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    Home/Original/inverse
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    Inverse View

    It is not the case that XL can have a strongly complete calculus

    ?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.Recursive enumerability of validities is necessary but not sufficient for strong completeness without a complete proof system being explicitly constructible.
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    • 2.XL's many-sorted quantification over heterogeneous domains may generate validity-preservation failures under certain sort-collapsing interpretations that standard enumeration procedures miss.
      ?

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    • 3.Lindström's theorem establishes that classical FOL is maximal for compactness plus Löwenheim-Skolem, but XL's sort structure introduces expressive additions that may escape this maximality boundary.
      ?

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    Reason for 2 of 2
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    • 1.Strong completeness requires that every semantically valid inference be derivable, but many-sorted logics with sort-restricted quantifiers can generate models where cross-sort inferences resist standard Henkin construction.
      ?

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    • 2.Henkin's completeness proof for FOL depends on the homogeneity of the domain, and many-sorted domains introduce partition constraints that can block canonical model construction for infinite sort hierarchies.
      ?

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    Reasons Against

    1 perspective
    Reason against
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    • 1.XL satisfies Compactness and Löwenheim-Skolem
      ?

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    • 2.XL has recursive enumerability of validities
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    • 3.These three properties together are sufficient for strong completeness
      ?

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