Set theoretic foundations for fuzzy set theory, and their applications

Abstract

The search for foundations for fuzzy set theory that are both mathematically elegant and general enough to encompass the practical applications of fuzzy sets has been a long process. Currently the most well-known formalisms are those based on category theory [3]. These category-theoretic formulations, in terms of quasi-topoi [4], are quite complex however, and may be difficult to use for those without expertise in category theory. In contrast, the more pragmatic approach of [11] leads to strong restrictions on the semantics that can be used, essentially requiring that formulae are always evaluated by using the Lukasiewicz valuation. This paper presents the results of work which has provided a simple, set-theoretic basis for the foundations of fuzzy sets. This is achieved by the use of residuated logic, which generalises intuitionistic predicate logic, and it encompasses a wide range of common semantic valuations for fuzzy logic. A set theory is built on top of this underlying logic, using the method of [6] to build a cumulative hierarchy of fuzzy sets. There is an interpretation of classical set theory into the resulting theory.

We give examples of applications of this theory to constraint-based reasoning, to the integration of probabilistic and possibilistic reasoning, and to proximity-based classification. Issues of the psychological credibility of fuzzy sets will also be addressed.