 This is what the entry on  second-order and higher-order logic  calls general semantics. The metatheory of second-order logic with Henkin models is very much like the metatheory of classical quantificational logic in that it is complete, compact, and subject to a Löwenheim-Skolem theorem. One reason for this is that second-order logic under the Henkin model theory is only a notational variant of a two-sorted first-order theory in which the axioms of quantificational logic are supplemented by suitable axioms of comprehension.