A natural interpretation of fuzzy sets and fuzzy relations

We present a new and natural interpretation of fuzzy sets and fuzzy relations where the basic notions and operations have quite natural meanings. We interpret fuzzy sets and fuzzy relations in a cumulative Heyting valued model for intuitionistic set theory, and define the basic notions and operations naturally in the model. As far as fuzzy sets and fuzzy relations are considered as extensions of crisp sets and relations, this interpretation seems to be most natural. In the interpretation the canonical embedding from the class of all sets into the model plays an important role. We distinguish generalized fuzzy sets, fuzzy subsets of crisp sets, and membership functions of fuzzy sets on crisp sets. Thus we present a foundation for developing a general theory of fuzzy sets where fuzzy subsets of different sets can be treated in a natural and uniform way. (C) 2002 Elsevier Science B.V. All rights reserved.