Three-Valued Random Fuzzy Sets
The theory of fuzzy sets was founded by Zadeh as an approach to cope with the pressing need to deal with phenomena which cannot be modelled properly by conventional mathematics because they contain factors which are fuzzy in nature. However, in this frame of the theory of fuzzy sets, we cannot discern two conflicting fuzzy conceptions properly, and therefore, the excluded middle law is violated. In this theory, one does not differ fuzzy sets from their membership functions. That is, a fuzzy set in the sense of Zadeh is equivalent to its membership function. This may be the main reason why the excluded middle law was violated. In this paper, we use the concept of three-valued fuzzy sets to provide a mathematical frame of fuzzy sets theory, such that it can eliminate the suspicion for the objective reality of membership functions of fuzzy sets. In our frame, fuzzy sets and their membership functions are not equivalent conceptions.
KeywordsThree-valued random fuzzy set Random set Fuzzy set Random variable
We thank the referee for several suggestions that improved the article.
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