Foundations of Physics

, Volume 30, Issue 10, pp 1663–1678 | Cite as

What Is Fuzzy Probability Theory?

  • S. Gudder
Article

Abstract

The article begins with a discussion of sets and fuzzy sets. It is observed that identifying a set with its indicator function makes it clear that a fuzzy set is a direct and natural generalization of a set. Making this identification also provides simplified proofs of various relationships between sets. Connectives for fuzzy sets that generalize those for sets are defined. The fundamentals of ordinary probability theory are reviewed and these ideas are used to motivate fuzzy probability theory. Observables (fuzzy random variables) and their distributions are defined. Some applications of fuzzy probability theory to quantum mechanics and computer science are briefly considered.

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REFERENCES

  1. 1.
    E.G. Beltrametti and S. Bugajski, “Quantum observables in classical frameworks,” Int.J.Theor.Phys. 34, 1221–1229 (1995).Google Scholar
  2. 2.
    E.G. Beltrametti and S. Bugajski, “A classical extension of quantum mechanics,” J.Phys.A: Math.Gen. 28, 3329–3334 (1995).Google Scholar
  3. 3.
    E.G.Beltrametti and S.Bugajski, “Effect algebras and statistical physical theories,” J.Math.Phys. (to appear).Google Scholar
  4. 4.
    S. Bugajski, “Fundamentals of fuzzy probability theory,” Int.J.Theor.Phys. 35, 2229–2244 (1996).Google Scholar
  5. 5.
    S.Bugajski, K.-E.Hellwig, and W.Stulpe, “On fuzzy random variables and statistical maps,” Rep.Math.Phys. (to appear).Google Scholar
  6. 6.
    A. Dvurecenskij and B. Riecan, “On joint distributions of observables for F-quantum spaces,” Fuzzy Sets Sys. 39, 65–73 (1991).Google Scholar
  7. 7.
    D. Foulis and M.K. Bennett, “Effect algebras and unsharp quantum logics,” Found.Phys. 24, 1331–1352 (1994).Google Scholar
  8. 8.
    R. Greechie and D. Foulis, “Transition to effect algebras,” Int.J.Theor.Phys. 34, 1369–1382 (1995).Google Scholar
  9. 9.
    S. Gudder, “Examples, problems and results in effect algebras,” Int.J.Theor.Phys. 35, 2365–2376 (1996).Google Scholar
  10. 10.
    S. Gudder, “Fuzzy probability theory,” Demon.Math. 31, 235–254 (1998).Google Scholar
  11. 11.
    F. Kôpka and F. Chovanec, “D-posets,” Math.Slovaca 44, 21–34 (1994).Google Scholar
  12. 12.
    G Ludwig, Foundations of Quantum Mechanics, Vols.vnI and II (Springer, Berlin, 1983-1985).Google Scholar
  13. 13.
    R. Mesiar, “Fuzzy observables,” J.Math.Anal.Appl. 48, 178–193 (1993).Google Scholar
  14. 14.
    R. Mesiar and K. Piasecki, “Fuzzy observables and fuzzy random variables,” Busefal 42, 62–76 (1990).Google Scholar
  15. 15.
    R. Yager, “A note on probabilities of fuzzy events,” Information Sci. 128, 113–129 (1979).Google Scholar
  16. 16.
    L.A. Zadeh, “Fuzzy sets,” Information Cont. 8, 338–353 (1965).Google Scholar
  17. 17.
    L.A. Zadeh, “Probability measures and fuzzy events,” J.Math.Anal.Appl. 23, 421–427 (1968).Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • S. Gudder
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of DenverDenver

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