Justified common knowledge

AbstractIn this paper we introduce the justified knowledge operator J with the intended meaning of Jϕ as ‘there is a justification for ϕ.’ Though justified knowledge appears here in a case study of common knowledge systems, a similar approach is applicable in more general situations. First we consider evidence-based common knowledge systems obtained by augmenting a multi-agent logic of knowledge with a system of evidence assertions t:ϕ, reflecting the notion ‘t is an evidence for ϕ,’ such that evidence is respected by all agents. Justified common knowledge is obtained by collapsing all evidence terms into one modality J. We show that in standard situations, when the base epistemic systems are T, S4, and S5, the resulting justified common knowledge systems are normal modal logics, which places them within the scope of well-developed machinery applicable to modal logic: Kripke-style epistemic models, normalized proofs, automated proof search, etc. In the aforementioned situations, the intended semantics of justified knowledge is supported by a realization theorem stating that for any valid fact about justified knowledge, one could recover its constructive meaning by realizing all occurrences of justified knowledge modalities Jϕ by appropriate evidence terms t:ϕ