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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
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    Home/Original/inverse
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    Inverse View

    It is not the case that When logic is understood proof-theoretically rather than model-theoretically, the validity problem coincides with derivability

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
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    • 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 for 2 of 2
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    • 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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    Reasons Against

    1 perspective
    Reason against
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    • 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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