Consequently quantification over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity ...

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).

He constructed the first propositional logic and the first predicate logic. A few years later, but independently, the Italian mathematician Giuseppe Peano also invented a version of predicate logic.