Algebraic structures identified with bivalent and non-bivalent semantics of experimental quantum propositions

  • Arkady BolotinEmail author
Regular Paper


The failure of distributivity in quantum logic is motivated by the principle of quantum superposition. However, this principle can be encoded differently, i.e., in different logico-algebraic objects. As a result, the logic of experimental quantum propositions might have various semantics. For example, it might have either a total semantics or a partial semantics (in which the valuation relation—i.e., a mapping from the set of atomic propositions to the set of two objects, 1 and 0—is not total), or a many-valued semantics (in which the gap between 1 and 0 is completed with truth degrees). Consequently, closed linear subspaces of the Hilbert space representing experimental quantum propositions may be organized differently. For instance, they could be organized in the structure of a Hilbert lattice (or its generalizations) identified with the bivalent semantics of quantum logic or in a structure identified with a non-bivalent semantics. On the other hand, one can only verify—at the same time—propositions represented by the closed linear subspaces corresponding to mutually commuting projection operators. This implies that to decide which semantics is proper—bivalent or non-bivalent—is not possible experimentally. Nevertheless, the latter allows simplification of certain no-go theorems in the foundation of quantum mechanics. In the present paper, the Kochen–Specker theorem asserting the impossibility to interpret, within the orthodox quantum formalism, projection operators as definite \(\{0,1\}\)-valued (pre-existent) properties, is taken as an example. This paper demonstrates that within the algebraic structure identified with supervaluationism (the form of a partial, non-bivalent semantics), the statement of this theorem gets deduced trivially.


Truth-value assignment Hilbert lattice Invariant-subspace lattices Quantum logic Supervaluationism Many-valued semantics Kochen–Specker theorem 



The author owes the anonymous referee a huge debt of gratitude for the incisive yet constructive comments which made possible to extensively improve this paper.


  1. 1.
    Fine, A., Teller, P.: Algebraic constraints on hidden variables. Found. Phys. 8, 629–636 (1978)MathSciNetCrossRefGoogle Scholar
  2. 2.
    van Dalen, Dirk: Logic and Structure. Springer, Berlin (1994)CrossRefGoogle Scholar
  3. 3.
    Michael Dunn, J., Hardegree, Gary: Algebraic Methods in Philosophical Logic. Oxford University Press, Oxford (2001)zbMATHGoogle Scholar
  4. 4.
    Kochen, S., Specker, E.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17(1), 59–87 (1967)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Peres, A.: Two simple proofs of the Kochen–Specker theorem. Phys. A Math. Gen. 24, L175–L178 (1991)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kernaghan, M., Peres, A.: Kochen–Specker theorem for eight-dimensional space. Phys. Lett. A 198, 1–5 (1995)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Cabello, A., Estebaranz, J., García-Alcaine, G.: Bell–Kochen–Specker theorem: a proof with 18 vectors. Phys. Lett. A 212, 183–187 (1996)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Yu, X.D., Guo, Y.Q., Tong, D.M.: A proof of the Kochen–Specker theorem can always be converted to a state-independent noncontextuality inequality. New J. Phys. 17, 093001 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Pavičić, M.: Classical logic and quantum logic with multiple and common lattice models. Adv. Math. Phys. 6830685, 2016 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Rédei, M.: Quantum Logic in Algebraic Approach. Springer, Dordrecht (1998)CrossRefGoogle Scholar
  11. 11.
    Birkhoff, G., von Neumann, J.: The logic of quantum mechanics. Ann. Math. 37, 823–843 (1936)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Mackey, G.: Quantum mechanics and hilbert space. Am. Math. Monthly 64, 45–57 (1957)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Varzi, A.: Supervaluationism and its logics. Mind 116, 633–676 (2007)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Keefe, R.: Theories of Vagueness. Cambridge University Press, Cambridge (2000)Google Scholar
  15. 15.
    Pykacz, J.: Quantum physics, fuzzy sets and logic. Steps towards a many-valued interpretation of quantum mechanics. Springer, Berlin (2015)CrossRefGoogle Scholar
  16. 16.
    Bèziau, J.-Y.: Bivalence, excluded middle and non contradiction. In: Behounek, L. (ed.) The Logica Yearbook 2003, pp. 73–84. Academy of Sciences, Prague (2003)Google Scholar
  17. 17.
    Kalmbach, G.: Orthomodular Lattices. Academic Press, London (1983)zbMATHGoogle Scholar
  18. 18.
    Radjavi, H., Rosenthal, P.: Invariant Subspaces. Dover Publications, Mineola (2003)zbMATHGoogle Scholar
  19. 19.
    Dichtl, M.: Astroids and pastings. Algebra Univers. 18, 380–385 (1984)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Navara, M., Rogalewicz, V.: The pasting constructions for orthomodular posets. Math. Nachr. 154, 157–168 (1991)MathSciNetCrossRefGoogle Scholar

Copyright information

© Chapman University 2019

Authors and Affiliations

  1. 1.Ben-Gurion University of the NegevBeershebaIsrael

Personalised recommendations