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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that 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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    Reasons For

    1 perspective
    Reason for
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    • 1.Compactness can fail even with first-order relations if the translation introduces infinitary constraints or loses semantic information during the encoding process.
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    • 2.S4 axioms (T, 4) are first-order expressible but S4 models sometimes require second-order conditions only for *completeness*, not semantic validity of the original logic.
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    • 3.The claim conflates expressibility of frame conditions with necessity for preserving compactness; restriction to FOL-definable relations may be artificially limiting.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.First-order definability ensures compactness preservation because FOL satisfies compactness, and translation fidelity requires expressible accessibility relations.
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    • 2.S4's reflexivity and transitivity are first-order expressible, but S4 with second-order frame conditions (like well-foundedness) exceeds FOL's expressiveness limits.
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    • 3.Many-sorted translation success depends on whether modal semantics can be captured in FOL; second-order conditions break this correspondence.
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