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Local and Fuzzy Logics

  • Chapter
Modern Uses of Multiple-Valued Logic

Part of the book series: Episteme ((EPIS,volume 2))

Abstract

Fuzzy logic differs from conventional logical systems in that it aims at providing a model for approximate rather than precise reasoning.

The fuzzy logic, FL, which is described in this paper has the following principal features, (a) The truth-values of FL are fuzzy subsets of the unit interval carrying labels such as true, very true, not very true, false, more or less true, etc.; (b) The truth-values of FL are structured in the sense that they may be generated by a grammar and interpreted by a semantic rule; (c) FL is a local logic in that, in FL, the truth-values as well as the connectives such as and, or, if… then have a variable rather than fixed meaning; and (d) The rules of inference in FL are approximate rather than exact.

The central concept in FL is that of a fuzzy restriction, by which is meant a fuzzy relation which acts as an elastic constraint on the values that may be assigned to a variable. Thus, a fuzzy proposition such as ‘Nina is young’ translates into a relational assignment equation of the form R (Age (Nina)) = young in which Age (Nina) is a variable, R (Age (Nina)) is a fuzzy restriction on the values of Age (Nina), and young is a fuzzy unary relation which is assigned as a value to R(Age(Nina)).

In general, a composite fuzzy proposition translates into a system of relational assignment equations. In this paper, translation rules are developed for propositions of four basic types: Type I, of the general form ‘X is mF,’ where X is the name of an object or a variable, m is a linguistic modifier, e.g., not, very, more or less, quite, etc., and F is a fuzzy subset of a universe of discourse. Type II, of the general form X is F Y is G or ‘AT is in relation R to Y,’ where * is a binary connective, e.g., and, or, if… then, etc., and R is a fuzzy relation, e.g., much greater. Type III, of the general form ‘QX are F,’ where Q is a fuzzy quantifier, e.g., some, most, many, several, etc., and F is a fuzzy subset of a universe of discourse. And, Type IV, of the general form ‘X is F is T,’ where T is a linguistic truth-value such as true, very true, more or less true, etc. These rules may be used in combination to translate composite propositions whose constituents are instances of some of the four types in question, e.g., “’Most tall men are stronger than most short men” is more or less true,’ where the italicized words denote labels of fuzzy sets.

The translation rules for fuzzy propositions of Types I, II, III and IV induce corresponding truth valuation rules which serve to express the fuzzy truth-value of a fuzzy proposition in terms of the truth-values of its constituents. In conjunction with linguistic approximation, these rules provide a basis for approximate inference from fuzzy premises, several forms of which are described and illustrated by examples.

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Authors and Affiliations

  1. Departments of Mathematics, Electrical Engineering and Medicine, University of Southern California, Los Angeles, CA, 90007, USA

    R. E. Bellman

  2. Division of Computer Science, Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory, University of California, Berkeley, CA, 94720, USA

    L. A. Zadeh

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  1. Dept. of Philosophy, Indiana University, 47401, Bloomington, Indiana, USA

    J. Michael Dunn

  2. Computer Science Dept., Indiana University, 47401, Bloomington, Indiana, USA

    George Epstein

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Bellman, R.E., Zadeh, L.A. (1977). Local and Fuzzy Logics. In: Dunn, J.M., Epstein, G. (eds) Modern Uses of Multiple-Valued Logic. Episteme, vol 2. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1161-7_6

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