×

Fuzzy set ideas in quantum logics. (English) Zbl 0789.03049

The paper describes some structures based on fuzzy set theory which are applicable to models of quantum mechanics. The author starts with the approaches of Łukasiewicz and of Zadeh, and comes through Giles’ ideas to fuzzy quantum logics based on Maczynski models.

MSC:

03G12 Quantum logic
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03E72 Theory of fuzzy sets, etc.
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Beltrametti, E. G., and Cassinelli, G. (1981).The Logic of Quantum Mechanics, Addison-Wesley, Reading, Massachusetts. · Zbl 0491.03023
[2] Birkhoff, G., and von Neumann, J. (1936). The logic of quantum mechanics,Annals of Mathematics,37, 823. · JFM 62.1061.04 · doi:10.2307/1968621
[3] Chapin, E. (1974). Set-valued set-theory, Part I,Notre Dame Journal of Formal Logic,4, 619. · doi:10.1305/ndjfl/1093891496
[4] Chapin, E. (1975). Set-valued set-theory, Part II,Notre Dame Journal of Formal Logic,5, 255. · doi:10.1305/ndjfl/1093891706
[5] Dubois, D., and Prade, H. (1980).Fuzzy Sets and Systems-Theory and Applications, Academic Press, New York. · Zbl 0444.94049
[6] Dvure?enskij, A., and Chovanec, F. (1988). Fuzzy quantum spaces and compatibility,International Journal of Theoretical Physics,27, 1069. · Zbl 0657.60004 · doi:10.1007/BF00674352
[7] Feyerabend, P. (1958). Reichenbach’s interpretation of quantum mechanics,Philosophical Studies,IX, 49. · doi:10.1007/BF00714346
[8] Giles, R. (1976). ?ukasiewicz logic and fuzzy set theory,International Journal of Man-Machine Studies,8, 315. · Zbl 0335.02037 · doi:10.1016/S0020-7373(76)80003-X
[9] Giles, R. (1977). A non-classical logic for physics, in:Selected Papers on ?ukasiewiczSentential Calculi, R. Wojcicki, Ossolineum, Wroc?aw, ed. [shortened version published inStudio Logica,33, 397 (1974)].
[10] Guz, W. (1984). Stochastic phase spaces, fuzzy sets, and statistical metric spaces,Foundations of Physics,14, 821. · doi:10.1007/BF00737552
[11] Guz, W. (1985). Fuzzy ?-algebras of physics,International Journal of Theoretical Physics,24, 481. · Zbl 0575.46052 · doi:10.1007/BF00669908
[12] Jauch, J. M. (1968).Foundations of Quantum Mechanics, Addison-Wesley, Reading, Massachusetts. · Zbl 0166.23301
[13] Kaufmann, A. (1975).Introduction to the Theory of Fuzzy Subsets, Vol. 1, Academic Press, New York. · Zbl 0332.02063
[14] ?ukasiewicz, J. (1970).Selected Works, D. Reidel, Amsterdam. · Zbl 0212.00902
[15] Mackey, G. W. (1963).Mathematical Foundations of Quantum Mechanics, Benjamin, New York. · Zbl 0114.44002
[16] Maczy?ski, M. J. (1973). The orthogonality postulate in axiomatic quantum mechanics,International Journal of Theoretical Physics,8, 353. · doi:10.1007/BF00687092
[17] Maczy?ski, M. J. (1974). Functional properties of quantum logics,International Journal of Theoretical Physics,11, 149. · doi:10.1007/BF01809565
[18] Novak, V. (1980). An attempt at Goedel-Bernays-like axiomatization of fuzzy sets,Fuzzy Sets and Systems,3, 323. · Zbl 0435.03036 · doi:10.1016/0165-0114(80)90027-5
[19] Piasecki, K. (1985). Probability of fuzzy events defined as denumerable additivity measure,Fuzzy Sets and Systems,17, 271. · Zbl 0604.60005 · doi:10.1016/0165-0114(85)90093-4
[20] Piron, C. (1976).Foundations of Quantum Mechanics, Benjamin, Reading, Massachusetts. · Zbl 0333.46050
[21] Post, E. L. (1921). Introduction to a general theory of elementary propositions,American Journal of Mathematics,43, 163. · JFM 48.1122.01 · doi:10.2307/2370324
[22] 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.
[23] Pykacz, J. (1987b). Quantum logics and soft fuzzy probability spaces,Bulletin pour les Sous-Ensembles Flous et leurs Applications,32, 150.
[24] Pykacz, J. (1988). Probability measures in the fuzzy set approach to quantum logics,Bulletin pour les Sous-Ensembles Flous et leurs Applications,37, 81.
[25] Pykacz, J. (1989). Fuzzy set description of physical systems and their dynamics,Bulletin pour les Sous-Ensembles Flous et leurs Applications,38, 102.
[26] Pykacz, J. (1990). Fuzzy quantum logics and the problem of connectives,Bulletin pour les Sous-Ensembles Flous et leurs Applications,43, 49.
[27] Reichenbach, H. (1944).Philosophic Foundations of Quantum Mechanics, University of California Press, Los Angeles.
[28] Suppes, P. (1966). The probabilistic argument for a non-classical logic of quantum mechanics,Philosophy of Science,33, 14. · Zbl 0202.29603 · doi:10.1086/288067
[29] Zadeh, L. A. (1965). Fuzzy sets,Information and Control,8, 338. · Zbl 0139.24606 · doi:10.1016/S0019-9958(65)90241-X
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.