Deterministic modal Bayesian Logic: derive the Bayesian within the modal logic T

In this paper a conditional logic is defined and studied. This conditional logic, DmBL, is constructed as close as possible to the Bayesian and is unrestricted, that is one is able to use any operator without restriction. A notion of logical independence is also defined within the logic itself. This logic is shown to be non trivial and is not reduced to classical propositions. A model is constructed for the logic. Completeness results are proved. It is shown that any unconditioned probability can be extended to the whole logic DmBL. The Bayesian is then recovered from the probabilistic DmBL. At last, it is shown why DmBL is compliant with Lewis triviality.