有限准概率与概率的重建：语法局部性的作用
Reconstruction of finite Quasi-Probability and Probability from Principles: The Role of Syntactic Locality
AI总结：
本文提出通过语法局部性原则，将准概率与概率的结构一致性要求推导为有限加法测度，从而建立准概率理论的唯一确定表示。
摘要：
准概率出现在物理的许多领域中，但其概念基础仍然不清楚：它们通常被视为计算工具，条件操作和贝叶斯定理变得模糊。我们通过发展一个原则性的框架来解决这两个问题，该框架从如何在不同的论域中对陈述进行估值的结构一致性要求中推导出准概率及其条件算术。我们从一个通用估值开始，为所有陈述分配确定的（可能复数）值。核心概念是语法局部性：每个论域可以嵌入到更大的环境论域中，并且通用估值必须在这种嵌入和限制下保持一致。从一组结构原则中，我们证明了一个表示定理，表明每个可接受的估值都可以重新表示为互斥陈述上的有限加法测度，这类似于通常的概率求和规则。我们称这样的加法代表为预概率。这种表示在加法再梯度自由度下是唯一的。当这种自由度可以被唯一地固定时，预概率减少为有限准概率，从而将准概率理论从计算工具提升为唯一确定的加法表示。经典有限概率作为准概率的子类出现，即在相对化下稳定的子类，即在限制到子论域下封闭。最后，相同的框架使我们能够定义条件语的协调理论，产生适用于预概率和准概率的明确广义贝叶斯定理。我们最后讨论了额外的规则条件，包括在这一设置中理性与无理概率的作用。
英文摘要：
Quasi-probabilities appear across diverse areas of physics, but their conceptual foundations remain unclear: they are often treated merely as computational tools, and operations like conditioning and Bayes' theorem become ambiguous. We address both issues by developing a principled framework that derives quasi-probabilities and their conditional calculus from structural consistency requirements on how statements are valued across different universes of discourse, understood as finite Boolean algebras of statements.We begin with a universal valuation that assigns definite (possibly complex) values to all statements. The central concept is Syntactic Locality: every universe can be embedded within a larger ambient one, and the universal valuation must behave coherently under such embeddings and restrictions. From a set of structural principles, we prove a representation theorem showing that every admissible valuation can be re-expressed as a finitely additive measure on mutually exclusive statements, mirroring the usual probability sum rule. We call such additive representatives pre-probabilities. This representation is unique up to an additive regraduation freedom. When this freedom can be fixed canonically, pre-probabilities reduce to finite quasi-probabilities, thereby elevating quasi-probability theory from a computational device to a uniquely determined additive representation of universal valuations. Classical finite probabilities arise as the subclass of quasi-probabilities stable under relativisation, i.e., closed under restriction to sub-universes. Finally, the same framework enables us to define a coherent theory of conditionals, yielding a well-defined generalized Bayes' theorem applicable to both pre-probabilities and quasi-probabilities. We conclude by discussing additional regularity conditions, including the role of rational versus irrational probabilities in this setting.
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