  We give a syntactic characterization of abstract elementary classes (AECs)
closed under intersections using a new logic with a quantifier for isomorphism
types that we call structural logic: we prove that AECs with intersections
correspond to classes of models of a universal theory in structural logic. This
generalizes Tarski's syntactic characterization of universal classes. As a
corollary, we obtain that any AEC with countable L\"owenheim-Skolem number is
axiomatizable in $\mathbb{L}_{\infty, \omega} (Q)$, where $Q$ is the quantifier
"there exists uncountably many".
