  We show in ZFC that the existence of completely separable maximal almost
disjoint families of subsets of $\omega$ implies that the modal logic S4.1.2 is
complete with respect to the \v{C}ech-Stone compactification of the natural
numbers, the space $\beta\omega$. In the same fashion we prove that the modal
logic S4 is complete with respect to the space
$\omega^*=\beta\omega\setminus\omega$. This improves the results of G.
Bezhanishvili and J. Harding who prove these theorems under stronger
assumptions ($\mathfrak{a}=\mathfrak{c}$). Our proof is also somewhat simpler.
