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The Paradoxical Success of Fuzzy Logic Charles Elkan, University of California, San Diego

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The Paradoxical Success of Fuzzy Logic Charles Elkan, University of California, San Diego I The Paradoxical Success of Fuzzy Logic Charles Elkan, University of California, San Diego Fuzzy logic methods have been used suc- Definition 1: Let A and B be arbitrary as- itively, and it is natural to apply it in rea- cessfully in many real-world applications, sertions. Then soning about a set of fuzzy rules, since but the foundations of fuzzy logic remain 7(A A 4)and B v (4A 4)are both t(A A B) = min [ t(A),t(B)) under attack. Taken together, these two reexpressions of the classical implication t(A v B) = max { t(A),t(B)] facts constitute a paradox. A second para- 4 4 B. It was chosen for this reason, but t(4)= 1 - t(A) dox is that almost all of the successful the same result can also be proved using t(A)= t(B)if A and B are logically fuzzy logic applications are embedded con- many other ostensibly reasonable logical equivalent. trollers, while most of the theoretical pa- aquivalences. pers on fuzzy methods deal with knowl- Depending how the phrase “logically equiv- It is important to be clear on what ex- edge representation and reasoning. I hope alent” is understood, Definition 1 yields actly Theorem 1 says, and what it does not here to resolve these paradoxes by identify- different formal systems. A fuzzy logic sys- say. On the one hand, the theorem applies ing which aspects of fuzzy logic render it tem is intended to allow an indefinite variety to any more general formal system that useful in practice, and which aspects are of numerical truth values. However, for includes the four postulates listed in Defin- inessential. My conclusions are based on a many notions of logical equivalence, only ition 1. Any extension of fuzzy logic to mathematical result, on a survey of litera- two different truth values are possible given accommodate first-order sentences, for ture on the use of fuzzy logic in heuristic the postulates of Definition 1. example, collapses to two truth values if it control and in expert systems, and on prac- admits the propositional fuzzy logic of Theorem 1: Given the formal system of Def- tical experience developing expert systems. Definition 1 and the equivalence used in inition 1, if l(A A 4)and B v (4A 4) the statement of Theorem 1 as a special are logically equivalent, then for any two An apparent paradox case. The theorem also applies to fuzzy set assertions A and B, either t(B)= t(A)or As is natural in a research area as active theory given the equation (A fl B‘)‘ = t(B) = 1-t(A). W as fuzzy logic, theoreticians have investi- B U (A‘ n BC),because Definition 1 can be gated many formal systems, and a variety A direct proof of Theorem 1 appears in the understood as axiomatizing degrees of of systems have been used in applications. sidebar, but it can also be proved using membership for fuzzy set intersections, Nevertheless, the basic intuitions have re- similar results couched in more abstract unions, and complements. mained relatively constant. At its simplest, On the other hand, the theorem does not fuzzy logic is a generalization of standard necessarily apply to versions of fuzzy logic Proposition: Let P be a finite Boolean al- propositional logic from two truth values, that modify or reject any of the postulates of gebra of propositions and let z be a truth- false and true, to degrees of truth between Definition 1 or the equivalence used in The- assignment function P + [0,1], supposedly 0 and 1. orem 1. However, it is possible to carry truth-functional via continuous connec- Formally, let A denote an assertion. In through the proof of the theorem in many tives. Then for all p E P, Q) E { 0, 1 ] W fuzzy logic, A is assigned a numerical value variant fuzzy logic systems. In particular, t(A),called the degree of truth of A, such The link between Theorem 1 and this propo- the theorem remains true when negation is that 0 5 t(A)I 1. For a sentence composed sition is that l(A A 4)= B v (4A -IB) is modeled by any operator in the Sugeno from simple assertions and the logical con- a valid equivalence of Boolean algebra. class,’ and when disjunction or conjunction nectives “and” (A), “or” (v), and “not” (1) Theorem 1 is stronger in that it relies on are modeled by operators in the Yager degree of truth is defined as follows: only one particular equivalence, while the classes! The theorem also does not depend proposition is stronger because it applies to on any particular definition of implication in any connectives that are truth-functional fuzzy logic. New definitions of fuzzy impli- and continuous (as defined in its authors’ cation are still being proposed as new appli- paper). cations of fuzzy logic are investigated.’ ~ The equivalence used in Theorem 1 is Of course, the last postulate of Definition An earlier version with the same title rather complicated, but it is plausible intu- 1 is the most controversial one. To preserve appeared in Proceedings of the Eleventh Na trona1 Conference on Artificial Intelligence (AAA1 ’93), MIT Press, 1993, pp 698-703 AUGUST 1994 3 Proof of Theorem 1 Theorem I; Given the formal system t(B)< 1 - r(B) < 1 - r(A), By the same reasoning as before, none of of Definition 1, if l(A A 4l) and B v the following can be true: that is if t(B) < 1 - t(B) and t(A) < r(B), (4A lB) are logically equivalent, then which happens if t(A) < t(B) < 0.5. So it 1 - r(A) < [(E)< 0.5 for any two assertions A and E, either t(B)= cannot be true that r(A) < t(B) < 0.5. !(A) < 1 - t(B)< 0.5 t(A) or r(B) = 1-t(A). Now note that the sentences -(A A 4) 1 - t(A)< 1 - t(B)< 0.5 Prmj Given the assumed equivalence, and E v (-A A 4)are both reexpressions of r(B) < t(A)< 0.5 (,(A A 4))= t(B v (-A A TB)). Now the material implication A 4B. One by one, 1 - t(B) < t(A) < 0.5 tf7(A A 7B)) = 1 - min [ r(A), 1 - t(B)] consider the seven other material implication t(B) < 1 - [(A)< 0.5 = 1 + max {-r(A), -1 + t(B)) sentences involving A and B, which are 1 - t(B) < I - t(A)< 0.5 = max [ 1 - (A), t(B)) 44B Now let x = min { r(A), 1 - r(A))and let A+yB and y = min [ t(B), 1 - t(B)].Clearly x I 0.5 and 434 y < 0.5 so if x # y. then one of the eight V A B+A f(B (4 iB)) = inequalities derived must he satisfied. Thus max {t(B),min { 1 - t(A), 1 - t(B))1. iB+A t(B)= t(A) or r(B) = 1 - r(A). B-4 The numerical expressions above are dif- -lB 44 ferent if a continuum of degrees of truth, one natu- Fuzzy logic in expert systems view of the extent of fuzzy logic applica- rally wants to restrict the notion of logical The basic motivation for fuzzy logic is tion in current commercial and industrial equivalence. In intuitive descriptions, fuzzy clear: While many ideas resemble tradi- knowledge-based systems. All the systems logic is often characterized as arising from tional assertions, they are not naturally in actual use described at the 1992 IEEE the rejection of the law of excluded middle: either true or false; uncertainty of some Intemational Conference on Fuzzy Sys- the assertion A v 4. Unfortunately, reject- sort is attached to them. Fuzzy logic is an tems are controllers, as opposed to reason- ing this law is not sufficient to avoid col- attempt to capture valid reasoning pattems ing systems. At the 1993 IEEE Conference lapse to just two truth values. Intuitionistic about uncertainty. The notion is now well on AI for Applications, no applications of logic rejects the law of excluded middle, accepted that there are many different types fuzzy logic in knowledge-based systems but the formal system of Definition 1 still of uncertainty, vagueness, and ign~rance.~ were reported. Of the 16 deployed systems collapses when logical equivalence means However, there is still debate as to what described at the 1993 AAA1 Conference on intuitionistic equivalence? (The Godel types of uncertainty are captured by fuzzy Innovative Applications of AI, three - the translations of classically equivalent sen- logic. Many papers have discussed (at a CAPE,* D~dger,~and DYCE'" systems - tences are intuitionistically equivalenL6 For high level of mathematical abstraction) the used fuzzy logic in some way. However, any sentence, the first three postulates of question of whether fuzzy logic provides none of these systems uses fuzzy logic op- Definition 1 make its degree of truth and suitable laws of thought for reasoning erators for reasoning about uncertainty. the degree of truth of its Godel translation about uncertainty - and if so, which vari- Input observations are assigned degrees of equal. Thus the proof in the sidebar can be eties of uncertainty. The question of inter- membership in fuzzy sets, but inference carried over directly.) Dubois and Prade est here is more empirical: whether or not with these degrees of membership uses note that if all the properties of a Boolean fuzzy logic is in practice an adequate for- other formalisms. algebra are preserved except for the law of malism for uncertain reasoning in knowl- In addition to DYCE, a team at IBM has excluded middle, their proposition no edge-based systems.
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