Soundness and completeness theorems between the Dempster-Shafer theory and logic of belief

Modelling of belief is one of most important issues in intelligent systems. Roughly speaking, one can find two approaches in the literature: one is a logical approach using modal logic; the other is a numerical approach such as Bayesian theory, Dempster-Shafer theory of evidence (or D-S theory). The former is mainly concerned with logical inference methodology with belief, while the latter is mainly concerned with belief formation using methods of updating such as conditional probability and Dempster´s rule of combination. Since characteristics of the two approaches are complementary to each other, one would have a more effective method of dealing with belief if they were unified. However, there are few studies of theoretical relationship connecting the two approaches. The purpose of this paper is to present a basis for unifying modal-logical and D-S-theory-based approaches by means of proving soundness and completeness theorems of several systems of modal logic with respect to classes of belief-function-based and plausibility-function-based models newly defined in this paper. The result shows that modal-logical structure is intrinsic in D-S theory, and would enable one (1) to introduce a concept of belief formation based on aggregation of evidence into a modal-logical approach to belief, (2) to decide rules of inference which are valid under available uncertain evidence, and (3) to introduce a concept of knowledge acquisition from a logical point of view into a belief formation method in D-S theory