Quantum Team Logic and Bell's Inequalities

TL;DR

This paper introduces quantum team logic, a modified probability logic where Bell's Inequalities are not provable, providing a formal framework to better understand quantum mechanics' violation of these inequalities.

Contribution

It develops quantum team semantics and proves a completeness theorem, extending dependence logic to a probabilistic and quantum context.

Findings

Quantum team logic prevents provability of Bell's Inequalities.

A completeness theorem for quantum team logic is established.

The approach generalizes team semantics to quantum probabilistic scenarios.

Abstract

A logical approach to Bell's Inequalities of quantum mechanics has been introduced by Abramsky and Hardy [2]. We point out that the logical Bell's Inequalities of [2] are provable in the probability logic of Fagin, Halpern and Megiddo [4]. 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 [7] first to probabilistic team semantics, and then to what we call quantum team semantics.