Epistemic Logics, Probability, and the Calculus of Evidence

Abstract

This paper presents results of the application to epistemic logic structures of the method proposed by Carnap for the development of logical foundations of probability theory. These results , which provide firm conceptual bases for the Dempster-Shafer calculus of evidence, are derived by exclusively using basic concepts from probability and modal logic theories, without resorting to any other theoretical notions or structures. A form of epistemic logic (equivalent in power to the modal system 85), is used to define a space of possible worlds or states of affairs. This space, called the epistemic universe, consists of all possible combined descriptions of the state of the real world and of the state of knowledge that certain rational agents have about it. These representations generalize those derived by Carnap, which were confined exclusively to descriptions of possible states of the real world. Probabilities defined on certain classes of sets of this universe, representing different states of knowledge about the world, have the properties of the major functions of the Dempster-Shafer calculus of evidence: belief functions and mass assignments. The importance of these epistemic probabilities lies in their ability to represent the effect of uncertain evidence in the states of knowledge of rational agents. Furthermore, if an epistemic probability is extended to a probability function defined over subsets of the epistemic universe that represent true states of the real world, then any such extension must satisfy the wellknown interval bounds derived from the Dempster-Shafer theory. Application of this logic-based approach to problems of knowledge integration results in a general expression, called the additive combination formula, which can be applied to a wide variety of problems of integration of dependent and independent knowledge. Under assumptions of probabilistic independence this formula is equivalent to Dempster's rule of combination.