Foundations of Physics

, Volume 30, Issue 9, pp 1503–1524 | Cite as

Łukasiewicz Operations in Fuzzy Set and Many-Valued Representations of Quantum Logics

  • Jarosław Pykacz


It, is shown that Birkhoff –von Neumann quantum logic (i.e., an orthomodular lattice or poset) possessing an ordering set of probability measures S can be isomorphically represented as a family of fuzzy subsets of S or, equivalently, as a family of propositional functions with arguments ranging over S and belonging to the domain of infinite-valued Łukasiewicz logic. This representation endows BvN quantum logic with a new pair of partially defined binary operations, different from the order-theoretic ones: Łukasiewicz intersection and union of fuzzy sets in the first case and Łukasiewicz conjunction and disjunction in the second. Relations between old and new operations are studied and it is shown that although they coincide whenever new operations are defined, they are not identical in general. The hypothesis that quantum-logical conjunction and disjunction should be represented by Łukasiewicz operations, not by order-theoretic join and meet is formulated and some of its possible consequences are considered.


Probability Measure Binary Operation Quantum Logic Fuzzy Subset Orthomodular Lattice 
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.
    G. Birkhoff and J. von Neumann, “The logic of quantum mechanics,” Ann. Math. 37, 823–843 (1936).Google Scholar
  2. 2.
    P. Burmeister and M. Mączyński, “Orthomodular ( partial ) algebras and their representations,” Demonstratio Mathematica 27, 701–722 (1994).Google Scholar
  3. 4.
    M. L. Dalla Chiara and R. Giuntini, “Partial and unsharp quantum logics,” Found. Phys. 24, 1161–1177 (1994).Google Scholar
  4. 5.
    M. L. Dalla Chiara and R. Giuntini, “Physical interpretation of the 4ukasiewicz quantum logical connectives,” in C. Garola and A. Rossi, eds., The Foundations of Quantum Mechanics––Historical Analysis and Open Questions (Kluwer Academic, Dordrecht, 1995).Google Scholar
  5. 6.
    P. Février, “Les relations d'incertitude de Heisenberg et la logique,” C. R. Acad. Sci. (Paris) 204, 481–483 (1937).Google Scholar
  6. 7.
    O. Frink, Jr., “New algebras of logic,” Amer. Math. Monthly 45, 210–219 (1938).Google Scholar
  7. 8.
    R. Giles, “Łukasiewicz logic and fuzzy set theory,” Int. J. Man-Machine Studies 8, 313–327 (1976).Google Scholar
  8. 9.
    K. Husimi, “Studies on the foundations of quantum mechanics I,” Proc. Physico-Math. Soc. Jap. 19, 766–789 (1937).Google Scholar
  9. 10.
    M. Jammer, The Philosophy of Quantum Mechanics (Wiley, New York, 1974).Google Scholar
  10. 11.
    F. Kôpka, “D-posets of fuzzy sets,” Tatra Mountains Math. Publ. 1, 83–87 (1992).Google Scholar
  11. 12.
    F. Kôpka, “Compatibility of D-posets of fuzzy sets,” Tatra Mountains Math. Publ. 6, 95–102 (1995).Google Scholar
  12. 13.
    J. Łukasiewicz, Selected Works (North-Holland, Amsterdam, 1970).Google Scholar
  13. 14.
    M. J. Mączyński, “The orthogonality postulate in axiomatic quantum mechanics,” Int. J. Theor. Phys. 8, 353–360 (1973).Google Scholar
  14. 15.
    M. J. Mączyński, “Functional properties of quantum logics,” Int. J. Theor. Phys. 11, 149–156 (1974).Google Scholar
  15. 16.
    A. Peres, “Unperformed experiments have no results,” Amer. J. Phys. 46, 745–747 (1978).Google Scholar
  16. 17.
    P. Pták and S. Pulmannová, Orthomodular Structures as Quantum Logics (Kluwer Academic, Dordrecht, 1991).Google Scholar
  17. 18.
    H. Putnam, “Three-valued logic,” Phil. Stud. 8, 73–80 (1957).Google Scholar
  18. 19.
    J. Pykacz, “Affine Máczyński logics on compact convex sets of states,” Int. J. Theor. Phys. 22, 97–106 (1983).Google Scholar
  19. 20.
    J. Pykacz, “Quantum logics as families of fuzzy subsets of the set of physical states,” Preprints of the Second IFSA Congress, Tokyo 1987, pp. 437–440.Google Scholar
  20. 21.
    J. Pykacz, “Fuzzy set ideas in quantum logics,” Int. J. Theor. Phys. 31, 1767–1783 (1992).Google Scholar
  21. 22.
    J. Pykacz, “Fuzzy quantum logics and infinite-valued Łukasiewicz logic,” Int. J. Theor. Phys. 33, 1403–1416 (1994).Google Scholar
  22. 23.
    J. Pykacz, “Attempt at the logical explanation of the wave-particle duality,” in M. L. Dalla Chiara et al., eds., Language, Quantum, Music (Kluwer Academic, Dordrecht, 1999), pp. 269–282.Google Scholar
  23. 24.
    H. Reichenbach, Philosophic Foundations of Quantum Mechanics (University of California Press, Berkeley, 1944).Google Scholar
  24. 25.
    B. Riečan, “A new approach to some notions of statistical quantum mechanics,” Bull. Sous-Ens. Flous Appl. 35, 4–6 (1988).Google Scholar
  25. 26.
    C. F. von Weizsäcker, “Die Quantentheorie der einfachen Alternative,” Z. Naturfarschung 13a, 245–253 (1958).Google Scholar
  26. 27.
    L. A. Zadeh, “Fuzzy sets,” Information and Control, 8, 338–353 (1965).Google Scholar
  27. 28.
    Z. Zawirski, “Attempts at application of many-valued logic to contemporary science” (in Polish), Sprawozdania Poznańskiego Towarzystwa Przyjaciół Nauk 2–4, 6–8 (1931).Google Scholar
  28. 29.
    Z. Zawirski, “Les logiques nouvelles et le champ de leur application,” Revue de Métaphysique et de Morale 39, 503–519 (1932).Google Scholar
  29. 30.
    Z. Zawirski, “Relations between many-valued logic and the calculus of probability” (in Polish), Prace Komisji Filozoficznej Poznańskiego Towarzystwa Przyjaciół Nauk 4, 155–240 (1934).Google Scholar
  30. 31.
    F. Zwicky, “On a new type of reasoning and some of its possible consequences,” Phys. Rev. 43, 1031–1033 (1933).Google Scholar

Copyright information

© Plenum Publishing Corporation 2000

Authors and Affiliations

  • Jarosław Pykacz
    • 1
  1. 1.Instytut MatematykiUniwersytet GdańskiGdańskPoland;

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