Algebraic structures of truth values in fuzzy control

Abstract

It is well known that in two-valued logic the set of truth values 0 and 1 with the logical operations AND(∧), OR(∀) and NOT(⌍) satisfy the axioms of Boolean algebra. The characteristics of Boolean algebra provide the fundamental properties of AI systems if they are developed based on the classical two-valued logic; for example, they can not treat ambiguous states which are frequent in the real world. Fuzzy logic was introduced to treat such ambiguous (or not certainly definable) propositions of daily life. Up until now in fuzzy logic different truth values have been used. For example, the typical truth values used in fuzzy logic are numeric values on the unit interval [0,1]. Another example of truth values used in fuzzy logic are interval truth values. These are pairs of numeric truth values (a, b), a, b ∈ [0,1], where a and b mean necessity and possibility degrees of the truth value, respectively. On the other hand, L. A. Zadeh introduced fuzzy truth values, which are fuzzy sets on [0,1]. Of course,-two-valued truth values 0 and 1 are special cases of numerical truth values, and numerical truth values are special cases of interval truth values. Furthermore, interval truth values are special cases of linguistic truth values[2].

In this paper, algebraic structures of numerical truth values, interval truth values and fuzzy truth values, are examined when three kinds of logic operations AND(∧),OR(∀) and NOT(⌍) are defined on these truth values and their fundamental properties are investigated. It will be very important to know the algebraic structures of truth values in fuzzy logic if fuzzy logic is to be used in AI systems to treat ambiguity, since the framework for treating degrees of truth or falsity is determined fundamentally by the algebraic structure of the truth values used in the fuzzy logic.