Fuzzy Sets in Approximate Reasoning: a Personal View

Abstract

Fuzzy rule-based approximate reasoning is attracting more and more interest from researchers and practitioners nowadays. Zadeh (1973, 1975, 1979a) provided the basic machinery for fuzzy set-based approximate reasoning more than fifteen years ago. In his approach, each granule of knowledge is represented by a fuzzy set or a fuzzy relation on the appropriate universe. Then the fuzzy set representations of the different granules are combined and the result of this combination is projected onto the universe(s) of interest. A well-known particular case of this method is the pattern of inference, named ‘generalized modus ponens’, which enables us to deduce a fuzzy conclusion from a fuzzy rule and a fuzzy fact pertaining to the universe of discourse associated with the condition part of the rule. However, this framework is rather general, and a proper application of it requires a correct understanding of the intended meaning of the pieces of knowledge to be represented by fuzzy sets. In particular, fuzzy rules may have very different semantics, which lead to different choices concerning the multiple-valued connective to be used to model the rule. In this paper we distinguish between purely gradual rules and rules with uncertain conclusion parts. Purely gradual rules are of the form ‘the more X is A, the more Y is B’, which qualitatively describes a relation between the values of X and Y, but which is not pervaded with uncertainty. Gradual rules of the form ‘the closer X is to..., the closer Y is to...’ express pieces of knowledge which are of interest for interpolative reasoning purposes.