 In words, L1 says that \(i\) knows \(A\)* at \(\omega\). L2 says that if \(i\) knows that \(A\)* obtains, then \(i\) knows that everyone knows that \(A\)* obtains. This axiom is meant to capture the idea that common knowledge is based upon a proposition \(A\)* that is publicly known, as is the case when agents hear a public announcement. If the agents’ knowledge is represented by partitions, then a typical basis for the agents’ common knowledge would be an element \(\mathcal{M}(\omega)\) in the  meet[14]  of their partitions. L3 says that \(i\) can infer from \(A\)* that \(E\). Lewis’ definition implies the entire common knowledge hierarchy, as is shown in the following result.