Dynamic Graded Epistemic Logic

Abstract

Graded epistemic logic is a logic for reasoning about uncertainties. Graded epistemic logic is interpreted on graded models. These models are generalizations of Kripke models. We obtain completeness of some graded epistemic logics. We further develop dynamic extensions of graded epistemic logics, along the framework of dynamic epistemic logic. We give an extension with public announcements, i.e., public events, and an extension with graded event models, a generalization also including nonpublic events. We present complete axiomatizations for both logics.

Neighbourhood semantics for graded modal logic

“…Graded semantics. In this subsection, we recall the graded semantics from Ma and van Ditmarsch [13]. The sum operation and the 'greater than or equal to' relation (≥) are defined over natural numbers N plus ω, the least ordinal number greater than any natural number, i.e., ∀n ∈ N, n < ω. Variables n, m, i, j range over the natural numbers N, not over N ∪ {ω}.…”

“…GrK is sound and complete with respect to the class of all Kripke frames. Theorem 2.2 (Theorem 3.2 of [13]). GrK is sound and complete with respect to the class of all graded frames.…”

“…(1) Using the approach developed in this paper, updating neighbourhood models [12] can be compared to updated graded models [13].…”

“…Van der Hoek and Meyer [16] proposed a graded modal logic GrS5, which is seen as a graded epistemic logic and is able to express 'accepting ϕ if there are at most n exceptions to ϕ'. Ma and van Ditmarsch [13] developed dynamic extensions of graded epistemic logics.…”

Neighbourhood Semantics for Graded Modal Logic

“…Fine [8], De Caro [6] and Cerrato [2] investigated the completeness of GrK and its extensions. Van der Hoek [15] investigated the expressibility, decidability and definability of graded modal logic and also correspondence theory. Cerrato [3] proved the decidability by filtration for graded modal logic.…”