Truth Values of Quantum Phenomena

Abstract

In the paper, the idea of describing not-yet-verified properties of quantum objects with logical many-valuedness is scrutinized. As it is argued, to promote such an idea, the following two foundational problems of many-valued quantum logic must be decided: the problem of choosing a proper system of many-valued logic and the problem of the emergence of bivalence from logical many-valuedness. Difficulties accompanying solutions of these problems are discussed. Introducing logical many-valuedness in quantum mechanicsThe argument claiming that quantum theory could not be comprehended on the grounds of classical two-valued logic is rather straightforward and goes like this.Let us consider a typical quantum interference experiment where a quantum particle being released from a source is absorbed by a screen after passing through a two-slit barrier 1 . Suppose that immediately behind that barrier are placed two which-way detectors able to verify (e.g., by way of clicking) the particle's passage through a corresponding slit. Let X 1 denote the proposition of the click of the detector placed behind slit 1 such that X 1 is true (denoted by "1") if the detector clicks and X 1 is false (denoted by "0") if the detector does not. Let X 2 in an analogous manner denote the proposition of the signal from the detector placed behind slit 2.Assume that the propositions X 1 and X 2 are in possession of not-yet-verified truth values -i.e., ones existing before the detectors can click -that are merely revealed by the act of verification of the particle's passage.Within the given assumption, let us accept that such values of the propositions X 1 and X 2 are either 1 or 0. Accordingly, exclusive disjunction on these values of X 1 and X 2 can be decided by * Email : arkadyv@bgu.ac.il 1 In the present paper, rather than being strictly restricted to spatially arranged slits, quantum interference is considered generally for any set of perfectly distinguishable alternatives.

Translation from Three-Valued Quantum Logic to Modal Logic

“…Let L q be the set of all the formulas of the many-valued quantum logic. 3 It consists of the propositional function α(|ψ ), which stands for the statement "The physical property α is observed in a pure state |ψ ." The definition of the logical connectives in the many-valued quantum logic does not play an essential role in this paper.…”

“…The closure is needed, because [[α]] may not be a closed subspace 3. The superscript "q" of L q represents "Quantum.…”

Obscure Qubits and Membership Amplitudes

“…Then, the result of the obscure-quantum computation consists of the quantum probabilities of the states together with the calculated "level of truth" for each of them (see, e.g. [18]).…”

“…Relations between truth values and probabilities were also given in [18]. The hardware realization of computations with vagueness was considered in [19,20].…”

Quantum contextuality implies a logic that does not obey the principle of bivalence

“…A semantics defined by set of these truth-value gaps in conjunction with the valuations (17) and ( 18) is gappy and yet two-valued. Accordingly, it can be called a supervaluationist semantics (for details of such semantics see [7,8] and also [9,10]).…”