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    Made withinDC&Austin
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    Inverse View

    It is not the case that Henkin's completeness proof presupposes a classical, set-theoretically robust metatheory, yet many-sorted logic is often motivated by contexts (e.g., predicative or constructive foundations) where such metatheory is unavailable.

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    Reasons For

    1 perspective
    Reason for
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    • 1.Henkin-style completeness can be reformulated using only intuitionistic logic and predicative resources; the classical metatheory is not essential to the core argument.
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    • 2.Using a classical metatheory to prove properties of constructive object-languages is standard practice and doesn't undermine the object-language's constructive credentials.
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    • 3.Many-sorted logic can be motivated pragmatically for expressive clarity regardless of foundational ideology, making metatheoretic rigor orthogonal to its utility.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Henkin's proof relies on set-theoretic constructions (maximal consistent sets, witness functions) that require classical logic and power set existence.
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    • 2.Predicative and constructive foundations explicitly reject impredicative definitions and excluded middle, making classical metatheory philosophically incoherent for them.
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    • 3.Many-sorted logic's motivation in type theory and proof assistants demands constructive or predicative justification, not post-hoc classical completeness.
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