Language Proof and Logic Multiple Quantifiers Chapter 11
- Slides: 6
Language, Proof and Logic Multiple Quantifiers Chapter 11
11. 1 Multiple uses of a single quantifier What do the following sentences say? 1. x y[Cube(x) Tet(y) Left. Of(x, y)] 2. x y[(Cube(x) Tet(y)) Left. Of(x, y)] 3. x[Cube(x) y(Tet(y) Left. Of(x, y))] 4. x[(Cube(x) y(Tet(y) Left. Of(x, y))] When evaluating a sentence with multiple quantifiers, don’t fall into the trap of thinking that distinct variables range over distinct objects. In fact, the sentence x y. P(x, y) logically implies x. P(x, x), and x. P(x, x) logically implies x y. P(x, y)! You try it, p. 299
11. 2 Mixed quantifiers What do the following sentences say? 1. x[Cube(x) y(Tet(y) Left. Of(x, y))] 2. x y. Likes(x, y) 3. x y. Likes(y, x) 4. y x. Likes(x, y) 5. y x. Likes(y, x) 6. x y[x y Cube(x) Cube(y)] 7. x[Cube(x) y(Cube(y) y=x)] You try it, p. 304
11. 3 The step-by-step method of translation “Each cube is to the left of a tetrahedron” A(x) = “x is to the left of a tetrahedron” “x is to the right of a tetrahedron” =
11. 4 Paraphrasing English “If a dog is hungry, then it is dangerous” Wrong translation: Paraphrasing: Right translation:
11. 5 Ambiguity and context sensitivity Every minute a man is mugged in NYC.
