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    Truth in second-order logic ('M ⊨_s φ') is not an absolut... — Carmelics
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    Truth in second-order logic ('M ⊨_s φ') is not an absolute property relative to ZFC.

    Philosophy of Language
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    Reasons For

    1 perspective
    Reason for
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    • 1.For first-order logic, the satisfaction relation 'M ⊨_s φ' is absolute relative to ZFC and can be written in Δ₁^ZFC form.
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    • 2.For second-order logic, while 'φ is a second-order formula', 'M is an L-structure', and 's is an assignment' are all absolute relative to ZFC, the satisfaction relation 'M ⊨_s φ' is not absolute.
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    • 3.The truth of the second-order sentence θ_CH in a sufficiently large model depends on whether the Continuum Hypothesis holds, and the Continuum Hypothesis is not settled by ZFC alone.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.The non-absoluteness of second-order truth relative to ZFC reflects a limitation of ZFC as a metatheory, not an intrinsic property of second-order truth itself.
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    • 2.Kreisel's squeezing argument and Shapiro's categoricity results suggest second-order logical consequence has a determinate extension fixed by the standard semantics, independent of set-theoretic indeterminacy.
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    • 3.If we adopt a background ontology of full powersets as primitive (as in Zermelo's 1930 quasi-categoricity results), CH has a determinate truth value, making second-order truth absolute relative to that stronger framework.
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    Reason against 2 of 2
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    • 1.The claim conflates syntactic provability within ZFC with semantic absoluteness, since a sentence can be semantically determinate even when ZFC cannot decide it.
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    • 2.Gödel's platonist position, endorsed in 'What is Cantor's Continuum Problem?', holds that set-theoretic statements like CH possess objective truth values that human axiom systems may simply fail to capture.
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    • 3.The ZFC-relative non-absoluteness of 'M ⊨_s φ' therefore demonstrates only epistemic incompleteness, not genuine ontological indeterminacy in second-order truth.
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    Topics

    Philosophy of LanguageTruth & Knowledge

    Connections

    1 topic

    Modality & Possibility1 linked

    Related

    For first-order logic, the satisfaction relation 'M ⊨_s φ' is absolute relative ...For second-order logic, while 'φ is a second-order formula', 'M is an L-structur...Gödel's platonist position, endorsed in 'What is Cantor's Continuum Problem?', h...If we adopt a background ontology of full powersets as primitive (as in Zermelo'...
    +5 moreShow less
    Kreisel's squeezing argument and Shapiro's categoricity results suggest second-o...The ZFC-relative non-absoluteness of 'M ⊨_s φ' therefore demonstrates only epist...The claim conflates syntactic provability within ZFC with semantic absoluteness,...The non-absoluteness of second-order truth relative to ZFC reflects a limitation...The truth of the second-order sentence θ_CH in a sufficiently large model depend...

    Similar

    For second-order logic, while 'φ is a second-order formula', 'M is an ...89%The Compactness Theorem does not hold for second-order logic in the fo...83%The property of a model satisfying a second-order sentence is not abso...82%The property of a second-order sentence having a model is not absolute...81%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-higher-order
    View source passageHide passage
    Let \(\theta_{\le}(P,R)\) be the formula \[ \exists F\left(\forall x\,\forall y\left( (F(x)=F(y)\to x=y) \land(P(x)\to R(F(x)) \right)\right). \] Now \(\mm\models_s\theta_\le(P,R)\) if and only if \(|s(P)|\le |s(R)|\). Let \(\theta_{\textrm{EQ}}(P,R)\) be the formula \(\theta_{{\le}}(P,R)\land \theta_{{\le}}(R,P)\). Now \(\mm\models_s\phi(P,R)\) if and only if \(|s(P)|=|s(R)|\). Let \(\theta'_{\textrm{EC}}(Y)\) be \[ \exists F\left( \forall x\,\forall y((F(x)=F(y)\to x=y)\land R(F(x)))
    Extraction notes

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

    Details

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