 Frege’s Begriffsschrift, recall, was designed to capture the Dedekind-Peano axioms for arithmetic, including the axiom of induction; see the entry on  Frege’s theorem and foundations for arithmetic.  This required quantification into positions occupiable by predicates, as well as positions occupiable by names. Using modern notation, Frege allowed for formulae like ‘\((Fa \land Fb) \rightarrow \exists X (Xa \land Xb)\)’ and ‘\(\forall x \forall y [x = y \leftrightarrow \forall X (Xx \leftrightarrow Xy)]\)’. And he took second-order quantification to be quantification over functions. This is to say, for example, that ‘\(\exists X (Xa \land Xb)\)’ is true iff: there is a function, \(X\), that maps both the individual called ‘\(a\)’ and the individual called ‘\(b\)’ onto the truth-value T. Frege also took it to be a truth of logic that for any predicate \(P\), there is a function such that for each individual \(x\), that function maps \(x\) to T iff \(x\) satisfies (or “falls under”) \(P\). In which case, for each predicate, there is the set of all and only the things that satisfy the predicate. The axioms for Frege’s logic thus generated  Russell’s paradox,   given predicates like ‘is not a member of itself’. This invited attempts to weaken the axioms, while preserving second-order quantification. But for various reasons, Quine and others advocated a restriction to a first-order fragment of Frege’s logic, disallowing quantification into positions occupied by predicates.  (Kurt Gödel had proved the completeness of first-order predicate calculus, thus providing a purely formal criterion for what followed from what in that language. Quine also held that second-order quantification illicitly treated predicates as names for sets, thereby spoiling Frege’s conception of propositions as unified by virtue of having unsaturated predicational constituents that are satisfied by things denoted by names.) On Quine’s view, we should replace