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    Mathematical induction does not function as an inference ... — Carmelics
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    Mathematical induction does not function as an inference that derives a universal conclusion from two premises

    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.The statement 'f(x) holds for all cardinal numbers' is not a conclusion inferred from the base case and the inductive step taken as independent truths
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    • 2.Rather, the universal statement just is the conjunction of the base case and the inductive step — it is a grammatical or definitional equivalence, not a logical inference
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Frege and Russell formalized mathematical induction as a genuine logical inference within second-order logic, deriving universal conclusions from well-formed axioms.
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    • 2.If the universal statement were merely definitionally equivalent to its base case and inductive step, induction would add no epistemic content — yet it genuinely extends our knowledge to infinite cases we cannot survey.
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    • 3.Peano's axiomatization treats the induction schema as a primitive axiom from which theorems are derived, not as a grammatical stipulation, and this framework successfully grounds arithmetic.
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    Reason against 2 of 2
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    • 1.Wittgenstein's grammatical equivalence thesis conflates the justificatory role of induction with its semantic content, which logicians like Boolos distinguish carefully.
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    • 2.If 'f(x) holds for all cardinals' simply IS the conjunction of base case and inductive step, then the infinity of arithmetic truths collapses into a finite linguistic act — an implausible consequence that Gödel's incompleteness results make formally precise.
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    Related

    Frege and Russell formalized mathematical induction as a genuine logical inferen...If 'f(x) holds for all cardinals' simply IS the conjunction of base case and ind...If the universal statement were merely definitionally equivalent to its base cas...Peano's axiomatization treats the induction schema as a primitive axiom from whi...
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    Rather, the universal statement just is the conjunction of the base case and the...The statement 'f(x) holds for all cardinal numbers' is not a conclusion inferred...Wittgenstein's grammatical equivalence thesis conflates the justificatory role o...

    Similar

    The principle of induction has no foundation in reasoning85%Enumerative induction is an inadequate method for establishing univers...84%Applying mathematical induction to F(x) using premises (i) and (ii) yi...83%Proving the truth of a conclusion is a sufficient condition for justif...82%

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    SEP: wittgenstein-mathematics
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    We are not saying that when \(f(1)\) holds and when \(f(c + 1)\) follows from \(f(c)\), the proposition \(f(x)\) is therefore true of all cardinal numbers: but: “the proposition \(f(x)\) holds for all cardinal numbers” means “it holds for \(x = 1\), and \(f(c + 1)\) follows from \(f(c)\)”. (PG 406)
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    Details

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    claim
    Perspectives
    3 (1 for, 2 against)
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