On the density of truth in modal logics

The aim of this paper is counting the probability that a random modal formula is a tautology. We examine {→, 2} fragment of two modal logics S5 and S4 over the language with one propositional variable. Any modal formula written in such a language may be interpreted as a unary binary tree. As it is known, there are finitely many different formulas written in one variable in the logic S5 and this is the key to count the proportion of tautologies of S5 among all formulas. Although the logic S4 does not have this property, there exist its normal extensions having finitely many non-equivalent formulas.