Characterizing consistency in probabilistic logic for a class of Horn clauses

The problem of consistently assigning probabilities to logical formulas is an important problem. In this paper a set of logical formulas will be identified for which the problem can be solved. For every directed graph the paper defines a set of logical formulas that it represents. If the underlying (undirected) graph is either perfect or t-perfect a closed form solution to the consistency problem can be given. A remarkable property of the class of formulas identified here is that it turns out to be closed under duality (if a set of formulas is represented by a digraph then the dual set of formulas is also represented by a digraph).