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Synthese

, Volume 30, Issue 3–4, pp 407–428 | Cite as

Fuzzy logic and approximate reasoning

In memory of Grigore Moisil
  • L. A. Zadeh
Article

Abstract

The term fuzzy logic is used in this paper to describe an imprecise logical system, FL, in which the truth-values are fuzzy subsets of the unit interval with linguistic labels such as true, false, not true, very true, quite true, not very true and not very false, etc. The truth-value set, ℐ, of FL is assumed to be generated by a context-free grammar, with a semantic rule providing a means of computing the meaning of each linguistic truth-value in ℐ as a fuzzy subset of [0, 1].

Since ℐ is not closed under the operations of negation, conjunction, disjunction and implication, the result of an operation on truth-values in ℐ requires, in general, a linguistic approximation by a truth-value in ℐ. As a consequence, the truth tables and the rules of inference in fuzzy logic are (i) inexact and (ii) dependent on the meaning associated with the primary truth-value true as well as the modifiers very, quite, more or less, etc.

Approximate reasoning is viewed as a process of approximate solution of a system of relational assignment equations. This process is formulated as a compositional rule of inference which subsumes modus ponens as a special case. A characteristic feature of approximate reasoning is the fuzziness and nonuniqueness of consequents of fuzzy premisses. Simple examples of approximate reasoning are: (a) Most men are vain; Socrates is a man; therefore, it is very likely that Socrates is vain. (b) x is small; x and y are approximately equal; therefore y is more or less small, where italicized words are labels of fuzzy sets.

Keywords

Approximate Solution Characteristic Feature Fuzzy Logic Unit Interval Truth Table 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References and Related Publications

  1. [1]
    J. Łukasiewicz, Aristotle's Syllogistic, Clarendon Press, Oxford, 1951.Google Scholar
  2. [2]
    J. Hintikka and P. Suppes (eds.), Aspects of Inductive Logic, North-Holland Publ. Co., Amsterdam, 1966.Google Scholar
  3. [3]
    N. Rescher, Many-Valued Logic, McGraw-Hill, New York, 1969.Google Scholar
  4. [4]
    L. A. Zadeh, ‘The Concept of a Linguistic Variable and Its Application to Approximate Reasoning’, Memorandum No. ERL-M411, Electronics Research Lab., Univ. of Calif., Berkeley, Calif., October 1973.Google Scholar
  5. [5]
    L. A. Zadeh, ‘Quantitative Fuzzy Semantics’, Information Sciences 3 (1971), 159–176.Google Scholar
  6. [6]
    D. Knuth, ‘Semantics of Context-Free Languages’, Math. Systems Theory 2 (1968), 127–145.Google Scholar
  7. [7]
    L. A. Zadeh, ‘Fuzzy Sets’, Information and Control 8 (1965), 338–353.Google Scholar
  8. [8]
    A. Kaufmann, Theory of Fuzzy Sets, Masson, Paris, 1972.Google Scholar
  9. [9]
    A. V. Aho and J. D. Ullman, The Theory of Parsing, Translation and Compiling, Prentice-Hall, Englewood Cliffs, N.J., 1973.Google Scholar
  10. [10]
    L. A. Zadeh, ‘A Fuzzy-Set-Theoretic Interpretation of Linguistic Hedges’, Jour. of Cybernetics 2 (1972), 4–34.Google Scholar
  11. [11]
    G. Lakoff, ‘Hedges: A Study of Meaning Criteria and the Logic of Fuzzy Concepts’, Jour. of Philosophical Logic 2 (1973), 458–508.Google Scholar
  12. [12]
    L. A. Zadeh, ‘A Fuzzy-Algorithmic Approach to the Definition of Complex or Imprecise Concepts’, Memorandum No. ERL-M474, Electronics Research Lab., Univ. of Calif., Berkeley, Calif., 1974.Google Scholar
  13. [13]
    M. Black, ‘Reasoning with Loose Concepts’, Dialogue 2 (1963), 1–12.Google Scholar
  14. [14]
    J. A. Goguen, ‘The Logic of Inexact Concepts’, Synthese 19 (1969), 325–373.Google Scholar
  15. [15]
    W. V. Quine, From a Logical Point of View, Harvard Univ. Press, Cambridge, 1953.Google Scholar
  16. [16]
    K. Fine, ‘Vagueness, Truth and Logic’, this issue, pp. 265–300.Google Scholar
  17. [17]
    B. C. van Fraassen, ‘Presuppositions, Supervaluations and Free Logic’, in The Logical Way of Doing Things, K. Lambert (ed.), Yale Univ. Press, New Haven, 1969.Google Scholar
  18. [18]
    A. Tarski, Logic, Semantics, Metamathematics, Clarendon Press, Oxford, 1956.Google Scholar
  19. [19]
    H. A. Simon and L. Siklossy (eds.), Representation and Meaning Experiments with Information Processing Systems, Prentice-Hall, Englewood Cliffs, N.J., 1972.Google Scholar
  20. [20]
    J. Hintikka, J. Moravcsik, and P. Suppes (eds.), Approaches to Natural Language, D. Reidel Publ. Co., Dordrecht, 1973.Google Scholar
  21. [21]
    J. A. Goguen, Jr., ‘Concept Representation in Natural and Artificial Languages: Axioms, Extensions and Applications for Fuzzy Sets’, Inter. Jour. of Man-Machine Studies 6 (1974), 513–561.Google Scholar
  22. [22]
    G. Lakoff, ‘Linguistic and Natural Logic’, in Semantics of Natural Languages, D. Davidson and G. Harman (eds.), D. Reidel Publ. Co., Dordrecht, 1971.Google Scholar
  23. [23]
    C. G. Hempel, ‘Fundamentals of Concept Formation in Empirical Science’, in International Encyclopedia of Unified Science, vol. 2, 1952.Google Scholar
  24. [24]
    C. J. Fillmore, ‘Toward A Modern Theory of Case’, pp. 361–375 in Modern Studies in English, Reibel and Schane (eds.), Prentice-Hall, Toronto, 1969.Google Scholar
  25. [25]
    W. A. Martin, ‘Translation of English into MAPL Using Winogard's Syntax, State Transition Networks, and a Semantic Case Grammar’, MIT APG Internal Memo 11, April 1973.Google Scholar
  26. [26]
    R. C. Schank, ‘Conceptual Dependency: A Theory of Natural Language Understanding’, Cognitive Psychology 3 (1972), 552–631.Google Scholar

Copyright information

© D. Reidel Publishing Company 1975

Authors and Affiliations

  • L. A. Zadeh
    • 1
  1. 1.University of CaliforniaBerkeley

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