Petr Hájek

Metamathematics of Fuzzy Logic (eBook, PDF)

• Format: PDF

Produktbeschreibung

This book presents a systematic treatment of deductive aspects and structures of fuzzy logic understood as many valued logic sui generis. It aims to show that fuzzy logic as a logic of imprecise (vague) propositions does have well-developed formal foundations and that most things usually named 'fuzzy inference' can be naturally understood as logical deduction. It is for mathematicians, logicians, computer scientists, specialists in artificial intelligence and knowledge engineering, and developers of fuzzy logic.

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Produktdetails

• Verlag: Springer Netherlands
• Seitenzahl: 299
• Erscheinungstermin: 1. Dezember 2013
• Englisch
• ISBN-13: 9789401153003
• Artikelnr.: 43984996

Inhaltsangabe

One / Preliminaries.- 1.1 Introduction.- 1.2 A survey of Boolean propositional logic.- 1.3 Boolean predicate calculus.- 1.4 Function symbols; varieties of algebras.- 1.5 Lattices and Boolean algebras.- 1.6 Ordered Abelian groups.- Two / Many-valued propositional calculi.- 2.1 Continuous t-norms and their residua.- 2.2 The basic many-valued logic.- 2.3 Residuated lattices; a completeness theorem.- 2.4 Some additional topics.- Three / ?ukasiewicz propositional logic.- 3.1 Getting ?ukasiewicz logic.- 3.2 MV-algebras; a completeness theorem.- 3.3 Rational Pavelka logic.- Four / Product logic, Gödel logic.- 4.1 Product logic.- 4.2 Gödel logic.- 4.3 Appendix: Boolean logic.- Five / Many-valued predicate logics.- 5.1 The basic many-valued predicate logic.- 5.2 Completeness.- 5.3 Axiomatizing Gödel logic.- 5.4 ?ukasiewicz and product predicate logic.- 5.5 Many-sorted fuzzy predicate calculi.- 5.6 Similarity and equality.- Six / Complexity and undecidability.- 6.1 Preliminaries.- 6.2 Complexity of fuzzy propositional calculi.- 6.3 Undecidability of fuzzy logics.- Seven / On approximate inference.- 7.1 The compositional rule of inference.- 7.2 Fuzzy functions and fuzzy controllers.- 7.3 An alternative approach to fuzzy rules.- Eight / Generalized quantifiers and modalities.- 8.1 Generalized quantifiers in Boolean logic.- 8.2 Two-valued modal logics.- 8.3 Fuzzy quantifiers and modalities.- 8.4 On "probably" and "many".- 8.5 More on "probably" and "many".- Nine / Miscellanea.- 9.1 Takeuti-Titani fuzzy logic.- 9.2 An abstract fuzzy logic.- 9.3 On the liar paradox.- 9.4 Concluding remarks.- Ten / Historical remarks.- 10.1 Until the forties.- 10.2 The fifties.- 10.3 The sixties.- 10.4 The seventies.- 10.5 The eighties.- 10.6 The nineties.- References.

Rezensionen

"...Moreover, it can be an excellent guideline for a postgraduate course in fuzzy logic because of its clear and well-organized content. Definitively, an excellent book that we enthusiastically recommend, and that fuzzy and mathematical logic (and probably philosophical logic as well) communities have been waiting for, for a long time."
(F. Esteva, L. Godo, International Journal of General Systems, 29:5 (2000)
"On whole this book occupies a unique place in the literature on fuzzy sets and fuzzy logic and fills a substantial gap. This is the only existing monograph offering a comprehensive study of triangular norms and their interconnections. Overall the book is very well written and should become the standard reference for triangular norms."
(Mathematical Reviews, 2002a)