We solve two fundamental problems of probabilistic reasoning: given finitely many conditional probability assessments, how to determine whether the assessments are mutually consistent, and how to determine what they imply about the conditional probabilities of other events? These problems were posed in 1854 by George Boole, who gave a partial solution using algebraic methods. The two problems are fundamental in applications of the Bayesian theory of probability; Bruno de Finetti solved the second problem for the special case of unconditional probability assessments in what he called ‘the fundamental theorem of probability’. We give examples to show that previous attempts to solve the two problems, using probabilistic logic and similar methods, can produce incorrect answers. Using ideas from the theory of imprecise probability, we show that the general problems have simple, direct solutions which can be implemented using linear programming algorithms. Unlike earlier proposals, our methods are formulated directly in terms of the assessments, without introducing unknown probabilities. Our methods work when any of the conditioning events may have probability zero, and they work when the assessments include imprecise (upper and lower) probabilities or previsions. The main methodological contribution of the paper is to provide general algorithms for making inferences from any finite collection of (possibly imprecise) conditional probabilities.
