 If \(E = \mathbf{K}^1_N (E)\), then \(E\) is a public event (Milgrom 1981) or a common truism (Binmore and Brandenburger 1989). Clearly, a common truism is common knowledge whenever it occurs, since in this case \(E = \mathbf{K}^1_N (E) = \mathbf{K}^2_N (E) =\ldots\) , so \(E = \mathbf{K}^*_N (E)\). The proof of Proposition 2.17 shows that the common truisms are precisely the elements of \(\mathcal{M}\) and unions of elements of \(\mathcal{M}\), so any commonly known event is the consequence of a common truism.