Fuzzy Sets and Fuzzy Logic: Theory and Applications

Fuzzy Sets and Fuzzy Logic is a true magnum opus. An enlargement of Fuzzy Sets, Uncertainty, and Information—an earlier work of Professor Klir and Tina Folger—Fuzzy Sets and Fuzzy Logic addresses practically every significant topic in the broad expanse of the union of fuzzy set theory and fuzzy logic. To me Fuzzy Sets and Fuzzy Logic is a remarkable achievement; it covers its vast territory with impeccable authority, deep insight and a meticulous attention to detail. To view Fuzzy Sets and Fuzzy Logic in a proper perspective, it is necessary to clarify a point of semantics which relates to the meanings of fuzzy sets and fuzzy logic. A frequent source of misunderstanding fias to do with the interpretation of fuzzy logic. The problem is that the term fuzzy logic has two different meanings. More specifically, in a narrow sense, fuzzy logic, FLn, is a logical system which may be viewed as an extension and generalization of classical multivalued logics. But in a wider sense, fuzzy logic, FL^ is almost synonymous with the theory of fuzzy sets. In this context, what is important to recognize is that: (a) FLW is much broader than FLn and subsumes FLn as one of its branches; (b) the agenda of FLn is very different from the agendas of classical multivalued logics; and (c) at this juncture, the term fuzzy logic is usually used in its wide rather than narrow sense, effectively equating fuzzy logic with FLW In Fuzzy Sets and Fuzzy Logic, fuzzy logic is interpreted in a sense that is close to FLW. However, to avoid misunderstanding, the title refers to both fuzzy sets and fuzzy logic. Underlying the organization of Fuzzy Sets and Fuzzy Logic is a fundamental fact, namely, that any field X and any theory Y can be fuzzified by replacing the concept of a crisp set in X and Y by that of a fuzzy set. In application to basic fields such as arithmetic, topology, graph theory, probability theory and logic, fuzzification leads to fuzzy arithmetic, fuzzy topology, fuzzy graph theory, fuzzy probability theory and FLn. Similarly, hi application to applied fields such as neural network theory, stability theory, pattern recognition and mathematical programming, fuzzification leads to fuzzy neural network theory, fuzzy stability theory, fuzzy pattern recognition and fuzzy mathematical programming. What is gained through fuzzification is greater generality, higher expressive power, an enhanced ability to model real-world problems and, most importantly, a methodology for exploiting the tolerance for imprecision—a methodology which serves to achieve tractability,