Discrete Mathematics & Theoretical Computer Science (Jan 2006)

Density of truth in modal logics

  • Zofia Kostrzycka

DOI
https://doi.org/10.46298/dmtcs.3500
Journal volume & issue
Vol. DMTCS Proceedings vol. AG,..., no. Proceedings

Abstract

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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.

Keywords