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    Hume's Principle does not require that the domain of numb... — Carmelics
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    Hume's Principle does not require that the domain of numbers be as large as the domain of concepts

    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.Hume's Principle only requires that the correlation between concepts and numbers be many-to-one, not one-to-one
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    • 2.Hume's Principle often correlates distinct concepts with the same number (e.g., 'author of Principia Mathematica' and 'positive integer between 1 and 4' are both correlated with the number 2)
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Hume's Principle, via Frege's Theorem, actually derives the full Dedekind-infinite sequence of natural numbers, making the number domain as large as any infinite concept-extension.
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    • 2.If every concept has a cardinal number and concepts can be defined over the natural numbers themselves, the number domain must be at least as large as the domain of number-concepts, collapsing the proposed asymmetry.
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    Reason against 2 of 2
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    • 1.George Boolos argued that Hume's Principle carries unacceptable ontological commitments by generating an object for every concept, including concepts with infinite extensions, yielding a domain of numbers co-extensive with the concept hierarchy.
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    • 2.The many-to-one mapping cited in support only holds for finite, co-equinumerous concepts, but Hume's Principle is unrestricted and applies to all concepts, including those individuating each distinct cardinal, forcing a domain at least as large as the full concept domain.
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    Related

    George Boolos argued that Hume's Principle carries unacceptable ontological comm...Hume's Principle often correlates distinct concepts with the same number (e.g., ...Hume's Principle only requires that the correlation between concepts and numbers...Hume's Principle, via Frege's Theorem, actually derives the full Dedekind-infini...
    +2 moreShow less
    If every concept has a cardinal number and concepts can be defined over the natu...The many-to-one mapping cited in support only holds for finite, co-equinumerous ...

    Similar

    Basic Law V requires that the domain of extensions be as large as the ...84%A domain cannot be simultaneously strictly larger than another domain ...80%There is no such thing as an infinite mathematical domain (i.e., total...78%Hume's Principle only requires that the correlation between concepts a...77%

    Source

    AI-extracted1/3 agreementValid
    SEP: frege-theorem
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    Notice that Hume’s Principle bears an obvious formal resemblance to Basic Law V. Both are biconditionals asserting the equivalence of an identity among singular terms (the left-side condition) with an equivalence relation on concepts (the right-side condition). Indeed, both correlate concepts with certain objects. In the case of Hume’s Principle, each concept \(F\) is correlated with \(\#F\). However, whereas Basic Law V problematically requires that the correlation between concepts and extensio
    Extraction notes

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    Details

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