Deduction Chains for Logic of Common Knowledge

Deduction chains represent a syntactic and in a certain sense constructive method for proving completeness of a formal system S with respect to the class of S-structures. Given a formula A, the deduction chains of A are built up by systematically decomposing A into its subformulae. In the case where A is a valid formula, the decomposition yields a (usually cut-free) proof of A in the system S. If A is not valid, the decomposition produces a countermodel for A. The method of deduction chains was originally devised by Schütte and is mainly used in the context of first order logic and semiformal systems for various forms of arithmetic. Schütte [2] has also adapted the method to modal logic. In the current study, we extend this approach to a semiformal system for the Logic of Common Knowledge, as studied by Alberucci and Jäger [1]. The presence of fixed point constructs in this logic leads to potentially infinite-length deduction chains of a non-valid formula, in which case fairness of decomposition requires special attention. An adequate order of decomposition also plays an important role in the reconstruction of the proof of a valid formula from the set of its deduction chains.