[论文解读] Reconstruction of finite Quasi-Probability and Probability from Principles: The Role of Syntactic Locality

Jacopo Surace|arXiv (Cornell University)|Feb 12, 2026

Logic, Reasoning, and Knowledge被引用 0

一句话总结

论文基于对有限布尔宇宙上普遍取值的结构性规则，构建了对(准)概率的 principled 重建，得到前概率、规范准概率范畴以及广义 Bayes 公式。

ABSTRACT

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.

研究动机与目标

• Motivate and clarify the conceptual foundations of quasi-probabilities beyond computational use in physics.
• Develop a framework of universal valuations on statements across finite Boolean universes.
• Prove representation results showing admissible valuations as finitely additive measures (pre-probabilities).
• Characterize the emergence of classical probabilities via relativisation and establish a generalized conditioning/Bayes framework.
• Explore regularity conditions and the role of rational vs irrational probabilities within the reconstruction.

提出的方法

• Formalize syntactic universes and sub-universes (relative universes) under Syntactic Locality.
• Specify five principles (compatibility with classical propositional logic, local deducibility, universality, maximum realisability, symmetry) and derive their consequences.
• Prove a representation theorem (Theorem 1) that every admissible universal valuation admits a finite-additive representation (pre-probability) and identify a gauge freedom (Lemma 2).
• Define a conditioning/synchronisation framework yielding well-defined total probability and a generalized Bayes’ theorem for both pre-probabilities and quasi-probabilities.
• Distinguish rational versus irrational probabilities and discuss regularity implications.

Figure 6: Two agents’ sub-universes embedded in an ambient shared universe.

实验结果

研究问题

• RQ1 How can universal valuations on statements over finite Boolean algebras be constrained to yield an additive probabilistic representation?
• RQ2 What is the relationship between quasi-probabilities and classical probabilities through relativisation (restriction to sub-universes)?
• RQ3 How can conditioning and Bayes’ theorem be coherently defined within the pre-probability/quasi-probability framework?
• RQ4 What regularity conditions determine when irrational probabilities arise and how do rational priors fit into this reconstruction?

主要发现

• Every admissible universal valuation can be reparametrised as a finitely additive representative on disjoint joins (a pre-probability) (Theorem 1).
• There is a canonical reparametrisation leading to finite quasi-probabilities; the representation is unique up to a gauge (additive regraduation).
• Classical finite probabilities arise as the subclass of quasi-probabilities stable under relativisation (restriction to sub-universes).
• A coherent theory of conditionals is obtained, delivering a generalized Bayes’ theorem applicable to pre-probabilities and quasi-probabilities.
• Symmetry considerations constrain valuational freedom and determine a valuation’s dependence on lattice level when atoms are indistinguishable (leading to a level-based collapse if atoms are identical).
• Appendices discuss regularity assumptions and the emergence of rational probabilities within this finite-additive framework.] ,
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• table_rows: []} {
• tldr
• 论文构建了有限布尔宇宙下从结构性规则推导出的(准)概率重建，得到前概率、规范的准概率范畴以及广义 Bayes 定理。
• meta_description
• principled reconstruction of finite quasi-probabilities from syntactic locality; introduces pre-probabilities, canonical quasi-probabilities, and a generalized Bayes’ theorem.
• objective…