The Logical Foundations of Evidential Reasoning (revised)

Abstract : The approach proposed by Carnap for the development of logical bases for probability theory is investigated by using formal structures that are based on epistemic logic. Epistemic logic are modal logic introduced to deal with issues that are relevant to the state of knowledge that rational agents have about the real world. The use of epistemic logic in problems of analysis of evidence is justified by the need to distinguish among such notions as the state of a real system, the state of knowledge possessed by rational agents, and the impact of information on that knowledge. Carnap's method for generating a universe of possible worlds is followed using an enhanced notion of possible world that encompasses descriptions of knowledge states. Within such generalized or epistemic universes, several classes of sets are identified in terms of the truth-values of propositions that describe either the state of the world or the state of knowledge about it. These classes of subsets have the structure of a sigma algebra. Probabilities defined over one of these sigma algebras, called the epistemic algebra, are then shown to have the properties of the belief and basic probability assignment functions of the Dempster-Shafer calculus of evidence. It also is shown that any extensions of a probability function defined on the epistemic algebra (representing different states of knowledge) to the truth algebra (representing true states of the real world) must satisfy the interval probability bounds derived from the Dempster-Shafer theory. These bounds also are shown to correspond to the classical notions of lower and upper probability. Furthermore, these constraints are shown to be the best possible bounds, given a specific state of knowledge. Finally, the problem of combining the knowledge that several agents have about a real-world system is addressed.