This book introduces the theory of graded consequence (GCT) and its mathematical formulation. It also compares the notion of graded consequence with other notions of consequence in fuzzy logics, and discusses possible applications of the theory in approximate reasoning and decision-support systems. One of the main points where this book emphasizes on is that GCT maintains the distinction between the three different levels of languages of a logic, namely object language, metalanguage and metametalanguage, and thus avoids the problem of violation of the principle of use and mention; it also shows, gathering evidences from existing fuzzy logics, that the problem of category mistake may arise as a result of not maintaining distinction between levels.
Erscheinungsdatum: 01.08.2019

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Chakraborty, Mihir Kumar; Dutta, Soma

Pagination_Cover

Springer Science + Business Media Singapore Pte Ltd

Springer Nature Singapore

Logic in Asia: Studia Logica Library, 1st ed. 2019, Singapore, 2019

Springer Nature, Singapore, 2019

Singapore, Singapore

Sep 15, 2019

2, 20190722

sm76411246

producers:
Acrobat Distiller 10.0.0 (Windows)

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Source title: Theory of Graded Consequence: A General Framework for Logics of Uncertainty (Logic in Asia: Studia Logica Library)

Preface 7
Contents 11
1 Introduction 14
1.1 Three Levels Inherent in a Logic Discourse 14
1.2 From Many-Valued Logics, Fuzzy Logics to Graded Consequence: A Brief Overview 18
1.3 Different Shades of Imprecision 20
1.4 Classical Consequence and Motivations for Lifting it to Many-Valuedness 24
References 28
2 Basics of the Theory of Graded Consequence 32
2.1 Characterization of Graded Consequence Relation 32
2.2 Semantic Consequence Relation Generalized 35
2.3 A Generalization of Hilbert-Type Axiom System 40
2.3.1 Graded Consequence by Fuzzy Axioms and Fuzzy Rules 41
2.3.2 Soundness and Examples of Hilbert-Type Axiomatic Consequence in GCT 44
2.4 Consistency, Inconsistency and Equivalence 49
2.4.1 Consistency and Inconsistency 49
2.4.2 Equivalence 50
2.5 Level Cuts of a Graded Consequence Relation 51
2.6 Appendix 54
References 56
3 Introducing Negation (¬) in the Object Language of the Theory of Graded Consequence 58
3.1 Extending the Notion Graded Consequence Relation in the Presence of Negation (¬) 58
3.2 Semantic Import of (GC4) and (GC5) on the Meta-structure 68
3.3 GC(¬) Algebra and its Properties 80
3.4 Possible Three-Valued and Four-Valued GC(¬)-Algebras 90
References 94
4 Proof Theoretic Rules in Graded Consequence: From Semantic Perspective 96
4.1 General Scheme for Proof Theory in GCT 96
4.2 Proof Theory of GCT: Links Between Object and Meta-level Algebras 108
4.2.1 Implication in the Object Language 111
4.2.2 Conjunction in the Object Language 114
4.2.3 Disjunction in the Object Language 116
4.2.4 Negation in the Object Language 120
4.3 Examples of Logics with Graded Notion of Consequence 125
4.3.1 Many-Valued Logics as a Special Case of GCT 127
4.4 A Comparative Analysis 132
References 136
5 Meta-logical Notions Generalized: Graded Consequence with Fuzzy Set of Premises 138
5.1 Consequence Operator and Consequence Relation in the Context of Fuzzy Logics 138
5.1.1 Pavelka's Fuzzy Consequence Operator and Chakraborty's Graded Consequence Relation 139
5.1.2 Fuzzy Consequence Relation: Castro et al. 141
5.2 Meta-Logic of Graded Consequence with Fuzzy Premises 141
5.2.1 Graded Inconsistency and Graded Consequence 146
5.2.2 Graded Consistency 150
5.2.3 Equivalence 153
5.3 Implicative Consequence Operator and Consequence Relation 155
5.4 Implicative Consequence Relation in the Light of Consistency-Generating Relation 157
5.4.1 From the Notion of Consistency-Generating Relation to the Notion of Consistency 161
References 163
6 Graded Consequence and Consequence in Different Approaches to Fuzzy Logics 165
6.1 Fuzzy Logic vis-á-vis Graded Consequence 165
6.1.1 Notion of Proof in Pavelka's Fuzzy Logic 167
6.1.2 The Expressed Meaning for Consequence: Goguen to Hájek 169
6.1.3 Pavelka's Notion of Proof Reframed Maintaining Distinction of Levels of Logic 170
6.2 Rewriting the Theory of Graded Consequence Maintaining Level Distinction 175
6.3 A Brief Revisit to Hájek's Logic RPL From the Perspective of Distinction of Levels 181
6.4 Łukasiewicz Fuzzy Propositional Logic 184
References 185
7 Graded Consequence in Decision-Making: A Few Applications 186
7.1 GCT with Interval Semantics: Different Approaches Towards Aggregating Information 186
7.1.1 Interval Mathematics: Some Basic Notions 188
7.1.2 Graded Consequence: Form (Σ) 189
7.1.3 Extension of subseteq as a Lattice Order Relation on Intervals 191
7.1.4 Graded Consequence: Form (Σ) 192
7.1.5 Graded Consequence: Form (Σ) 195
7.2 GCT in the Context of Logic Infomorphism: A Case for Decision-Making in a Distributed Network 199
7.2.1 Barwise and Seligman's Logic for Distributed System 201
7.2.2 Paraconsistent Set 206
7.2.3 Graded Consequence: Logic for Distributed System 207
7.3 Sorites Paradox in the Light of GCT 222
7.3.1 Different Approaches Towards Sorites Paradox 223
7.3.2 Approach to Sorites Paradox in the Context of GCT 225
References 227
Index 230

Front Matter ....Pages i-xiii
Introduction (Mihir Kumar Chakraborty, Soma Dutta)....Pages 1-18
Basics of the Theory of Graded Consequence (Mihir Kumar Chakraborty, Soma Dutta)....Pages 19-44
Introducing Negation (\(\lnot \)) in the Object Language of the Theory of Graded Consequence (Mihir Kumar Chakraborty, Soma Dutta)....Pages 45-82
Proof Theoretic Rules in Graded Consequence: From Semantic Perspective (Mihir Kumar Chakraborty, Soma Dutta)....Pages 83-124
Meta-logical Notions Generalized: Graded Consequence with Fuzzy Set of Premises (Mihir Kumar Chakraborty, Soma Dutta)....Pages 125-151
Graded Consequence and Consequence in Different Approaches to Fuzzy Logics (Mihir Kumar Chakraborty, Soma Dutta)....Pages 153-173
Graded Consequence in Decision-Making: A Few Applications (Mihir Kumar Chakraborty, Soma Dutta)....Pages 175-218
Back Matter ....Pages 219-224

2019-07-23