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    The Henkin construction assumes all sorts are non-empty, ... — Carmelics
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    Challenges→Each consistent set of many-sorted formulas has a model, making syntactic consistency and semantic satisfiability equivalent

    The Henkin construction assumes all sorts are non-empty, but many-sorted logic permits empty sorts in some formulations.

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    1 reason for
    1 reason against

    Reasons For

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    Reason for
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    • 1.Henkin's completeness proof requires witnesses for existential formulas, necessitating non-empty domains for each sort to guarantee such witnesses exist.
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    • 2.Many-sorted logic's greater generality permits empty sorts to model scenarios where certain categories lack instantiation, expanding its expressiveness.
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    • 3.Standard first-order logic allows empty domains; restricting many-sorted logic to non-empty sorts artificially constrains its formal flexibility without gain.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Empty sorts create logical pathologies: universal quantification over empty domains becomes vacuously true, complicating semantic interpretation and proof theory.
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    • 2.The Henkin construction's non-emptiness assumption reflects standard model-theoretic practice; many formulations of many-sorted logic actually require non-empty sorts by default.
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    • 3.Empty sorts are rarely necessary in practice; most formal systems and applications work equivalently by simply omitting unused sorts rather than allowing emptiness.
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    Truth & Knowledge1 linkedPhilosophy of Language1 linked

    Related

    Each consistent set of many-sorted formulas has a model, making syntactic consis...Empty sorts are rarely necessary in practice; most formal systems and applicatio...Empty sorts create logical pathologies: universal quantification over empty doma...Henkin's completeness proof requires witnesses for existential formulas, necessi...
    +3 moreShow less
    Many-sorted logic's greater generality permits empty sorts to model scenarios wh...Standard first-order logic allows empty domains; restricting many-sorted logic t...The Henkin construction's non-emptiness assumption reflects standard model-theor...

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