https://www.jstor.org/stable/2273786?seq=1#page_scan_tab_contents Is the article where it is from. It seems to have never been added to the library, which would be my fault.

https://projecteuclid.org/download/pdf_1/euclid.pl/1235417266 is the first chapter of the textbook where I personally found it, although I would use the JSTOR article in the library of Cantor's Attic after this post.

The massive textbook I used is called "Model-Theoretic
Logics" by the "Ω-Group" (the coolest pen-name for a group of mathematicians). In Part F, this equivalence is proven.

Chapter XVII is "Set-theoretic definability of logics" (written by Väänänen, one of my heroes) which is where this comes from. The definition is quite nuanced, but it is a great read. I recommend this textbook.

## What Is a Logic? (loose definition)

We start with a "vocabulary." $\tau$ is often used for this type of object; it is defined as a (nonempty) class of constant symbols, finitary predicate symbols, and finitary function symbols (simple enough). However, you also have to have "sort symbols" for every function symbol and every constant symbol, and furthermore you have to have sort symbols for every argument of every relation symbol and every input of every function symbol.

The sort symbols are just for well-defining "terms" and their syntactics. The "$\tau$-terms" are built up of just function symbols applied to constant symbols, each of which are then given a "sort" to help define things later.

Using the sort, you can well-define the number of inputs in a given "formula" made out of $\tau$.

An "abstract logic" $\mathcal{L}$ gives every vocabulary $\tau$ a class $\mathcal{L}(\tau)$ of formulae and "atomic $\tau$-formulae of $\mathcal{L}$." More important is that it gives every $\tau$ a relation $\models^\tau$ which defines semantics: how a structure interprets a formula of $\mathcal{L}$.

The kind of logic we are talking about has been formalized by Väänänen, but it suggests that the definitions of $\models^\tau$ and deciding what things are atomic formulae are recursive somewhat (synctatic) and furthermore there is a set $A$ such that every formula of $\mathcal{L}$ is in $A$ and that $\mathcal{L}$ can easily describe this set $A$ (the syntax of $\mathcal{L}$ is representable).

There is a lot more to this definition that I haven't talked about, so this is a very rough summary. I urge you to take a look at the book instead of just taking my word (I am in no ways an expert).

Model Theoryor in the bookModel Theoretic Logicsedited by Barwise and Feferman. $\endgroup$7more comments