A Qualitative Logic for Uncertain Evidence and Belief Comparison

We introduce a qualitative logic for comparing strengths of belief and evidential support explicitly, discerning these two comparative notions both syntactically and semantically within a modal logical framework. More precisely, we employ Dempster-Shafer theory (DST) of belief functions to represent uncertain, possibly mutually inconsistent, and incomplete evidence, as well as evidence-based degrees of beliefs. We propose a bi-modal logic that compares propositions in two ways: (1) based on the strengths of belief an evidence-possessing agent has in them and (2) based on the degrees of certainty of the evidence supporting them. (2) is the novel component of the proposed logic, designed to capture a notion of certainty-dominance among sets of evidence, modeled via an Egli-Milner-like order lifting on individual pieces of evidence. We justify this modeling choice, provide key (in)validities of our logic, and establish links to existing modal logics of evidence and belief (functions).