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    The translation function Trans presupposes a fixed domain... — Carmelics
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    Challenges→Modal logic S4 is deductively embeddable into many-sorted logic: if Π ⊢_S4 φ then Trans(Π) ∪ ΔS4 ⊢ Trans(φ).

    The translation function Trans presupposes a fixed domain across modal contexts, smuggling in a Barcan-style assumption that S4 itself does not require.

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    Reasons For

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    Reason for
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    • 1.S4 modal logic is axiomatically neutral on domain variation; it permits but doesn't mandate fixed domains across possible worlds.
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    • 2.Translation functions that map modal formulas to first-order logic must make domain assumptions explicit; hidden assumptions violate logical transparency.
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    • 3.The Barcan formula's validity depends on constant domains, yet S4 remains consistent with variable domain semantics, showing they're independent commitments.
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    Reasons Against

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    Reason against
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    • 1.Standard S4 semantics typically employ constant domains in Kripke models; Trans simply reflects the dominant metalogical convention, not a hidden assumption.
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    • 2.Any effective translation between modal and first-order logic requires *some* domain specification; calling this 'smuggling in' conflates necessary with problematic presuppositions.
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    Proof of definition segments1 linkedModality & Possibility1 linked

    Related

    Any effective translation between modal and first-order logic requires *some* do...Modal logic S4 is deductively embeddable into many-sorted logic: if Π ⊢_S4 φ the...S4 modal logic is axiomatically neutral on domain variation; it permits but does...Standard S4 semantics typically employ constant domains in Kripke models; Trans ...
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    The Barcan formula's validity depends on constant domains, yet S4 remains consis...Translation functions that map modal formulas to first-order logic must make dom...

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