Combining implicational quantifiers for equivalence ones by fuzzy connectives

Abstract

Relations between two Boolean attributes derived from data can be quantified by truth functions defined on fourfold tables corresponding to pairs of the attributes. Several classes of such quantifiers~implicational, double implicational, and equivalence ones! with truth values in the unit interval were investigated in the frame of the theory of data mining methods. The definition of double implicational quantifiers is based on the idea of conjunction of both directions implications~similarly for equivalence!. In the fuzzy logic theory, there are well-defined classes of fuzzy operators, namely t-norms representing various types of evaluations of fuzzy conjunctioñ and t-conorms representing fuzzy disjunction!. I assert that each t-norm applied to an implicational quantifier gives a double implicational quantifier. Analogously, each t-conorm applied to a double implicational quantifier gives an equivalence quantifier. Logical properties of obtained quantifiers are discussed. The method is illustrated by examples of well-known quantifiers and operators.

Construction of implicational quantifiers from fuzzy implications

“…Several results on this topic are included in [8]. A general research idea is to represent useful four-fold table quantifiers by fuzzy formulae and apply fuzzy logic methodology for dealing with them.…”