Introduction: Bridging the Gap between Multiple-Valued Logics, Fuzzy Logic, Uncertain Reasoning and Reasoning about Knowledge

Abstract

The term ‘Fuzzy Logic’, although very well-known, is ambiguous as it refers to several only loosely related concerns ranging from rule-based system control to various multiple-valued logics. The most popular acception of the term ‘fuzzy logic’ is generally not related to logic proper, since it is primarily employed by control engineers that use fuzzy rule-based systems like a sort of neural network capable of approximating non-linear functions. However in the recent past, it has been stressed that fuzzy logic in the narrow sense can be envisaged from the point of view of logic provided that fuzzy sets are considered as stemming from the multiple-valued logic tradition. The relationship between multiple-valued logic and fuzzy sets had been noticed by Moisil [1972] in the late sixties. At the same period, in Eastern Germany, Klaua independently built up a multiple-valued set theory (see Gottwald [1984] for an extensive bibliography of Klaua’s papers). The current trend relating fuzzy sets and multiple-valued logic actually dates back to a seminal paper by Goguen [1969]. In this paper Goguen insists on an algebraic structure he calls a ‘closg’ (for commutative lattice ordered semi-group) and shows that the lattice-theoretical concept of residuation can be generalized to operations other than the minimum. Following Goguen’s program, Pavelka [1979] has definitely anchored fuzzy logic in the multiple-valued tradition, emphasizing a link with Lukasiewicz logic already pointed out by Giles [1976]. Since then, a large amount of work has been carried out whose aim is to equip fuzzy logic with a syntactic component, and several algebraic structures have been laid bare as potential candidates for supporting fuzzy logics.