Contextuality is a key signature of quantum non-classicality, which has been shown to play a central role in enabling quantum advantage for a wide range of information-processing and computational tasks. We study the logic of contextuality from a structural point of view, in the setting of partial Boolean algebras introduced by Kochen and Specker in their seminal work. These contrast with traditional quantum logic \`a la Birkhoff and von Neumann in that operations such as conjunction and disjunction are partial, only being defined in the domain where they are physically meaningful. We study how this setting relates to current work on contextuality such as the sheaf-theoretic and graph-theoretic approaches. We introduce a general free construction extending the commeasurability relation on a partial Boolean algebra, i.e. the domain of definition of the binary logical operations. This construction has a surprisingly broad range of uses. We apply it in the study of a number of issues, including: - establishing the connection between the abstract measurement scenarios studied in the contextuality literature and the setting of partial Boolean algebras; - formulating various contextuality properties in this setting, including probabilistic contextuality as well as the strong, state-independent notion of contextuality given by Kochen-Specker paradoxes, which are logically contradictory statements validated by partial Boolean algebras, specifically those arising from quantum mechanics; - investigating a Logical Exclusivity Principle, and its relation to the Probabilistic Exclusivity Principle widely studied in recent work on contextuality as a step towards closing in on the set of quantum-realisable correlations; - developing some work towards a logical presentation of the Hilbert space tensor product, using logical exclusivity to capture some of its salient quantum features.

中文翻译：

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语境逻辑

语境性是量子非经典性的一个关键特征，它已被证明在为广泛的信息处理和计算任务实现量子优势方面发挥着核心作用。我们从结构的角度研究上下文的逻辑，在 Kochen 和 Specker 在他们开创性工作中引入的部分布尔代数的设置中。这些与传统的量子逻辑 \`a la Birkhoff 和 von Neumann 形成对比，因为诸如合取和析取之类的运算是部分的，仅在它们具有物理意义的领域中被定义。我们研究这种设置如何与当前关于上下文的工作相关，例如层理论和图论方法。我们引入了扩展部分布尔代数上的可测性关系的一般自由构造，即 二元逻辑运算的定义域。这种结构具有惊人的广泛用途。我们将其应用于许多问题的研究，包括： - 在上下文文献中研究的抽象测量场景与部分布尔代数的设置之间建立联系；- 在此设置中制定各种上下文属性，包括概率上下文以及由 Kochen-Specker 悖论给出的强的、与状态无关的上下文概念，这是由部分布尔代数验证的逻辑上矛盾的陈述，特别是那些源自量子力学的陈述；- 研究逻辑排他性原则，以及它与最近关于上下文的工作中广泛研究的概率排他性原则的关系，作为接近量子可实现相关性集合的一步；- 为 Hilbert 空间张量积的逻辑表示开发一些工作，使用逻辑排他性来捕获其一些显着的量子特征。