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

    The Curry-Howard correspondence maps proof *structures*, but classical logic proofs lack the constructive witnesses required for type inhabitation.

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    Reasons For

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    Reason for
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    • 1.Classical proofs use excluded middle and indirect arguments that don't construct explicit witnesses needed for computational type inhabitation.
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    • 2.Intuitionistic logic naturally produces lambda terms; classical logic requires double-negation elimination, which lacks direct computational content.
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    • 3.Type theory interprets proofs as programs, but classical tautologies like (P∨¬P) don't yield computable functions without additional axioms.
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    Reasons Against

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    Reason against
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    • 1.Classical logic can be embedded in intuitionistic logic via CPS translation; proofs still correspond to types, just indirectly.
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    • 2.Curry-Howard applies to classical systems too (e.g., call/cc in typed languages), mapping to control operators rather than pure lambdas.
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    • 3.The claim conflates 'lacks direct witnesses' with 'lacks type inhabitation'—these are different properties; classical proofs inhabit different type structures.
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    Truth & Knowledge1 linkedPhilosophy of Language1 linked

    Related

    Classical logic can be embedded in intuitionistic logic via CPS translation; pro...Classical proofs use excluded middle and indirect arguments that don't construct...Curry-Howard applies to classical systems too (e.g., call/cc in typed languages)...Intuitionistic logic naturally produces lambda terms; classical logic requires d...
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    The Curry-Howard correspondence holds not only between provable formulae and typ...The claim conflates 'lacks direct witnesses' with 'lacks type inhabitation'—thes...Type theory interprets proofs as programs, but classical tautologies like (P∨¬P)...

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