Properties of quantifiers ·x Vy is the same as Vy Vx 3x 3y is the same as y x ● ·3 x Vy is not the same as∀yx ·3 x Vy Loves(xy) -"There is a person who loves everyone in the world" ·y3 x Loves(x,y) "Everyone in the world is loved by at least one person" 。 Quantifier duality:each can be expressed using the other ● ·∀x Likes(x,IceCream) x-Likes(x,IceCream) Properties of quantifiers • x y is the same as y x • • x y is the same as y x • • x y is not the same as y x • • x y Loves(x,y) – “There is a person who loves everyone in the world” – • y x Loves(x,y) – “Everyone in the world is loved by at least one person” – • Quantifier duality: each can be expressed using the other • • x Likes(x,IceCream) x Likes(x,IceCream) •