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

Jacopo Surace

TL;DR

This work develops a principled reconstruction of quasi-probability theory from structural principles applied to finite Boolean algebras of statements. It introduces a Universal Valuation $V$ and the concept of Syntactic Locality to derive an additive representation of $V$ as a pre-probability $R=oldsymbol{\phi}\circ V$, unique up to additive regraduation. When a canonical reparametrisation is possible, quasi-probabilities emerge; probabilities arise as the subset stable under relativisation, and a coherent generalized Bayes' theorem applies to both pre-probabilities and quasi-probabilities. The framework clarifies conditioning in quasi-probability theory, resolves well-known conditioning issues, and reveals a rational versus irrational structure via semantic frames; holomorphic regularity collapses the gauge freedom to a global scalar, recovering standard probability theory within a broader algebraic setting. The results have foundational significance for quantum theory and probabilistic reasoning, offering a unifying account of additive representations, conditioning, and the emergence of classical probabilities from a more general quasi-probabilistic formalism.

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.

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

TL;DR

This work develops a principled reconstruction of quasi-probability theory from structural principles applied to finite Boolean algebras of statements. It introduces a Universal Valuation and the concept of Syntactic Locality to derive an additive representation of as a pre-probability , unique up to additive regraduation. When a canonical reparametrisation is possible, quasi-probabilities emerge; probabilities arise as the subset stable under relativisation, and a coherent generalized Bayes' theorem applies to both pre-probabilities and quasi-probabilities. The framework clarifies conditioning in quasi-probability theory, resolves well-known conditioning issues, and reveals a rational versus irrational structure via semantic frames; holomorphic regularity collapses the gauge freedom to a global scalar, recovering standard probability theory within a broader algebraic setting. The results have foundational significance for quantum theory and probabilistic reasoning, offering a unifying account of additive representations, conditioning, and the emergence of classical probabilities from a more general quasi-probabilistic formalism.

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.
Paper Structure (79 sections, 22 theorems, 189 equations, 12 figures)

This paper contains 79 sections, 22 theorems, 189 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organisation.
  4. Universal valuations and reconstruction of quasi-probability theory
  5. Universes and sub-universes: the idea of Syntactic Locality
  6. Universal Valuations of statements
  7. Formalisation of the principles
  8. Compatibility with classical propositional logic
  9. Local Deducibility
  10. Universality
  11. Maximum Realisability
  12. Symmetry removes valuational freedom
  13. Pre-probabilities
  14. Semantic frames and semantic dimension
  15. Embedding sub-universes in ambient universes
  16. ...and 64 more sections

Key Result

Lemma 1

Let $\mathcal{L}$ be a universe with $n\ge 2$ atoms and let $V:\mathcal{L}\to\mathbb C$ satisfy Local Deducibility and universality. If there exists $x\in\mathbb C$ such that then $V$ is constant on each level of the universe: for every $l=1,\dots,n$ there exists $v_l\in\mathbb C$ such that for every statement $s$ at level $l$.

Figures (12)

  • Figure 1: A universe $\mathcal{L}_4$ with highlighted in blue the relative (sub-)universe $\mathcal{L}_{a\vee b}$.
  • Figure 2: Correspondence between valuation and classical truth assignment. On the left in different colours are represented the values $\varphi$ and $\tau$ for $V$.
  • Figure 3: From the principle of Local Deducibility one can deduce $V(a\vee c)$ from the valuation of all the green statements.
  • Figure 4: The levels of a universe with three atomic statement labelled.
  • Figure 5: Imposing a symmetry of the valuation with respect to all Boolean isomorphisms collapses the Boolean structure of universes into a chain structure.
  • ...and 7 more figures

Theorems & Definitions (49)

  • Definition 1: Syntactic universe
  • Definition 2: Sub-universe
  • Definition 3: Relative universe
  • Definition 4: Consistent partial valuation
  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Definition 5: Semantic frames and semantic dimension
  • Lemma 3
  • Definition 6: Synchronisation of pre-probability valuations
  • ...and 39 more