On the Purportedly Paradoxical Nature of Fuzzy Logic

Elkan’s original paper purportedly showed that fuzzy logic was paradoxical in nature due to its reliance on formal bases that preclude truth values other than 0 and 1, Elkan has now modified some of his claims and arguments slightly, although he still depends on that result as his major source of insight into the technology. We are now told, for example, that fuzzy logic is paradoxical because it is successful in many applications while its foundations remain under attack. Perplexing as this situation might be from a sociological viewpoint, it is hardly a logical self-contradiction, and describing it as a paradox is totally inappropriate. Nor is a paradox implied by the claim that most theoretical fuzzy logic papers seem to deal with representation and reasoning methods, while most fuzzy logic applications have resulted in embedded controllers. The embedded controllers have been developed, of course, upon foundations provided by the representation and inferential methods of fuzzy logic. Elkan is not only unaware of this fact, but his overall analysis of the technology is colored by the strange notion that the depth and quality of deductive procedures in a controller are inferior to those in “sophisticated’ reasoning systems. For reasons of space, I will not discuss here Elkan’s statements about application of fuzzy logic to control and other intelligent reasoning systems, but will confine my comments to the formal result (Theorem 1) that remains the major basis of his claims about the purported paradoxical nature of fuzzy logic. Other assertions about the methodology -arising in some cases from superficial analyses of relevant literature and issues, but mostly out of ignorance or plain confusion are appropriately addressed by other respondents. Starting from an axiomatic characterization of fuzzy logic proposed by Gaines, and assuming that logical equivalence in fuzzy logic means equivalence in the sense of classical logic (thus implying that all classical logic theorems are also fuzzy logic theorems), Elkan shows that Gaines’ axioms imply that the only possible truth values are 0 and 1: fuzzy logic collapses into conventional logic. Anybody acquainted with fuzzy logic, however, would not have much difficulty questioning Elkan’s notion of logical equivalence; it is well known that many theorems of propositional logic are not valid in fuzzy logic. Assuming otherwise immediately leads to the result that Elkan finds so paradoxical. Applying, for example, the axioms of fuzzy logic to the law of the excluded middle a v +x, which is not a theorem of fuzzy logic, leads to the equation