Full Idea
The counterparts of Completeness, Compactness and the Löwenheim-Skolem theorems all fail for second-order languages with standard semantics, but hold for Henkin or first-order semantics. Hence such logics are much like first-order logic.
Gist of Idea
Completeness, Compactness and Löwenheim-Skolem fail in second-order standard semantics
Source
Stewart Shapiro (Foundations without Foundationalism [1991], 4.1)
Book Reference
Shapiro,Stewart: 'Foundations without Foundationalism' [OUP 1991], p.80
A Reaction
Shapiro votes for the standard semantics, because he wants the greater expressive power, especially for the characterization of infinite structures.
Related Ideas
Idea 13648 The Löwenheim-Skolem theorems show an explosion of infinite models, so 1st-order is useless for infinity [Shapiro]
Idea 13645 In standard semantics for second-order logic, a single domain fixes the ranges for the variables [Shapiro]