 Aumann (1976) gives the first mathematically rigorous formulation of common knowledge using set theory. Schiffer (1972) uses the formal vocabulary of epistemic logic (Hintikka 1962) to state his definition of common knowledge. Schiffer’s general approach is to augment a system of sentential logic with a set of knowledge operators corresponding to a set of agents, and then to define common knowledge as a hierarchy of propositions in the augmented system. Bacharach (1992), Bicchieri (1993) and Fagin, et al. (1995) adopt this approach, and develop logical theories of common knowledge which include soundness and completeness theorems. Fagin, et al. show that the syntactic and set-theoretic approaches to developing common knowledge are logically equivalent.