Completely separably MAD families and the modal logic of βω

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We show in ZFC that the existence of completely separable maximal almost disjoint families of subsets of ω implies that the modal logic S4.1.2 is complete with respect to the Čech-Stone compactification of the natural numbers, the space βω. In the same fashion we prove that the modal logic S4 is complete with respect to the space ω*=βωω. This improves the results of G. Bezhanishvili and J. Harding who prove these theorems under stronger assumptions (a=c). Our proof is also somewhat simpler.