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    The addition of Basic Law V to second-order logic implies... — Carmelics
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    The addition of Basic Law V to second-order logic implies a contradiction.

    Modality & Possibility
    ?Rate how convincing each reason is below to see the overall strength.
    2 reasons for
    1 reason against

    Reasons For

    2 perspectives
    Reason for 1 of 2
    ?
    • 1.Russell's paradox derives directly from Basic Law V: the concept 'x is an extension that does not contain itself' yields an extension that both contains and does not contain itself.
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    • 2.Frege himself acknowledged this contradiction in his 1902 correspondence with Russell, confirming Basic Law V's inconsistency is not merely technical but foundational.
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    • 3.Any system entailing a statement of the form 'P and not-P' is logically trivial, making the conjunction of Basic Law V with second-order logic deductively worthless.
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    Reason for 2 of 2
    ?
    • 1.Boolos demonstrated that second-order comprehension axioms license forming the concept of all non-self-membered extensions, which Basic Law V then maps to a paradoxical object.
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    • 2.The inconsistency is provable in a finite number of steps within the combined system, satisfying Frege's own standard of rigorous gapless proof that he applied throughout Grundgesetze.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Basic Law V requires that the domain of extensions be as large as the domain of concepts, since every concept maps to an extension.
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    • 2.Second-order logic requires that the domain of concepts be strictly larger than the domain of extensions, since concepts range over all subsets of the domain.
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    • 3.A domain cannot be simultaneously strictly larger than another domain and equal in size to that same domain.
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    Topics

    Modality & PossibilityTruth & Knowledge

    Connections

    2 topics

    Proof of definition segments2 linkedPhilosophy of Language1 linked

    Related

    A domain cannot be simultaneously strictly larger than another domain and equal ...Any system entailing a statement of the form 'P and not-P' is logically trivial,...Basic Law V requires that the domain of extensions be as large as the domain of ...Boolos demonstrated that second-order comprehension axioms license forming the c...
    +4 moreShow less
    Frege himself acknowledged this contradiction in his 1902 correspondence with Ru...Russell's paradox derives directly from Basic Law V: the concept 'x is an extens...

    Similar

    Hume's Principle can be consistently added to second-order logic88%The Downward Löwenheim-Skolem Theorem fails for second-order logic85%The Compactness Theorem does not hold for second-order logic in the fo...83%The same proof strategy used to establish Prenex Normal Form in first-...83%

    Source

    AI-extracted1/3 agreementValid
    SEP: frege-theorem
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    Thus, the addition of Basic Law V to second-order logic implies an impossible situation in which the domain of concepts has to be strictly larger than the domain of extensions while at the same time the domain of extensions has to be as large as the domain of concepts.
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

    Type
    claim
    Second-order logic requires that the domain of concepts be strictly larger than ...
    The inconsistency is provable in a finite number of steps within the combined sy...
    Perspectives
    3 (2 for, 1 against)
    Edits
    1 edit