COMMON BELIEF IN MONOTONIC EPISTEMIC LOGIC

We study the extent to which the notion of common belief may be expres sed by a finitary logic. We devise a set of axioms for common belief i n a system where beliefs are only required to be monotonic. These axio ms are generally less restrictive than those in the existing literatur e. We prove completeness with respect to monotonic neighborhood models , in which the iterative definition for common belief may involve tran sfinite levels of mutual belief. We show that this definition is equiv alent to the fixed-point type definition that Monderer and Samet elabo rated in a probabilistic framework. We show further, that in systems a s least as strong as the K-system, our axiomatization for common belie f coincides with other existing axiomatizations. In such systems, howe ver, there are consistent sets of formulas that have no model. We conc lude that the full contents of common belief cannot be expressed by a logic that admits only finite conjunctions.