r-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 try to provide a mathematical frame of fuzzy sets theory, such that it can not only eliminate the suspicion for the objective reality of membership functions of fuzzy sets, but also avoid violating the excluded middle law. In our frame, fuzzy sets and their membership functions are not equivalent conceptions.