Probabilities are always axiomatizable

TL;DR

This paper explores the axiomatizability of probabilities within logic, demonstrating that the convex hull of evaluations is always effectively axiomatizable and providing a calculus with compactness and finite completeness, along with a counterexample.

Contribution

It introduces a Birkhoff-style calculus for probability axioms and proves key properties, advancing understanding of probabilistic logic's axiomatizability.

Findings

Convex hull of evaluations is always effectively axiomatizable.

A Birkhoff-style calculus for probability axioms is developed with compactness.

An example of a logic where probabilities are not finitely axiomatizable is provided.

Abstract

In this paper we study the interaction between logic and probability. In particular, we show that the convex hull of evaluations of a broad class of logics is always effectively axiomatizable. We define a Birkhoff-style calculus for probability axioms for which compactness, and finite completeness is proved. We give example for a logic for which probabilities are not finitely axiomatizable.