Metamathematics of Fuzzy Logic

Erscheinungsjahr 2001
ISBN: 978-1-4020-0370-7
Verlag: Springer

Buch, Englisch, 299 Seiten, Format (B × H): 156 mm x 234 mm, Gewicht: 477 g

Reihe: Trends in Logic

This book presents a systematic treatment of deductive aspects and structures of fuzzy logic understood as many valued logic . 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.

Zielgruppe

Research

Autoren/Hrsg.

Hájek, Petr

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Grundlagen der Mathematik
• Geisteswissenschaften Philosophie Philosophische Logik, Argumentationstheorie

Inhaltsverzeichnis

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.