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    Fagin's theorem and Immerman-Vardi establish co-extension... — Carmelics
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    Challenges→P ≠ NP if and only if there exists a class of ordered structures definable in existential second-order logic which is not definable by a formula of FO(LFP)

    Fagin's theorem and Immerman-Vardi establish co-extensionality of classes, not identity of properties, so the biconditional is weaker than it appears.

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    1 reason for
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    Reasons For

    1 perspective
    Reason for
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    • 1.Co-extensionality (same truth conditions) differs from identity (same intrinsic nature); biconditionals only guarantee the former.
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    • 2.FO+LFP and SO capture identical problem sets extensionally, but may involve fundamentally different computational properties or mechanisms.
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    • 3.Properties concern how something works; classes concern what gets classified. The theorems address only classification equivalence.
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    Reasons Against

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    Reason against
    ?
    • 1.In logic, a true biconditional between formal definitions is precisely what identity of logical properties means; extensionality is identity here.
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    • 2.The distinction between co-extensionality and property-identity presumes an unexplained gap between syntax and semantics in formal systems.
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    • 3.If FO+LFP and SO express identical computable classes, distinguishing their 'properties' lacks clear meaning without external metaphysical commitment.
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    Related

    Co-extensionality (same truth conditions) differs from identity (same intrinsic ...FO+LFP and SO capture identical problem sets extensionally, but may involve fund...If FO+LFP and SO express identical computable classes, distinguishing their 'pro...In logic, a true biconditional between formal definitions is precisely what iden...
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    P ≠ NP if and only if there exists a class of ordered structures definable in ex...Properties concern how something works; classes concern what gets classified. Th...The distinction between co-extensionality and property-identity presumes an unex...

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