Full Idea
Second-order logic raises doubts because of its ontological commitment to the set-theoretic hierarchy, and the allegedly problematic epistemic status of the second-order consequence relation.
Gist of Idea
Second-order logic needs the sets, and its consequence has epistemological problems
Source
Marcus Rossberg (First-order Logic, 2nd-order, Completeness [2004], §1)
A Reaction
The 'epistemic' problem is whether you can know the truths, given that the logic is incomplete, and so they cannot all be proved. Rossberg defends second-order logic against the second problem. A third problem is that it may be mathematics.