A Treatise on Many-Valued Logics

A Treatise on Many-Valued Logics

by Siegfried Gottwald, Prof Siegfried Gottwald
     
 

ISBN-10: 0863802621

ISBN-13: 9780863802621

Pub. Date: 11/01/2000

Publisher: Research Studies Press Limited

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,

Overview

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.

Product Details

ISBN-13:
9780863802621
Publisher:
Research Studies Press Limited
Publication date:
11/01/2000
Series:
Studies in Logic and Computation
Pages:
616
Product dimensions:
6.38(w) x 9.35(h) x 1.68(d)

Related Subjects

Table of Contents

Part I.Basic Notions
1.General Background3
1.1Classical and Many-Valued Logic3
1.2Preliminary Notions6
2.The Formalized Language and its Interpretations15
2.1Propositional Syntax15
2.2Propositional Semantics17
2.3First-Order Syntax21
2.4Many-Valued Predicates24
2.5First-Order Semantics26
3.Logical Validity and Entailment29
3.1Designated Truth Degrees29
3.2The Propositional Situation31
3.3The First-Order Situation38
3.4Elementary Model Theory40
4.Outline of the History of Many-Valued Logic55
Part II.General Theory
5.Particular Connectives and Truth Degree Sets63
5.1Conjunction Connectives65
5.2Negation Connectives84
5.3Disjunction Connectives88
5.4Implication Connectives91
5.5The J-Connectives104
6.Axiomatizability107
6.1The Axiomatizability Problem107
6.2Axiomatizing Propositional Systems108
6.3Axiomatizing First-Order Systems120
6.4Axiomatizing the Entailment Relation128
7.Sequent and Tableau Calculi137
7.1Tableau Calculi for Many-Valued Logic138
7.2Sequent Calculi for Many-Valued Logic149
8.Some Further Topics161
8.1Functional Completeness161
8.2Decidability of Propositional Systems171
8.3Product Systems173
Part III.Particular Systems of Many-Valued Logic
9.The Lukasiewicz Systems179
9.1The Propositional Systems179
9.1.1Important tautologies of the Lukasiewicz systems181
9.1.2Characterizing the number of truth degrees185
9.1.3Axiomatizability193
9.1.4Decidability of the system L[subscript infinity]199
9.1.5Representability of truth degree functions201
9.2Algebraic Structures for Lukasiewicz Systems214
9.2.1MV-algebras215
9.2.2MV-algebras and axiomatizations of the L-systems234
9.2.3Wajsberg algebras242
9.2.4Lukasiewicz algebras247
9.3The First-Order Systems249
9.3.1Important logically valid formulas250
9.3.2Theoretical results for the L-systems253
9.3.3The infinitely many-valued L-system259
10.The Godel Systems267
10.1The Propositional Systems267
10.2The First-Order Systems284
11.Product Logic291
11.1The Propositional System291
11.2The First-Order System308
12.The Post Systems313
12.1The Original Presentation313
12.2The Present Form318
13.t-Norm Based Systems327
13.1The Propositional Systems327
13.2The First-Order Systems338
14.Axiomatizing t-Norm Based Logics345
14.1The Propositional Systems345
14.1.1Some particular cases345
14.1.2A global approach346
14.1.3Monoidal logic352
14.1.4Monoidal t-norm logic362
14.1.5Basic t-norm logic367
14.1.6Completeness under continuous t-norms370
14.2The First-Order Systems374
15.Some Three- and Four-Valued Systems385
15.1Three-Valued Systems385
15.2Four-Valued Systems393
16.Systems with Graded Identity401
16.1Graded Identity Relations401
16.2Identity: the Absolute Point of View403
16.3Identity: the Liberal Point of View406
16.4Identity and Extent of Existence413
Part IV.Applications of Many-Valued Logic
17.The Problem of Applications419
18.Fuzzy Sets, Vague Notions, and Many-Valued Logic423
18.1Vagueness of Notions and Fuzzy Sets423
18.2Basic Theory of Fuzzy Sets425
18.2.1Elementary set algebraic operations426
18.2.2Graded inclusion of fuzzy sets429
18.2.3Particular fuzzy sets431
18.2.4Generalized set algebraic operations433
18.2.5Fuzzy cartesian products435
18.2.6The extension principle437
18.3Fuzzy Relations438
18.4The Full Image Under a Relation442
18.5Special Types of Fuzzy Relations445
18.5.1Fuzzy equivalence relations446
18.5.2Fuzzy partitions of fuzzy sets448
18.5.3Transitive hulls452
18.5.4Fuzzy ordering relations454
18.6Graded Properties of Fuzzy Relations460
19.Fuzzy Logic471
19.1Many-Valued Logic with Graded Consequences472
19.2The Semantic Approach473
19.3The Syntactic Approach475
19.4Axiomatizing Fuzzy Logic477
19.5Partial Soundness of Inference Rules480
19.5.1Formalizing the problem480
19.5.2Partially sound rules in many-valued and fuzzy logics482
19.6Some Theoretical Results484
19.7The Algebraic Approach486
20.Treating Presuppositions with Many-Valued Logic493
20.1The Phenomenon of Presuppositions493
20.2Three-Valued Approaches496
20.3Four-Valued Approaches499
21.Truth Degrees and Alethic Modalities503
21.1Interpreting Modal Logic as Many-Valued Logic503
21.2Graded Modalities512
22.Approximating Intuitionistic and Other Logics525
22.1Many-Valued Approaches toward Intuitionistic Logic525
22.2Approximating Logics by Many-Valued Logics527
23.Independence Proofs535
23.1The Propositional Case535
23.2The First-Order Case538
24.Consistency Considerations for Set Theory557
References567
Subject Index595
Index of Names601
Index of Symbols603

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