Reading on Infinitary languages I'm only seeing first order infinitary languages $\mathcal L\_{\kappa, \lambda}$, i.e. in all of these languages no quantification over predicate and function symbols is allowed.



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> Are there second (or even higher) order infinitary languages? I mean something like $\mathcal L^2\_{\kappa, \lambda}$ (more generally $\mathcal L^n\_{\kappa, \lambda}$)rendering of the infinitary first order language $\mathcal L\_{\kappa, \lambda}$. Or, is it the case that all of those are reducible to first order infinitary languages, and so dispense with all of them?
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> If there are, what are the recommended sources on those?
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Are there second (or higher) order infinitary logic languages? References?
Sure there are. They even come up in practice from time to time; e.g. to show that assuming Vopenka's Principle the modal analogue of *second-order* logic (a la [Hamkins/Woloszyn](https://arxiv.org/abs/2009.09394)) has definable-in-$V$ semantics, the only argument I'm aware of goes through $\mathcal{L}\_{\theta,\theta}^2$ where $\theta$ is the smallest limit of extendible cardinals. *(This is basically a Los-Tarski-type argument, see [here](https://mathoverflow.net/questions/428896/how-strong-is-this-modal-definability-principle).)*


That said, even finitary second-order logic is incredibly complicated. In particular, it doesn't have any good metatheorems (compare forcing absoluteness for $\mathcal{L}\_{\infty,\omega}$ or [complicated analogues of Barwise compactness](https://www.sciencedirect.com/science/article/pii/0168007284900125?via%3Dihub) for $\mathcal{L}\_{\infty,\omega\_1}$). So I'm not aware of any source treating infinitary extensions of $\mathsf{SOL}$ or higher-order logics in detail; it's more something that is treated as it shows up.


It's also worth noting that there's a finitary second-order sentence not equivalent even to any $\mathcal{L}\_{\infty,\infty}$ sentence:



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>  "The cardinality of the universe is a successor cardinal."
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