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    The Curry-Howard correspondence holds not only between pr... — Carmelics
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    The Curry-Howard correspondence holds not only between provable formulae and type ascriptions, but also between proof terms and proofs of corresponding formulae

    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.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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    Reasons Against

    2 perspectives
    Reason against 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 against 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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    Related

    Classical tautologies like double negation elimination have no canonical computa...Howard additionally showed a correspondence between terms in type ascriptions an...Howard showed a correspondence between provable formulae in the sequent calculus...If proofs of a proposition are identified up to propositional equality (as in Ho...
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    Proof identity in the correspondence presupposes that syntactically distinct pro...The Curry-Howard correspondence maps proof *structures*, but classical logic pro...

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    Howard additionally showed a correspondence between terms in type ascr...89%The Curry-Howard correspondence allows rephrasing the intuitionist pos...85%Howard showed a correspondence between provable formulae in the sequen...84%The Curry-Howard correspondence can be extended beyond propositional l...80%

    Source

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    SEP: formalism-mathematics
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    Curry and Feys (1958) extended the correspondence idea to one between type theory and Gentzen’s sequent calculus. In the paper already cited, circulated in 1969, but only published in a volume in a Festschrift for Curry in 1980, W.A. Howard (1969) deepened the CH correspondence by demonstrating a correspondence between intuitionistic sequent form natural deduction and type theory in \(\lambda\)-calculus format, generalising to encompass intuitionist arithmetic- ‘Heyting arithmetic’ (HA)- (thus r
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    Details

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    claim
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    3 (1 for, 2 against)
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    1 edit