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Truth Values of Quantum Phenomena

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Abstract

In the paper, the idea of describing not-yet-verified properties of quantum objects with logical many-valuedness is scrutinized. As it is argued, to promote such an idea, the following two foundational problems of many-valued quantum logic must be decided: the problem of choosing a proper system of many-valued logic and the problem of the emergence of bivalence from logical many-valuedness. Difficulties accompanying solutions of these problems are discussed.

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Notes

  1. In the present paper, rather than being strictly restricted to spatially arranged slits, quantum interference is considered generally for any set of perfectly distinguishable alternatives.

  2. Obviously, it is possible to avoid this conclusion merely by accepting nonlocal realism (i.e., an interpretation of quantum theory in terms of ‘hidden variables’ such as Bohmian mechanics [1,2,3]). But in doing so one would confront with additional deficiencies that plague the ‘hidden variables’ approach (the analysis of those deficiencies can be found, e.g., in [4, 5])

  3. That might be such three-valued logical systems as the Kleene (strong) logic K3 or the 3-valued Łukasiewicz system [6, 7].

  4. In [8], the relation between the functions v(y ⊔ z) and F∨ ([[Y ]] v , [[Z]] v ) as well as v(y ⊓ z) and F∧ ([[Y ]] v , [[Z]] v ) is studied to examine whether Łukasiewicz operations can also be used to model conjunctions and disjunctions. As it is stated in the paper, Łukasiewicz disjunction and conjunction coincide with the truth-functions of joins and meets, namely, v(y ⊔ z) = min {v(y) + v(z), 1} and v(y ⊓ z) = max {v(y) + v(z) − 1, 0}, whenever these Łukasiewicz connectives can be defined.

  5. In fact, this idea – called many-valued quantum logic or fuzzy quantum logic – has already been developed in a series of papers [9,10,11,12,13,14]; however, for the aim of this discussion, it is not necessary to follow those papers precisely. Also, for the discussion it is immaterial to present in its entirety the generally accepted interpretation of the elements of a quantum logic – an interested reader can be referred to any textbook on quantum logic: see, for example, [15] or [16].

  6. At the same time, \(v(\sim \!\hat {P}_{x}) = 1-\langle {\Psi }|\hat {P}_{x}|{\Psi }\rangle \) represents the degree to which the not-yet-verified truth value of the proposition X is not true (that is, the degree to which the system does not possess the mentioned property prior to the verification)

  7. As stated by Bayesian approach to probability theory, probabilities are degrees of belief, not facts. Probabilities cannot be derived from facts alone. Two agents who agree on the facts can legitimately assign different prior probabilities. In this sense, probabilities are not objective, but subjective (see, e.g., [18,19,20,21]).

  8. This is reminiscent of the logical system of intuitionistic logic that lacks a complete set of truth values because its semantics is specified in terms of provability conditions.

  9. This inference concurs with the conclusion drawn in the paper [24] arguing that the major transformation from classical to quantum physics lies in the shift from intrinsic to extrinsic properties. In consequence, a compound property such as X ∨ Y may have a truth value, even though neither X nor Y has one.

References

  1. Bohm, D.: A suggested interpretation of the quantum theory in terms of “Hidden” variables. I. Phys. Rev. 85, 166–179 (1952)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Bohm, D.: A suggested interpretation of the quantum theory in terms of “Hidden” variables. II. Phys. Rev. 85, 180–193 (1952)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Bohm, D., Bub, J.: A proposed solution to the measurement problem in quantum mechanics by a hidden variable theory. Rev. Mod. Phys. 38(3), 453–469 (1966)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Rainer, D.: Advanced Quantum Mechanics: Materials and Photons. Springer-Verlag, New York (2012)

    MATH  Google Scholar 

  5. Jung, K.: Is the de Broglie-Bohm interpretation of quantum mechanics really plausible?. J. Phys.: Conf. Ser. 442(012060), 1–9 (2013)

    Google Scholar 

  6. Gottwald, S.: A Treatise on Many-Valued Logics. Research Studies Press Ltd., Baldock. Hertfordshire, England (2001)

    Google Scholar 

  7. Miller, D., Thornton, M.: Multiple valued logic: Concepts and representations. Morgan & Claypool Publishers, San Rafael CA (2008)

  8. Pykacz, J., Fra̧ckiewicz, P.: The Problem of Conjunction and Disjunction in Quantum Logics. Int. J. Theor. Phys. (2017). https://doi.org/10.1007/s10773-017-3402-y

  9. Pykacz, J. : Fuzzy quantum logics and infinite-valued Łukasiewicz logic. Int. J. Theor Phys. 33, 1403–1416 (1994)

    Article  MATH  Google Scholar 

  10. Pykacz, J.: Quantum logic as partial infinite-valued Łukasiewicz logic. Int. J. Theor. Phys. 34(8), 1697–1710 (1995)

    Article  MATH  Google Scholar 

  11. Pykacz, J.: Łukasiewicz operations in fuzzy set and many-valued representations of quantum logics. Found. Phys. 30, 1503–1524 (2000)

    Article  MathSciNet  Google Scholar 

  12. Pykacz, J.: Unification of two approaches to quantum logic: Every Birkhoff - von Neumann quantum logic is a partial infinite-valued Łukasiewicz logic. Stud. Logica 95, 5–20 (2010)

    Article  MATH  Google Scholar 

  13. Pykacz, J.: Towards many-valued/fuzzy interpretation of quantum mechanics. Int. J. Gen. Syst. 40, 11–21 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pykacz, J.: Can Many-Valued logic help to comprehend quantum phenomena. Int. J. Theor. Phys. 54, 4367–4375 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Beltrametti, E., Cassinelli, G.: The Logic of Quantum Mechanics. Addison-Wesley, Reading MA (1981)

    MATH  Google Scholar 

  16. Pitowsky, I.: Quantum Probability – Quantum Logic Lecture Notes in Physics, vol. 321. Springer-Verlag, Berlin (1989)

    MATH  Google Scholar 

  17. Caves, C., Fuchs, C., Schack, R.: Subjective probability and quantum certainty. arXiv:quant-ph/0608190 (2007)

  18. Morgan, C., Leblanc, H.: Probability theory, intuitionism, semantics, and the dutch book argument. Notre Dame Journal of Formal Logic 24(3), 289–304 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. Savage, L.: Foundations of Statistics, 2nd ed. Dover, New York (1972)

    MATH  Google Scholar 

  20. de Finetti, B.: Theory of Probability. Wiley, New York (1990)

    MATH  Google Scholar 

  21. Bernardo, J., Smith, A.: Bayesian Theory. Wiley, Chichester (1994)

    Book  MATH  Google Scholar 

  22. Losada, M., Fortin, S., Holik, F.: Classical limit and quantum logic. Int. J. Theor. Phys. 57(2), 465–475 (2017). https://doi.org/10.1007/s10773-017-3579-0

    Article  MathSciNet  Google Scholar 

  23. Landsman, N.: Between classical and quantum” arXiv:quant-ph/0506082 (2008)

  24. Kochen, S.: A reconstruction of quantum mechanics. Found. Phys. 45(5), 557–590 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. Sorkin, R.: An exercise in “anhomomorphic logic” arXiv:quant-ph/0703276 (2007)

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Acknowledgment

The author would like to thank the anonymous referee for the inspiring feedback and the insights.

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  1. Ben-Gurion University of the Negev, Beersheba, Israel

    Arkady Bolotin

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  1. Arkady Bolotin
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Bolotin, A. Truth Values of Quantum Phenomena. Int J Theor Phys 57, 2124–2132 (2018). https://doi.org/10.1007/s10773-018-3737-z

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