Fuzzy logic or lukasiewicz logic: A clarification

Abstract

Pavelka [10] had shown in 1979 that the only natural way of formalizing fuzzy logic for truth-value in the unit interval [0,1] is by using Lukasiewicz's implication operator a→b = min{1, 1−a+b} or some isomorphic form of it A considerable number of other papers around the same time had attempted to formulate alternative definitions for a→b by giving intuitive justifications for them. There continues to be some confusion, however, even today about the right notion of fuzzy logic. Much of this has its origin in the use of improper “and” (“or”) and the “not” operations and a misunderstanding of some of the key differences between “proofs” or inferencing in fuzzy logic and those in Lukasiewicz's logic. We point out the need for defining the strong conjunction operator “⊗” in connection with fuzzy Modus-ponens rule and why we do not need the fuzzy Syllogism rule. We also point out the shortcomings of many of the alternative definitions of a→b, which indicate further support for Pavelka's result. We hope that these discussions help to clarify the misconceptions about fuzzy logic.