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Density of truth in modal logics

Abstract : The aim of this paper is counting the probability that a random modal formula is a tautology. We examine $\{ \to,\Box \}$ fragment of two modal logics $\mathbf{S5}$ and $\mathbf{S4}$ 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 $\mathbf{S5}$ and this is the key to count the proportion of tautologies of $\mathbf{S5}$ among all formulas. Although the logic $\mathbf{S4}$ does not have this property, there exist its normal extensions having finitely many non-equivalent formulas.
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https://hal.inria.fr/hal-01184704
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Zofia Kostrzycka. Density of truth in modal logics. Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, 2006, Nancy, France. pp.161-170, ⟨10.46298/dmtcs.3500⟩. ⟨hal-01184704⟩

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