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    Strong completeness holds for many-sorted logic: if Γ ⊨ φ... — Carmelics
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    Strong completeness holds for many-sorted logic: if Γ ⊨ φ then Γ ⊢ φ

    Philosophy of Language
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.If Γ ⊨ φ, then Γ ∪ {¬φ} is not satisfiable and has no model
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    • 2.By Henkin's theorem, if Γ ∪ {¬φ} has no model then Γ ∪ {¬φ} is contradictory
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    • 3.The calculus rules allow elimination of ¬φ to infer Γ ⊢ φ
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.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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    • 2.If the metatheory is weakened to predicative or intuitionistic set theory, the Henkin model construction fails because maximal consistent extensions require non-constructive choice principles.
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    • 3.Therefore, strong completeness for many-sorted logic holds only relative to a classical metatheory that many-sorted frameworks were partly designed to avoid presupposing.
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    Reason against 2 of 2
    ?
    • 1.Strong completeness requires that every semantically valid inference over all many-sorted structures is provable, but Lindström's theorem shows any logic stronger than first-order that gains expressive power loses completeness or compactness.
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    • 2.Many-sorted logic, when sorts are allowed to range over proper classes or when sort predicates are defined second-order, exceeds the expressive boundary at which Henkin-style completeness is guaranteed.
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    • 3.The claim therefore conflates the well-behaved finitary fragment of many-sorted logic with richer formulations where incompleteness results analogous to those established by Lindström and later Barwise genuinely apply.
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    Related

    By Henkin's theorem, if Γ ∪ {¬φ} has no model then Γ ∪ {¬φ} is contradictoryHenkin's completeness proof presupposes a classical, set-theoretically robust me...If the metatheory is weakened to predicative or intuitionistic set theory, the H...If Γ ⊨ φ, then Γ ∪ {¬φ} is not satisfiable and has no model
    +5 moreShow less
    Many-sorted logic, when sorts are allowed to range over proper classes or when s...Strong completeness requires that every semantically valid inference over all ma...The calculus rules allow elimination of ¬φ to infer Γ ⊢ φThe claim therefore conflates the well-behaved finitary fragment of many-sorted ...Therefore, strong completeness for many-sorted logic holds only relative to a cl...

    Similar

    Strong completeness (if Γ ⊨ φ then Γ ⊢ φ) holds for many-sorted logic.99%Translations of a logic into many-sorted logic yield completeness resu...89%Many-sorted logic also possesses these properties (Compactness and Löw...87%Modal logic S4 is deductively embeddable into many-sorted logic: if Π ...85%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-many-sorted
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    , determining validity, or equivalently, testing for satisfiability of given formulas) for many-sorted logic is undecidable. So, we are in the same situation encountered in one-sorted first-order logic. Of course, if a calculus is to be helpful it would never allow erroneous reasonings: it is not going to drive us from true hypotheses to false conclusions. It must be a sound calculus. Further, it is highly desirable that all the consequences of a set \(\Gamma\) of hypotheses could be derived fr
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit