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    The Curry-Howard correspondence can be extended beyond pr... — Carmelics
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    The Curry-Howard correspondence can be extended beyond propositional logic to encompass predicate logic, specifically Heyting arithmetic

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    Reasons For

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    Reason for
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    • 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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    Reasons Against

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    Reason against 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 against 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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    Related

    Conflating Howard's propositional result with predicate-level extensions obscure...Dependent type theories (e.g., Martin-Löf type theory) are not merely extensions...Heyting arithmetic's quantifiers range over an open-ended domain of natural numb...Howard demonstrated a correspondence between intuitionistic sequent form natural...
    +3 moreShow less
    Kreisel's informal rigour argument establishes that the intended meaning of intu...The extension to predicate logic requires dependent types, which introduce ontol...This correspondence generalises to encompass intuitionist arithmetic (Heyting ar...

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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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