  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 Vop\v{e}nka'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.
