Common Belief in Monotonic Epistemic Logic

To what extent maya finitary logic express the notion of common belief? We devise a set of axioms for common belief in a system where beliefs are only required to be monotonic. These axioms are generally less restrictive than those suggested by Lismont-Mongin (1993) and Halpern-Vardi (1992). We prove completeness with respect to monotonic neighbourhood models, in which the iterative definition for common belief may involve transfinite levels of mutual belief. We show that this definition is equivalent to the fixed-point type definition that Monderer and Samet (1989) elaborated in a probabilistic framework. We show further, that in systems as least at strong as the K-system, the three axiomatizations for common belief coincide) as do their semantic counterparts. In such systems, however, there are consistent sets of formulas that have no model. We conclude that the full contents of common belief cannot be expressed by a logic that admits only finite conjunctions .