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    A formula can be proof-theoretically valid without being ... — Carmelics
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    Challenges→When logic is understood proof-theoretically rather than model-theoretically, the validity problem coincides with derivability

    A formula can be proof-theoretically valid without being derivable in any given formal system, as shown by incompleteness phenomena affecting extensions of PA.

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    1 reason for
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    Reasons For

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    Reason for
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    • 1.Gödel's incompleteness theorems prove true sentences exist unprovable in any consistent formal system extending PA.
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    • 2.Proof-theoretic validity concerns what follows from axioms; derivability concerns what a specific system can derive—these diverge.
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    • 3.We can semantically recognize formulas as valid through model-theoretic means even when no proof exists in formal systems.
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    Reasons Against

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    Reason against
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    • 1.The distinction between 'proof-theoretically valid' and 'derivable' conflates semantic truth with proof-theoretic properties confusingly.
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    • 2.Any coherent notion of 'proof-theoretic validity' must ultimately be definable within or relative to some formal system's rules.
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    • 3.Incompleteness shows limits of specific systems, not that validity exists outside all possible formal derivation frameworks.
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    2 topics

    Truth & Knowledge1 linkedPhilosophy of Language1 linked

    Related

    Any coherent notion of 'proof-theoretic validity' must ultimately be definable w...Gödel's incompleteness theorems prove true sentences exist unprovable in any con...Incompleteness shows limits of specific systems, not that validity exists outsid...Proof-theoretic validity concerns what follows from axioms; derivability concern...
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    The distinction between 'proof-theoretically valid' and 'derivable' conflates se...We can semantically recognize formulas as valid through model-theoretic means ev...When logic is understood proof-theoretically rather than model-theoretically, th...

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