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    Carmelics

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    Made withinDC&Austin
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    Inverse View

    It is not the case that As Henkin (1950) demonstrated, weakening the semantics of second-order logic to general models preserves completeness but loses categoricity, so validity in full models and validity relative to θ come apart.

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    Reasons For

    1 perspective
    Reason for
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    • 1.Henkin's completeness theorem doesn't 'lose' categoricity—full second-order logic was never categorical for all theories, only for specific axiomatizations like PA.
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    • 2.The distinction between full and general model validity may reflect our incomplete understanding of quantification over properties, not a genuine semantic difference.
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    • 3.Preserving completeness while accepting non-categoricity might be desirable: it trades metaphysical assumptions about properties for proof-theoretic tractability.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Henkin's general models achieve completeness because they allow non-standard interpretations that validate exactly the theorems provable in SOL.
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    • 2.Categoricity requires full models only; general models necessarily admit multiple non-isomorphic models of the same theory, making categoricity impossible.
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    • 3.The divergence between full and general model validity demonstrates that standard semantics captures intended meaning while Henkin semantics captures proof-theoretic structure.
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