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    Made withinDC&Austin
    The reverse deductive correspondence holds: translations ... — Carmelics
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Modality & Possibility
    HistoryEditSee Inverse

    The reverse deductive correspondence holds: translations that are theorems of MSL correspond to theorems of the modal calculus.

    Modality & PossibilityProof of definition segments
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.The canonical model B_K (or B_S4) can be used to build the general structure B_K𝔊 (or B_S4𝔊)
      ?

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    • 2.The translation of a modal formula φ is true at a world of this general structure if and only if φ belongs to that world
      ?

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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The canonical model construction presupposes completeness of the base modal calculus, which fails for some normal modal logics (e.g., certain logics above S4).
      ?

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    • 2.If the modal calculus is incomplete with respect to Kripke semantics, the canonical model B_K cannot faithfully represent all theorems, breaking the correspondence.
      ?

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    Reason against 2 of 2
    ?
    • 1.Many-sorted logic translations collapse intensional distinctions when modal operators are encoded as predicates over sort-restricted domains (cf. Fitting's concerns about intensional contexts).
      ?

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    • 2.A formula provable in MSL via extensional reasoning about sorted domains may not correspond to a genuinely modal theorem but to an artifact of the translation scheme.
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    Topics

    Modality & PossibilityProof of definition segments

    Connections

    1 topic

    Truth & Knowledge1 linked

    Related

    A formula provable in MSL via extensional reasoning about sorted domains may not...If the modal calculus is incomplete with respect to Kripke semantics, the canoni...Many-sorted logic translations collapse intensional distinctions when modal oper...The canonical model B_K (or B_S4) can be used to build the general structure B_K...
    +2 moreShow less
    The canonical model construction presupposes completeness of the base modal calc...The translation of a modal formula φ is true at a world of this general structur...

    Similar

    The Main Theorem establishes that consequence in XL is equivalent to c...78%The translation of a modal formula φ is true at a world of this genera...77%Compactness and Löwenheim-Skolem properties hold for modal logics K an...76%Validity of φ in propositional modal logic (PML) is equivalent to vali...75%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-many-sorted
    View source passageHide passage
    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