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 F if the probability that it is valid in the F-closure of a random 𝑛-point frame tends to 1 as 𝑛 → ∞. Such formulas constitute a normal modal logic. We show that for pseudotransitive and pseudoeuclidean closures this logic equals S5, and the zero-one law holds.