Structural logic and abstract elementary classes with intersections

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 prove that any AEC closed under intersections with countable Löwenheim–Skolem–Tarski number is axiomatizable in $\mathbb {L}_{\infty , \omega } (Q)$, where $Q$ is the quantifier “there exist uncountably many”.