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    Carmelics

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    Made withinDC&Austin
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    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that Mathematical induction does not function as an inference that derives a universal conclusion from two premises

    ?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.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 for 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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    Reasons Against

    1 perspective
    Reason against
    ?
    • 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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