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Tyrrell_McAllister comments on Completeness, incompleteness, and what it all means: first versus second order logic - Less Wrong
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Right. There exist deductive systems, plural. Are they equivalent, like the ones for first-order logic are? As I understand it, If you want to do deduction in second-order logic, you need to specify a deductive system; you can't just do deduction in second-order logic alone. Whereas in first-order logic there's no need to specify the deductive system because they're all equivalent.
Good question. I don't know the answer. If they're not equivalent, then I see your point.