Abstract.
We study the modal logic M L r of the countable random frame, which is contained in and `approximates' the modal logic of almost sure frame validity, i.e. the logic of those modal principles which are valid with asymptotic probability 1 in a randomly chosen finite frame. We give a sound and complete axiomatization of M L r and show that it is not finitely axiomatizable. Then we describe the finite frames of that logic and show that it has the finite frame property and its satisfiability problem is in EXPTIME. All these results easily extend to temporal and other multi-modal logics. Finally, we show that there are modal formulas which are almost surely valid in the finite, yet fail in the countable random frame, and hence do not follow from the extension axioms. Therefore the analog of Fagin's transfer theorem for almost sure validity in first-order logic fails for modal logic.
Similar content being viewed by others
Author information
Authors and Affiliations
Additional information
Received: 1 May 2000 / Revised version: 29 July 2001 / Published online: 2 September 2002
Mathematics Subject Classification (2000): 03B45, 03B70, 03C99
Key words or phrases: Modal logic – Random frames – Almost sure frame validity – Countable random frame – Axiomatization – Completeness
Rights and permissions
About this article
Cite this article
Goranko, V., Kapron, B. The modal logic of the countable random frame. Arch. Math. Logic 42, 221–243 (2003). https://doi.org/10.1007/s001530100135
Issue Date:
DOI: https://doi.org/10.1007/s001530100135