Dempster-Shafer vs. Probabilistic Logic

For example, Nilsson in [6] and Grosof in [3] have considered methods for reasoning with sets of probability assignments generated by probabilistic equality and inequality constraints1. Following Nilsson, I use the expression "Probabilistic Logic" to denote the collection of such methods. The aim of these methods is to compute a set of possible probabilities for a given statement from the specified set of probability assignments. If the set of probability assignments is generated by probabilistic equality and inequality constraints, the possible probabilities for a given statement form an interval. Since Dempster-Shafer also associates an interval with each statement A, namely the interval bounded by Bel(A) and Pls(A), the question arises as to the connection between Dempster-Shafer belief functions and sets of probability assignments defined by equality and inequality constraints. Grosof [3] has shown that the latter is a generalization of the former: every Dempster-Shafer belief function is representable by a set of probability assignments arising from equality and inequality constraints, but not vice-versa. A related issue concerns the connection between Dempster's rule of combination and the combination of evidence statements in probabilistic logic. Grosof [2] states some results concerning conditions under which these two methods of combining evidence yield the same result. The aim of this paper is to generalize Grosof's results and to investigate how divergent the two