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    ∀x[P(x) → D(x)] is equivalent to ¬∃x[P(x) ∧ ¬D(x)] — Carmelics
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    ∀x[P(x) → D(x)] is equivalent to ¬∃x[P(x) ∧ ¬D(x)]

    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.P(x) → D(x) is equivalent to ¬P(x) ∨ D(x) by Frege's definitions
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    • 2.∀x is equivalent to ¬∃¬
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    • 3.By de Morgan's Laws, ¬∃x¬[¬P(x) ∨ D(x)] iff ¬∃x[P(x) ∧ ¬D(x)]
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    Reasons Against

    2 perspectives
    Reason against 1 of 2
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    • 1.Intuitionistic logic, following Brouwer and Heyting, rejects the classical equivalence of ∀x[P(x) → D(x)] and ¬∃x[P(x) ∧ ¬D(x)] because the latter requires a constructive witness for negation that the former does not.
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    • 2.In intuitionistic systems, ¬∃xφ(x) means no construction can verify ∃xφ(x), while ∀x¬φ(x) requires a uniform construction for each x, and these are not interderivable without the law of excluded middle.
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    • 3.The supporting argument's use of de Morgan's Laws implicitly invokes double negation elimination, which is intuitionistically inadmissible and thus smuggles in a non-neutral logical commitment.
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    Reason against 2 of 2
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    • 1.Classical equivalences like ¬∃x[P(x) ∧ ¬D(x)] presuppose a non-empty domain, but free logic permits empty domains where ∀x[P(x) → D(x)] holds vacuously while ¬∃x[P(x) ∧ ¬D(x)] may differ in truth value.
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    • 2.Karel Lambert and Jaakko Hintikka showed that classical quantifier interdefinability fails in free logic systems designed to handle non-denoting terms and empty universes of discourse.
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    Philosophy of LanguageTruth & Knowledge

    Related

    By de Morgan's Laws, ¬∃x¬[¬P(x) ∨ D(x)] iff ¬∃x[P(x) ∧ ¬D(x)]Classical equivalences like ¬∃x[P(x) ∧ ¬D(x)] presuppose a non-empty domain, but...In intuitionistic systems, ¬∃xφ(x) means no construction can verify ∃xφ(x), whil...Intuitionistic logic, following Brouwer and Heyting, rejects the classical equiv...
    +4 moreShow less
    Karel Lambert and Jaakko Hintikka showed that classical quantifier interdefinabi...P(x) → D(x) is equivalent to ¬P(x) ∨ D(x) by Frege's definitionsThe supporting argument's use of de Morgan's Laws implicitly invokes double nega...∀x is equivalent to ¬∃¬

    Similar

    P(x) → D(x) is equivalent to ¬P(x) ∨ D(x) by Frege's definitions93%∀x[P(x) → D(x)] iff ¬∃x[P(x) ∧ ¬D(x)]82%Combined with R ∈ R ⊃ R ∈ R, this yields R ∈ R ⊃ (R ∈ R ∧ ~(R ∈ R))80%By de Morgan's Laws, ¬∃x¬[¬P(x) ∨ D(x)] iff ¬∃x[P(x) ∧ ¬D(x)]78%

    Source

    AI-extracted1/3 agreementValid
    SEP: logical-form
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    With regard to the proposition that some politician is deceitful, traditional grammar suggests the division ‘Some politician / is deceitful’, with the noun ‘politician’ combining with the quantificational word to form a complex subject. But on a Fregean view, grammar masks the logical division between the existential quantifier and the rest: \(\exists x [P(x) \land D(x)]\). With regard to the proposition that every politician is deceitful, Frege also stresses the logical division between the qua
    Extraction notes

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

    Details

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