 In many formal analyses of knowledge in the literature, possibility sets are taken as primitive and Proposition 2.2 is given as the definition of knowledge. If one adopts this viewpoint, then the axioms K1–K5 follow as consequences of the definition of knowledge. In many applications, the agents’ possibility sets are assumed to  partition[11]  the set, in which case \(\mathcal{H}_i\) is called i’s private information partition. Notice that if axioms K1–K5 hold, then the possibility sets of each agent always partition the state set, and vice versa.