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Non-reflexive logical foundation for quantum mechanics. (English) Zbl 1314.81021

Summary: 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.

MSC:

81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
03A10 Logic in the philosophy of science
81P05 General and philosophical questions in quantum theory
81S05 Commutation relations and statistics as related to quantum mechanics (general)
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References:

[1] Arenhart, J.R.B., Krasue, D.: A discussion on quantum non-individuality. J. Appl. Non-Class. Log. 22, 105-124 (2012) · Zbl 1400.81009 · doi:10.1080/11663081.2012.682447
[2] Bohm, D.: Quantum Theory. Dover, New York (1951) · Zbl 0048.21802
[3] Brignole, D., da Costa, N.: On supernominal Ehresmann-Dedcker universes. Math. Z 122, 342-350 (1971) · Zbl 0212.02101 · doi:10.1007/BF01110169
[4] da Costa, N.C.A.: Ensaio sobre os Fundamentos da Lógica, HUCITEC (1979). · Zbl 0215.32603
[5] Costa, NCA; Ioda, J. (ed.); Melmick, J. (ed.); Melmick, S. (ed.), Aspectos de la lógica atual, 221-240 (1986), Santiago de Chile
[6] da Costa, N.C.A.: Logique Classique et Non-Classique. Masson, Paris (1997)
[7] da Costa, N.C.A., Bueno, O.: Non reflexive logics. Rev. Bras. Filos. 232, 181-196 (2009)
[8] da Costa, N.C.A., de Ronde, C.: The paraconsistent logic of quantum superpositions. Found. Phys. 43, 845-858 (2013) · Zbl 1272.81014 · doi:10.1007/s10701-013-9721-9
[9] Costa, NCA; Krause, D.; Dalla Chiara, ML (ed.); Giuntin, R. (ed.); Laudisa, F. (ed.), Set theoretical models for quantum systems, 171-181 (1999), Dordrecht · Zbl 1024.03054 · doi:10.1007/978-94-017-2043-4_16
[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) · doi:10.1590/S1678-31662012000100004
[11] da Costa, N.C.A., Rodrigues, A.A.M.: Definability and Invariance. Stud. log. 86, 1-30 (2007) · Zbl 1125.03030 · doi:10.1007/s11225-007-9049-6
[12] da Costa, N.C.A., Routley, R.: Cause as an implication. Stud. Log. 47, 413-428 (1987)
[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).
[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) · Zbl 1282.81018 · doi:10.1016/j.shpsb.2013.10.003
[15] Domenech, G., Holik, F., Krause, D.: Q-spaces and the foundations of quantum mechanics. Found. Phys. 38, 969-994 (2008) · Zbl 1161.81386 · doi:10.1007/s10701-008-9246-9
[16] Fraenkel, A.A., Bar-Hillel, Y.: Foundations of Set Theory. North-Holland, Amsterdam (1958) · Zbl 0082.26203
[17] French, S., Krause, D.: Identity in Physics. Clarendon Press, Oxford (2006) · doi:10.1093/0199278245.001.0001
[18] Ghirardi, G.: Sneaking a Look to God’s Cards. Princeton University Press, Princeton (2005) · Zbl 1138.81006
[19] Griffiths, D.: Introduction to Quantum Mechanics. Mc Graw-Hill, New York (1996)
[20] Holland, P.R.: The Quantum Theory of Motion. Cambridge University Press, Cambridge (1995) · Zbl 0854.00009
[21] Kleene, S.: Introduction to Metamathematics. North Holland, Amsterdam (1952) · Zbl 0047.00703
[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) · Zbl 0156.23302
[23] Krause, D., Arenhart, J.R.B.: “Classical or non-reflexive logics? A case of semantic underdetermination”, forthcoming. · Zbl 1272.81014
[24] Liboff, R.L.: Introductory Quantum Mechanics. Addison Wesley, Reading, MA (1997) · Zbl 0891.00009
[25] Schoenfield, J.: Mathematical Logic. Addison Wesley, Reading, Boston (1967)
[26] Whitehead, A.N., Russell, B.: Principia Mathematica, vol. 1. Cambridge University Press, Cambridge (1910) · JFM 41.0083.02
[27] Wittgenstein, L.: Tractatus Logico-Philosophicus, (transl. by C. K. Odgen). Routledge and Kegan Paul, London (1988,). · JFM 48.1128.13
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