Interpolating between fuzzy rules using improper S-implications Original Research Article

A strong implication operator (S-implication) is a fuzzy material implication operator defined in terms of a t-norm and a strong negation as a generalization of the classical equivalence between “IF A then B” and “Not(A and not B)”. This paper is concerned with a family of improper S-implications derived from an extension of the Schweizer–Sklar family of parameterized improper t-norms defined over (−∞, +1]. Analysis of fuzzy and defuzzified interpolation outputs includes proper and improper fuzzy set outputs and exact defuzzified solutions for important special cases such as Kleene–Dienes, Reichenbach, and Łukasiewicz implications. The effect of the Schweizer–Sklar parameter on interpolation between fuzzy points is outlined. The final section of the paper illustrates the use of the method with two examples, one in political geography and the other in rule-based control.