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

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