ABSTRACT
Fuzzy Logic was initiated in 1965 by Lotfi A. Zadeh, professor for computer science at the University of California in Berkeley. Basically, Fuzzy Logic (FL) is a multivalued logic that defines intermediate values between traditional evaluations like true/false, yes/no, high/low, etc. These intermediate values can be formulated mathematically and processed by computers, in order to apply a more human like way of thinking. Based on Aristotle and other later mathematicians, the so called “Laws of Thought” was posited. One of these, the “Law of the Excluded Middle,” states that every proposition must either be True or False. Even when Parminedes proposed the first version of this law (around 400 B.C.) there were strong and immediate objections: for example, Heraclitus proposed that things could be simultaneously True and not True. It was Plato who laid the foundation for what would become fuzzy logic, indicating that there was a third region (beyond True and False) where these opposites “tumbled about.” Other, more modern philosophers echoed his sentiments, notably Hegel, Marx, and Engels. An alternative approach to the bivalued logic of Aristotle was proposed by Lukasiewicz. Fuzzy Logic has been developed as a profitable tool for the controlling and steering of systems and complex industrial processes, as well as for household and entertainment electronics. In viewing the evolution of fuzzy logic, three principal phases may
be discerned. The first phase, from 1965 to 1973, was concerned in the main with fuzzification, that is, with generalization of the concept of a set, with two-valued characteristic function generalized to a membership function taking values in the unit interval or, more generally, in a lattice. The basic issues and applications that were addressed were, for the most part, set-theoretic in nature, and logic and reasoning were not at the center of the stage. The second phase, 1973 to 1999, began with two key concepts: (a) the concept of a linguistic variable; and (b) the concept of a fuzzy if-then rule. Today, almost all applications of fuzzy set theory
is applied in various applications in two different senses: A narrow sense — In a narrow sense fuzzy logic, abbreviated as FL,
is a logical system which is a generalization of multivalued logic A wide sense — In a wide sense fuzzy logic is abbreviated as FL, is
a union of FLn, fuzzy set theory, possibility theory, calculus of fuzzy if-then rules, fuzzy arithmetic, calculus of fuzzy quantifiers, and related concepts and calculi The distinctive feature of FL is that in FL everything is, or is allowed
to be, a matter of degree. Possibly the most salient growth during the second phase of the evolution was the rapid growth of fuzzy control, alongside the boom in fuzzy logic applications, especially in Japan. There were many other major developments in fuzzy-logic-related ba-
sic and applied theories, among them the genesis of possibility theory and possibilistic logic, knowledge representation, decision analysis, cluster analysis, pattern recognition, fuzzy arithmetic, fuzzy mathematical programming, fuzzy topology and, more generally, fuzzy mathematics. Fuzzy control applications proliferated but their dominance in the literature became less pronounced. An important development making the beginning of the third phase
was “From Computing with Numbers to Computing with Words” in 1999. Basically, development of computing with words and perceptions brings together earlier strands of fuzzy logic and suggests that scientific theories should be based on fuzzy logic rather than on Aristotelian, bivalent logic, as they are at present. The concept of Precisiated Natural Language (PNL) is the key constituent in the area of computing words. PNL gives room to a major enlargement of the purpose of natural languages in technological hypotheses. It may well turn out to be the case that, in coming years, one of the most important application-areas of fuzzy logic, and especially PNL, will be the Internet, centering on the conception and design of search engines and question-answering systems. From its inception, fuzzy logic has been and to some degree still an
object of skepticism and controversy. The disbelief about fuzzy logic is a manifestation of the reality that, in English, the term ’fuzzy’ is usually used in a uncomplimentary sense. Merely, fuzzy logic is hard to accept as abandoning bivalence breaks with centuries-old tradition of basing scientific theories on bivalent logic. It may take some time for this to happen, but eventually abandonment
of bivalence will be viewed as a logical development in the evolution of science and human thought. This chapter will discuss the basic fuzzy sets, operations on fuzzy sets, relations between fuzzy sets, composition, and fuzzy arithmetic. A few MATLAB programs are also illustrated on topics such as membership functions, fuzzy operations, fuzzy arithmetic, relations, and composition.
