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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))]
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
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
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.