We present characterizations of large cardinals in terms of properties of strong logics, such as Löwenheim-Skolem and weak compactness properties of certain fragments of infinitary second-order logic. For example, a cardinal kappa is strong if and only if the Löwenheim-Skolem property holds at kappa for a certain fragment of infinitary second-order logic, namely for the sentences whose negation normal form has only <kappa-length conjunctions and finitary existential quantifiers (but arbitrary-length disjunctions and universal quantifiers.)
We also present recent work with Jonathan Osinski obtaining a related result for a kind of weak compactness property. Namely, a cardinal kappa is Shelah if and only if the following property holds: for every theory of size kappa in the dual of the aforementioned logic fragment, if every subtheory of size less than kappa has a model of size less than kappa, then the theory itself has a model.