Only knowing with degrees of confidence

A new logic of belief (in the “only knowing” family) with confidence levels is presented. The logic allows a natural distinction between explicit and implicit belief representations, where the explicit form directly expresses its models. The explicit form can be found by applying a set of equivalence preserving rewriting rules to the implicit form. The rewriting process is performed entirely within the logic, on the object level, provided we supply an explicit formalization of the logical space. We prove that the problem of deciding whether there exists a consistent explicit form is Σ2p-complete, a complexity class to which many problems of nonmonotonic reasoning belong. The article also contains a conceptual analysis of basic notions like belief, co-belief and degrees of confidence.