Full Idea
In addition to standard and Henkin semantics for second-order logic, one might also employ substitutional or game-theoretical or topological semantics, or Boolos's plural interpretation, or even a semantics inspired by Lesniewski.
Clarification
Lesniewski is the father of mereology
Gist of Idea
There are at least seven possible systems of semantics for second-order logic
Source
Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §3)
A Reaction
This is helpful in seeing the full picture of what is going on in these logical systems.