Skip to content
Carmelics
TopicsThinkersChangesContributorsLoading account…

    Carmelics

    A reasoning platform. Break down any belief into clear reasons, explore both sides, and weigh the evidence honestly.

    Navigate

    • Topics
    • Search
    • Recent Changes
    • Contribute
    • How It Works
    • Glossary
    • Thinkers
    • Contributors
    • About
    • Statistics
    • Terms
    • Privacy

    Database

    Statements
    —
    Perspectives
    —
    Topics
    —

    Press ? for keyboard shortcuts

    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that Compactness and Löwenheim-Skolem properties hold for modal logics K and S4.

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 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.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    Reason for 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.
      ?

      Think about whether this reason is strong or weak

    • 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).
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.First-order axioms for reflexivity and transitivity are equivalent to the many-sorted translated sentences MS(T) and MS(4)
      ?

      Think about whether this reason is strong or weak

    • 2.These properties are inherited from many-sorted logic via the translation
      ?

      Think about whether this reason is strong or weak

    Next step

    Based on where you are in your exploration

    Strongest counterpoint
    Explore the most compelling reason on the other side.