# Candidate "AEC-yielding" fragments of bad logics

Given a logic $$\mathcal{L}$$ and a signature $$\Sigma$$, let the $$\Sigma$$-system of $$\mathcal{L}$$ be the pair $$Sys_\Sigma(\mathcal{L})=(Struc(\Sigma),\preccurlyeq_\mathcal{L})$$ of structures partially ordered by the $$\mathcal{L}$$-elementary substructure relation. This is sometimes, but not always, an abstract elementary class; the first four AEC axioms always hold, but the Tarski-Vaught or Lowenheim-Skolem axioms could easily fail. Say that a logic yields AECs iff its $$\Sigma$$-system is an AEC for every signature $$\Sigma$$. Examples of logics which yield AECs include first-order logic and the "small" infinitary logic $$\mathcal{L}_{\omega_1,\omega}$$, while the most natural examples of logics not yielding AECs in my opinion are second-order logic $$\mathsf{SOL}$$ and the "big" infinitary logic $$\mathcal{L}_{\omega_1,\omega_1}$$.

Quick caveat: note that the Lowenheim-Skolem axiom for AECs is stronger than what one might expect coming from non-AEC land. Also, the Tarski-Vaught axiom has nothing to do with the Tarski-Vaught test.

Now suppose we have a logic which doesn't yield AECs, but we want to find a reasonably large and canonically definable fragment of it which does. There's a trick which seems promising, at least as far as the Tarski-Vaught axiom goes. For a logic $$\mathcal{J}$$, let $$\hat{\mathcal{J}}$$ be the set of $$\mathcal{J}$$-formulas stable under elementary chains, that is, the set of $$\varphi\in\mathcal{J}$$ such that whenever $$(\mathfrak{A}_i)_{i<\eta}$$ are structures with $$\mathfrak{A}_i\preccurlyeq_\mathcal{L}\mathfrak{A}_j$$ for all $$i we have $$\varphi^{\mathfrak{A}_0}=\varphi^{\bigcup_{i<\eta}\mathfrak{A}_i}\cap\mathfrak{A}_0^{arity(\varphi)}$$. Now given a logic $$\mathcal{L}$$ we can iterate this process through the ordinals, starting with $$\mathcal{L}$$ itself and taking intersections at limit stages as usual; we eventually hit a fixed point $$\mathcal{L}^*$$, and it's easy to show that $$(i)$$ for each $$\Sigma$$, if $$Sys_\Sigma(\mathcal{L}^*)$$ satisfies the Lowenheim-Skolem axiom then it is an AEC and $$(ii)$$ every sublogic of $$\mathcal{L}$$ which yields AECs is a sublogic of $$\mathcal{L}^*$$.

I'd like to know what happens to the two most natural logics, in my mind, which fail to yield AECs:

Does $$\mathsf{SOL}^*$$ yield AECs? Does $$\mathcal{L}_{\omega_1,\omega_1}^*$$ yield AECs?

I strongly suspect the answer to the first question is negative: failures of the Lowenheim-Skolem axiom for $$Sys_\emptyset(\mathsf{SOL})$$ are easy to come by, e.g. there is a second-order sentence true in exactly those structures of infinite limit cardinality. However, while there are lots of such failures, each one I can think of goes away when we pass to $$\mathsf{SOL}^*$$. So I don't immediately see an obstacle to $$\mathsf{SOL}^*$$ yielding AECs.

I'm much more ambivalent about $$\mathcal{L}_{\omega_1,\omega_1}^*$$. Tentatively I'll suspect that it does, but I really don't know.