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We show in ZFC that the existence of completely separable maximal almostdisjoint families of subsets of $\omega$ implies that the modal logic S4.1.2 iscomplete with respect to the \v{C}ech-Stone compactification of the naturalnumbers, the space $\beta\omega$. In the same fashion we prove that the modallogic 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 strongerassumptions ($\mathfrak{a}=\mathfrak{c}$). Our proof is also somewhat simpler.

Completely separably MAD families and the modal logic of $\beta\omega$