Deterministic Bayesian Logic

In this paper a conditional logic is defined and studied. This conditional logic, Deterministic Bayesian Logic, is constructed as a deterministic counterpart to the (probabilistic) Bayesian conditional. The logic is unrestricted, so that any logical operations are allowed. This logic is shown to be non-trivial and is not reduced to classical propositions. The Bayesian conditional of DBL implies a definition of logical independence. Interesting results are derived about the interactions between the logical independence and the proofs. 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 DBL. The Bayesian conditional is then recovered from the probabilistic DBL. At last, it is shown why DBL is compliant with Lewis triviality.