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    When logic is understood proof-theoretically rather than ... — Carmelics
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    When logic is understood proof-theoretically rather than model-theoretically, the validity problem coincides with derivability

    Philosophy of LanguageTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.Under a proof-theoretic interpretation, a logic is understood as the set of formulas derivable from some set of axioms
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    • 2.Under this interpretation, the validity problem becomes the problem of deciding whether a formula is derivable from the axioms
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.Proof-theoretic validity, as developed by Prawitz and Dummett, is defined via validity of proofs in all possible extensions, not mere derivability from axioms.
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    • 2.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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    • 3.Therefore the coincidence of validity and derivability holds only for specific well-behaved logics like propositional logic, not as a general proof-theoretic thesis.
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    Reason against 2 of 2
    ?
    • 1.Kreisel's squeezing argument demonstrates that informal provability and formal derivability can come apart even when both fall under a proof-theoretic framework.
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    • 2.The proof-theoretic validity of a formula in Gentzen-style systems depends on normalization properties that are not reducible to axiomatic derivability in the relevant system.
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    Philosophy of LanguageTruth & Knowledge

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    Modality & Possibility1 linked

    Related

    A formula can be proof-theoretically valid without being derivable in any given ...Kreisel's squeezing argument demonstrates that informal provability and formal d...Proof-theoretic validity, as developed by Prawitz and Dummett, is defined via va...The proof-theoretic validity of a formula in Gentzen-style systems depends on no...
    +3 moreShow less
    Therefore the coincidence of validity and derivability holds only for specific w...Under a proof-theoretic interpretation, a logic is understood as the set of form...Under this interpretation, the validity problem becomes the problem of deciding ...

    Similar

    Under a proof-theoretic interpretation, a logic is understood as the s...85%What matters for logical argument is not derivability from arbitrary p...83%Hoare logic is a true logic of programs, not merely a proof method82%Soundness entails that if an argument is deducible (proof-theoreticall...81%

    Source

    AI-extracted1/3 agreementValid
    SEP: computational-complexity
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    It thus seems reasonable to summarize the current status of the \(\textbf{P} \neq \textbf{NP}\)? problem as follows: (i) \(\textbf{P} \neq \textbf{NP}\) is widely believed to be true on the basis of convergent inductive and heuristic evidence; (ii) we currently have no reason to suspect that this statement is formally independent of the mathematical theories which we accept in practice; but (iii) a proof \(\textbf{P} \neq \textbf{NP}\) is still considered to be beyond the reach of current techni
    Extraction notes

    Validity: Extracted via Max plan + API grounding/validity checks

    Details

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