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.
This chapter investigates the rationale for having the lexical categories or features mass and count. Some theories make the features be syntactic; others make it be semantic. It is concluded here that none of the standard accounts of their function actually serve the purpose for which they are adopted, and that we should instead remove these features from the lexicon and have lexical nouns be neither +mass nor +count. But on the other hand, if every lexical noun could be characterized by both +mass and +count, then various of the desiderata would be captured. So we conclude that lexical nouns should be neither +mass nor +count, and both +mass and +count. Although this investigation is carried out in English, the moral holds for any ‘number marking’ language. Furthermore, the resulting theory is the one that is naturally congenial to classifier languages, showing a hithertofore unnoticed similarity between the two language classes. (However, languages that are of neither type … such as Dene Su ̨łiné, Yudja, and Karitianan … require some totally distinct vision of a mass-count distinction.)
This study examines the problem of belief revision, defined as deciding which of several initially accepted sentences to disbelieve, when new information presents a logical inconsistency with the initial set. In the first three experiments, the initial sentence set included a conditional sentence, a non-conditional (ground) sentence, and an inferred conclusion drawn from the first two. The new information contradicted the inferred conclusion. Results indicated that conditional sentences were more readily abandoned than ground sentences, even when either choice would lead to a consistent belief state, and that this preference was more pronounced when problems used natural language cover stories rather than symbols. The pattern of belief revision choices differed depending on whether the contradicted conclusion from the initial belief set had been a modus ponens or modus tollens inference. Two additional experiments examined alternative model-theoretic definitions of minimal change to a belief state, using problems that contained multiple models of the initial belief state and of the new information that provided the contradiction. The results indicated that people did not follow any of four formal definitions of minimal change on these problems. The new information and the contradiction it offered was not, for example, used to select a particular model of the initial belief state as a way of reconciling the contradiction.The preferred revision was to retain only those initial sentences that had the same, unambiguous truth value within and across both the initial and new information sets. The study and results are presented in the context of certain logicbased formalizations of belief revision, syntactic and model-theoretic representations of belief states, and performance models of human deduction. Principles by which some types of sentences might be more "entrenched" than others in the face of contradiction are also discussed from the perspective of induction and theory revision.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.