Probability Logic: A Model Theoretic Perspective

TL;DR

This paper explores the model theoretic properties of propositional probability logic, adapting ultraproducts for different classes of probability models and analyzing their compactness and L"owenheim-Skolem numbers.

Contribution

It adapts ultraproduct constructions for probability models and establishes compactness results for various fragments of probability logic.

Findings

Compactness holds for basic probability logic in $\sigma$-additive models.

Extended compactness applies to positive probability logic in finitely-additive models.

L"owenheim-Skolem number is uncountable for $\sigma$-additive models, countable for finitely-additive models.

Abstract

In this paper (propositional) probability logic ($PL$) is investigated from model theoretic point of view. First of all, the ultraproduct construction is adapted for $\sigma$-additive probability models, and subsequently when this class of models is considered it is shown that the compactness property holds with respect to a fragment of $PL$ called basic probability logic ($BPL$). On the other hand, when dealing with finitely-additive probability models, one may extend the compactness property for a larger fragment of probability logic, namely positive probability logic ($PPL$). We finally prove that while the L\"owenheim-Skolem number of the class of $\sigma$-additive probability models is uncountable, it is $\aleph_0$ for the class of finitely additive probability models.