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According to this article

Quine has criticized higher-order logic (with standard semantics) as "set theory in sheep's clothing". Quine's criticism focuses on the lack of an effective, sound, complete proof theory; he argues that this makes HOL not a "logic". Shapiro has responded to this criticism, arguing that the additional semantic expressiveness can offset the lack of a proof theory, and arguing that a "logic" need only have a deductive system or a semantical system, but perhaps may not have both.

a. Why does Quine say "set theory in sheep's clothing"?

I tend to agree with Shapiro on the tension between deductiveness & expressiveness.

b. But are there useful results which naturally use the higher-order expressiveness either in their proof or statement, apart from the categoricity of 2-logic as opposed to 1-logic.

Since higher logic with Henkin semantics reduces to typed 1-logic it seems essential to keep full semantics.

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The idea is that, in set theory, one is developing a formal language to talk about sets. In particular, one is quantifying over sets in set theory. Second-order logic, when it introduces predicate variables, seems to be doing something very similar, since the interpretation of these predicates is meant to be a set. You can't quantify over sets of sets in second-order logic, but as you ascend into higher and higher orders, you come closer and closer to set theory. Quine's objection, then, is that second-order "logic" isn't a logic at all: it's a restricted theory of sets.

If you want to learn more about these things, you really ought to read Shapiro's book Foundations without Foundationalism, which is a fantastic introductory source for both the technical and philosophical material regarding second-order logic. I would also suggest looking at Quine's thoughts on these things in Philosophy of Logic. Boolos also has some interesting articles defending (monadic) second-order logic philosophically as a kind of plural logic (i.e. a logic with plural terms).

With regard to using second-order logic in mathematics, you might want to look into the field of reverse mathematics which deals a lot with second-order PA (I don't know much about it myself). The issue of finding proofs that use second-order logic is a bit tricky, as I understand it, since when you do want to use second-order logic, you can often just get away with using first-order set theory.

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    Quines objection can be turned to advantage. Category theorists have come up with toposes to capture category-theoretic properties of Set Theory. In short, they call toposes generalised set theories. It turns out each topos has its own internal language which can be used to reason about the topos - the language being is higher-order intuitionistic typed logic. It works the other way round too. Its interesting that Quine saw the possibility of interpreting logic as set theory! – Mozibur Ullah Jun 3 '13 at 4:58
  • when you say 'since the interpretation of these predicates is meant to be a set' is that in a model of the logic? I can't see clearly how one interprets quantifying a predicate as a set. – Mozibur Ullah Jun 3 '13 at 5:02
  • The interpretation of predicate variables in a model is a set of elements or a set of tuples of elements. It's the same way variables in models are interpreted as elements of the domain. – Alex Kocurek Jun 3 '13 at 16:42

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