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.
Author(s): Petr Hájek
Edition: Softcover reprint of the original 1st ed. 1998
Publisher: Springer
Year: 2001
Language: English
Pages: 307
Contents
PREFACE
CHAPTER 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
CHAPTER 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
CHAPTER THREE / LUKASIEWICZ PROPOSITIONAL LOGIC
3.1 Getting Lukasiewicz logic
3.2 MV-algebras; a completeness theorem
3.3 Rational Pavelka logic
CHAPTER FOUR / PRODUCT LOGIC, GODEL LOGIC
4.1 Product logic
4.2 Godellogic
4.3 Appendix: Boolean logic
CHAPTER FIVE / MANY-VALUED PREDICATE LOGICS
5.1 The basic many-valued predicate logic
5.2 Completeness 119 5.3 Axiomatizing Godellogic
5.4 Lukasiewicz and product predicate logic
5.5 Many-sorted fuzzy predicate calculi
5.6 Similarity and equality
CHAPTER SIX / COMPLEXITY AND UNDECIDABILITY
6.1 Preliminaries
6.2 Complexity of fuzzy propositional calculi
6.3 Undecidability of fuzzy logics
CHAPTER 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
CHAPTER 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
CHAPTER NINE / MISCELLANEA
9.1 Takeuti-Titani fuzzy logic
9.2 An abstract fuzzy logic
9.3 On the liar paradox
9.4 Concluding remarks
CHAPTER 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
INDEX
