Commutativity and Quantifiers

Created on March 5, 2013, 9:28 p.m. by Hevok & updated by Hevok on May 2, 2013, 5:31 p.m.

With Quantifiers one has to be careful. Although, Quantifiers of the same sort are commutative, Quantifiers that are not of the same sort (Essential and Universal) are not. Switching Quantifiers which are not the same does not mean the same.

  • Quantifiers (of the same sort) are commutative

    (∀X)(∀Y)F ≡ (∀Y)(∀X)F
    (∃X)(∃Y)F ≡ (∃Y)(∃X)F
  • But

    (∃X)(∀Y)F ≢ (∀Y)(∃X)F

  • Example:

  • (∃x)(∀y): loves(x,y)

    "There exists somebody, who loves everybody."

  • (∀y)(∃x): loves(x,y)

    "Everybody" is loved by somebody."


Tags: logic, unequivalence
Parent: Canonical Form

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