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    Carmelics

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    LoyalLoyalJusticeJustice
    Made withinDC&Austin
    Statements
    321,452
    Perspectives
    108,905
    Topics
    42
    Home/Original/inverse
    See Original
    Inverse View

    It is not the case that ∀x[P(x) → D(x)] is equivalent to ¬∃x[P(x) ∧ ¬D(x)]

    ?Set your confidence on the premises below to see your aggregate.

    Reasons For

    2 perspectives
    Reason for 1 of 2
    ?
    • 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.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    Reason for 2 of 2
    ?
    • 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.
      ?

      Think about whether this reason is strong or weak

    • 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.
      ?

      Think about whether this reason is strong or weak

    Reasons Against

    1 perspective
    Reason against
    ?
    • 1.P(x) → D(x) is equivalent to ¬P(x) ∨ D(x) by Frege's definitions
      ?

      Think about whether this reason is strong or weak

    • 2.∀x is equivalent to ¬∃¬
      ?

      Think about whether this reason is strong or weak

    • 3.By de Morgan's Laws, ¬∃x¬[¬P(x) ∨ D(x)] iff ¬∃x[P(x) ∧ ¬D(x)]
      ?

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    Strongest counterpoint
    Explore the most compelling reason on the other side.