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

Abstract

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.

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Acknowledgements

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.

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Correspondence to Arkady Bolotin.

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Bolotin, A. Algebraic structures identified with bivalent and non-bivalent semantics of experimental quantum propositions. Quantum Stud.: Math. Found. 7, 363–371 (2020). https://doi.org/10.1007/s40509-019-00212-7

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Keywords

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