Foundations of Probabilistic Logic

We formalize a mathematical approach to probabilistic logic for zero-order logic and derive new inequalities that are necessary and sufficient for consistent probability assignments to propositions. We prove that a complete theory of probabilistic logic requires the a priori assignment of probabilities for a system with k basic propositions. We also show that a proposal due to Cheeseman, namely, to regard measures of confidence in knowledge systems as expectations that are conditioned on unknown distributions, does not work in general. We demonstrate this by showing that several certainty measures proposed for expert systems are not consistent with the derived inequalities for probabilistic logics.