On the semantics of fuzzy logic

This paper presents a formal characterization of the major concepts and constructs o f fuzzy logic in terms of notions o f distance, closeness, and similarity between pairs o f possible worlds. The formalism is a direct extension (by recognition of multiple degrees o f accessibility, conceivability, or reachability) of the major modal logic concepts of possible and necessary truth. Given a function that maps pairs of possible worlds into a number between 0 and 1, generalizing the conventional concept of an equivalence relation, the major constructs of fuzzy logic (conditional and unconditioned possibility distributions) are defined in terms of this similarity relation using familiar concepts from the mathematical theory of metric spaces. This interpretation is dO~ferent in nature and character from the typical, chance-oriented, meanings associated with probabilistic concepts, which are grounded on the mathematical notion of set measure. The similarity structure defines a topological notion of continuity in the space of possible worlds (and in that of its subsets, i.e., propositions) that allows a form of logical "'extrapolation'" between possible worlds. This logical extrapolation operation corresponds to the major deductive rule of fuzzy logicthe compositional rule o f inference or generalized modus ponens of Zadehan inferential operation that generalizes its classical counterpart by virtue of its ability to be utilized when propositions representing available evidence match only approximately the antecedents of conditional propositions. The relations between the similarity-based interpretation of the role of conditional possibility distributions and the approximate inferential procedures of Baldwin are also discussed. A straightforward extension of the theory to the case where the similarity scale is symbolic rather than numeric is described. The problem of generating similarity functions from a given set of possibility distributions, with the latter interpreted as defining a number o f (graded) discernibility relations and the former as the result *To my friends Nadal Batle, Francesc Esteva, Ram6n L6pez de M~ntaras, Enric Trillas, and LlorenG Valverde. Address correspondence to Enrique H. Ruspini, AI Center, SRI International, 333 Ravenswood Avenue, Menlo Park, CA 94025. International Journal of Approximate Reasoning 1991; 5:45-88 © 1991 Elsevier Science Publishing Co., Inc. 655 Avenue of the Americas, New York, NY 10010 0888-613X/91/$3.50 45 46 Enrique H. Ruspini of combining them into a joint measure of distinguishability between possible worlds, is briefly discussed.