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    Carmelics

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

    It is not the case that Intuitionistic logic, following Brouwer and Heyting, rejects the classical equivalence of ∀x[P(x) → D(x)] and ¬∃x[P(x) ∧ ¬D(x)] because the latter requires a constructive witness for negation that the former does not.

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    Reasons For

    1 perspective
    Reason for
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    • 1.If ∀x[P(x)→D(x)] is constructively proven, we have a procedure converting P-proofs to D-proofs, making counterexample-refutation immediate.
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    • 2.The claim conflates informal 'witness requirement' with formal logical equivalence; both formulas express the same content formally.
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    • 3.Intuitionistic logic still validates the equivalence under appropriate proof-theoretic interpretation; the difference is epistemic, not logical.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Negation in constructivism means providing an algorithm deriving contradiction, not merely assuming falsehood holds somewhere.
      ?

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    • 2.∀x[P(x)→D(x)] requires only constructing D(x) given any constructive proof of P(x), while ¬∃x[P(x)∧¬D(x)] requires actively refuting all counterexamples.
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    • 3.Classical logic's law of excluded middle lacks computational content; intuitionistic logic tracks what we can explicitly demonstrate.
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