Several extensions of truth values in fuzzy logic

  • Masao Mukaidono
Communications 3B Logics for AI
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1609)


Truth values in fuzzy logic are extended into interval truth values {[a, b]: 0≤ab≤1} from the numerical truth values of [0, 1]. Next we extend this interval truth values into two directions; one is to n-dimensional case such as [a 1,...,a n ] where a 1≤...≤a n , and the other is to dropping the condition ab in [a, b]. We examine the algebraic structures of these truth values on which three kinds of logic operations AND(·), OR(∨), NOT(⊃ are defined, where we interpret (x·y), (xy), 0 and (⊃x) as min(x, y), max(x, y), and (1−x), respectively. It is shown that we can find out structures of Kleene algebra and De Morgan algebra in these truth values.


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  1. 1.
    Zadeh, L. A.: Fuzzy Sets, Inf. and Control, 8, pp.304–311, 1965MathSciNetCrossRefGoogle Scholar
  2. 2.
    Mukaidono, M.: On some properties of fuzzy logic. Systems Computers Controls, Vol. 6, No. 2, pp.36–43, 1975.MathSciNetGoogle Scholar
  3. 3.
    Mukaidono, M.: A set of independent and complete axioms for a fuzzy algebra (Kleene algebra), Proceedings of the 11th International Symposium on Multiple-Valued Logic, IEEE, pp.27–34, 1981.Google Scholar
  4. 4.
    Mukaidono, M.: Algebraic structures of fuzzy interval truth values, Proceedings of FUZZ-IEEE’97, Vol. II, pp.699–705, 1997.Google Scholar
  5. 5.
    Zadeh, L.A.: The concept of a linguistic variable and its application to approximate reasoning (I), (II), (III). Inf. and Science, 8, pp.199–249; 8, pp.301–359;9 pp.43–80 (1975).MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Mukaidono, M. and H. Kikuchi: A Proposal on Fuzzy Interval Logic, Journal of Japan Society for Fuzzy Theory and Systems, vol. 2, No. 2, 1990, pp.97–11Google Scholar
  7. 7.
    Mukaidono, M.: Algebraic Structures of Truth Values in Fuzzy Logic, Fuzzy Logic and Fuzzy, Control, Lecture Notes in Artificial Intelligence 833, Springer-Verlag, pp.15–21, 1994Google Scholar
  8. 8.
    Mukaidono, M.: Kleene algebras in fuzzy truth values, Proceedings of International Workshop on Rough Sets, Fuzzy Sets, and Machine Discovery, pp.37–43, 1996Google Scholar
  9. 9.
    Ginsberg, M. L.: Multi-valued logics: a uniform approach to reasoning in artificial intelligence, Computational Intelligence, Vol. 4, pp.265–319, 1988.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Masao Mukaidono
    • 1
  1. 1.Department of Computer ScienceMeiji UniversityKawasaki-shiJapan

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