Modal logics of almost sure validities and zero-one laws in Horn classes

In this paper we develop a method to study Horn classes of Kripke frames from a probabilistic perspective. We consider the uniform distribution on the set of all $n$-point Kripke frames. A formula is almost surely valid in a Horn class $\mathcal{F}$ if the probability that it is valid in the $\mathcal{F}$-closure of a random $n$-point frame tends to $1$ as $n\to\infty$. Such formulas constitute a normal modal logic. We show that for pseudotransitive and pseudoeuclidean closures this logic equals $\mathrm{S}5$, and the zero-one law holds.