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    ΔS4 must therefore either beg the question by encoding mo... — Carmelics
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    Part of a larger discussion

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

    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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    Reasons For

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    Reason for
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    • 1.Modal axioms like T and 4 encode facts about accessibility relations that cannot be derived from non-modal logical resources alone.
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    • 2.Any embedding that claims completeness without explicit modal axioms must smuggle modal content into supposedly neutral sort-theoretic apparatus.
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    • 3.The dilemma is genuine: either modal facts are primitive (begging the question) or the embedding is incomplete for S4-validity.
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    Reasons Against

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    Reason against
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    • 1.Sort-theoretic embeddings can encode modal properties through structural constraints on domains without treating modality as primitive.
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    • 2.Completeness may be achieved by deriving S4-valid sequents via logical rules applied to sort-structure, not requiring explicit modal axioms.
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    • 3.The disjunction ignores hybrid approaches where modal content emerges from interaction of logical rules and domain structure systematically.
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    Proof of definition segments1 linkedModality & Possibility1 linked

    Related

    Any embedding that claims completeness without explicit modal axioms must smuggl...Completeness may be achieved by deriving S4-valid sequents via logical rules app...Modal axioms like T and 4 encode facts about accessibility relations that cannot...Modal logic S4 is deductively embeddable into many-sorted logic: if Π ⊢_S4 φ the...
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    Sort-theoretic embeddings can encode modal properties through structural constra...The dilemma is genuine: either modal facts are primitive (begging the question) ...The disjunction ignores hybrid approaches where modal content emerges from inter...

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    claim
    Perspectives
    2 (1 for, 1 against)
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