Uncertainty Theories by Modal Logic

Abstract

In a series of papers initiated by Resconi et al. [1], interpretations for various uncertainty theories were proposed, including fuzzy set theory, Dempster-Shafer theory, and possible theory, using models of modal logic [1, 2, 3]. There were two main reasons for pursuing research in this direction. The first reason was to offer the standard semantics of modal logic as a unifying framework within which it would be possible to compare and relate uncertainty theories to each other. Since, from time to time, some of the uncertainty theories are questioned regarding their internal adequateness, the second reason was to support them by developing interpretations for them in a relatively well-established area: in our case, modal logic. This paper is a summary of these efforts. To avoid unnecessary repetition of previous material, we will not repeat all the basic definitions and properties; the reader is referred to relevant literature for fuzzy set theory [5], for possibility theory [3], for Dempster-Shafer theory [4], and for modal logic [8]. A more thorough treatment of the material summarised in this paper is covered in the above-mentioned papers [4,5].