Skip to main content
Log in

The Paraconsistent Logic of Quantum Superpositions

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

Physical superpositions exist both in classical and in quantum physics. However, what is exactly meant by ‘superposition’ in each case is extremely different. In this paper we discuss some of the multiple interpretations which exist in the literature regarding superpositions in quantum mechanics. We argue that all these interpretations have something in common: they all attempt to avoid ‘contradiction’. We argue in this paper, in favor of the importance of developing a new interpretation of superpositions which takes into account contradiction, as a key element of the formal structure of the theory, “right from the start”. In order to show the feasibility of our interpretational project we present an outline of a paraconsistent approach to quantum superpositions which attempts to account for the contradictory properties present in general within quantum superpositions. This approach must not be understood as a closed formal and conceptual scheme but rather as a first step towards a different type of understanding regarding quantum superpositions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. According to Bohr [35, p. 338] the Schrödinger wave equation is just an abstract method of calculus and it does not designate in itself any phenomena. See also [3] for discussion.

  2. What is special for a classical system, is that ‘observables’ can be described by functions on the state space. This is the main reason that, a measurement corresponding to such an observable, can be left out of the description of the theory ‘in case one is not interested in the change of state provoked by the measurement’, but ‘only interested in the values of the observables’. It is in this respect that the situation is very different for a quantum system. Observables can also be described, as projection valued measures on the Hilbert space, but ‘no definite values can be attributed to such a specific observable for a substantial part of the states of the system’. For a quantum system, contrary to a classical system, it is not true that ‘either a property or its negation is actual’.

  3. Einstein designed, in the by now famous EPR ‘paper’ [13], a definition of when a physical quantity could be considered an element of physical reality within quantum mechanics. By using this definition Einstein, Podolsky and Rosen argued against the completeness of the quantum theory. For a general discussion see [16].

  4. In an analogous fashion as Décio Krause has developed a Q-set theory which accounts for indistinguishable particles with a formal calculus “right from the start” [21].

  5. These words, according to Heisenberg himself, led him to the development of the indetermination principle in his foundational paper of 1927.

References

  1. Akama, S., Abe, J.M.: The role of inconsistency in information systems. In: Proc. of the 5th World Multiconference on Systemics, Cybernetics and Informatics (SCI’2001), Orlando, pp. 355–360 (2001)

    Google Scholar 

  2. Bacciagaluppi, G.: Topics in the modal interpretation of quantum mechanics. Doctoral dissertation, University of Cambridge, Cambridge (1996)

  3. Bokulich, P., Bokulich, A.: Niels Bohr’s generalization of classical mechanics. Found. Phys. 35, 347–371 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. da Costa, N.C.A., French, S.: Partial Truth: A Unitary Approach to Models and Scientific Reasoning. Oxford University Press, Oxford (2003)

    Google Scholar 

  5. da Costa, N.C.A., Krause, D., Bueno, O.: Paraconsistent logics and paraconsistency. In: Jacquette, D. (ed.) Handbook of the Philosophy of Science (Philosophy of Logic), pp. 791–911. Elsevier, Amsterdam (2007)

    Google Scholar 

  6. de Ronde, C.: For and against metaphysics in the modal interpretation of quantum mechanics. Philosophica 83, 85–117 (2010)

    Google Scholar 

  7. de Ronde, C.: The contextual and modal character of quantum mechanics: a formal and philosophical analysis in the foundations of physics. Ph.D. dissertation, Utrecht University (2011)

  8. DeWitt, B., Graham, N.: The Many-Worlds Interpretation of Quantum Mechanics. Princeton University Press, Princeton (1973)

    Google Scholar 

  9. Dickson, M., Dieks, D.: Modal interpretations of quantum mechanics. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Winter 2002 Edition) (2002)

    Google Scholar 

  10. Dieks, D.: The formalism of quantum theory: an objective description of reality. Ann. Phys. 7, 174–190 (1988)

    Article  MathSciNet  Google Scholar 

  11. Dieks, D.: Quantum mechanics, chance and modality. Philosophica 82, 117–137 (2010)

    Google Scholar 

  12. Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn. Oxford University Press, London (1974)

    Google Scholar 

  13. Einstein, A., Podolsky, B., Rosen, N.: Can quantum-mechanical description be considered complete? Phys. Rev. 47, 777–780 (1935)

    Article  ADS  MATH  Google Scholar 

  14. Everett, H.: ‘Relative state’ formulation of quantum mechanics. Rev. Mod. Phys. 29, 454–462 (1957)

    Article  MathSciNet  ADS  Google Scholar 

  15. Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals. McGraw-Hill, New York (1965)

    MATH  Google Scholar 

  16. Fine, A.: The Einstein-Podolsky-Rosen argument in quantum theory. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (2001). http://plato.stanford.edu/archives/win2011/entries/qt-epr/

    Google Scholar 

  17. Foulis, D.J., Piron, C., Randall, C.H.: Realism, operationalism and quantum mechanics. Found. Phys. 13, 813–841 (1983)

    Article  MathSciNet  ADS  Google Scholar 

  18. Fuchs, C., Peres, A.: Quantum theory needs no ‘interpretation’. Phys. Today 53, 70 (2000)

    Article  Google Scholar 

  19. Ghirardi, G.C., Rimini, A., Weber, T.: Unified dynamics for microscopic and macroscopic systems. Phys. Rev. D 34, 470–491 (1986)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  20. Kleene, S.C.: Introduction to Metamathematics. Van Nostrand, Princeton (1964)

    Google Scholar 

  21. Krause, D.: On a quasi-set theory. Notre Dame J. Form. Log. 33, 402–411 (1992)

    Article  MATH  Google Scholar 

  22. Leibfried, D., Knill, E., Seidelin, S., Britton, J., Blakestad, R.B., Chiaverini, J., Hume, D.B., Itano, W.M., Jost, J.D., Langer, C., Ozeri, R., Reichle, R., Wineland, D.J.: Creation of a six-atom ‘Schrödinger cat’ state. Nature 438, 639–642 (2005)

    Article  ADS  Google Scholar 

  23. Mendelson, E.: Introduction to Mathematical Logic. Taylor & Francis, New York (2010)

    Google Scholar 

  24. Nakamatsu, K., Abe, J.M., Suzuki, A.: Annotated semantics for defeasible deontic reasoning. In: Ziarko, W., Yao, Y. (eds.) RCCTC 2000, pp. 470–478. Springer, Berlin (2001)

    Google Scholar 

  25. Nakamatsu, K., Abe, J.M., Suzuki, A.: A railway interlocking safety verification systems based on abductive paraconsistent logic programming. In: Abraham, A., Ruiz-del-Solar, J., Koppen, M. (eds.) Soft Computing Systems: Design, Management and Applications, pp. 775–784. Ohmsha, Tokyo (2002)

    Google Scholar 

  26. Nakamatsu, K., Abe, J.M., Suzuki, A.: Defeasible deontic robot control based on extended vector annotated logic programming. In: Dubois, D.M. (ed.) Computer Anticipatory Systems: SASYS 2001—Fifth International Conference, pp. 490–500. Am. Inst. of Phys., New York (2002)

    Google Scholar 

  27. Ourjoumtsev, A., Jeong, H., Tualle-Brouri, R., Grangier, P.: Generation of optical ‘Schrödinger cats’ from photon number states. Nature 448, 784–786 (2007)

    Article  ADS  Google Scholar 

  28. Piron, C.: Foundations of Quantum Physics. Benjamin, Elmsford (1976)

    MATH  Google Scholar 

  29. Piron, C.: Le realisme en physique quantique: une approche selon Aristote. In: The Concept of Physical Reality (1983). Proceedings of a conference organized by the Interdisciplinary Research Group, University of Athens

    Google Scholar 

  30. Schrödinger, E.: The present situation in quantum mechanics. Naturwissenschaften 23, 807 (1935). Translated to English in Wheeler, J.A., Zurek, W.H. (eds.) Quantum Theory and Measurement. Princeton University Press, Princeton (1983)

    Article  ADS  Google Scholar 

  31. Smets, S.: The modes of physical properties in the logical foundations of physics. Log. Log. Philos. 14, 37–53 (2005)

    MathSciNet  MATH  Google Scholar 

  32. Van Fraassen, B.C.: The Scientific Image. Clarendon, Oxford (1980)

    Book  Google Scholar 

  33. Van Fraassen, B.C.: Quantum Mechanics: An Empiricist View. Clarendon, Oxford (1991)

    Book  Google Scholar 

  34. Vermaas, P.E.: A Philosophers Understanding of Quantum Mechanics. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  35. Von Weizsäcker, C.F.: La Imagen Física del Mundo. Biblioteca de Autores Cristianos, Madrid (1974)

    Google Scholar 

Download references

Acknowledgements

The authors wish to thank an anonymous referee for his/her careful reading of our manuscript and useful comments. This work was partially supported by the following grants: Ubacyt 2011/2014 635, FWO project G.0405.08 and FWO-research community W0.030.06. CONICET RES. 4541-12 (2013–2014).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to C. de Ronde.

Rights and permissions

Reprints and permissions

About this article

Cite this article

da Costa, N., de Ronde, C. The Paraconsistent Logic of Quantum Superpositions. Found Phys 43, 845–858 (2013). https://doi.org/10.1007/s10701-013-9721-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-013-9721-9

Keywords

Navigation