The paradoxical success of fuzzy logic

Abstract

Fuzzy logic methods have been used successfully in many real-world applications, but the foundations of fuzzy logic remain under attack. Taken together, these two facts constitute a paradox. A second paradox is that almost all of the successful fuzzy logic applications are embedded controllers, while most of the theoretical papers on fuzzy methods deal with knowledge representation and reasoning. I hope here to resolve these paradoxes by identifying which aspects of fuzzy logic render it useful in practice, and which aspects are inessential. My conclusions are based on a mathematical result, on a survey of literature on the use of fuzzy logic in heuristic control and in expert systems, and on practical experience developing expert systems. An apparent paradoxAs is natural in a research area as active as fuzzy logic, theoreticians have investigated many formal systems, and a variety of systems have been used in applications. Nevertheless, the basic intuitions have remained relatively constant. At its simplest, fuzzy logic is a generalization of standard propositional logic from two truth values, false and true, to degrees of truth between 0 and 1.Formally, let A denote an assertion. In fuzzy logic, A is assigned a numerical value t(A), called the degree of truth of A , such that 0 5 t(A) I 1. For a sentence composed from simple assertions and the logical connectives "and" (A), "or" (v), and "not" ( 1 ) degree of truth is defined as follows: MIT Press, 1993, pp 698-703 Definition 1: Let A and B be arbitrary as- sertions. Then t ( A A B ) = min [ t(A), t(B)) t(A v B ) = max { t ( A ) , t ( B ) ] t(A) = t(B) if , either t ( B ) = t ( A ) or t(B) = 1-t(A). WA direct proof of Theorem 1 appears in the sidebar, but it can also be proved using similar results couched in more abstractProposition: Let P be a finite Boolean algebra of propositions and let z be a truthassignment function P + [0,1], supposedly truth-functional via continuous connectives. Then for all p E P, Q) E { 0, 1 ] WThe link between Theorem 1 and this proposition is that l ( A A 4) = B v (4 A -IB) is a valid equivalence of Boolean algebra. Theorem 1 is stronger in that it relies on only one particular equivalence, while the proposition is stronger because it applies to any connectives that are truth-functional and continuous (as defined in its authors'The equivalence used in Theorem 1 is rather complicated, but it is plausible intupaper).itively, and it is natural to apply it in reasoning about a set of fuzzy rules, since 7 ( A A 4 ) and B v (4 A 4 ) are both reexpressions of the classical implication 4 4 B. It was chosen for this reason, but the same result can also be proved using many other ostensibly reasonable logical aquivalences.It is important to be clear on what exactly Theorem 1 says, and what it does not say. On the one hand, the theorem applies to any more general formal system that includes the four postulates listed in Definition 1. Any extension of fuzzy logic to accommodate first-order sentences, for example, collapses to two trut...

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