Advertisement

On Fuzzy Quantifiers

  • Helmut Thiele
Chapter
Part of the Theory and Decision Library book series (TDLD, volume 16)

Abstract

First, A.Mostowski’s concept of generalized quantifiers introduced in 1957 has been translated into fuzzy logic and has been compared with the concept of fuzzy quantifiers that was introduced by L.A.Zadeh in 1983 and has been investigated by him, by R.R.Yager and others in the following years. Secondly, using the notions of T-norm and S-norm a new class of fuzzy universal and existential quantifiers, respectively, has been created.

Thirdly, there are approaches for defining fuzzy quantifiers like “almost all,” “most,” and “many” in arbitrary universes, where the quantifier “almost-all” is used as basic quantifier.

Keywords

Fuzzy Logic Cardinal Number Existential Quantifier Syllogistic Reasoning Basic Quantifier 
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.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. Barwise and R. Cooper: Generalized Quantifiers and Natural Language. Linguistics and Philosophy 4 (1980), 159–219.CrossRefGoogle Scholar
  2. [2]
    J. Barwise and F. Feferman (Eds.): Model-Theoretic Logics. Series: Perspectives in Mathematical Logics. Springer-Verlag, 1985.Google Scholar
  3. [3]
    D. Dubois and H. Prade: On fuzzy syllogisms. Computational Intelligence 4 (1988), 171–179.CrossRefGoogle Scholar
  4. [4]
    K. Goedel: Die Vollstaendigkeit der Axiome des logischen Funktionenkalkuels. Monatshefte fuer Mathematik und Physik 37 (1930), 349–360.CrossRefzbMATHGoogle Scholar
  5. [5]
    S. Gottwald: A Note on Fuzzy Cardinals. Kybernetika 16 (1980), 156–158.zbMATHMathSciNetGoogle Scholar
  6. [6]
    S. Gottwald: Fuzzy Sets and Fuzzy Logic. Foundations of Application — from a Mathematical Point of View. Vieweg 1993.Google Scholar
  7. [7]
    A. De Luca and S. Termini: A definition of non-probabilistic entropy in the setting of fuzzy sets. Information and Control 20 (1972), 301–312.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [8]
    L. Kalmár: Constributions to the reduction theory of the decision problem. Fourth paper: Reduction to the case of a finite set of individuals. Acta Mathematica Academiae Scientiarum Hungaricae 2 (1951), 125–142.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    H.J. Keisler: Logic with the Quantifier “there exists uncountably many”. Ann. Math. Logic, 1 (1970), 1–93.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    A. Mostowski: On a generalization of quantifiers. Fundamenta mathematicae 44 (1957), 12–36.MathSciNetGoogle Scholar
  11. [11]
    V. Novák: Fuzzy Sets and their Applications. Adam Hilger 1989.Google Scholar
  12. [12]
    P. Peterson: On the Logic of Few, Many and Most. Notre Dame J. Formal Logic 20 (1979), 155–179.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    N. Rescher: Many-valued Logic. McGraw-Hill Book Company 1969.Google Scholar
  14. [14]
    H. Thiele: On Fuzzy Quantifiers. Fifth International Fuzzy Systems Association World Congress ’93, July 4–9, 1993, Seoul, Korea. Conference Proceedings. Volume I, 395–398.Google Scholar
  15. [15]
    H. Thiele: On T-Quantifiers and S-Quantifiers. Twenty-Fourth International Symposium on Multiple-Valued Logic, May 25–27, 1994, Boston, Massachusetts. Conference Proceedings, 264–269.Google Scholar
  16. [16]
    H. Thiele: On the Concept of Cardinal Number for Fuzzy Sets. Invited Paper, EUFIT’94 (European Congress on Fuzzy and Intelligent Technologies), Aachen, Germany, September 20–23, 1994. Conference Proceedings, vol. 1, 504–516.Google Scholar
  17. [17]
    B.A. Trakhtenbrot: On the algorithmic unsolvability of the decision problem in finite domains, (in Russian). Dokl. Akad. Nauk SSSR 70 (1950), 569–572.Google Scholar
  18. [18]
    R.R. Yager: Reasoning with fuzzy quantified statements, part I. Kybernetes 14 (1985), 233–240.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [19]
    R.R. Yager: Connectives and Quantifiers in Fuzzy Sets. Fuzzy Sets and Systems 40 (1991), 39–75.CrossRefzbMATHMathSciNetGoogle Scholar
  20. [20]
    L.A. Zadeh: A computational approach to fuzzy quantifiers in natural language. Comp. Math. Appl. 9 (1983), 149–184.CrossRefzbMATHMathSciNetGoogle Scholar
  21. [21]
    L.A. Zadeh: A computational theory of dispositions. In: Proc. 1984 Int. Conference Computational Linguistics (1984), 312–318. See also: [ZAD87A].Google Scholar
  22. [22]
    L.A. Zadeh: Syllogistic reasoning as a basis for combination of evidence in expert systems. In: Proceedings of IJCAL, Los Angeles, CA (1985), 417–419.Google Scholar
  23. [23]
    L.A. Zadeh: Syllogistic reasoning in fuzzy logic and its application to usuality and reasoning with dispositions. IEEE Transactions on Systems, Man, and Cybernetics 15 (1985), 754–763.zbMATHMathSciNetGoogle Scholar
  24. [24]
    L.A. Zadeh: On computational theory of dispositions. International Journal of Intelligent Systems 2 (1987), 39–63.zbMATHGoogle Scholar
  25. [25]
    L.A. Zadeh: Dispositional logic and commonsense reasoning. In: Proceedings of the Second Annual Artificial Intelligence Forum, NASA-Ames Research Center, Moffett-Field, CA (1987), 375–389.Google Scholar
  26. [26]
    L.A. Zadeh: Dispositional logic. Appl. Math. Lett. 1 (1988), 95–99.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Helmut Thiele
    • 1
  1. 1.Department of Computer Science 1University of DortmundDortmundGermany

Personalised recommendations

Cite chapter

Buy options