# Existing literature on logics "describing their own equivalence notions"

Say that a regular logic $$\mathcal{L}$$ is self-equivalence-describing (SED) iff for every finite language $$\Sigma$$ there is a larger language $$\Sigma'$$ containing at least $$\Sigma$$ and two new unary relation symbols $$A,B$$ and a $$\mathcal{L}[\Sigma']$$-sentence $$\eta$$ such that the following are equivalent for any pair of $$\Sigma$$-structures $$\mathcal{A},\mathcal{B}$$:

• $$\mathcal{A}\equiv_\mathcal{L}\mathcal{B}$$.

• There is a model $$\mathcal{M}$$ of $$\eta$$ such that $$A^\mathcal{M}\cong\mathcal{A}$$ and $$B^\mathcal{M}\cong\mathcal{B}$$.

For example, first-order logic is SED via Fraisse's theorem (this is a crucial point in the proof of Lindstrom's theorem), and in fact Fraisse's theorem also implies the SED-ness of second-order logic since - roughly speaking - powersethood is second-order definable and second-order equivalence corresponds to first-order equivalence of "powerstructures." On the other hand, infinitary first-order logics are usually not SED; Farmer S. has shown that neither $$\mathcal{L}_{\omega_1,\omega}$$ nor $$\mathcal{L}_{\omega_2,\omega}$$ is SED, the latter proof starting similarly to but being substantially harder than the former, and it seems that there is a decently strong negative result implicit in these arguments.

SED seems like a reasonable property to consider, but I haven't been able to find existing literature on the topic (although to be fair my literature search has largely been limited to the book Model-theoretic logics and its bibliography). So I would like to ask:

Are there any sources on (non-)SED-ness in the existing abstract-model-theoretic literature?

Here are some concrete-ish questions which I am interested in and suspect have been looked at already:

• Which $$\mathcal{L}_{\kappa,\omega}$$s are SED? Which $$\mathcal{L}_{\kappa,\kappa}$$s are SED? What about infinitary second-order logics?

• Are there any "natural" logics which are SED for reasons not having to do with Fraisse's theorem?

• Is there (or when is there) a reasonably-natural way to take a non-SED logic $$\mathcal{L}$$ and construct a stronger logic $$\mathcal{L}'$$ which does have SED, or at least can characterize $$\mathcal{L}$$-equivalence with a single sentence?