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...

One of the goals of a certain brand of philosopher has been to give an account of language and linguistic phenomena by means of showing how sentences are to be translated into a "logically perspicuous notation" (or an "ideal language"-to use pass~ terminology). The usual reason given by such philosophers for this activity is that such a notational system will somehow illustrate the "logical form" of these sentences. There are many candidates for this notational system: (almost)ordinary first-order predicate logic (see Quine [1960]), higher-order predicate logic (see Parsons [1968, 1970]), intensional logic (see Montague [1969, 1970a, 1970b, 1971]), and transformational grammar (see Harrnan [1971]), to mention some of the more popular ones. I donor propose to discuss the general question of the correctness of this approach to the philosophy of language, nor do 1 wish to adjudicate among the notational systems mentioned here. Rather, I want to focus on one problem which must be faced by all such systems-a problem that must be discussed before one decides upon a notational system and tries to demontrate that it in fact can account for all linguistic phenomena. The general problem is to determine what we shall allow as linguistic data; in this paper I shall restrict my attention to this general problem as it appears when we try to account for certain words with non-singular reference, in particular, the words that are classified by the count/ mass and sortal/non-sortal distinctions. Nouns are normally divided into two classes: proper and common. Proper nouns themselves faU into two classes: those in one very rarely occtir with a determiner, and those in the other usually with 'the' (Connecticut is a state, The Connecticut is a river). 2 In the case of common nouns, there is general recognition that there are two quite distinct classes-at least "quite distinct"

In an interesting experimental study, Bonini et al. (1999) present partial support for truth-gap theories of vagueness. We say this despite their claim to find theoretical and empirical reasons to dismiss gap theories and despite the fact that they favor an alternative, epistemic account, which they call 'vagueness as ignorance'.We present yet more experimental evidence that supports gap theories, and argue for a semantic/pragmatic alternative that unifies the gappy supervaluationary approach together with its glutty relative, the subvaluationary approach.

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