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    The claim that the Church–Rosser theorem guarantees a uni... — Carmelics
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    The claim that the Church–Rosser theorem guarantees a unique final result is a slight overstatement

    Philosophy of LanguageTruth & Knowledge
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    2 reasons for
    1 reason against

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 1.The Church–Rosser property guarantees confluence: if a term has a normal form, that normal form is unique.
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    • 2.Guaranteeing uniqueness only under the condition of existence is logically weaker than guaranteeing a unique final result simpliciter.
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    • 3.Conflating conditional uniqueness with unconditional uniqueness is a semantic overstatement, as Barendregt's lambda calculus treatment explicitly distinguishes these properties.
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    Reason for 2 of 2
    ?
    • 1.In proof-theoretic semantics, a 'result' presupposes termination; Church–Rosser is silent on whether reduction terminates for arbitrary terms.
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    • 2.Curry and Feys's original combinatory logic literature treats normalization and confluence as independent properties requiring separate proofs.
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    • 3.Describing Church–Rosser as guaranteeing a 'unique final result' elides this independence, making the claim a genuine overstatement rather than mere imprecision.
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    Reasons Against

    1 perspective
    Reason against
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    • 1.Uniqueness of a final result implies termination of every reduction sequence
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    • 2.The Church–Rosser theorem does not itself guarantee termination—a term may diverge rather than reach a normal form
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    Related

    Conflating conditional uniqueness with unconditional uniqueness is a semantic ov...Curry and Feys's original combinatory logic literature treats normalization and ...Describing Church–Rosser as guaranteeing a 'unique final result' elides this ind...Guaranteeing uniqueness only under the condition of existence is logically weake...
    +4 moreShow less
    In proof-theoretic semantics, a 'result' presupposes termination; Church–Rosser ...The Church–Rosser property guarantees confluence: if a term has a normal form, t...The Church–Rosser theorem does not itself guarantee termination—a term may diver...Uniqueness of a final result implies termination of every reduction sequence

    Similar

    The Church–Rosser theorem guarantees that the final result of a series...78%F ⊢ Prf_F(n̲, ⌈G_F⌉) would contradict Gödel's incompleteness theorem75%No satisfactory proof of Prawitz's conjecture has been given.75%Goodstein's theorem is true but unprovable in PA74%

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    SEP: logic-combinatory
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    If we think that reduction is like computing the value of a function, then the Church–Rosser theorem—in a first approximation—can be thought to state that the final result of a series of calculations with a term is unique—independently of the order of the steps. This is a slight overstatement though, because uniqueness implies that each series of calculations ends (or “loops” on a term). That is, if there is a unique final term, then only finitely many distinct consecutive calculation steps are
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    Details

    Type
    claim
    Perspectives
    3 (2 for, 1 against)
    Edits
    1 edit