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    Made withinDC&Austin
    Checking the validity of an arbitrary second-order senten... — Carmelics
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    Home/Truth & Knowledge
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    Checking the validity of an arbitrary second-order sentence φ can be recursively reduced to checking the validity of a Σ¹₁-sentence.

    Proof of definition segmentsTruth & Knowledge
    ?Rate how convincing each reason is below to see the overall strength.
    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
    ?
    • 1.The Π¹₁-formula θ axiomatizes structures that interpret second-order quantification over a base set U as first-order quantification over the power-set expansion of U.
      ?

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    • 2.Any second-order sentence φ translates to a first-order sentence φ* relative to models of θ.
      ?

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    • 3.The original sentence φ is valid if and only if the Σ¹₁-sentence (θ → φ*) is valid, completing the reduction.
      ?

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    Reasons Against

    2 perspectives
    Reason against 1 of 2
    ?
    • 1.The reduction presupposes that θ faithfully captures full second-order semantics, but Henkin models satisfying θ need not validate full comprehension, undermining the biconditional in P3.
      ?

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    • 2.As Henkin (1950) demonstrated, weakening the semantics of second-order logic to general models preserves completeness but loses categoricity, so validity in full models and validity relative to θ come apart.
      ?

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    Reason against 2 of 2
    ?
    • 1.Σ¹₁-validity is itself not recursively enumerable, as established by results tracing to Gödel and elaborated by Kreisel, so the reduction preserves undecidability rather than achieving any effective proof-theoretic gain.
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    • 2.A recursive reduction to an equally undecidable class transfers computational intractability without reduction in logical complexity, making the claim technically correct but epistemically idle as a foundation for proof procedures.
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    Topics

    Truth & KnowledgeProof of definition segments

    Connections

    1 topic

    Philosophy of Language2 linked

    Related

    A recursive reduction to an equally undecidable class transfers computational in...Any second-order sentence φ translates to a first-order sentence φ* relative to ...As Henkin (1950) demonstrated, weakening the semantics of second-order logic to ...The original sentence φ is valid if and only if the Σ¹₁-sentence (θ → φ*) is val...
    +3 moreShow less
    The reduction presupposes that θ faithfully captures full second-order semantics...The Π¹₁-formula θ axiomatizes structures that interpret second-order quantificat...Σ¹₁-validity is itself not recursively enumerable, as established by results tra...

    Similar

    The original sentence φ is valid if and only if the Σ¹₁-sentence (θ → ...80%With this closure condition, the set of validities coincides with sent...78%Any second-order sentence φ translates to a first-order sentence φ* re...77%There are only countably many second-order sentences.76%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-higher-order
    View source passageHide passage
    During round i of the game player I can pick a relation \(A_i\) on A (or an element \(a_i\) of A) and then player II has to pick a relation \(B_i\) on B of the same arity as \(A_i\) (or an element \(b_i\) of on B) and vice versa: Player I can instead pick a relation \(B_i\) on B (or an element \(b_i\) of B) and then II picks a relation \(A_i\) on A of the same arity as \(B_i\) (or an element \(a_i\)) of A. After n rounds the pairs of played elements \((a_i,b_i)\) form a binary relation R on \(A\
    Extraction notes

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

    Details

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