Translating classical probability logics into modal fuzzy logics

This paper is a contribution to the study of two distinct kinds of modal logics for modeling uncertainty. Both approaches use logics with a two-layered syntax, but while one employs classical logic on both levels [6], the other involves a suitable system of fuzzy logic in the upper layer [9]. We take two prominent examples of the former approach, probability logics Prlin and Prpol, and build explicit faithful translations into, respectively, the two-layered modal fuzzy logics given by Łukasiewicz logic with Δ and its expansion with the product connective. We first prove the faithfulness of both translations using semantics of all four involved logics. Then, we use the axiomatization of Prlin and a hypersequent presentation of the two-layered system over Łukasiewicz logic to obtain an alternative syntactical proof.