HIERARCHICAL UNCERTAINTY METATHEORY BASED UPON MODAL LOGIC

Abstract

This paper is intended to contribute to the formal study of uncertainty from a broad perspective. Its aim is to demonstrate that, in addition to propositional calculus and probability theory, both fuzzy set theory and Dempster-Shafer evidence theory can be represented by the formal and semantic structures of modal logic. In particular, it is shown that the concept of multiple worlds in modal logic can be employed for constructing membership-grade functions of fuzzy sets, as well as belief and plausibility measures of evidence theory. It is also shown that additional theories of uncertainty, which have not been considered as yet, emerge naturally from the framework of modal logic. When looking at these various uncertainty theories emerging from modal logic from a metatheoretical perspective, a hierarchical ordering of the theories is recognized. We refer to the hierarchically ordered collection of uncertainty theories captured within the realm of modal logic as hierarchical uncertainty metatheory. Although we introduce relevant notation and key concepts of fuzzy set theory and evidence theory, we assume that the reader is familiar with the fundamentals of these theories. We do not assume knowledge of modal logic, but we overview only those of its aspects that we need for our purpose in this paper.