Fuzzy-Set Based Logics — an History-Oriented Presentation of Their Main Developments

FUZZY-SET BASED LOGICS — AN HISTORY-ORIENTED PRESENTATION OF THEIR MAIN DEVELOPMENTS Didier Dubois, Francesc Esteva, Llu´ıs Godo and Henri Prade 1 INTRODUCTION: A HISTORICAL PERSPECTIVE The representation of human-originated information and the formalization of com- monsense reasoning has motivated different schools of research in Artificial or Computational Intelligence in the second half of the 20th century. This new trend has also put formal logic, originally developed in connection with the foundations of mathematics, in a completely new perspective, as a tool for processing information on computers. Logic has traditionally put emphasis on symbolic processing at the syntactical level and binary truth-values at the semantical level. The idea of fuzzy sets introduced in the early sixties [Zadeh, 1965] and the development of fuzzy logic later on [Zadeh, 1975a] has brought forward a new formal framework for capturing graded imprecision in information representation and reasoning de- vices. Indeed, fuzzy sets membership grades can be interpreted in various ways which play a role in human reasoning, such as levels of intensity, similarity degrees, levels of uncertainty, and degrees of preference. Of course, the development of fuzzy sets and fuzzy logic takes its roots in concerns already encountered in non-classical logics in the first half of the century, when the need for intermediary truth-values and modalities emerged. We start by briefly surveying some of the main issues raised by this research line before describing the historical development of fuzzy sets, fuzzy logic and related issues. Jan Luk" asiewicz (1878-1956) and his followers have developed three-valued logics, and other many-valued systems, since 1920 [Luk" asiewicz, 1920]. This research was motivated by philosophical concerns as well as some technical problems in logic but not so much by issues in knowledge representation, leaving the interpretation of intermediate truth-values unclear. This issue can be related to a misunder- standing regarding the law of excluded middle and the law of non-contradiction, and the connections between many-valued logics and modal logics. The principle of bivalence, Every proposition is either true or false, Handbook of the History of Logic. Volume 8 Dov M. Gabbay and John Woods (Editors) c 2007 Elsevier BV. All rights reserved. ! 326 Didier Dubois, Francesc Esteva, Llu´ıs Godo and Henri Prade formulated and strongly defended by Chrisippus and his school in antique Greece, was for instance questioned by Epicureans, and even rejected by them in the case of propositions referring to future contingencies. Let us take an example considered already by Aristotle, namely the proposition: “There will be a sea battle to-morrow (p) and there will not be a sea battle to-morrow ( p)” ¬ This proposition “p and p” is ever false, because of the non-contradiction law and the proposition “p or p¬” is ever true, because tertium non datur. But we may fail to know the truth ¬of both propositions “there will be a sea battle to-morrow” and “there will not be a sea battle to-morrow”. In this case, at least intuitively, it seems reasonable to say that it is possible that there will be a sea battle to-morrow but at the same time, it is possible that there will not be a sea battle to-morrow. There has been a recurrent tendency, until the twentieth century many-valued logic tradition, to claim the failure of the bivalence principle on such grounds, and to consider the modality possible as a third truth value. This was apparently (unfortunately) the starting motivation of Luk" asiewicz for introducing his three- valued logic. Indeed, the introduction of a third truth-value was interpreted by Luk" asiewicz as standing for possible. However the proposition “possible p” is not the same as p, and “possible p” is not the negation of “possible p”. Hence the fact that the proposition ¬ “possible p” “possible p” ∧ ¬ may be true does not question the law of non-contradiction since “possible p” and “possible p” are not mutually exclusive. This situation leads to interpretation problems¬for a fully truth-functional calculus of possibility, since even if p is “possible” and p is “possible”, still p p is ever false. On the contrary¬ , vague or fuzzy prop∧ ¬ositions are ones such that, due to the gradual boundary of their sets of models, proposition “p and p” is not completely false in some interpretations. This is why Moisil [1972] speaks¬ of fuzzy logics as Non-Chrisippean logics. A similar confusion seems to have prevailed in the first half of the century between probability and partial truth. Trying to develop a quantitative concept of truth, H. Reichenbach [1949] proposed his probability logic in which the alterna- tive true-false is replaced by a continuous scale of truth values. In this logic he introduces probability propositions to which probabilities are assigned, interpreted as grades of truth. In a simple illustrative example, he considers the statement “I shall hit the center”. As a measure of the degree of truth of this statement, Reichenbach proposes to measure the distance r of the hit to the center and to take the truth-value as equal to 1/(1 + r). But, of course, this can be done only after the shot. However, quantifying the proposition after the hit is not a matter of belief assessment when the distance to the center is known. It is easy to figure out retrospectively that this method is actually evaluating the fuzzy proposition “I hit close to the center”. Of course we cannot evaluate the truth of the above Fuzzy Logic 327 sentence before the shot, because now it is a matter of belief assessment, for which probability can be suitable. Very early, when many-valued logics came to light, some scholars in the foundations of probability became aware that probabilities differ from what logicians call truth-values. De Finetti [1936], witnessing the emergence of many-valued logics (especially the works of Luk" asiewicz, see [Luk" asiewicz, 1970]), pointed out that uncertainty, or partial belief, as captured by probability, is a meta-concept with respect to truth degrees, and goes along with the idea that a proposition, in its usual acceptance, is a binary notion. On the contrary, the notion of partial truth (i.e. allowing for intermediary degrees of truth between true -1- and false -0-) as put forward by Luk" asiewicz [1930], leads to changing the very notion of proposition. Indeed, the definition of a proposition is a matter of convention. This remark clearly points out the fact that many-valued logics deal with many-valuedness in the logical status of propositions (as opposed to Boolean status), not with belief or probability of propositions. On the contrary, uncertainty pertains to the beliefs held by an agent, who is not totally sure whether a proposition of interest is true or false, without questioning the fact that ultimately this proposition cannot be but true or false. Probabilistic logic, contrary to many-valued logics, is not a substitute of binary logic. It is only superposed to it. However this point is not always clearly made by the forefunners of many-valued logics. Carnap [1949] also points out the difference in nature between truth-values and probability values (hence degrees thereof), precisely because “true” (resp: false) is not synonymous to “known to be true” (resp: known to be false), that is to say, verified (resp: falsified). He criticizes Reichenbach on his claim that probability values should supersede the two usual truth-values. In the same vein, H. Weyl [1946] introduced a calculus of vague predicates treated as functions defined on a fixed universe of discourse U, with values in the unit interval. Operations on such predicates f : U [0, 1] have been defined as follows: → f g = min(f, g) (conjunction); f ∩ g = max(f, g) (disjunction); ∪ f c = 1 f (negation). − Clearly, this is one ancestor of the fuzzy set calculus. However, one of the ap- proaches discussed by him for interpreting these connectives again considers truth values as probabilities. As shown above, this interpretation is dubious, first because probability and truth address different issues, and especially because probabilities are not compositional for all logical connectives (in fact, only for negation). The history of fuzzy logic starts with the foundational 1965 paper by Lotfi Zadeh entitled “Fuzzy Sets” [Zadeh, 1965]. In this paper, motivated by problems in pat- tern classification and information processing, Zadeh proposes the idea of fuzzy sets as generalized sets having elements with intermediary membership grades. In this view, a fuzzy set is characterized by its membership function, allocating a 328 Didier Dubois, Francesc Esteva, Llu´ıs Godo and Henri Prade membership grade to any element of the referential domain. The unit interval is usually taken as the range of these membership grades, although any suitable par- tially ordered set could also be used (typically: a complete lattice [Goguen, 1967]. Then, extended set theoretic operations on membership functions are defined by means of many-valued connectives, such as minimum and maximum for the intersection and the union respectively. Later, due to other researchers, it has been recognised that the appropriate connectives for defining generalized intersection and union operations was a class of associative monotonic connectives known as triangular norms (t-norms for short), together with their De Morgan dual triangular co-norms (t-conorms for short) (see Section 2.1). These operations are at the basis of the semantics of a class of mathematical fuzzy logical systems that have been thoroughly studied in the recent past, as it will be reported later in Section 3. While the many-valued logic stream has mainly been developed in a mathematical logic style, the notion of fuzzy set-based approximate reasoning as imagined by Zadeh in the seventies is much more related to information processing: he wrote in 1979 that “the theory of approximate reasoning is concerned with the deduction of possibly imprecise conclusions from a set of imprecise premises” [Zadeh, 1979a].

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