Properties of Trapezoid Truth Values in Fuzzy Logic

Abstract

In this paper, we consider trapezoid truth values which are truth values between interval truth values and fuzzy truth values. In trapezoid truth value, their membership values are 1 between two tops of a trepezoid are 0 at outside of two end-sides of a trapezoids. We defined the new logical operations on the set of trapezoid truth values, which are different from the definitions based on the extension principle. Because the result of operations based on the extension principle may not be trapezoid truth values, in the other word trapezoid truth values are not closed under these operations. The algebra which is composed of the trapezoid truth values and these operators satisfies De Morgan algebra like interval truth values. Furthermore, we show a necessary and sufficient condition for the trapezoid truth values to satisfy Kleene's laws. Finally, we show the properties of the sets of trapezoid truth values which are generated from finite trapezoid truth values and our logical operations. Especially, we show the maximum number of elements in the set which is generated from two trapezoid truth values, and the maximum number of elements when the generators satisfy Kleene's laws.