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    Each consistent set of many-sorted formulas has a model, ... — Carmelics
    Home/Philosophy of Language
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    Each consistent set of many-sorted formulas has a model, making syntactic consistency and semantic satisfiability equivalent

    Philosophy of LanguageTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Henkin's strategy for first-order logic can be applied to many-sorted calculus
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    • 2.Every consistent set of formulas can be extended to a maximal consistent set with witnesses
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    • 3.A maximally consistent set with witnesses can be used to build a precise model, because it is a detailed description of a structure
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The Henkin construction assumes all sorts are non-empty, but many-sorted logic permits empty sorts in some formulations.
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    • 2.If any sort in a many-sorted signature is empty, the canonical Henkin model construction fails to produce a legitimate interpretation for quantifiers over that sort.
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    • 3.Therefore, completeness holds only under the non-empty sort assumption, making the equivalence conditional rather than universal.
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    Reason against 2 of 2
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    • 1.Kreisel's squeezing argument shows that informal notions of validity and formal provability align only when the semantics is itself precisely fixed.
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    • 2.In many-sorted logic, the choice of whether sorts must be non-empty, disjoint, or allow subsort relations introduces semantic underdetermination not present in single-sorted first-order logic.
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    • 3.Syntactic consistency cannot be equivalent to satisfiability when 'satisfiability' picks out different model classes depending on unresolved foundational choices about sort ontology.
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    Philosophy of LanguageTruth & Knowledge

    Connections

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    Proof of definition segments4 linked

    Related

    A maximally consistent set with witnesses can be used to build a precise model, ...Every consistent set of formulas can be extended to a maximal consistent set wit...Henkin's strategy for first-order logic can be applied to many-sorted calculusIf any sort in a many-sorted signature is empty, the canonical Henkin model cons...
    +6 moreShow less
    In many-sorted logic, the choice of whether sorts must be non-empty, disjoint, o...Kreisel's squeezing argument shows that informal notions of validity and formal ...Soundness is assumedSyntactic consistency cannot be equivalent to satisfiability when 'satisfiabilit...The Henkin construction assumes all sorts are non-empty, but many-sorted logic p...Therefore, completeness holds only under the non-empty sort assumption, making t...

    Similar

    Henkin's theorem establishes that every consistent set of formulas has...92%Second-order logic satisfies the Completeness Theorem when Henkin mode...79%The Gödel sentence G_F is true (when F is consistent and the provabili...78%Given a many-sorted structure A, every many-sorted sentence true at A ...77%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-many-sorted
    View source passageHide passage
    , 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