Algebraic structures identified with bivalent and non-bivalent semantics of experimental quantum propositions
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Algebraic structures identified with bivalent and non-bivalent semantics of experimental quantum propositions Arkady Bolotin
Received: 24 June 2019 / Accepted: 16 October 2019 © Chapman University 2019
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. Keywords Truth-value assignment · Hilbert lattice · Invariant-subspace lattices · Quantum logic · Supervaluationism · Many-valued semantics · Kochen–Specker theorem 1 Introduction To understand quantum mechanics from a logico-algebraic perspective, an assignment of truth values to experimental propositions—i.e., meaningful declarative sentences that are (or make) statements about a physical system—plays an essential role. Let us elucidate this point. Assume that any experimental proposition associated with a quantum system is represented by a closed linear subspace of the Hilbert space H characterizing the system. If all results of unperformed experiments on the quantum A. Bolotin (B) Ben-Gurion University of the Negev, Beersheba, Israel e-mail: [email protected]
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system are classically pre-determined, then each not-yet-proven experimental proposition pertaining to the system is either true or false. Suppose that A and B are compatible experimental propositions, i.e., ones that can
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