Zero-One Laws for Modal Logic

We show that a O-l 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 O-l 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 Z7: formula, the O-l law for frame validity helps delineate when O-l laws exist for second-order logics.