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Sniffnoy 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.
The different deductive systems described there (can't access the link, wikiped closed) all seem the same - they differ only in the axioms they use, which isn't really a difference in deductive systems.
But the question is, starting from the same axioms -- not logical axioms, not axioms of the deductive systems, but the axioms of whatever it is you're trying to reason about -- would they produce the same theorems?
Wikipedia is accessible if you disable JavaScript (or use a mobile app, or just Google cache).
If anyone is curious, I'm downvoting everyone in this thread - not only is this a terrible place to discuss SOPA and blackout circumventions (seriously, we can't wait a day and get on with our lives?), there's already a SOPA post in Discussion.