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    Inverse View

    It is not the case that Quine argued that second-order logic is 'set theory in sheep's clothing,' meaning purportedly second-order quantification over relations reduces to first-order quantification over sets.

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    Reasons For

    1 perspective
    Reason for
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    • 1.Second-order quantification has a distinct logical character—quantifying over functions or relations directly—independent of set representations.
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    • 2.First-order set theory itself requires second-order principles (like comprehension) to formalize, so reduction creates circularity rather than clarity.
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    • 3.Second-order logic captures structural properties (like categoricity) that first-order logic cannot, suggesting genuine logical expressiveness beyond set theory.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Second-order quantifiers range over all subsets of a domain, which requires set-theoretic machinery to define properly.
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    • 2.Without reducing second-order logic to set theory, we cannot formally specify what counts as a valid second-order interpretation.
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    • 3.First-order logic with set variables can express everything second-order logic expresses, making the apparent distinction ontologically redundant.
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