Slides / Bullets

• Mixed quantifiers
  • The order of quantifiers makes a difference:
    • ∀x∃y(Hates(x,y)) vs ∃x∀y(Hates(x,y))
    • ∀x∃y(SameCol(x,y)) vs ∃x∀y(SameCol(x,y))
• The step-by-step method of translation
  • This is very useful for sentences that contain several quantified noun phrases.
  • EG: ‘Every dog lives in some kennel’
    • Step one: ∀x(Dog(x) → lives-in-some-kennel(x))
    • Step two: ∀x(Dog(x) → ∃y(Kennel(y) ∧ LivesIn(x,y))
• • EG: ‘Every dog who lives in a kennel has an owner who lives in a house’.
    • Step One: ∀x(Dog(x)∧lives-in-a-kennel(x) → has-an-owner-who-lives-in-a-house(x)).
    • Step Two: ∀x(Dog(x)∧∃y(Kennel(y) ∧ LivesIn(x,y)) → ∃z(Owns(x,z) ∧ lives-in-a-house(z)).
    • Step Three: ∀x(Dog(x)∧∃y(Kennel(y) ∧ LivesIn(x,y)) → ∃z(Owns(x,z) ∧ ∃y(House(y) ∧ LivesIn(z,y))).
• Ambiguity
  • Quantifiers in English are a rich source of ambiguity.
    • ‘Some poor sucker is mugged every minute’.
      • ? ∃x(Poor(x)∧Sucker(x)∧mugged-every-minute(x))
      • ∀x(Minute(x)→some-poor-sucker-is-mugged-during(x))
  • Often this ambiguity is resolved by context.