Logic of Propositions in Topos Quantum Theory

  • Cecilia Flori
Part of the Lecture Notes in Physics book series (LNP, volume 944)


In Chap.  10 of the first series of lecture notes on topos quantum theory [26] we showed that quantum propositions were represented by clopen sub-objects of the spectral presheaf \( \underline {\Sigma }\) [26, Ch.10, Sec.1]. The collection of all such clopen sub-object, which we denoted by \(Sub_{cl}( \underline {\Sigma })\), was shown to form a Heyting algebra [26, Th.10.2], hence the logic of quantum theory derived from the topos approach is an intuitionistic logic. In this chapter we will explain some recent results obtained in [15] in which it is shown that \(Sub_{cl}( \underline {\Sigma })\) is not only a complete Heyting algebra , but also a complete co-Heyting algebra , therefore quantum logic is represented by a complete bi-Heyting algebra where two types of implications and negations are present.


  1. 15.
    A. Doering, Topos-based logic for quantum systems and bi-Heyting algebras (2012). arXiv:1202.2750 [quant-ph]Google Scholar
  2. 22.
    A. Doering, C.J. Isham, A topos foundation for theories of physics. II. Daseinisation and the liberation of quantum theory. J. Math. Phys. 49, 053516 (2008). quant-ph/0703062 [quant-ph]Google Scholar
  3. 26.
    C. Flori, A First Course in Topos Quantum Theory. Lecture Notes in Physics, vol. 868 (Springer, Heidelberg, 2013)Google Scholar

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© Springer International Publishing AG 2018

Authors and Affiliations

  • Cecilia Flori
    • 1
  1. 1.Computing and Mathematical SciencesThe Waikato UniversityHamiltonNew Zealand

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