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    Hume's Principle is consistent (can be consistently added... — Carmelics
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    Hume's Principle is consistent (can be consistently added to second-order logic), unlike Basic Law V

    Philosophy of LanguageTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Hume's Principle only requires a many-to-one correlation between concepts and numbers, not a one-to-one correlation
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    • 2.Basic Law V's requirement of a one-to-one correlation between concepts and extensions is problematic (leads to contradiction)
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    • 3.Several authors have independently developed models demonstrating that Hume's Principle can be consistently added to second-order logic
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Hume's Principle, while consistent, is not logically innocent: it carries substantial ontological commitments to infinitely many abstract objects.
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    • 2.A principle that entails the existence of infinitely many objects cannot be treated as a mere logical truth or analytic definition.
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    • 3.Frege's Neo-Logicist program fails if its foundational principle is synthetic rather than analytic, undermining the epistemological motivation for the consistency result.
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    Reason against 2 of 2
    ?
    • 1.George Boolos argued that second-order logic is itself ontologically committal, smuggling set-theoretic assumptions into the metatheory.
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    • 2.Demonstrating consistency of Hume's Principle within second-order logic inherits whatever foundational problems afflict second-order logic itself.
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    • 3.The consistency proof thus presupposes a mathematical framework of comparable strength to what it aims to ground, making the result epistemically circular.
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    Related

    A principle that entails the existence of infinitely many objects cannot be trea...Basic Law V's requirement of a one-to-one correlation between concepts and exten...Demonstrating consistency of Hume's Principle within second-order logic inherits...Frege's Neo-Logicist program fails if its foundational principle is synthetic ra...
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    George Boolos argued that second-order logic is itself ontologically committal, ...Hume's Principle only requires a many-to-one correlation between concepts and nu...Hume's Principle, while consistent, is not logically innocent: it carries substa...Several authors have independently developed models demonstrating that Hume's Pr...The consistency proof thus presupposes a mathematical framework of comparable st...

    Similar

    Hume's Principle can be consistently added to second-order logic90%The consistency of second-order arithmetic (Z_2) is entailed by Takeut...85%Peano Arithmetic (PA) is consistent.85%Arithmetic is consistent (contains no contradictions).85%

    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

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

    Details

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