Consider a set of logical sentences together with probabilities that they are true. These probabilities must satisfy certain conditions for this system to be consistent. It is shown that an analytical form of these conditions can be obtained by enumerating the extreme rays of a polyhedron. We also consider the cases when (i) intervals of probabilities are given, instead of single values; and (ii) best lower and upper bounds on the probability of an additional logical sentence to be true are sought. Enumeration of vertices and extreme rays is used. Each vertex defines a finear expression and the maximum (minimum) of these defines a best possible lower (upper) bound on the probability of the additional logical sentence to be true. Each extreme ray leads to a constraint on the probabilities assigned to the initial set of logical sentences. Redundancy in these expressions is studied. Illustrations are provided in the domain of reasoning under uncertainty. © 1995.601-3181193Boole, (1854) An Investigation of the Laws of Thought, on which are Founded the Mathematical Theories of Logic and Probabilities, , Walton and Maberley, London, reprint (Dover, New York, 1958)Boole, On the conditions by which solutions of questions in the theory of probabilities are limited (1854) The London, Edinburgh and Dublin Philos. Magazine and J. Sci. Ser. 4, 8, pp. 91-98Boole, On a general method in the theory of probabilities (1854) The London, Edinburgh and Dublin Philos. Magazine and J. Sci. Ser. 4, 8, pp. 431-444Boole, On certain propositions in algebra connected to the theory of probabilities (1855) The London, Edinburgh and Dublin Philos. Magazine and J. Sci. Ser. 4, 9, pp. 165-179Boole, On propositions numerically definite (read posthumously by De Morgan 16 March, 1868) (1871) Trans. Cambridge Philos. Soc., 11, pp. 396-411Buchanan, Shortliffe, (1984) Rule-based Expert Systems, the Mycin Experiments of the Stanford Heuristic Programming Project, , Addison-Wesley, Reading, MAChen, Hansen, Jaumard, Partial pivoting in vertex enumeration (1992) GERAD Research Report 92-15, , MontréalFourier, Solution d'une question particulière du calcul d'inégalités (1826) Histoire de l'Académie, pp. 317-328. , 2nd ed., French Academy of Sciences, (1823, 1824) Oeuvres IIGeorgakopoulos, Kavvadias, Papadimitriou, Probabilistic satisfiability (1988) J. Complexity, 4, pp. 1-11Hailperin, Best possible inequalities for the probability of a logical function of events (1965) The American Mathematical Monthly, 72, pp. 343-359Hailperin, Boole's logic and probability (1986) Studies in Logic and the Foundations of Mathematics, 85. , 2nd ed., North-Holland, New YorkJaumard, Hansen, Poggi de Aragão, Column generation methods for probabilistic logic (1991) ORSA J. Comput., 3, pp. 135-148Kavvadias, Papadimitriou, A linear programming approach to reasoning about probabilities (1990) Ann. Math. Artificial Intelligence, 1, pp. 189-205Kounias, Marin, Best linear Bonferroni bounds (1976) SIAM J. Appl. Math., 30, pp. 307-323Nemhauser, Wolsey, (1988) Integer and Combinatorial Optimization, , Wiley, New YorkNilsson, Probabilistic logic (1986) Artificial Intelligence, 28, pp. 71-87Zemel, Polynomial algorithms for estimating network reliability (1982) Networks, 12, pp. 439-452