 Some preliminary notions must be stated first. Following C. I. Lewis (1943–1944) and Carnap (1947), propositions are formally subsets of a set \(\Omega\) of state descriptions or possible worlds. One can think of the elements of \(\Omega\) as representing Leibniz’s possible worlds or Wittgenstein’s possible states of affairs. Some results in the common knowledge literature presuppose that \(\Omega\) is of finite cardinality. If this admittedly unrealistic assumption is needed in any context, this will be explicitly stated in this essay, and otherwise one may assume that \(\Omega\) may be either a finite or an infinite set. A distinguished actual world \(\omega_{\alpha}\) is an element of \(\Omega\). A proposition \(A \subseteq \Omega\) obtains (or is true) if the actual world \(\omega_{\alpha} \in A\). In general, we say that \(A\) obtains at a world \(\omega \in \Omega\) if \(\omega \in A\). What an agent \(i\) knows about the possible worlds is stated formally in terms of a knowledge operator \(\mathbf{K}_i\). Given a proposition \(A \subseteq \Omega , \mathbf{K}_i (A)\) denotes a new proposition, corresponding to the set of possible worlds at which agent \(i\) knows that A obtains. \(\mathbf{K}_i (A)\) is read as ‘\(i\) knows (that) \(A\) (is the case)’. The knowledge operator \(\mathbf{K}_i\) satisfies certain axioms, including: