Interacting Explicit Evidence Systems

Logic of proofs , introduced by S. Artemov, originally designed for describing properties of formal proofs, now became a basis for the theory of knowledge with justification (cf. S. Artemov, Evidence-based common knowledge, Technical report TR–2004018, CUNY Ph.D. Program in Computer Science, 2005). So far, in epistemic systems with justification the corresponding “evidence part”, even for multi-agent systems, consisted of a single explicit evidence logic. In this paper we introduce logics describing two interacting explicit evidence systems. We find an appropriate formalization of the intended semantics and prove the completeness of these logics with respect to both symbolic and arithmetical models. Also, we find the forgetful projections for the two-agent justification logics which are extensions of the bimodal logic .