Multiple%20Quantifiers

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Multiple Quantifiers
Language, Proof and Logic
Chapter 11
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Multiple uses of a single quantifier
11.1
What do the following sentences say?
1. ?x?yCube(x)?Tet(y)?LeftOf(x,y)
2. ?x?y(Cube(x)?Tet(y))?LeftOf(x,y)
3. ?xCube(x)? ?y(Tet(y)?LeftOf(x,y))
4. ?x(Cube(x) ? ?y(Tet(y)?LeftOf(x,y))
When evaluating a sentence with multiple
quantifiers, dont 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
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Mixed quantifiers
11.2
What do the following sentences say?
1. ?xCube(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?yx?y ? Cube(x) ? Cube(y)
7. ?xCube(x) ? ?y(Cube(y) ? yx)
You try it, p. 304
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The step-by-step method of translation
11.3
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
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Paraphrasing English
11.4
If a dog is hungry, then it is dangerous
Wrong translation Paraphrasing Right
translation
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Ambiguity and context sensitivity
11.5
Every minute a man is mugged in NYC.