 \(A\)* is a basis for the agents’ common knowledge. \(\mathbf{L}*_N (E)\) denotes the proposition defined by L1–L3 for a set \(N\) of \(A\)*-symmetric reasoners, so we can say that \(E\) is Lewis-common knowledge for the agents of \(N\) iff \(\omega \in \mathbf{L}*_N (E)\).