 Definition 2.7  A proposition \(E\) is Lewis-common knowledge at \(\omega \in \Omega\) among the agents of a set \(N = \{1, \ldots ,n\}\) iff there is a proposition \(A\)* such that \(\omega \in A\)*, the agents of \(N\) are \(A\)*-symmetric reasoners, and for every \(i \in N\),  \[\begin{align} \tag{L1} &amp;\omega \in \mathbf{K}_i (A^*) \\ \tag{L2} &amp;\mathbf{K}_i(A*) \subseteq \mathbf{K}_i(\bigcap_{j\in N} \mathbf{K}_j(A^*)) \\ \tag{L3} &amp;\mathbf{K}_i (A*) \subseteq \mathbf{K}_i (E) \end{align}\]   \(A\)* 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 \(\omega \in \mathbf{L}*_N (E)\). 