Transcript Section
Language, Proof and
Logic
Multiple Quantifiers
Chapter 11
11.1
Multiple uses of a single quantifier
What do the following sentences say?
1. xy[Cube(x)Tet(y)LeftOf(x,y)]
2. xy[(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 xyP(x,y) logically implies xP(x,x), and
xP(x,x) logically implies xyP(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. xyLikes(x,y)
3. xyLikes(y,x)
4. yxLikes(x,y)
5. yxLikes(y,x)
6. xy[xy 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.