A propositional system of modal logic is second-order if it contains quantifiers ∀p and ∃p, which, in the standard interpretation, are construed as ranging over sets of possible worlds (propositions).

Richard Zach, Completeness before Post: Bernays, Hilbert, and the Development of Propositional Logic, The Bulletin of Symbolic Logic, Vol. 5, No. 3 (Sep., 1999), pp ...