The scope of quantifiers

As far as scope is concerned, quantifiers behave like "¬".

So, in

            [$x x is a cow Ù x likes Buttercup]

the third "x" is free. (The whole expression is a 1-place predicate).

Note that we are not allowed to have one quantifier with an "x" in it (say) in the scope of another quantifier with an "x" in it.

So, whereas the following is all right:

            [$x x is a cow Ù $x x likes Buttercup]

(It translates "something is a cow and something likes Buttercup")

The following is forbidden:

            $x[x is a cow Ù $x x likes Buttercup].

The following, however, is all right:

            $x[x is a cow Ù $y y likes Buttercup].

So is:

            $x$y[x is a cow Ù y likes Buttercup].

They are in fact equivalent to each other, and are also equivalent to "[$x x is a cow Ù $x x likes Buttercup]". You should not however think that it is safe simply to move all one's quantifiers to the front. Even if the resulting expression is well formed, it will not necessarily be equivalent to the expression one started with.

Note also that vacuous quantifiers are not allowed: that is, quantifiers which do not bind any variables. So the following is not permitted:

            [$x y is a cow Ù x is a cow]

because "$x" does not bind the "x" – its scope is just "$x y is a cow".

The following is all right, however:

            $x[y is a cow Ù x is a cow]

(It is, of course, a 1-place predicate.)