Set-Valued Extensions of Fuzzy Logic: Classi cation Theorems

—Experts are often not 100% con dent in their statements. In traditional fuzzy logic, the expert's degree of con dence in each of his or her statements is described by a number from the interval [0, 1]. However, due to similar uncertainty, an expert often cannot describe his or her degree by a single number. It is therefore reasonable to describe this degree by, e.g., a set of numbers. In this paper, we show that under reasonable conditions, the class of such sets coincides either with the class of all 1-point sets (i.e., with the traditional fuzzy set set of all numbers), or with the class of all subintervals of the interval [0, 1], or with the class of all closed subsets of the interval [0, 1]. Thus, if we want to go beyond standard fuzzy logic and still avoid sets of arbitrary complexity, we have to use intervals. These classi cation results shows the importance of interval-valued fuzzy logics. I. FORMULATION OF THE PROBLEM A. Fuzzy Logic: Brief Reminder In classical (2-valued) logic, every statement is either true or false. Such a 2-valued logic is often not adequate in describing expert knowledge, because experts are usually not fully con dent about their statements. To formally describe this uncertainty in human reasoning, L. A. Zadeh introduced the notion of fuzzy logic; see, e.g., [2], [4]. In fuzzy logic, a person's degree of con dence is described by a number from the interval [0, 1], so that absolute con dence in a statement corresponds to 1, absolute con dence in its negation corresponds to 0, and intermediate values correspond to intermediate degrees of con dence. In fuzzy logic, once we know the degree of con dence a in a statement A and the degree of con dence b in a statement B, we usually estimate the degree of con dence in composite statements A ∧B and A ∨B as, correspondingly, a ∧ b def = min(a, b)