Set-Valued Extensions of Fuzzy Logic: Classification Theorems

Experts are often not 100% confident in their statements. In traditional fuzzy logic, the expert's degree of confidence 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 classification results shows the importance of interval-valued fuzzy logics.