I'm looking for a good introduction to second order logic. Any recommendations?
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$\begingroup$ The most common academic characterization (by certainly not the only one) of foundations of mathematics is ZFC axioms in first order logic. I mention this in case 2nd order logic for you is a means rather than a goal. Also, 1st and 2nd order logic aren't logics themselves, they are sets of logics. There is no one "second order logic" just like there is no one "first order logic". $\endgroup$– DanielVJun 21, 2016 at 20:10
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1$\begingroup$ If you're interested in second-order arithmetic, an excellent reference I found online is sas.upenn.edu/~htowsner/prooftheory/ReverseMath.pdf. $\endgroup$– user21820Jun 23, 2016 at 3:29
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4$\begingroup$ @DanielV: As far as I know, "first-order logic" is used to refer to the unique logical consequence relation, which by completeness turns out to be also the provability relation. So one can justifiably say that there is only one first-order (classical) logic. Of course, there are different first-order theories, but they all use the same first-order logic. It is a different matter for second-order logic, where you might have Henkin semantics (and hence essentially reducible to first-order logic) or full semantics (but no recursive deductive system). $\endgroup$– user21820Jun 23, 2016 at 3:37
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For a book-long introduction to SOL, both philosophical and mathematical, the place to go is:
Stewart Shapiro, Foundations without Foundationalism: A Case for Second Order Logic, Oxford Logic Guides 17, OUP.
For a more concise treatment try the Stanford Encyclopedia of Philosophy article.