TITLE: Henkin vs. "Full" Semantics for Second-order Logic and Multi-Sorted First Order Interpretations
QUESTION [8 upvotes]: In this paper by Jeff Ketland, he notes:

With Henkin semantics, the Completeness, Compactness and Löwenheim-Skolem Theorems all hold, because Henkin structures can be re-interpreted as many-sorted first-order structures.

What is it about the full semantics for second-order logic that defies re-interpretation into a many-sorted first-order logic?

REPLY [6 votes]: Full semantics differ from Henkin semantics In only one germane way: rather than considering arbitrary Henkin structures, full semantics considers only full structures. That shows immediately why e.g. the completeness theorem fails for full semantics:

Completeness: A formula is provable in second order logic if and only if the formula holds in all Henkin structures
Not every Henkin structure is a full structure, and there are formulas true in every full structure but not every Henkin structure

From the right point of view, it really is that straightforward.