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    There must be at least as many extensions as there are co... — Carmelics
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    Home/Philosophy of Language
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    There must be at least as many extensions as there are concepts.

    Philosophy of LanguageTruth & Knowledge
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Every concept is correlated with some extension (the correlation is total).
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    • 2.The correlation between concepts and extensions is one-to-one (distinct concepts map to distinct extensions).
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    • 3.In a total one-to-one mapping from concepts to extensions, the cardinality of extensions is at least as large as the cardinality of concepts.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Russell's paradox shows that not every concept determines a well-defined extension (e.g., the concept 'is not a member of itself' yields contradiction).
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    • 2.If some concepts fail to determine any extension, the mapping from concepts to extensions is not total, undermining P1.
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    • 3.A non-total mapping cannot guarantee that the cardinality of extensions meets or exceeds the cardinality of concepts.
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    Reason against 2 of 2
    ?
    • 1.Frege's Basic Law V, which grounds the one-to-one correlation in P2, was proven inconsistent by Russell and abandoned even by Frege himself in 1902.
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    • 2.Without a consistent principle licensing distinct extensions for distinct concepts, P2 lacks the logical foundation required to derive the cardinality claim.
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    Philosophy of LanguageTruth & Knowledge

    Related

    A non-total mapping cannot guarantee that the cardinality of extensions meets or...Every concept is correlated with some extension (the correlation is total).Frege's Basic Law V, which grounds the one-to-one correlation in P2, was proven ...If some concepts fail to determine any extension, the mapping from concepts to e...
    +4 moreShow less
    In a total one-to-one mapping from concepts to extensions, the cardinality of ex...Russell's paradox shows that not every concept determines a well-defined extensi...The correlation between concepts and extensions is one-to-one (distinct concepts...Without a consistent principle licensing distinct extensions for distinct concep...

    Similar

    Basic Law V requires that the domain of extensions be as large as the ...83%In a total one-to-one mapping from concepts to extensions, the cardina...81%If concepts F and G differ, then the extensions of F and G differ.79%Vb correlates distinct concepts with distinct extensions.78%

    Source

    AI-extracted1/3 agreementValid
    SEP: frege-theorem
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    So the contrapositive asserts that if \(\neg \forall x(Fx \equiv Gx)\) then \(\epsilon F \neq \epsilon G\). But in the case where the material equivalence of \(F\) and \(G\) is a sufficient condition for \(F = G\), i.e., in the case where \(F \neq G\) implies \(\neg \forall x(Fx \equiv Gx)\), then Vb implies \(F \neq G \to \epsilon F \neq \epsilon G\), i.e., that if concepts \(F\) and \(G\) differ, the extensions of \(F\) and \(G\) differ. So, the correlation that Basic Law V sets up between co
    Extraction notes

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

    Details

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