Set-valued extensions of fuzzy logic: Classification theorems
Experts are often not 100% confident in their statements. One of the most widely used approaches to describe the different degrees of confidence is the approach of fuzzy logic. 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 thesis, we show that under reasonable conditions, the class of such sets coincides: (1) either with the class of all 1-point sets (corresponding to the traditional fuzzy logic), (2) or with the class of all subintervals of the interval [0, 1] (corresponding to the interval-valued fuzzy logic), (3) or with the class of all closed subsets of the interval [0, 1]. ^ Thus, if we want to go beyond the traditional fuzzy logic and still avoid sets of arbitrary complexity, we have to use intervals. This classification result shows the importance of interval-valued fuzzy logics. ^
Ornelas, Gilbert, "Set-valued extensions of fuzzy logic: Classification theorems" (2007). ETD Collection for University of Texas, El Paso. AAI1449742.