The empty domain

What if one's domain of quantification is empty?

Then anything that begins "$x" must be false, because it says that there is something such that…, which must be false if there isn't anything at all. That means that anything that begins "¬$x" must be true.

Equally, anything that begins ""x" must be true. (It is equivalent to something beginning "¬$x¬".) And anything which begins "¬"x" must be false. (It is equivalent to something beginning "$x¬".)

So, notice that it is possible for ""x x is brown" to be true but "$x x is brown" false: when the domain is empty. (Just as it is possible, as we have seen for ""x[x is a cow ® x is brown]" to be true and "$x[x is a cow Ù x is brown]" false: when there are no cows.)