Reconstruction of finite Quasi-Probability and Probability from Principles: The Role of Syntactic Locality

TL;DR

This paper develops a principled framework for deriving and understanding quasi-probabilities from structural consistency principles, clarifying their conceptual foundations and operational rules in finite Boolean algebras.

Contribution

It introduces a universal valuation approach, proves a representation theorem linking valuations to finitely additive measures, and defines a coherent conditional calculus for quasi-probabilities.

Findings

Quasi-probabilities can be uniquely represented as finitely additive measures.

Classical probabilities are a special case of stable quasi-probabilities under restriction.

A generalized Bayes' theorem is established for both pre-probabilities and quasi-probabilities.

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.