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E-grāmata: Fuzzy Logic of Quasi-Truth: An Algebraic Treatment

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Fuzzy Logic of Quasi-Truth: An Algebraic TreatmentPalielināt
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This book presents the first algebraic treatment of quasi-truth fuzzylogic and covers the algebraic foundations of many-valued logic. Itoffers a comprehensive account of basic techniques and reports on importantresults showing the pivotal role played by perfect many-valued algebras(MV-algebras). It is well known that the first-order predicate Lukasiewiczlogic is not complete with respect to the canonical set of truth values. However, it is complete with respect to alllinearly ordered MV -algebras. As there are no simple linearly orderedMV-algebras in this case, infinitesimal elements of an MV-algebra are allowedto be truth values. The book presents perfect algebras as an interestingsubclass of local MV-algebras and provides readers with the necessary knowledgeand tools for formalizing the fuzzy concept of quasi true and quasi false. Allbasic concepts are introduced in detail to promote a better understanding ofthe more complex ones. It is an advanced and inspiring refe

rence-guide forgraduate students and researchers in the field of non-classical many-valuedlogics.

Introduction.- Basic Notions.- Classical Sentential Calculus and Lukasiewicz Sentential Calculus.- MV -Algebras: Generalities.- Local MV -algebras.- Perfect MV -algebras.- The Variety Generated by Perfect MV -algebras.- Representations of Perfect MV -algebras.- The Logic of Perfect Algebras.- The Logic of Quasi True.- Perfect Pavelka Logic.

Recenzijas

This book studies many-valued logics and their algebraic counterparts which are suitable for formalizing and modelling the concept of quasi-true. This work is a valuable source of information for all logicians and philosophers interested in mathematical models of vagueness, and their application to modelling many-valued truth-degrees in an algebraically coherent manner. (Tomį Kroupa, Mathematical Reviews, March, 2017)

1 Introduction
1(6)
References
5(2)
2 Basic Notions
7(12)
2.1 Ordered Sets and Lattices
7(1)
2.2 Topological Spaces
8(1)
2.3 Universal Algebras
8(8)
2.4 Categories
16(3)
References
18(1)
3 Classical Sentential Calculus and Lukasiewicz Sentential Calculus
19(8)
3.1 Classical Sentential Calculus
19(3)
3.2 Lukasiewicz Sentential Calculus
22(5)
References
25(2)
4 MV-Algebras: Generalities
27(6)
4.1 MV-Algebras
27(1)
4.2 Examples of MV-Algebras
28(2)
4.3 Properties of MV-Algebras
30(1)
4.4 Ideals, Filters, Congruence Relations
31(2)
References
32(1)
5 Local MV-Algebras
33(4)
Reference
36(1)
6 Perfect MV-Algebras
37(10)
6.1 The Category of Perfect MV-Algebras
40(3)
6.2 Ultraproduct of Perfect MV-Algebras
43(4)
References
46(1)
7 The Variety Generated by Perfect MV-Algebras
47(6)
7.1 Quasi Variety Generated by C
50(3)
References
51(2)
8 Representations of Perfect MV-Algebras
53(18)
8.1 Godel Spaces
53(3)
8.2 M-Generated Free MV(C)-Algebra
56(3)
8.3 Spectral Duality
59(1)
8.4 Belluce's Functor
60(2)
8.5 A Weak Duality
62(2)
8.6 Coproduct in MV(C)G
64(7)
References
70(1)
9 The Logic of Perfect Algebras
71(14)
9.1 Finitely Generated Projective MV(C)-Algebras
73(3)
9.2 Projective Formulas
76(2)
9.3 Unification Problem
78(2)
9.4 Structural Completeness
80(5)
References
83(2)
10 The Logic of Quasi True
85(6)
10.1 Introduction
85(1)
10.2 CL-Algebras
85(3)
10.3 Logics CL and CL+
88(3)
Reference
89(2)
11 Perfect Pavelka Logic
91(20)
11.1 Introduction
91(1)
11.2 The Language F of Perfect Pavelka Logic
92(1)
11.3 Semantics: Valuations
93(1)
11.4 Syntax: Axioms and Rules of Inference
94(4)
11.5 T-complete Formulas in Perfect Pavelka Logic
98(3)
11.6 Examples and New Rules of Inference
101(7)
11.7 What Can and What Cannot Be Expressed in Rational or Perfect Pavelka Logic
108(3)
References
109(2)
Index 111
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