 Russell and Whitehead thus possessed a notation for the two quantifiers, as well as a distinction between quantifications of the first and higher types. But this is not the same as possessing a conception of first-order logic, conceived as a free-standing logical system, worthy of study in its own right. There were essentially two things blocking the way. First (and in contrast to Peirce), their object of study was not multiple logical systems, but logic tout court: they show no interest in splitting off a fragment for separate study, let alone in arguing that the first-order fragment enjoys a privileged status. On the contrary: as with Frege, the ambition of Principia was to demonstrate that mathematics can be reduced to logic, and for Whitehead and Russell logic encompassed the full apparatus of ramified type theory (together with the axioms of infinity, choice, and reducibility). Secondly, although Principia provided an axiomatization of type theory (and thus can be viewed as specifying a conception of deductive consequence), Whitehead and Russell thought of their system as an interpreted system, stating the truths of logic, rather than as a formal calculus in the sense of Hilbert. Hilbert was to use their axiomatization as the starting-point for his own axiomatizations of various systems of logic; but until the distinction between logic and metalogic had been formulated, it did not naturally occur to anybody to pose the metalogical questions of completeness, consistency, and decidability, or to investigate such matters as the relationship between deductive and semantic completeness, or failures of categoricity; and it was only once such notions became the focus of attention that the significance of first-order logic became apparent.