Non-convex Fuzzy Truth Values and Their Properties

Abstract

There are many numbers of studies that discuss algebraic structures on fuzzy truth values. But, their main interest is in convex fuzzy truth values. This paper concerns with non-convex fuzzy truth values. A multi-interval truth value is defied as a collection of interval truth values. Therefore, it is not convex in general. This paper focuses on the simplest multi-interval truth values, ternary multi-interval truth values. A ternary multi-interval truth value is a non-empty subset of {0, 1, 2}. Thus, multi-interval truth value {0, 2} is not convex. Min, Max, and involution are defined over multi-interval truth values by Zadeh's extension principle. Then, it is known that a set of multi-interval truth values with the three operations is a de Morgan bi-lattices. This means that non-convex fuzzy truth values do not form a lattice any more. This paper describes functions over ternary multi-interval truth values which are expressed by logic formulas. Then, the paper clarifies that the functions are determined by the four multi-interval truth values {0}, {1}, {2}, and {0, 2}.