Pairing Traditional and Generic Common Knowledge

Common Knowledge C is a standard tool in epistemic logics. Generic Common Knowledge J is an alternative which has desirable logical behavior such as cut-elmination and which can be used in place of C in the analysis of many games and epistemic senarios. In order to compare their deductive strengths directly we define the multi-agent logic S4 n built on a language with both C and J operators in addition to agents’ Kis so that any finite prefix of modal operators is acceptable. We prove S4 n is complete, decidable, and that Jφ → Cφ though not Cφ → Jφ. Additional epistemic scenarios may be investigated which take advantage of this dual layer of common knowledge agents.