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    Modal logic S4 is deductively embeddable into many-sorted... — Carmelics
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    Modal logic S4 is deductively embeddable into many-sorted logic: if Π ⊢_S4 φ then Trans(Π) ∪ ΔS4 ⊢ Trans(φ).

    Modality & PossibilityProof of definition segments
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 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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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 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.
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    Reason against 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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    Topics

    Modality & PossibilityProof of definition segments

    Related

    Many-sorted logic is an extensional framework where sort-membership is a static,...S4 permits variable domain semantics (as Kripke's 1963 semantics shows), so Tran...S4's accessibility relation encodes iterated epistemic or metaphysical possibili...The translation function Trans presupposes a fixed domain across modal contexts,...
    +3 moreShow less
    The translation preserves the deductive structure of S4Translations of axioms T and 4 are theorems of ΔS4ΔS4 must therefore either beg the question by encoding modal facts as primitive ...

    Similar

    Modal logic K is deductively embeddable into many-sorted logic: if Π ⊢...100%Strong completeness holds for many-sorted logic: if Γ ⊨ φ then Γ ⊢ φ85%Many-sorted logic satisfies the Compactness theorem, Enumerability the...82%When a logic is successfully translated into many-sorted logic, only a...82%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-many-sorted
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    Given a Kripke structure \[\mathcal{A}=\langle \mathbf{W},\mathbf{R},\langle P^{\mathcal{A}}\rangle _{P\in \Atom}\rangle\] we say that \(\mathcal{AG}\) is a general structure built on \(\mathcal{A}\) if and only if \[\mathcal{AG}=\langle \mathbf{W},\mathbf{W}^{\prime },\mathbf{R},\epsilon _{1}^{\mathcal{A}},\langle P^{\mathcal{A}}\rangle _{P\in \Atom}\rangle\] where \(\Def \subseteq \mathbf{W}^{\prime }\subseteq \wp (\mathbf{W})\). [22] It can be proved that the set of worlds where a moda
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Type
    claim
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit