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    Preserving completeness while accepting non-categoricity ... — Carmelics
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    Challenges→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.

    Preserving completeness while accepting non-categoricity might be desirable: it trades metaphysical assumptions about properties for proof-theoretic tractability.

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