Only knowing with degrees of confidence

Abstract

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 p 2 -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.

Logical Spaces in Multi-agent Only Knowing Systems

“…Intuitively, the function of a logical space is to explicate every logical possibility of the logic. This is what has been done in the singleagent only knowing system AE presented in the paper by Lian et al [8] and further analyzed in [11]. A logical space in AE is a formula λ such that for each purely Boolean ϕ, either λ ♦ϕ or λ ¬♦ϕ.…”

“…The size of a logical space grows quickly beyond any tractable level. However, in a particular situation one will not need to span the entire space syntactically, exactly like one in AE can provide an implicit definition of a logical space by means of a characteristic formula [8,11]. We plan to address this question in a subsequent paper.…”

Trustworthiness by Default

“…is a special instance of the system AE ρ introduced in [8] and further analyzed and motivated in [12]; the references contain in particular an axiomati- …”

“…The proof uses simple properties from the model theory of AE , cf. [12]. In an AE model M all points agree on the truth value of every completely modalized formula.…”

“…For the theory of trustworthiness, the most important is the work of John Cantwell [1,2], in which the basic relation of trustworthiness is de ned in a way that is close to the one given here. For the aspects that relate to default inference and belief, the prioritized belief logic AE [9,10,12], which is closely related to that of [7], has been the primary source of reference.…”