An algebraic synthesis of the foundations of logic and probability (Q1096615)

This paper gives the foundations of a unified theory of probability and logic, in which the controversial slogan, ``the probability of a conditional is the conditional probability of the consequent given the antecedent'' is satisfied. The author shows that no truth-functional compound of `or', `not', and `and', will serve as this conditional. ``Probability logic is part of modal logic.'' (p. 196) He introduces a conditional proposition, `q given p', (q\(| p)\), in terms of the ideal generated by q in the conditional logic L/(p), where (p) is the smallest ideal containing p. The author goes on to provide rules for combining the conditional propositions with `and', `or', and `not', and even with the new operator `\(| '\). He explores the connections with the so-called `material conditional' (q\(\vee \sim p)\), as well as with fuzzy membership and degrees of truth. Probabilities are derived from measures of sets of models in a universe of models. This is a rich and accessible paper, tying together important questions in the foundations of probability, in model theory, in logic, and in abstract algebra.