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Fuzzy set ideas in quantum logics

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Abstract

Fuzzy set theory language and ideas are used to express basic quantum logic notions. The possibility of replacing probabilistic interpretation of quantum mechanics by interpretation based on infinite-valued logics and fuzzy set theory is outlined. Short review of various structures encountered in the fuzzy set approach to quantum logics is given.

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References

  • Beltrametti, E. G., and Cassinelli, G. (1981).The Logic of Quantum Mechanics, Addison-Wesley, Reading, Massachusetts.

    Google Scholar 

  • Birkhoff, G., and von Neumann, J. (1936). The logic of quantum mechanics,Annals of Mathematics,37, 823.

    Google Scholar 

  • Chapin, E. (1974). Set-valued set-theory, Part I,Notre Dame Journal of Formal Logic,4, 619.

    Google Scholar 

  • Chapin, E. (1975). Set-valued set-theory, Part II,Notre Dame Journal of Formal Logic,5, 255.

    Google Scholar 

  • Dubois, D., and Prade, H. (1980).Fuzzy Sets and Systems-Theory and Applications, Academic Press, New York.

    Google Scholar 

  • Dvurečenskij, A., and Chovanec, F. (1988). Fuzzy quantum spaces and compatibility,International Journal of Theoretical Physics,27, 1069.

    Google Scholar 

  • Feyerabend, P. (1958). Reichenbach's interpretation of quantum mechanics,Philosophical Studies,IX, 49.

    Google Scholar 

  • Giles, R. (1976). Łukasiewicz logic and fuzzy set theory,International Journal of Man-Machine Studies,8, 315.

    Google Scholar 

  • Giles, R. (1977). A non-classical logic for physics, in:Selected Papers on ŁukasiewiczSentential Calculi, R. Wojcicki, Ossolineum, Wrocław, ed. [shortened version published inStudio Logica,33, 397 (1974)].

  • Guz, W. (1984). Stochastic phase spaces, fuzzy sets, and statistical metric spaces,Foundations of Physics,14, 821.

    Google Scholar 

  • Guz, W. (1985). Fuzzy σ-algebras of physics,International Journal of Theoretical Physics,24, 481.

    Google Scholar 

  • Jauch, J. M. (1968).Foundations of Quantum Mechanics, Addison-Wesley, Reading, Massachusetts.

    Google Scholar 

  • Kaufmann, A. (1975).Introduction to the Theory of Fuzzy Subsets, Vol. 1, Academic Press, New York.

    Google Scholar 

  • Łukasiewicz, J. (1970).Selected Works, D. Reidel, Amsterdam.

    Google Scholar 

  • Mackey, G. W. (1963).Mathematical Foundations of Quantum Mechanics, Benjamin, New York.

    Google Scholar 

  • Maczyñski, M. J. (1973). The orthogonality postulate in axiomatic quantum mechanics,International Journal of Theoretical Physics,8, 353.

    Google Scholar 

  • Maczyñski, M. J. (1974). Functional properties of quantum logics,International Journal of Theoretical Physics,11, 149.

    Google Scholar 

  • Novak, V. (1980). An attempt at Goedel-Bernays-like axiomatization of fuzzy sets,Fuzzy Sets and Systems,3, 323.

    Google Scholar 

  • Piasecki, K. (1985). Probability of fuzzy events defined as denumerable additivity measure,Fuzzy Sets and Systems,17, 271.

    Google Scholar 

  • Piron, C. (1976).Foundations of Quantum Mechanics, Benjamin, Reading, Massachusetts.

    Google Scholar 

  • Post, E. L. (1921). Introduction to a general theory of elementary propositions,American Journal of Mathematics,43, 163.

    Google Scholar 

  • Pykacz, J. (1987a). Quantum logics as families of fuzzy subsets of the set of physical states, inPreprints of the Second IFSA Congress, Tokyo, Vol. 2, p. 437.

    Google Scholar 

  • Pykacz, J. (1987b). Quantum logics and soft fuzzy probability spaces,Bulletin pour les Sous-Ensembles Flous et leurs Applications,32, 150.

    Google Scholar 

  • Pykacz, J. (1988). Probability measures in the fuzzy set approach to quantum logics,Bulletin pour les Sous-Ensembles Flous et leurs Applications,37, 81.

    Google Scholar 

  • Pykacz, J. (1989). Fuzzy set description of physical systems and their dynamics,Bulletin pour les Sous-Ensembles Flous et leurs Applications,38, 102.

    Google Scholar 

  • Pykacz, J. (1990). Fuzzy quantum logics and the problem of connectives,Bulletin pour les Sous-Ensembles Flous et leurs Applications,43, 49.

    Google Scholar 

  • Reichenbach, H. (1944).Philosophic Foundations of Quantum Mechanics, University of California Press, Los Angeles.

    Google Scholar 

  • Suppes, P. (1966). The probabilistic argument for a non-classical logic of quantum mechanics,Philosophy of Science,33, 14.

    Google Scholar 

  • Zadeh, L. A. (1965). Fuzzy sets,Information and Control,8, 338.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, University of Gdañsk, Gdañsk, Poland

    Jarosław Pykacz

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  1. Jarosław Pykacz
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Pykacz, J. Fuzzy set ideas in quantum logics. Int J Theor Phys 31, 1767–1783 (1992). https://doi.org/10.1007/BF00671785

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  • DOI: https://doi.org/10.1007/BF00671785

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