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 pseudo-Euclidean closures this logic equals S5, and the zero-one law holds.