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    Many-sorted logic, when sorts are allowed to range over p... — Carmelics
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    Challenges→Strong completeness holds for many-sorted logic: if Γ ⊨ φ then Γ ⊢ φ

    Many-sorted logic, when sorts are allowed to range over proper classes or when sort predicates are defined second-order, exceeds the expressive boundary at which Henkin-style completeness is guaranteed.

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    Key Terms

    Expressive boundary(as used in logic)
    The limit of how much meaning or complexity a logical system can capture or describe.
    Henkin-style completeness(as used in mathematical logic)
    A mathematical guarantee (named after logician Leon Henkin) that every statement that should logically be true can actually be proven true using the system's formal rules.
    Proper classes(as used in set theory)
    In set theory, collections of things that are too large or too weird to be treated as ordinary 'sets' (think of them as super-big collections).
    Sort predicates(as used in mathematical logic)
    Rules that identify which category or type something belongs to in a logical system.
    Sorts(as used in mathematical logic)

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    Categories or types that organize what things you're allowed to talk about in a logical system—like separating 'numbers' from 'colors' so you don't mix them up.
    many-sorted logic(Logic foundations and translations)
    A logic that accommodates reasoning about more than one sort (type) of objects, generalizing first-order logic by allowing multiple base types.
    second-order(as used in logic and philosophy of language)
    Referring to thinking about thinking itself, or rules about rules—a step removed from the basic level.

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    Proof of definition segments1 linkedPhilosophy of Language1 linked

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    Strong completeness holds for many-sorted logic: if Γ ⊨ φ then Γ ⊢ φ

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