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    Frege's Basic Law V cannot be true — Carmelics
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    Frege's Basic Law V cannot be true

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    2 reasons for
    1 reason against

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.Frege's own logicism required extensions to be logical objects, but Russell's paradox shows no coherent domain of logical objects can satisfy unrestricted comprehension.
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    • 2.Dummett's analysis in 'Frege: Philosophy of Mathematics' demonstrates Basic Law V cannot be rescued by restricting extensions without abandoning the logicist program entirely.
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    • 3.Any consistent fragment of Basic Law V (e.g., New Foundations or Hume's Principle) requires abandoning the universality that made Basic Law V philosophically significant to Frege.
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    Reason for 2 of 2
    ?
    • 1.Boolos showed in 'The Consistency of Frege's Foundations of Arithmetic' that arithmetic can be recovered from Hume's Principle without Basic Law V, confirming Basic Law V is the locus of contradiction.
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    • 2.If Basic Law V were true, the formal system of Grundgesetze would be consistent, but Frege himself conceded in his appendix to Grundgesetze Vol. II that Russell's paradox destroys the system's foundations.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Frege's Basic Law V commits to the existence of a set for every predicate
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    • 2.The predicate 'x is not a member of itself' is a well-formed predicate
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    • 3.If Basic Law V holds, there must exist a set R of all sets that are not members of themselves
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    2 topics

    Philosophy of Language3 linkedModality & Possibility2 linked

    Related

    A set cannot both be and not be a member of itself simultaneouslyAny consistent fragment of Basic Law V (e.g., New Foundations or Hume's Principl...Boolos showed in 'The Consistency of Frege's Foundations of Arithmetic' that ari...Dummett's analysis in 'Frege: Philosophy of Mathematics' demonstrates Basic Law ...
    +6 moreShow less
    Frege's Basic Law V commits to the existence of a set for every predicateFrege's own logicism required extensions to be logical objects, but Russell's pa...If Basic Law V holds, there must exist a set R of all sets that are not members ...

    Similar

    Beliefs can be false88%Therefore E cannot be false whenever D is true86%The Relational Analysis is false.86%Sentence (s) is false86%

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    The most immediately calamitous challenge to Logicism was, however, the famous paradox Russell raised for one of Frege’s crucial axioms, his prima facie plausible “Basic Law V” (sometimes called “the unrestricted Comprehension Axiom”), which had committed him to the existence of a set for every predicate. But what, asked Russell, of the predicate x is not a member of itself? If there were a set for that predicate, that set itself would be a member of itself if and only if it wasn’t; consequently
    Extraction notes

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    Details

    If Basic Law V were true, the formal system of Grundgesetze would be consistent,...
    Set R is a member of itself if and only if R is not a member of itself
    The predicate 'x is not a member of itself' is a well-formed predicate
    Type
    claim
    Perspectives
    3 (2 for, 1 against)
    Edits
    1 edit