Petr Hájek

Metamathematics of Fuzzy Logic

• Gebundenes Buch

Produktbeschreibung

This book presents a systematic treatment of deductive aspects and structures of fuzzy logic understood as many valued logic sui generis. Some important systems of real-valued propositional and predicate calculus are defined and investigated. The aim is 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.

Produktdetails

• Trends in Logic 4
• Verlag: Springer / Springer Netherlands
• Artikelnr. des Verlages: 978-0-7923-5238-9
• 1998.
• Seitenzahl: 312
• Erscheinungstermin: 31. August 1998
• Englisch
• Abmessung: 241mm x 160mm x 23mm
• Gewicht: 631g
• ISBN-13: 9780792352389
• ISBN-10: 0792352386
• Artikelnr.: 21517595

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)