  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.
