Deterministic Bayesian Logic

TL;DR

This paper introduces Deterministic Bayesian Logic (DBL), a new logical framework that models Bayesian conditionals deterministically, allowing for logical independence, completeness, and extension of probabilities within the logic.

Contribution

It defines and studies a deterministic conditional logic, DBL, that extends classical logic to incorporate Bayesian conditionals and logical independence, with proven completeness and probabilistic extension.

Findings

DBL is non-trivial and not reducible to classical logic.

A model for DBL is constructed and completeness is proved.

Any unconditioned probability extends to DBL, recovering Bayesian conditionals.

Abstract

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.