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    The standard Kripke semantics for S4 admits frames with i... — Carmelics
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    Challenges→Compactness and Löwenheim-Skolem properties hold for modal logics K and S4.

    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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    Key Terms

    Downward reduction(in logic and mathematics)
    The process of taking a larger, more complex mathematical structure and simplifying it to a smaller version while keeping the essential properties.
    Frames(Second-order logic semantics)
    Second-order structures in which set and relation variables range over universes that are subsets of standard universes, without the extra closure condition required by general structures
    Infinite ascending chains(in logic and mathematics)
    A sequence that keeps going forever, where each step is 'higher' or 'stronger' in some way and never loops back on itself.
    Kripke semantics(in modal logic and philosophy of language)
    A formal system (created by philosopher Saul Kripke) for understanding how words and concepts work across different possible scenarios or worlds.

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    Löwenheim-Skolem(as used in mathematical logic)
    A famous theorem in logic stating that if a set of logical rules can describe something, then those same rules can also describe things of wildly different sizes (like describing both tiny and infinite versions of the same structure).
    S4(as used in the statement)
    A specific system of modal logic with particular rules about how possibility and necessity work; it's named S4 just like a product model number.
    models(models of global democracy)
    idealized theoretical constructions designed to express the normative qualities of a democratic system as well as its constitutive institutions
    uncountable(Examples include the real numbers and the power set of the natural numbers.)
    An infinite set that cannot be put into one-to-one correspondence with the natural numbers.

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    2 topics

    Proof of definition segments1 linkedModality & Possibility1 linked

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

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