Fuzzy Possibility Space and Type-2 Fuzzy Variable

In this paper, we present an axiomatic approach to developing the theory of type-2 (T2) fuzziness, called fuzzy possibility theory. We first introduce some fundamental concepts in this theory, such as fuzzy possibility measure, fuzzy possibility space, and T2 fuzzy variable. The fuzzy possibility space includes three parts: the universe, an ample field, and a fuzzy possibility measure; and the fuzzy possibility measure is defined as a set function on the ample field taking on regular fuzzy variable (RFV) values. Then, we define a T2 fuzzy vector as a measurable map from a fuzzy possibility space (FPS) to the space of real vectors, and present several concepts associated with T2 fuzzy vectors, such as secondary possibility distribution function and T2 possibility distribution function. Finally, to characterize the properties of T2 fuzzy vectors via possibility distributions, we propose the marginal secondary possibility distribution function and mutually independent T2 fuzzy variables