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    Carmelics

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    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that Modal logic S4 is deductively embeddable into many-sorted logic: if Π ⊢_S4 φ then Trans(Π) ∪ ΔS4 ⊢ Trans(φ).

    ?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.Many-sorted logic is an extensional framework where sort-membership is a static, non-modal classification of objects.
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    • 2.S4's accessibility relation encodes iterated epistemic or metaphysical possibility, a hyperintensional structure that extensional sort distinctions cannot fully capture without circularity.
      ?

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    • 3.ΔS4 must therefore either beg the question by encoding modal facts as primitive sort axioms, or fail to derive all S4-valid sequents under the proposed embedding.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 1.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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    • 2.S4 permits variable domain semantics (as Kripke's 1963 semantics shows), so Trans(Π) ∪ ΔS4 may validate inferences S4 itself rejects under anti-Barcan conditions.
      ?

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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Translations of axioms T and 4 are theorems of ΔS4
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    • 2.The translation preserves the deductive structure of S4
      ?

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