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Studia Logica

Semantics of the minimal logic of quantum mechanics

Herman Dishkant

Published Dec 1, 1972• 75 citations
Publisher
DOI
10.1007/bf02120818

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Semantics of the Minimal Logic of Quantum Mechanics

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In this paper I formalize the notion of minimal disturbance, as this seems to be required by usual interpretations of the theory of quantum mechanics, and construct a quantum logical (lattice) model of the type of situation that seems to be at the root of the problem of the interpretation of Luders’ projection rule as a criterion of minimal disturbance for individual state transformations. What is particularly interesting in the situation to be depicted here is that, on the basis of a simple model, which depends only on some very general features of the lattice structure of the theory (and its semantical interpretation), usual interpretive assumptions on minimal disturbance appear to be wanting. If we restrict our attention to the statistics of measurement results, Luders’ rule can, easily be interpreted as a formula describing ‘minimal change’ for statistical states.

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Abstract A logical approach to Bell’s Inequalities of quantum mechanics has been introduced by Abramsky and Hardy (Abramsky & Hardy, 2012). We point out that the logical Bell’s Inequalities of Abramsky & Hardy (2012) are provable in the probability logic of Fagin, Halpern and Megiddo (Fagin et al., 1990). Since it is now considered empirically established that quantum mechanics violates Bell’s Inequalities, we introduce a modified probability logic, that we call quantum team logic, in which Bell’s Inequalities are not provable, and prove a Completeness theorem for this logic. For this end we generalise the team semantics of dependence logic (Väänänen, 2007) first to probabilistic team semantics, and then to what we call quantum team semantics.

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Logic of Quantum Mechanics

Herman Dishkant

Springer eBooks•Jan 1, 1976•100 citations
In this paper I formalize the notion of minimal disturbance, as this seems to be required by usual interpretations of the theory of quantum mechanics, and construct a quantum logical (lattice) model of the type of situation that seems to be at the root of the problem of the interpretation of Luders’ projection rule as a criterion of minimal disturbance for individual state transformations. What is particularly interesting in the situation to be depicted here is that, on the basis of a simple model, which depends only on some very general features of the lattice structure of the theory (and its semantical interpretation), usual interpretive assumptions on minimal disturbance appear to be wanting. If we restrict our attention to the statistics of measurement results, Luders’ rule can, easily be interpreted as a formula describing ‘minimal change’ for statistical states.
Abstract A logical approach to Bell’s Inequalities of quantum mechanics has been introduced by Abramsky and Hardy (Abramsky & Hardy, 2012). We point out that the logical Bell’s Inequalities of Abramsky & Hardy (2012) are provable in the probability logic of Fagin, Halpern and Megiddo (Fagin et al., 1990). Since it is now considered empirically established that quantum mechanics violates Bell’s Inequalities, we introduce a modified probability logic, that we call quantum team logic, in which Bell’s Inequalities are not provable, and prove a Completeness theorem for this logic. For this end we generalise the team semantics of dependence logic (Väänänen, 2007) first to probabilistic team semantics, and then to what we call quantum team semantics.