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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
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    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that The addition of Basic Law V to second-order logic implies a contradiction.

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    1 perspective
    Reason for
    ?
    • 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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    Reasons Against

    2 perspectives
    Reason against 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.
      ?

      Think about whether this reason is strong or weak

    • 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 against 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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