 When considering only a single agent, it is typically not relevant to include more worlds in W than there are possible valuations of atoms. In multi-agent cases, this is not the case: to express the different forms of available higher-order knowledge, many copies of “the same” world are needed. Let us exemplify for \(\mathcal{A}=\{a,b\}\), \(\textit{Atom}=\{p\}\) and each \(R_{i},i\in\mathcal{A},\) an equivalence relation. Let us represent that both a and b know p, but b does not know that a knows p , i.e., \(K_{a}p\wedge K_{b}p\wedge\neg K_{b}K_{a}p\). Then we need three worlds: