Fuzzy logic or Lukasiewicz logic: A clarification

Abstract

We discuss some issues on fuzzy logic developed by Pavelka [Z. Math. Logic 25 (1979) 45–52, 119–134, 447–464] vs. the many-valued Lukasiewicz's logic. The focus is on the proper choice of fuzzy implication operations, a question which has been addressed many times in the fuzzy research literature. Pavelka had shown in 1979 that the only natural way of formalizing fuzzy logic for truth values 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. Many other papers around the same time had attempted to formulate alternative definitions for a → b by giving intuitive justifications. There continues to be some confusion, however, even today about the right notion of fuzzy logic. Much of this confusion lies in the use of improper “an” (“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 inference rule. We formulate two requirements of the fuzzy implication operator, which are satisfied by Lukasiewicz's a → b, but which fail for many of the alternative definitions for “→”.