A note on I prototype theory and fuzzy s

In a recent paper, Osherson and Smith (198 1 i present an insig of some of the contending approaches to prototype theory and arrive a.l: the conclusiii3n that the theory of fuzzy sets does not provide a satisfactory solution to the problems that stand in the v~ay of constructing an adequate theory of prototypes. In what follows, the issues raised by Osherson and Smith are commented upon and an alternative definition of the concept of a prototype is proposed. In contrast to some of the conventional definitions, the proposed definition does not associate with a given set A a ur)lique prototypical element of A. Rather, the prototypes of A constitute a fuzzy set, PI’(A), whose elements, in general, are not elements of A. Viewed in this perspective, the concept of a prototype is a fuzzy concept, and the theory of fuzzy sets provides an appropriate framework for its formulation and applications. The first issue raised by Osherson and Smith relates to the observation that, in the case of conjunctive concepts, e.g., a striped apple, the rules of combination of fuzzy sets lead to a contradiction when an object is more prototypical of a conjunction than of its constituents. The same point was raised earlier by Paul Kay (1973, who observed that, in some cases, the grade of membership of an object, u, in the intersection of two fuzzy sets A and I3 may be greater than its grade of membership in A (or B). In explaining this phenomenon (Zadeh, 1978), it was pointed out that when (a) the intersection of A and B is a subnormal fuzzy set (i.e., a fuzzy set whose maximal grade of membership is less than unity); and (b) we focus our attention on A n B by giving it a label, say C, we are, in effect, tacitly normalizing C by relativizing the grades of membership in C with respect to the maximal grade of membership in A n B. By so doing, we are generating a normalized fuzzy set Norm (A n B) which is not a subset of A and B. Consequently, an object, u, may ha-‘*. :, higher grade of membership in Norm (A n B) than in A or B.