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New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic

机译:古典逻辑,概率逻辑和量子逻辑的分类逻辑新方向

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Intuitionistic logic, in which the double negation law not-not-P = P fails,is dominant in categorical logic, notably in topos theory. This paper follows adifferent direction in which double negation does hold. The algebraic notionsof effect algebra/module that emerged in theoretical physics form thecornerstone. It is shown that under mild conditions on a category, its maps ofthe form X -> 1+1 carry such effect module structure, and can be used aspredicates. Predicates are identified in many different situations, and capturefor instance ordinary subsets, fuzzy predicates in a probabilistic setting,idempotents in a ring, and effects (positive elements below the unit) in aC*-algebra or Hilbert space. In quantum foundations the duality between states and effects plays animportant role. It appears here in the form of an adjunction, where we use maps1 -> X as states. For such a state s and a predicate p, the validityprobability s |= p is defined, as an abstract Born rule. It captures many formsof (Boolean or probabilistic) validity known from the literature. Measurement from quantum mechanics is formalised categorically in terms of`instruments', using Lüders rule in the quantum case. These instruments arespecial maps associated with predicates (more generally, with tests), whichperform the act of measurement and may have a side-effect that disturbs thesystem under observation. This abstract description of side-effects is one ofthe main achievements of the current approach. It is shown that in the specialcase of C*-algebras, side-effect appear exclusively in the non-commutativecase. Also, these instruments are used for test operators in a dynamic logicthat can be used for reasoning about quantum programs/protocols. The paper describes four successive assumptions, towards a categoricalaxiomatisation of quantitative logic for probabilistic and quantum systems.
机译:直觉逻辑在分类逻辑中占主导地位,其中双重否定法则not-not-P = P失效。本文遵循双重否定的不同方向。理论物理学中出现的效果代数/模块的代数概念形成了角石。结果表明,在一个类别的温和条件下,其X-> 1 + 1形式的映射具有这种效应模块结构,可以用作谓词。谓词在许多不同的情况下被识别,并捕获例如普通子集,概率设置中的模糊谓词,环中的幂等以及在C *-代数或希尔伯特空间中的效应(单元下方的正元素)。在量子基础中,状态和效应之间的二元性起着重要的作用。它在这里以附加形式出现,我们使用maps1-> X作为状态。对于这样的状态s和谓词p,将有效性概率s | = p定义为抽象的Born规则。它捕获了文献中已知的多种形式的(布尔或概率)有效性。量子力学中的测量是根据“仪器”分类形式化的,在量子情况下使用Lüders规则。这些仪器是与谓词(更一般而言,与测试)相关联的特殊映射,它们执行测量的行为并且可能具有干扰所观察系统的副作用。对副作用的这种抽象描述是当前方法的主要成就之一。结果表明,在C *代数的特殊情况下,副作用仅在非交换情况下出现。而且,这些仪器还用于动态逻辑中的测试运算符,可用于推理量子程序/协议。本文描述了四个连续的假设,以实现概率系统和量子系统的定量逻辑分类。

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