Zero-One Laws for Modal Logic

Abstract We show that a 0–1 law holds for propositional modal logic, both for structure validity and frame validity. In the case of structure validity, the result follows easily from the well-known 0–1 law for first-order logic. However, our proof gives considerably more information. It leads to an elegant axiomatization for almost-sure structure validity and to sharper complexity bounds. Since frame validity can be reduced to a Π 1 1 formula, the 0–1 law for frame validity helps delineate when 0–1 laws exist for second-order logics.

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