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 Pr-lin and Pr-pol, and build explicit faithful translations into, respectively, the two-layered modal fuzzy logics given by Lukasiewicz logic with Delta 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 Pr-lin and a hypersequent presentation of the two-layered system over Lukasiewicz logic to obtain an alternative syntactical proof.