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

    It is not the case that The Curry-Howard correspondence holds not only between provable formulae and type ascriptions, but also between proof terms and proofs of corresponding formulae

    ?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 Curry-Howard correspondence maps proof *structures*, but classical logic proofs lack the constructive witnesses required for type inhabitation.
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    • 2.Classical tautologies like double negation elimination have no canonical computational term, undermining the claim's scope beyond intuitionistic systems.
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    Reason for 2 of 2
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    • 1.Proof identity in the correspondence presupposes that syntactically distinct proofs of the same formula are genuinely different, but proof-irrelevance principles in some type theories collapse this distinction.
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    • 2.If proofs of a proposition are identified up to propositional equality (as in HoTT's h-propositions), the term-to-proof mapping loses the granularity the correspondence requires.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Howard showed a correspondence between provable formulae in the sequent calculus and type ascriptions
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    • 2.Howard additionally showed a correspondence between terms in type ascriptions and proofs of the corresponding formulae
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