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    XL satisfies Compactness and Löwenheim-Skolem — Carmelics
    Home/Philosophy of Language
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    Supports→XL can have a strongly complete calculus
    Supports→XL satisfies Compactness and Löwenheim-Skolem

    XL satisfies Compactness and Löwenheim-Skolem

    Philosophy of LanguageTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.The Main Theorem establishes that consequence in XL is equivalent to consequence of translations in MSL modulo theory Delta
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    • 2.XL satisfies Compactness and Löwenheim-Skolem
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    • 3.The Main Theorem transfers these properties from MSL to XL
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.The translation from XL to MSL via Delta may not preserve all semantic content if sort-relativized quantifiers lack exact MSL correlates.
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    • 2.Löwenheim-Skolem for MSL depends on Skolem functions ranging over unsorted domains, but XL's sort constraints may block required witness construction.
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    • 3.If Delta itself is an infinite theory, compactness transfer requires Delta's own compactness, creating a circularity in the proof strategy.
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    Reason against 2 of 2
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    • 1.Barwise and Feferman's work on generalized logics shows that adding expressive mechanisms to first-order logic routinely sacrifices Löwenheim-Skolem.
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    • 2.XL's sort-indexed expressions constitute a genuine increase in expressive power over unsorted FOL, making metalogical property inheritance non-trivial.
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    Philosophy of LanguageTruth & Knowledge

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    Proof of definition segments4 linked

    Related

    Barwise and Feferman's work on generalized logics shows that adding expressive m...If Delta itself is an infinite theory, compactness transfer requires Delta's own...Löwenheim-Skolem for MSL depends on Skolem functions ranging over unsorted domai...MSL satisfies Compactness and Löwenheim-Skolem
    +7 moreShow less
    The Main Theorem establishes that consequence in XL is equivalent to consequence...The Main Theorem transfers these properties from MSL to XLThe translation from XL to MSL via Delta may not preserve all semantic content i...These three properties together are sufficient for strong completenessXL can have a strongly complete calculusXL has recursive enumerability of validitiesXL's sort-indexed expressions constitute a genuine increase in expressive power ...

    Similar

    MSL satisfies Compactness and Löwenheim-Skolem100%If PH = PSPACE, then TWO PLAYER SAT would be complete for PH (since it...76%BHP (Bounded Halting Problem) is NP-complete76%Propositional logic satisfiability is NP-complete76%

    Source

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    SEP: logic-many-sorted
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    Namely, the set of validities of \(\XL\) is recursively enumerable. Therefore, \(\XL\) is complete in an abstract sense. Remark: So, we learn that a calculus for \(\XL\) is a natural demand, but we also learn that in MSL we can simulate such a calculus and then we could use a theorem prover for MSL. 5 Level Two: the Main Theorem When the \(\XL\) logic under scrutiny has a concept of logical consequence, we may try to prove the Main theorem; that is, that consequence in \(\XL\) (\(\Pi \models _

    Details

    Type
    premise
    Perspectives
    3 (1 for, 2 against)
    Edits
    1 edit