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    Carmelics

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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 claim that FO 'cannot express properties in P without extensions' trivially conflates the base logic with its natural and well-motivated closure operations, which logicians since Kleene have treated as intrinsic to logical expressibility.

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    Reasons For

    1 perspective
    Reason for
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    • 1.First-order logic's expressive limits are precisely what motivate studying extensions; conflating them obscures important boundaries.
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    • 2.Calling closure operations 'intrinsic' begs the question: they're added precisely because base FO cannot express those properties.
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    • 3.Historical convention among logicians doesn't settle whether expressibility should include non-elementary closure mechanisms.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Closure operations (like least fixed points) have been foundational to logic since Kleene's recursion theory, not afterthoughts.
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    • 2.Distinguishing 'base logic' from 'closure operations' artificially fragments what logicians treat as unified expressibility.
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    • 3.Properties definable only through natural closure operations are legitimately expressible within the extended logical system.
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