## Relations and Multiple Quantification

The statements we have dealt with so far have propositional connectives occurring within the scope of the quantifier. But connectives may also occur outside its scope.

 ~ (X)Gx There are no ghosts. (x) ~ Gx There is something that is not a ghost. (x)Px ~ (y)Gy If everything is physical, then there are no ghosts. (x)(Gx Bx) v (y)(Cy My) Either all the gears are broken, or a cylinder is missing.

To tell whether connectives belong inside or outside the scope of the quantifiers, we need to think carefully about what the statement means.

Comprehension Questions

 1. Which one of the following is the correct symbolization of the following proposition: Someone loves everyone. a) (x) (y) [(Px.Py) > Lxy] b) (x) (y) [(Px.Py) > Lxy] c) (x) (y) [(Px.Py).Lxy] d) (x) (y) [(Px.Py).Lxy] 2. Which one of the following is the correct symbolization of the following proposition: Everyone loves someone. a) (x) (y) [(Px.Py) > Lxy] b) (x) (y) [(Px.Py) > Lxy] c) (x) (y) [(Px.Py).Lxy] d) (x) (y) [(Px.Py).Lxy] 3. Which one of the following is the correct symbolization of the following proposition: Everyone loves everyone. a) (x) (y) [(Px.Py) > Lxy] b) (x) (y) [(Px.Py) > Lxy] c) (x) (y) [(Px.Py).Lxy] d) (x) (y) [(Px.Py).Lxy] 4. Which one of the following is the correct symbolization of the following proposition: Someone loves somebody. a) (x) (y) [(Px.Py) > Lxy] b) (x) (y) [(Px.Py) > Lxy] c) (x) (y) [(Px.Py).Lxy] d) (x) (y) [(Px.Py).Lxy]

Proof