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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
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    Inverse View

    It is not the case that An inductive proof enables us to perceive that a direct proof of any particular proposition can be constructed, even though it cannot prove the infinite possibility of application.

    ?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.Mathematical induction proves a universal generalization ∀n φ(n) as a single logical act, not merely a procedure for constructing finite proofs.
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    • 2.Frege and Russell demonstrated that the logical content of '∀n φ(n)' is irreducible to any finite conjunction or constructive recipe for particular instances.
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    • 3.If induction only licenses perception of constructibility without proving the infinite generalization, then transfinite arithmetic and completed infinities lose any rigorous foundation.
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    Reason for 2 of 2
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    • 1.Wittgenstein's eliminability thesis treats φ(m) as a pseudo-proposition, but this collapses the distinction between a proof and a decision procedure for checking proofs.
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    • 2.Gödel's incompleteness results demonstrate that syntactic provability within a system cannot be identified with the semantic truth of the proposition proven, undermining any purely procedural account of what induction establishes.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.An inductive proof proves the inductive base φ(1) and the inductive step φ(n) → φ(n+1).
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    • 2.Once the inductive base and inductive step are proved, we need not reiterate modus ponens m-1 times to prove a particular proposition φ(m).
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    • 3.The proxy statement φ(m) is an eliminable pseudo-proposition standing proxy for the proved inductive base and inductive step, not a proposition that asserts its own generality.
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