Vopenka's principle is equivalent to the existence of a strong compactness cardinal for any "logic"? According to Cantor's attic, Vopenka's principle is equivalent to the existence of a strong compactness cardinal for any "logic". But I can't find a definition of what a "logic" is either there or in any of the cited references.
Question: What is a "logic" in the sense of this statement?
Some examples are given: apparently infinitary logic $L_{\kappa\kappa}$ is a "logic", and infinitary higher(but finite)-order logic $L_{\kappa\kappa}^n$ is a "logic". Thus Vopenka's principle should imply the existence of a proper class of strongly compact cardinals, and even a proper class of extendible cardinals.
But I don't think it's supposed to be as simple as "Vopenka's principle is equivalent to a proper class of extendibles", so there must be more "logics" than these. The next thing I can think of is some sort of infinitary-order infinitary logic $L^\alpha_{\kappa\kappa}$. It would also make sense to consider structures with infinitary operations. I don't know if there's a large cardinal principle associated to strong compactness for either of these sorts of logic.
And then of course there are "logics" such as various flavors of type theory (which higher order logic starts to resemble anyway!) but just for cultural reasons, I doubt that "logic" is meant to encompass anything along these lines.
 A: 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).
