 Peirce drew these various distinctions—and, in particular, the distinction between first-order and second-order logic—with greater clarity than any logician until Hilbert’s lectures in 1917. And, unlike Hilbert, Peirce was steeped in the writings of the medieval logicians. He fully appreciated the philosophical significance of the arguments about the reality of universals: this is clearly why he drew such a sharp differentiation between the logic of §2 and that of §3. It was thus open to him to make (or, at least, to consider) a nominalist argument on behalf of first-order logic, and against second-order logic. But apart from a few incidental remarks, he himself did not further develop his observations on second-intentional logic, and it seems likely that the modern distinction between first-order and higher-order logic was a re-discovery made independently in 1917/18 by Hilbert, rather than being directly inspired by Peirce.