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    Home/Original/inverse
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    Inverse View

    It is not the case that The Curry-Howard correspondence can be extended beyond propositional logic to encompass predicate logic, specifically Heyting arithmetic

    ?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 extension to predicate logic requires dependent types, which introduce ontological commitments absent in the propositional case.
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    • 2.Dependent type theories (e.g., Martin-Löf type theory) are not merely extensions of Howard's original correspondence but constitute distinct foundational frameworks.
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    • 3.Conflating Howard's propositional result with predicate-level extensions obscures the non-trivial philosophical gap between proof-functional and proof-object semantics.
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    Reason for 2 of 2
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    • 1.Heyting arithmetic's quantifiers range over an open-ended domain of natural numbers, but the Curry-Howard correspondence treats proofs as closed, syntactically defined objects.
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    • 2.Kreisel's informal rigour argument establishes that the intended meaning of intuitionistic quantifiers cannot be fully captured by any formal recursive proof calculus.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Howard demonstrated a correspondence between intuitionistic sequent form natural deduction and type theory in lambda-calculus format
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    • 2.This correspondence generalises to encompass intuitionist arithmetic (Heyting arithmetic), which requires an extension from propositional to predicate logic
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