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    Carmelics

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    Made withinDC&Austin
    Home/Original/inverse
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    Inverse View

    It is not the case that Each consistent set of many-sorted formulas has a model, making syntactic consistency and semantic satisfiability equivalent

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 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 for 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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    Reasons Against

    1 perspective
    Reason against
    ?
    • 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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