Probabilities of Fuzzy Events Based on Scalar Cardinalities

  • Jaume Casasnovas
  • Francesc Rosselló
Part of the Advances in Intelligent and Soft Computing book series (AINSC, volume 16)


The standard set of Bell-type inequalities is satisfied by the extension of probabilities to fuzzy events based on the axiomatic definition of scalar cardinality of a fuzzy set, even though the lattice defined by the intersection, union and negation of fuzzy sets in the sense of Zadeh is not a boolean algebra.


Fuzzy Subset Probability Calculus Fuzzy Probability Fuzzy Event Axiomatic Definition 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E.G. Beltrametti and M.J. Maczinski. On a characterization of classical and nonclassical probabilities. Journal of Mathematical Physics 32 (1991), 1280–1286MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    D. Dubois. A new definition of the fuzzy cardinality of finite sets preserving the classical additivity property. Bull. Stud. Ecxch. Fuzziness Appl. (BUSEFAL) 5 (1981), 11–12.Google Scholar
  3. 3.
    R. Mesiar and M. Navara. Ts-tribes and Ts-measures. J. Math. Anal. Appl. 201 (1996), 91–102.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    J. Pycacz and E. Santos. Hidden variables in quantum logic approach reexamined. Journal of Mathematical Physics 32 (1991), 1287–1292.MathSciNetCrossRefGoogle Scholar
  5. 5.
    J. Pycacz and B. d’Hooghe. Bell-Type inequalities in fuzzy probability calculus. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 9 (2000), 263–275.Google Scholar
  6. 6.
    S. Pulmanova and V. Majernik. Bell inequalities on quantim logics. Journal of Mathematical Physics 33 (1992), 2173–2178.MathSciNetCrossRefGoogle Scholar
  7. 7.
    M. Wygralak. Questions of cardinality of finite fuzzy sets. Fuzzy Sets and Systems 102 (1999), 185–210.MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    M. Wygralak. An axiomatic approach to scalar cardinalities of fuzzy sets. Fuzzy Sets and Systems 110 (2000), 175–179.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    L.A. Zadeh. Fuzzy sets. Information and Control 8 (1965), 338–353.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    L.A. Zadeh. Probability measures of fuzzy events. J. Math. Anal. Appl. 23 (1968), 421–427.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Jaume Casasnovas
    • 1
  • Francesc Rosselló
    • 1
  1. 1.Departament de Matemàtiques i InformàticaUniversitat de les Illes BalearsPalma de MallorcaSpain

Personalised recommendations