Qualitative Approximations of Fuzzy Sets and Non-classical Three-Valued Logics (II)

Abstract

(α,β)-qualitative approximations of fuzzy sets are studied by using a pair of subsets defined by an α-cut and a strong β-cut. This setting naturally leads to three disjoint regions and an analysis based on a three-valued logic. This study combines both an algebra view and a logic view. From the algebra view, the mathematical definition of an (α,β)-approximation of fuzzy sets is given, and algebraic operations based on various t-norms and fuzzy implications are established. From the logic view, two non-classical three-valued logics are introduced. Corresponding to these new non-classical three-valued logics, the related origins of t-norms and fuzzy implications are examined.