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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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Hold on a minute. What does it even mean to speak of proving something in second-order logic? First order has various deduction systems for it, which we usually don't bother to mention by name because they're all equivalent. How does one actually perform deductions in proofs in second-order logic? I was under the impression that second-order logic was purely descriptive (i.e. a language to write precise statements which then may judged true or false) and did not allow for deduction. After all, to perform deductions about sets, one will need some sort of theory of how the sets work, no? And then you may as well just do set theory -- which is after all how we generally do things...
There exist deductive systems for second order logic.
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.