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    Made withinDC&Austin
    Statements
    321,452
    Perspectives
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    42
    If Precedes(n,a) and n is a natural number, then a is a n... — Carmelics
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    If Precedes(n,a) and n is a natural number, then a is a natural number

    Proof of definition segments
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Precedes(n,a) is assumed
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    • 2.n is a natural number, which means Precedes⁺(0,n) holds by definition
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    • 3.By the property of the strong ancestral R⁺ (Fact 3), Precedes(n,a) and Precedes⁺(0,n) together entail Precedes*(0,a)
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The strong ancestral R⁺ in Frege's system is defined within second-order logic, which carries ontological commitments that are not logically innocent.
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    • 2.If second-order quantification over concepts is not purely logical (as Quine argues in 'Philosophy of Logic'), then Precedes⁺ inherits empirical or mathematical presuppositions.
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    • 3.A proof of natural number closure under Precedes that relies on impredicative second-order comprehension cannot establish the claim on purely logical grounds alone.
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    Reason against 2 of 2
    ?
    • 1.Hume's Principle, which grounds the Precedes relation via equinumerosity, is a material abstraction principle, not a logical truth, as George Boolos argued in 'Is Hume's Principle Analytic?'.
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    • 2.If Precedes is defined through a non-logical principle, then the closure property depends on the mathematical content of that principle rather than on logical form.
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    • 3.The claim that a is a natural number follows not from logical necessity but from the substantive mathematical axiom embedded in the ancestry construction.
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    Topics

    Truth & KnowledgeProof of definition segments

    Connections

    1 topic

    Modality & Possibility4 linked

    Related

    A number a satisfying Precedes⁺(0,a) is by definition a natural numberA proof of natural number closure under Precedes that relies on impredicative se...By the definition of Precedes⁺, Precedes*(0,a) implies Precedes⁺(0,a)By the property of the strong ancestral R⁺ (Fact 3), Precedes(n,a) and Precedes⁺...
    +7 moreShow less
    Hume's Principle, which grounds the Precedes relation via equinumerosity, is a m...If Precedes is defined through a non-logical principle, then the closure propert...If second-order quantification over concepts is not purely logical (as Quine arg...Precedes(n,a) is assumedThe claim that a is a natural number follows not from logical necessity but from...The strong ancestral R⁺ in Frege's system is defined within second-order logic, ...n is a natural number, which means Precedes⁺(0,n) holds by definition

    Similar

    n is a natural number, which means Precedes⁺(0,n) holds by definition84%A number a satisfying Precedes⁺(0,a) is by definition a natural number83%Precedes(n,a) is assumed79%No natural number n is the Gödel number of a proof of G_F in F79%

    Source

    AI-extracted1/3 agreementValid
    SEP: frege-theorem
    View source passageHide passage
    Proof: Suppose that \(\mathit{Precedes}(n,a)\). Then, by definition, since \(n\) is a natural number, \(\mathit{Precedes}^{+}(0,n)\). So by Fact (3) about \(R^{+}\) (in the subsection on the Weak Ancestral in §4), it follows that \(\mathit{Precedes}^*(0,a)\), and so by the definition of \(\mathit{Precedes}^{+}\), it follows that \(\mathit{Precedes}^{+}(0,a)\); i.e., \(a\) is a natural number.
    Extraction notes

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

    Details

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