach 486
20. Treating Presuppositions with Many-Valued Logic 493
20.1 The Phenomenon of Presuppositions 493
20.2 Three-Valued Approaches 496
20.3 Four-Valued Approaches 499
21. Truth Degrees and Alethic Modalities 503
21.1 Interpreting Modal Logic as Many-Valued Logic 503
21.2 Graded Modalities 512
22. Approximating Intuitionistic and Other Logics 525
22.1 Many-Valued Approaches toward Intuitionistic Logic 525
22.2 Approximating Logics by Many-Valued Logics 527
23. Independence Proofs 535
23.1 The Propositional Case 535
23.2 The First-Order Case 538
24. Consistency Considerations for Set Theory 557
References 567
Subject Index 595
Index of Names 601
Index of Symbols 603
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