 From this it has been concluded that derivability (in any calculus) must be incomplete with respect to logical truth. But a fundamental problem is that this conclusion is based on two assumptions that will not necessarily be granted by the champion of derivability: first, the assumption that the expressions typically cataloged as logical in higher-order languages, and in particular the quantifiers in quantifications of the form \(\forall X\) (where \(X\) is a higher-order variable), are in fact logical expressions; and second, the assumption that being universally valid is a sufficient condition for logical truth. On these assumptions it is certainly very reasonable to think that derivability, in any calculus satisfying (4), must be incomplete with respect to logical truth. But in the absence of additional considerations, a critic may question the assumptions, and deny relevance to the argument. The second assumption would probably be questioned e.g. from the point of view that logical truths must be analytic, for there is no conclusive reason to think that universally valid formulae must be analytic. The first assumption actually underlies any conviction one may have that (4) holds for any one particular higher-order calculus. (Note that if we denied that the higher-order quantifiers are logical expressions we could equally deny that the arguments presented above against the soundness of model-theoretic validity with respect to logical truth are relevant at all.) That the higher-order quantifiers are logical has often been denied on the grounds that they are semantically too “substantive”. It is often pointed out in this connection that higher-order quantifications can be used to define sophisticated set-theoretic properties that one cannot define just with the help of first-order quantifiers. (Defenders of the logical status of higher-order quantifications, on the other hand, point to the wide applicability of the higher-order quantifiers, to the fact that they are analogous to the first-order quantifiers, to the fact that they are typically needed to provide categorical axiomatizations of mathematical structures, etc. See Quine (1970), ch. 5, for the standard exponent of the restrictive view, and Boolos (1975) and Shapiro (1991) for standard exponents of the liberal view.) 