  This paper discusses the semantics and proof theory of Nilsson's
probabilistic logic, outlining both the benefits of its well-defined model
theory and the drawbacks of its proof theory. Within Nilsson's semantic
framework, we derive a set of inference rules which are provably sound. The
resulting proof system, in contrast to Nilsson's approach, has the important
feature of convergence - that is, the inference process proceeds by computing
increasingly narrow probability intervals which converge from above and below
on the smallest entailed probability interval. Thus the procedure can be
stopped at any time to yield partial information concerning the smallest
entailed interval.
