In this paper, it said that given $\omega$ is the current state of the world and $P_1(\omega)$ is the unique element from the information partition $\textit{P_1}$ of actor 1 in which the player 1 is informed that $P_1(\omega)$ contains $\omega$. According to the aforementioned paper (p.1237), it says that "To say that 1 knows that 2 knows (Event) E means that E includes all $P_2$ in the information partition $\textit{P_2}$ that intersect $P_1(\omega)$." I wonder why is the aforementioned intersection in this passage so important, since intersection here only mean that two information partition elements share some states.
I think there are two thing I need to understand: The interpretation of the shared states of two information partition elements from different people; the interpretation of the case when two informational partition elements intersect.
Also why is it seem to be inapplicable for the case when the partition elements mentioned are partially in $E$