Probability Logic: A Model Theoretic Perspective

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.