 In Henkin semantics, each interpretation specifies a specific range of the higher-order variables. For monadic second-order variables, each interpretation specifies a non-empty subset of the powerset of the domain, for two-place second-order variables, a non-empty set of ordered pairs of members of the domain, etc. The system has all of the above limitative meta-theoretic results. There is a deductive system that is sound and complete for Henkin semantics; the logic is compact; and the downward and upward Löwenheim-Skolem theorems all hold.