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    Home/Original/inverse
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    Inverse View

    It is not the case that If Precedes(n,a) and n is a natural number, then a is a natural number

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 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 for 2 of 2
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    • 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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    Reasons Against

    1 perspective
    Reason against
    ?
    • 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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