Zero-one laws for modal logic

It is shown that a 0-1 law holds for propositional modal logic, both for structure validity and for frame validity. In the case of structure validity, the result follows easily from the well-known 0-1 law for first-order logic. However, the proof gives considerably more information. It leads to an elegant axiomatization for almost-sure structure validity, and sharper complexity bounds. Since frame validity can be reduced to a II₁¹ formula, the 0-1 law for frame validity helps delineate when 0-1 laws exist for second-order logics