 involving generalized quantifiers are syntactically first-order, insofar as the quantified variables \(z_{i1},\ldots ,z_{in_i} = [z_i]\) are first-order \((1 \le i \le k)\). The semantics of a generalized quantifier \(Q\) is formulated by associating with each domain M a \(k\)-ary relation \(Q_M\) on M, with \(Q_M \subseteq\) M\(^{n_1} \times \ldots \times\) M\(^{n_k}\). E.g., for any infinite cardinal \(\kappa\), there is a generalized quantifier \(Q_{\ge \kappa}\) such that \(Q_{\ge \kappa}z P(z)\) is true in a model \(M\) iff there are at least \(\kappa\) elements that satisfy the predicate \(P\). Hence generalized quantifiers can be semantically higher-order. (The notion of cardinality is a higher-order notion.) The fact that the variables in a formula range over individuals only, does not offer a reliable criterion for the logic’s first-order status. 