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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that If a second-order formulation of F can fix the standard model categorically, then the existence of non-standard models of first-order F reflects an expressive limitation of first-order logic, not an ineliminable feature of F's intended semantics.

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    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Second-order categoricity requires strong set-theoretic assumptions that are themselves contentious and not part of F's original intended semantics.
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    • 2.Non-standard models may reflect genuine semantic indeterminacy in F itself, not merely first-order logic's weakness—second-order capture doesn't establish otherwise.
      ?

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    • 3.Categoricity in second-order logic is relative to background set theory; different set theories yield different 'standard' models, undercutting the argument.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Second-order logic can quantify over all subsets, enabling categorical axiomatization impossible in first-order logic's limited expressiveness.
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    • 2.Non-standard models arise precisely where first-order logic fails to express constraints that second-order formulations capture naturally.
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    • 3.If F's intended semantics uniquely determines its models, then expressive limitations—not F itself—explain why first-order formulations admit alternatives.
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