Skip to main content
Log in

Quantum Logic and Quantum Reconstruction

  • Published:
Foundations of Physics Aims and scope Submit manuscript


Quantum logic understood as a reconstruction program had real successes and genuine limitations. This paper offers a synopsis of both and suggests a way of seeing quantum logic in a larger, still thriving context.

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.

Similar content being viewed by others


  1. Note that the term as used here is not the usual lattice-theoretic term.

  2. i.e., if \(\alpha \) is any non-zero element of \(\mathfrak {L}_A\) and \(\beta \) is any non-zero element of \(\mathfrak {L}_A\), then \(i_A(\alpha )\wedge i_B(\beta )\) is a non-zero element of \(\mathfrak {L}_A\otimes \mathfrak {L}_B\).


  1. Piron, Constantin: Axiomatique quantique. Helv. Phys. Acta 37, 439–468 (1964)

    MATH  MathSciNet  Google Scholar 

  2. Keller, H.A.: Ein nicht-klassischer Hilbertscher raum. Mathematische Zeitschrift 172, 41–49 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  3. Solèr, M.P.: Characterization of Hilbert spaces with orthomodular spaces. Commun. Algebra 23, 219–243 (1995)

    Article  MATH  Google Scholar 

  4. Holland, S.S. Jr. Orthomodularity in infinite dimensions; a theorem of M. Solèr. Bull. Am. Math. Soc. 32(2), (1995)

  5. Pitowsky, I.: Quantum mechanics as a theory of probability. In: Demopoulos, W., Pitowsky, I. (eds.) Physical Theory and Its Interpretation: Essay in Honor of Jeffrey Bub. Springer, Western Ontario Series in Philosophy of Science, New York (2006)

  6. Schrödinger, Erwin: Discussion of probability relations between separated systems. Proc. Camb. Philos. Soc. 31, 555–563 (1935)

    Article  ADS  Google Scholar 

  7. Stairs, Allen: On the logic of pairs of quantum systems. Synthese 56, 47–60 (1983)

    MATH  MathSciNet  Google Scholar 

  8. Tsirelson, B.S.: Quantum generalizations of Bell’s inequality. Lett. Math. Phys. 4, 93–100 (1980)

    Article  MathSciNet  ADS  Google Scholar 

  9. Popescu, Sandu, Rohrlich, Daniel: Quantum nonlocality as an axiom. Found. Phys. 24, 379 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  10. Stairs, Allen, Bub, Jeffrey: Correlations, contextuality and quantum logic. J. Philos. Log. 42, 483–499 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Holland, S.S. Jr.: The current interest in orthomodular lattices. In Abbott, J.C. (ed.) Trends in Lattice Theory, pp. 41–126. Van Nostrand Reinhold, New York, (1970). Reprinted in The Logico-Algebraic Approach to Quantum Mechanics, Vol. 1, pp. 437–496. C.A. Hooker, D. Reidel Publishing Company, Dodrecht (1975)

  12. Halvorson, H., Bub, J.: Can quantum cryptography imply quantum mechanics? Reply to Smolin (2009). arXiv:quant-ph/0412063v1

  13. Pitowsky, Itamar: Betting on the outcomes of measurements: a Bayesian theory of quantum probability. Stud. Hist. Philos. Mod. Phys. 34, 395–414 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gleason, A.N.: Measures on the closed sub-spaces of Hilbert spaces. J. Math. Mech. 6, 885–893 (1957)

    MATH  MathSciNet  Google Scholar 

  15. Jauch, J.M., Piron, C.: On the structure of quantal propositional systems. Helv. Phys. Acta 42, 842–848 (1969)

    MATH  MathSciNet  Google Scholar 

  16. Wilce, A.: Quantum logic and probability theory. In Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (2012).

  17. Howard, Mark, Wallman, Joel, Veitch, Victor, Emerson, Joseph: Contextuality supplies the ‘magic’ for quantum computation. Nature 510, 351–355 (2014)

    ADS  Google Scholar 

  18. Cabello, A., Severini, S., Winter, A.: Graph-theoretic approach to quantum correlations. Phys. Rev. Lett. 112, 040401 (2014)

    Article  ADS  Google Scholar 

  19. Busch, Paul: Quantum states and generalized observables: a simple proof of Gleason’s theorem. Phys. Rev. Lett. 91(12), 120403 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  20. Caves, Carlton M., Fuchs, Christopher A., Manne, Kiran K., Renes, Joseph M.: Gleason-type derivations of the quantum probability rule for generalized measurements. Found. Phys. 34(2), 193–209 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  21. Cabello, Adán: Kochen-Specker theorem for a single qubit using positive operator-valued measures. Phys. Rev. Lett. 90(19), 190401 (2003)

    Article  MathSciNet  ADS  Google Scholar 

  22. Gianpiero Cattaneo, Maria Luisa Dalla Chiara, Roberto Giuntini, and Francesco Paoli. Quantum logic and nonclassical logics. In Kurt Engesser, Dov M. Gabbay, and Daniel Lehrman, editors, Handbook of Quantum Logic and Quantum Structures: Quantum Logic, pages 127–226. North Holland, Amsterdam, 2009

  23. Busch, Paul, Jaeger, Gregg: Unsharp quantum reality. Found. Phys. 40, 1341–1367 (2010)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. Stairs, Allen: POVMs and hidden variables. Phys. Lett. A 365, 268–272 (2007)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  25. Lucien Hardy. Quantum theory from five reasonable axioms (2001). arXiv e-print arXiv:quant-ph/0101012

  26. Maria Luisa Dalla Chiara and Roberto Giuntini. Quantum logics (2008). arXiv:quant-ph/0101028v2

  27. Cabello, A., Danielsen, L.E., López-Tarrida, A.J., Portillo, J.R.: Basic exclusivity graphs in quantum correlations. Phys. Rev. A 88, 032104 (2013)

    Article  ADS  Google Scholar 

  28. Cabello, Adán: Simple explanation of the quantum violation of a fundamental inequality. Phys. Rev. Lett. 110(6), 060402 (2013)

    Article  MathSciNet  ADS  Google Scholar 

  29. Bub, J.: Quantum probabilities: an information-theoretic interpretation. In: Beisbart, C., Hartmann, S. (eds.) Probabilities in Physics, pp. 231–262. Oxford University Press, Oxford (2011)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Allen Stairs.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Stairs, A. Quantum Logic and Quantum Reconstruction. Found Phys 45, 1351–1361 (2015).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: