On the Semantics of Fuzzy Linguistic Quantifiers

  • Helmut Thiele
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 33)


For definiteness we start with some notations and definitions.


Fuzzy Logic Generalize Quantifier Cardinal Number Syllogistic Reasoning Finite Universe 
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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  1. 1.
    J. Barwise and R. Cooper. Generalized Quantifiers and Natural Language. Linguistics and Philosophy 4, 159–219, 1980.CrossRefGoogle Scholar
  2. 2.
    J. Barwise and F. Feferman (editors). Model-Theoretic Logics. Perspectives in Mathematical Logics. Springer-Verlag, 1985.Google Scholar
  3. 3.
    P. Bosc and L. Lietard. Monotonic quantified statements and fuzzy integrals. In: Proc. 1st Internat. Joint Conf. of Nafips, Ifis and Nasa, pages 8–12, 1994.Google Scholar
  4. 4.
    Didier Dubois and Henri Prade. Fuzzy cardinality and the modeling of imprecise quantification. Fuzzy Sets and Systems 16, 199–230, 1985.Google Scholar
  5. 5.
    Didier Dubois and Henri Prade. On fuzzy syllogisms. Computational Intelligence 4,171–179, 1988.CrossRefGoogle Scholar
  6. 6.
    J. Van Eijck. Generalized quantifiers and traditional logic. In: J. Van Ben-Them et al. (editors), Generalized Quantifiers, Theory and Applications. Foris, 1985.Google Scholar
  7. 7.
    KurtDel. Die Vollständigkeit der Axiome des logischen Funktionenkalkiils. Monatshefte für Mathematik und Physik 37, 349–360, 1930.MATHCrossRefGoogle Scholar
  8. 8.
    Siegfried Gottwald. A Note on Fuzzy Cardinals. Kybernetika 16, 156–158, 1980.Google Scholar
  9. 9.
    Siegfried Gottwald. Fuzzy Sets and Fuzzy Logic. Foundations of Application from a Mathematical Point of View. Vieweg, Braunschweig, Wiesbaden, 1993.Google Scholar
  10. 10.
    R. HÄHnle. Commodious axiomatization of quantifiers in multiple-valued logic. In: The Twenty-Sixth International Symposium on Multiple-Valued Logic, Santiago de Compostela, Spain, May 29–31, 1996.Google Scholar
  11. 11.
    K. Hartig. Über einen Quantifikator mit zwei Wirkungsbereichen. In: Proceedings, pages 31–36, Tihany, Hungary, 1962.Google Scholar
  12. 12.
    Kurt Hauschild. Zum Vergleich von Härtigquantor und Rescherquantor. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 27, 255264, 1981.Google Scholar
  13. 13.
    L. Hella. Definability hierarchies of generalized quantifiers. Ann. Math. Logic 1990.Google Scholar
  14. 14.
    L. Kalmar. Contributions 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, 125–142, 1951.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    H. J. Keisler. Logic with the Quantifier “there exists uncountably many”. Ann. Math. Logic 1, 1–93, 1970.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    P. LindstrÖM. First order predicate logic with generalized quantifiers. Theoria 32, 186–195, 1966.MathSciNetMATHGoogle Scholar
  17. 17.
    Yaxin Liu and Etienne E. Kerre. An overview of fuzzy quantifiers. (I). Interpretations. Fuzzy Sets and Systems 95, 1–21, 1998.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Yaxin LIu and Etienne E. Kerre. An overview of fuzzy quantifiers. (II). Reasoning and applications. Fuzzy Sets and Systems 95, 135–146, 1998.MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Aldo DE Luca and Settimo Termini. A definition of a non-probabilistic entropy in the setting of fuzzy sets theory. Information and Control 20, 301–312, 1972.Google Scholar
  20. 20.
    Radko Mesiar and Helmut Thiele. On T-quantifiers and S-quantifiers. To appear.Google Scholar
  21. 21.
    A. MostcwSki. On a generalization of quantifiers. Fundamenta mathematicae 44, 12–36, 1957.MathSciNetGoogle Scholar
  22. 22.
    VilÉM Novak. Fuzzy Sets and Their Applications. Adam Hilger, Bristol, Philadelphia, 1989.Google Scholar
  23. 23.
    P. Peterson. On the Logic of Few, Many and Most. Notre Dame J. Formal Logic 20, 155–179, 1979.MATHCrossRefGoogle Scholar
  24. 24.
    Henri Prade. A note on the evaluation of conditions involving vague quantifiers in presence of imprecise or uncertain information. Bull. Stud. Exchanges Fuzziness Appl. (Busefal) 32, 1987.Google Scholar
  25. 25.
    D. Ralescu. Cardinality, quantifiers, and the aggregation of fuzzy criteria. Fuzzy Sets and Systems 69, 355–365, 1995.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    N. Rescher. Plurality-quantification (abstract). The Journal of Symbolic Logic 27, 373–374, 1962.MathSciNetGoogle Scholar
  27. 27.
    N. Rescher. Many-valued Logic. McGraw-Hill Book Company, 1969.Google Scholar
  28. 28.
    R. Rovatti and C. Fantuzzi. s-norm aggregation of infinite collections. Fuzzy Sets and Systems 84, 255–269, 1996.MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Helmut Thiele. On Fuzzy Quantifiers. In: Proceedings: Fifth International Fuzzy Systems Association World Congress ’83,volume I, pages 395–398, Seoul, Korea, July 4–9, 1993.Google Scholar
  30. 30.
    Helmut Thiele On T-Quantifiers and S-Quantifiers. In: The Twenty-Fourth International Symposium on Multiple-Valued Logic,pages 264–269, Boston, Massachusetts, May 22–25, 1994.Google Scholar
  31. 31.
    Helmut Thiele. On the Concept of Cardinal Number for Fuzzy Sets. In: EU-Fit ’84—European Congress on Fuzzy and Intelligent Technologies,volume 1, pages 504–516, Aachen, Germany, September 20–23, 1994. Invited Paper.Google Scholar
  32. 32.
    Helmut Thiele. On Fuzzy Quantifiers. In: Z. Bien and K. C. Min (editors), Fuzzy Logic and its Applications, Information Systems, and Intelligent Systems, pages 343–352. Kluwer Academic Publishers, Dordrecht, 1995.Google Scholar
  33. 33.
    B. A. Trakhtenbrot. On the algorithmic unsolvability of the decision problem in finite domains (in Russian). Dokl. Akad. Nauk Sssr 70, 569–572, 1950.Google Scholar
  34. 34.
    Dag Westerstahl. Quantifiers in Formal and Natural Languages. In: D. Gabbay and F. Guenthner (editors), Handbook of Philosophical Logic, volume IV, pages 1–131. Reidel, Dordrecht, 1989.Google Scholar
  35. 35.
    Maciej Wygralak. Vaguely Defined Objects. Kluwer Academic Publishers, Dordrecht, 1995.Google Scholar
  36. 36.
    Ronald R. Yager. Quantified propositions in a linguistic logic. International Journal of Man Machine Studies 19, 195–227, 1983.MATHCrossRefGoogle Scholar
  37. 37.
    Ronald R. Yager. Reasoning with fuzzy quantified statements, part I. Kybernetes 14, 233–240, 1985.MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Ronald R. Yager. Connectives and Quantifiers in Fuzzy Sets. Fuzzy Sets and Systems 40, 39–75, 1991.MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Ronald R. Yager.Families of Owa operators. Fuzzy Sets and Systems 59, 125–148, 1993.MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Ronald R. P Ager, S. Ovchinnikov,R. M. Tong and H. T. Nguyen (ed-itors). Fuzzy Sets and ApplicationsSelected Papers by L. A. Zadeh. John Wiley and Sons, New York, 1987.Google Scholar
  41. 41.
    Lotfi A. Zadeh. A computational approach to fuzzy quantifiers in natural language. Comp. Math. Appl. 9, 149–184, 1983.MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Lotfi A. Zadeh. A computational theory of dispositions. In: Proc. 1984 Int. Conference Computational Linguistics,pages 312–318, 1984. See also [47].Google Scholar
  43. 43.
    Lotfi A. Zadeh. A theory of commonsense knowledge. In: H. J. Skala et al. (editors), Aspects of Vagueness, pages 257–295. Reidel, Dordrecht, 1984.CrossRefGoogle Scholar
  44. 44.
    Lotfi A. Zadeh. Syllogistic reasoning as a basis for combination of evidence in expert systems. In: Proceedings of Ijcal, pages 417–419, Los Angeles, CA, 1985.Google Scholar
  45. 45.
    Lotfi A. Zadeh. Syllogistic reasoning in fuzzy logic and its application to usuality and reasoning with dispositions Ieee Transactions on Systems, Man, and Cybernetics 15, 754–763, 1985.CrossRefGoogle Scholar
  46. 46.
    Lotfi A. Zadeh. Dispositional logic and commonsense reasoning. In: Proceedings of the Second Annual Artificial Intelligence Forum, pages 375–389, Moffett-Field, CA, 1987. Nasa—Ames Research Center.Google Scholar
  47. 47.
    Lotfi A. Zadeh. On computational theory of dispositions. International Journal of Intelligent Systems 2, 39–63, 1987.MATHGoogle Scholar
  48. 48.
    Lotfi A. Zadeh.Pruf — A Meaning Representation Language for Natural Languages. In: Yager et al. [40], pages 499–568.Google Scholar
  49. 49.
    Lotfi A. Zadeh. Test-score semantics as a basis for a computational approach to the representation of meaning. In: Yager et al. [40], pages 655–684.Google Scholar
  50. 50.
    Lotfi A. Zadeh. Dispositional logic. Appl. Math. Lett. 1, 95–99, 1988.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

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

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