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    Any system requiring an oracle for first-order arithmetic... — Carmelics
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    Challenges→Hoare logic is a true logic of programs, not merely a proof method

    Any system requiring an oracle for first-order arithmetic to achieve completeness lacks the self-contained inferential closure definitive of a logic.

    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    1 reason against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Gödel's incompleteness theorems show FOA-complete systems transcend first-order expressiveness, violating closure within a single logical framework.
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    • 2.Self-contained inference requires the system to justify all completeness-enabling resources internally; external oracles violate this requirement.
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    • 3.Logics are characterized by decidable proof procedures; oracle-dependence makes completeness verification non-algorithmic and non-logical.
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    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.Many mathematical logics (second-order, type theory) are standardly considered logics despite requiring semantic resources beyond FOA decidability.
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    • 2.Oracles are formal mathematical objects; oracle-relative completeness is rigorous and internal to recursion theory—no mysterious external addition.
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    • 3.Self-containment conflates proof-theoretic completeness with semantic adequacy; they're distinct properties and both can hold with oracles.
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    Connections

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    Truth & Knowledge1 linkedPhilosophy of Language1 linked

    Related

    Gödel's incompleteness theorems show FOA-complete systems transcend first-order ...Hoare logic is a true logic of programs, not merely a proof methodLogics are characterized by decidable proof procedures; oracle-dependence makes ...Many mathematical logics (second-order, type theory) are standardly considered l...
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    Oracles are formal mathematical objects; oracle-relative completeness is rigorou...Self-contained inference requires the system to justify all completeness-enablin...Self-containment conflates proof-theoretic completeness with semantic adequacy; ...

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    Perspectives
    2 (1 for, 1 against)
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