Synopses & Reviews
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. There are two main groups of intended readers. First, logicians: they can see that fuzzy logic is indeed a branch of logic and may find several very interesting open problems. Second, equally important, researchers involved in fuzzy logic applications and soft computing. As a matter of fact, most of these are not professional logicians so that it can easily happen that an application, clever and successful as it may be, is presented in a way which is logically not entirely correct or may appear simple-minded. (Standard presentations of the logical aspects of fuzzy controllers are the most typical example.) This fact would not be very important if only the bon ton of logicians were harmed; but it is the opinion of the author (who is a mathematical logician) that a better understanding of the strictly logical basis of fuzzy logic (in the usual broad sense) is very useful for fuzzy logic appliers since if they know better what they are doing, they may hope to do it better. In addition, a better mutual understanding between (classical) logicians and researchers in fuzzy logic, promises to lead to deeper cooperation and new results.
Review
"...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)
Synopsis
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.
Table of Contents
Preface.
1. Preliminaries.
2. Many-Valued Propositional Calculi.
3. Lukasiewicz Propositional Logic.
4. Product Logic, Gödel Logic.
5. Many-Valued Predicate Logics.
6. Complexity and Undecidability.
7. On Approximate Inference.
8. Generalized Quantifiers and Modalities.
9. Miscellanea.
10. Historical Remarks. References. Index.