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Foundations of Physics

, Volume 44, Issue 12, pp 1369–1380 | Cite as

Non-reflexive Logical Foundation for Quantum Mechanics

  • N. C. A. da Costa
  • C. de Ronde
Article

Abstract

On the one hand, non-reflexive logics are logics in which the principle of identity does not hold in general. On the other hand, quantum mechanics has difficulties regarding the interpretation of ‘particles’ and their identity, also known in the literature as ‘the problem of indistinguishable particles’. In this article, we will argue that non-reflexive logics can be a useful tool to account for such quantum indistinguishability. In particular, we will provide a particular non-reflexive logic that can help us to analyze and discuss this problem. From a more general physical perspective, we will also analyze the limits imposed by the orthodox quantum formalism to consider the existence of indistinguishable particles in the first place, and argue that non-reflexive logics can also help us to think beyond the limits of classical identity.

Keywords

Identity Non-reflexive logic Interpretation of quantum mechanics 

Notes

Acknowledgments

This work was partially supported by the following grants: VUB Project GOA67; FWO-research community W0.030.06; CONICET RES. 4541-12 (2013-2014).

References

  1. 1.
    Arenhart, J.R.B., Krasue, D.: A discussion on quantum non-individuality. J. Appl. Non-Class. Log. 22, 105–124 (2012)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bohm, D.: Quantum Theory. Dover, New York (1951)Google Scholar
  3. 3.
    Brignole, D., da Costa, N.: On supernominal Ehresmann-Dedcker universes. Math. Z 122, 342–350 (1971)CrossRefMathSciNetGoogle Scholar
  4. 4.
    da Costa, N.C.A.: Ensaio sobre os Fundamentos da Lógica, HUCITEC (1979).Google Scholar
  5. 5.
    da Costa, N.C.A.: Aspectos de la lógica atual. In: Ioda, J., Melmick, J., Melmick, S. (eds.) En Chile También hay Ciencia, pp. 221–240. Universidad de Chile, Santiago de Chile (1986)Google Scholar
  6. 6.
    da Costa, N.C.A.: Logique Classique et Non-Classique. Masson, Paris (1997)Google Scholar
  7. 7.
    da Costa, N.C.A., Bueno, O.: Non reflexive logics. Rev. Bras. Filos. 232, 181–196 (2009)Google Scholar
  8. 8.
    da Costa, N.C.A., de Ronde, C.: The paraconsistent logic of quantum superpositions. Found. Phys. 43, 845–858 (2013)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    da Costa, N.C.A., Krause, D.: Set theoretical models for quantum systems. In: Dalla Chiara, M.L., Giuntin, R., Laudisa, F. (eds.) Language, Quantum, Music, pp. 171–181. Kluwer, Dordrecht (1999)CrossRefGoogle Scholar
  10. 10.
    da Costa, N.C.A., Krause, D., Arenhart, J.R.B., Schinaider, J.: Sobre uma fundamentacao nao-reflexiva da mecanica cuantica. Sci. Stud. 10, 71–104 (2012)CrossRefGoogle Scholar
  11. 11.
    da Costa, N.C.A., Rodrigues, A.A.M.: Definability and Invariance. Stud. log. 86, 1–30 (2007)CrossRefzbMATHGoogle Scholar
  12. 12.
    da Costa, N.C.A., Routley, R.: Cause as an implication. Stud. Log. 47, 413–428 (1987)Google Scholar
  13. 13.
    de Ronde, C.: The contextual and modal character of quantum mechanics: a formal and philosophical analysis in the foundations of physics, PhD Dissertation, Utrecht University (2011).Google Scholar
  14. 14.
    de Ronde, C., Freytes, H., Domenech, G.: Interpreting the modal Kochen–Specker theorem: possibility and many worlds in quantum mechanics. Stud. Hist. Philos. Mod. Sci. 45, 11–18 (2014)CrossRefzbMATHGoogle Scholar
  15. 15.
    Domenech, G., Holik, F., Krause, D.: Q-spaces and the foundations of quantum mechanics. Found. Phys. 38, 969–994 (2008)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Fraenkel, A.A., Bar-Hillel, Y.: Foundations of Set Theory. North-Holland, Amsterdam (1958)zbMATHGoogle Scholar
  17. 17.
    French, S., Krause, D.: Identity in Physics. Clarendon Press, Oxford (2006)CrossRefGoogle Scholar
  18. 18.
    Ghirardi, G.: Sneaking a Look to God’s Cards. Princeton University Press, Princeton (2005)Google Scholar
  19. 19.
    Griffiths, D.: Introduction to Quantum Mechanics. Mc Graw-Hill, New York (1996)Google Scholar
  20. 20.
    Holland, P.R.: The Quantum Theory of Motion. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  21. 21.
    Kleene, S.: Introduction to Metamathematics. North Holland, Amsterdam (1952)zbMATHGoogle Scholar
  22. 22.
    Kochen, S., Specker, E.: “On the problem of Hidden Variables in Quantum Mechanics”. J. Math. Mech. 17, 59–87. Reprinted in Hooker 1975, 293–328 (1967)Google Scholar
  23. 23.
    Krause, D., Arenhart, J.R.B.: “Classical or non-reflexive logics? A case of semantic underdetermination”, forthcoming.Google Scholar
  24. 24.
    Liboff, R.L.: Introductory Quantum Mechanics. Addison Wesley, Reading, MA (1997)Google Scholar
  25. 25.
    Schoenfield, J.: Mathematical Logic. Addison Wesley, Reading, Boston (1967)Google Scholar
  26. 26.
    Whitehead, A.N., Russell, B.: Principia Mathematica, vol. 1. Cambridge University Press, Cambridge (1910)zbMATHGoogle Scholar
  27. 27.
    Wittgenstein, L.: Tractatus Logico-Philosophicus, (transl. by C. K. Odgen). Routledge and Kegan Paul, London (1988,).Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Universidade Federal de Santa CatarinaFlorianopolisBrazil
  2. 2.University of Buenos Aires, CONICETBuenos AiresArgentina
  3. 3.Center Leo Apostel (CLEA) and Foundations of the Exact Sciences (FUND)Brussels Free UniversityBrusselsBelgium

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