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    The semantics of second-order logic depend on the metathe... — Carmelics
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    The semantics of second-order logic depend on the metatheory with respect to the Axiom of Choice.

    Philosophy of LanguageTruth & Knowledge
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    1 reason for
    2 reasons against

    Reasons For

    1 perspective
    Reason for
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    • 1.By Cohen's results, there exist models N and N' both satisfying the axioms of ZF without the Axiom of Choice such that the sentence θ expressing the Axiom of Choice for continuum-size families of subsets of the reals holds in N but not in N'.
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    • 2.Whether 'N ⊨ θ' holds therefore depends on which model of ZF is taken as the metatheory, not on the sentence θ alone.
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Standard semantics for second-order logic fixes the power set interpretation absolutely, making the range of predicate variables a matter of logic, not set theory.
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    • 2.Shapiro's full semantics treats second-order quantifiers as ranging over the actual power set of the domain, so metatheoretic AC-independence reflects set-theoretic indeterminacy, not semantic ambiguity in the logic itself.
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    • 3.The dependence on AC exposed by Cohen's models is a feature of Henkin semantics, where comprehension is restricted, not of second-order logic under its intended, unrestricted interpretation.
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    Reason against 2 of 2
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    • 1.Kreisel's squeezing argument supports the view that our pre-theoretic notion of logical validity is determinate even when formal metatheories disagree, so Cohen-style independence results do not destabilize the semantics of second-order logic.
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    • 2.The metatheoretic variation Cohen's forcing reveals is a pathology of first-order set-theoretic surrogates for second-order semantics, not evidence that second-order semantic content is itself indeterminate.
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    Related

    By Cohen's results, there exist models N and N' both satisfying the axioms of ZF...Kreisel's squeezing argument supports the view that our pre-theoretic notion of ...Shapiro's full semantics treats second-order quantifiers as ranging over the act...Standard semantics for second-order logic fixes the power set interpretation abs...
    +3 moreShow less
    The dependence on AC exposed by Cohen's models is a feature of Henkin semantics,...The metatheoretic variation Cohen's forcing reveals is a pathology of first-orde...Whether 'N ⊨ θ' holds therefore depends on which model of ZF is taken as the met...

    Similar

    The semantics of second-order logic depend on metatheoretic set theory...89%The semantics of second-order logic depends so deeply on metatheoretic...86%Absoluteness of first-order semantics relative to ZFC gives assurance ...84%Quine's characterization of second-order logic as 'set theory in sheep...80%

    Source

    AI-extracted1/3 agreementValid
    SEP: logic-higher-order
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    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