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    Compactness and Löwenheim-Skolem properties hold for moda... — Carmelics
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    Home/Modality & Possibility
    HistoryEditSee Inverse

    Compactness and Löwenheim-Skolem properties hold for modal logics K and S4.

    Modality & Possibility
    ?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.First-order axioms for reflexivity and transitivity are equivalent to the many-sorted translated sentences MS(T) and MS(4)
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    • 2.These properties are inherited from many-sorted logic via the translation
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The standard Kripke semantics for S4 admits frames with infinite ascending chains, enabling formulas that force uncountable models resistant to Löwenheim-Skolem downward reduction.
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    • 2.Fine's 1975 incompleteness results demonstrate that not all modal logics are complete with respect to first-order definable frame classes, undermining the translation-inheritance argument.
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    Reason against 2 of 2
    ?
    • 1.The many-sorted translation preserves compactness only when modal operators are interpreted over first-order definable accessibility relations, but S4 permits second-order frame conditions.
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    • 2.Van Benthem's correspondence theory shows that some S4-valid formulas correspond to non-elementary frame properties, which are invisible to the first-order many-sorted translation MS(4).
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    Topics

    Modality & PossibilityProof of definition segments

    Connections

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    Related

    Fine's 1975 incompleteness results demonstrate that not all modal logics are com...First-order axioms for reflexivity and transitivity are equivalent to the many-s...The many-sorted translation preserves compactness only when modal operators are ...The standard Kripke semantics for S4 admits frames with infinite ascending chain...
    +2 moreShow less
    These properties are inherited from many-sorted logic via the translationVan Benthem's correspondence theory shows that some S4-valid formulas correspond...

    Similar

    In S5 modal logic (or the weaker system B), possibly necessary that A ...84%Validity and satisfiability for modal logics S4 and S5 are PSPACE-comp...83%Tense logic is a species of modal logic83%Validity and satisfiability problems for modal logics S4 and S5 are PS...82%

    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