Beliefs and bilattices

Abstract

A bilattice is a structure which can be viewed as a class of truth values that can accommodate incomplete and inconsistent information. In bilattice theory, knowledge are ordered along two dimensions: truth/ falsity and certainty/ uncertainty. Bilattice theory has been applied in areas such as non-monotonic reasoning, truth maintenance systems and more recently to logic programming. In this paper, I will attempt to connect bilattice theory to epistemic logic. In the last decade there has been a resurgence of work on epistemic logic in artificial intelligence. Among them were the logic of implicit and explicit beliefs and the logic of awareness. Most of these approaches attempted to alleviate the problem of logical omniscience. Although these different models have surface dissimilarities, a closer examination shows that they have strong resemblance. Further, given the current proliferation of epistemic logic, it seems prudent to establish comparisons among the various approaches and to consider the issue of unification among them. The main contribution of this research is to show that a uniform framework for reasoning about knowledge can be defined in the context of bilattice theory. This formulation is based on the observation that epistemic notions such as implicit belief, explicit belief and awareness can be associated with subsets of truth values in a bilattice. It is hoped that the proposed work will lead to new insights in both bilattice theory and epistemic logic.