 Any answer can only be speculative. One factor, a minor one, is that Peirce was not himself a nominalist. Another is that he operated within a variety of logical systems: he was temperamentally eclectic, and not disposed to search for the “one true logic”. There are also technical considerations. Peirce, unlike Hilbert, does not present first-intentional logic as an axiomatized system, nor does he urge it as a vehicle for studying the foundations of mathematics. He does not possess the distinction between an uninterpreted, formal, axiomatic calculus and its metalanguage. As a result, he does not ask about questions of decidability, or completeness, or categoricity; and without the metamathematical results a full understanding of the differences in expressive power between first-order and second-order logic was not available to him. One of the strongest arguments against second-order logic—that quantification over all subsets of a denumerable collection entails quantification over a non-denumerable totality—could not even have been formulated until Cantor’s Theorem was known. The logical and set-theoretical paradoxes were not yet discovered, and Zermelo had not yet axiomatized the theory of sets: so Peirce lacked the acute sense of motivation to discover a “secure foundation for mathematics”. And of course Peirce had no inkling of the Löwenheim-Skolem theorems, or the Skolem paradox, or the sequence of metalogical theorems that were to bring first-order logic into sharp focus. He provided a flexible and suggestive notation that was to prove enormously fertile, and he was the first to distinguish clearly between first-order and second-order logic: but the tools for understanding the mathematical significance of the distinction did not yet exist. (As Henri Pirenne once remarked, the Vikings discovered America, but they forgot about it, because they did not yet need it.)