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Multiple Quantifiers

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)LeftOf(x,y)]. 2. xy[(Cube(x)Tet(y))LeftOf(x,y)]. 3. x[Cube(x) y(Tet(y)LeftOf(x,y))].

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Multiple Quantifiers

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  1. Language, Proof and Logic Multiple Quantifiers Chapter 11

  2. 11.1 Multiple uses of a single quantifier What do the following sentences say? 1. xy[Cube(x)Tet(y)LeftOf(x,y)] 2. xy[(Cube(x)Tet(y))LeftOf(x,y)] 3. x[Cube(x) y(Tet(y)LeftOf(x,y))] 4. x[(Cube(x) y(Tet(y)LeftOf(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yP(x,y) logically implies xP(x,x), and xP(x,x) logically implies xyP(x,y)! You try it, p. 299

  3. 11.2 Mixed quantifiers What do the following sentences say? 1. x[Cube(x)  y(Tet(y)LeftOf(x,y))] 2. xyLikes(x,y) 3. xyLikes(y,x) 4. yxLikes(x,y) 5. yxLikes(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

  4. 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” =

  5. 11.4 Paraphrasing English “If a dog is hungry, then it is dangerous” Wrong translation: Paraphrasing: Right translation:

  6. 11.5 Ambiguity and context sensitivity Every minute a man is mugged in NYC.

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