ABSTRACT
The phrase “fuzzy logic” has two meanings. On the one hand, it refers to the use of fuzzy sets in the representation and manipulation of vague information for the purpose of making decisions or taking actions. It is the theory of fuzzy sets, the study of the system of all mappings of a set U into the unit interval. This involves not only the usual connectives of max and min on fuzzy sets, but the theory of t-norms, t-conorms, negations, and many other related concepts. Also there are generalizations of ordinary set theoretical concepts to the fuzzy setting, such as that of equivalence relation. Many topics can be “fuzzified”. Some of these appear throughout this book. On the other hand, fuzzy logic means the extension of ordinary logic
with truth values in the two-element Boolean algebra ({0, 1},∨,∧,0 , 0, 1) to the case where they are in the Kleene algebra ([0, 1],∨,∧,0 , 0, 1). There are of course many extensions of two-valued logic to multivalued ones, generally with the truth values being finite in number. A standard reference for this is [132]. This chapter focuses on fuzzy logic in this second sense. An impor-
tant reference is [67]. First, we present the basics of the two-valued propositional calculus, next the corresponding material for a well-known three-valued propositional calculus due to Lukasiewicz, and then for the propositional calculus in which the truth values consist of the Kleene algebra above, that is, for fuzzy propositional calculus. The fact that these last two propositional calculi are equal does not seem to be widely known. But it is a useful fact. It enables one to determine in finitely many steps, following a specific algorithm, whether two expressions in fuzzy sets are the
tives max, min, and the complement x→ 1−x. An analogous fact holds for fuzzy sets whose values are themselves intervals in [0, 1].
