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    Extensional adequacy, properly construed, already entails... — Carmelics
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    Challenges→The second incompleteness theorem is an intensional result, not merely an extensional one.

    Extensional adequacy, properly construed, already entails that a provability predicate captures the intended proof relation, making intensionality redundant.

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    1 reason for
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    Reasons For

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    Reason for
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    • 1.If a provability predicate correctly identifies all and only provable sentences, it captures the proof relation's extensional identity regardless of mechanism.
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    • 2.Intensional criteria (deriving from specific proof rules) add nothing beyond extensional adequacy if they yield identical outputs on all inputs.
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    • 3.Scientific parsimony favors extensional criteria: we need not specify internal structure when behavioral equivalence is established and sufficient.
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    Reasons Against

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    Reason against
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    • 1.Extensional adequacy on finite or standard models underdetermines behavior on non-standard models, where intensional structure becomes evidentially crucial.
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    • 2.A predicate matching proof-theoretic output may still fail to respect proof-theoretic properties like constructivity or computational complexity differences.
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    • 3.Understanding *why* something is provable (intensional content) serves explanatory purposes extensional adequacy alone cannot fulfill.
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    Connections

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    Truth & Knowledge1 linkedPhilosophy of Language1 linked

    Related

    A predicate matching proof-theoretic output may still fail to respect proof-theo...Extensional adequacy on finite or standard models underdetermines behavior on no...If a provability predicate correctly identifies all and only provable sentences,...Intensional criteria (deriving from specific proof rules) add nothing beyond ext...
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    Scientific parsimony favors extensional criteria: we need not specify internal s...The second incompleteness theorem is an intensional result, not merely an extens...Understanding *why* something is provable (intensional content) serves explanato...

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