Outward compactness

We introduce and study a new type of compactness principle for strong logics that, roughly speaking, infers the consistency of a theory from the consistency of its small fragments in certain outer models of the set-theoretic universe. We refer to this type of compactness property as outward compactness, and we show that instances of this type of principle for second-order logic can be used to characterize various large cardinal notions between measurability and extendibility, directly generalizing a classical result of Magidor that characterizes extendible cardinals as the strong compactness cardinals of second-order logic. In addition, we generalize a result of Makowsky that shows that Vopěnka's Principle is equivalent to the existence of compactness cardinals for all abstract logics by characterizing the principle "Ord is Woodin" through outward compactness properties of abstract logics.