The domain of quantification

The domain of quantification is the set of things one's quantifiers range over; that is to say, the things one is talking about when one uses quantifiers.

So, if the domain is the set of cows, "$x x is brown" translates "Some cow is brown" or "There is a cow which is brown", and ""x x is brown" translates "All cows are brown").

If the domain is people, "$x x is brown" translates "Some person is brown". If the domain is numbers, "$x x>5" translates "There is a number greater than 5". And so on.

Note that for any sentence (or set of sentences being considered together) one can specify only one domain of quantification. One may not specify one domain for one quantifier and another for another.

If one restricts one's domain to cows, one can translate "Some cows are brown" as "$x x is brown". (It is true that this says only that there is at least one cow that is brown, and maybe the original says that more than one is; but let us not be fussy.)

To translate "Some cows are brown" where one's domain contains other things as well as cows one will have to say instead:

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

To translate "All cows are brown", where the domain contains other things one can say:

            "x[x is a cow ® x is brown]

(i.e. "Take anything: if it is a cow, it is brown".)

To translate "No cows are brown" one can say:

            ¬$x[x is a cow Ù x is brown]  (i.e. it's not true that some cows are brown.)

or

            "x[x is a cow ® ¬x is brown] (i.e. all cows are non-brown.)

Notice two things about these translations.

1. ""x[x is a cow ® x is brown]" comes out true if there are no cows at all. This is because "[j®y]" is equivalent to "[¬jÚy]". (It is true whenever "j" is false and whenever "y" is true.). This means that ""x[x is a cow ® x is brown]" is equivalent to ""x[¬x is a cow Ú x is brown]", which means that each thing is either not a cow or is brown. But this will be true if nothing is a cow. Equally, if there are no cows, the translation of "No cows are brown" will also be true. And the translation of "Some cows are brown" will be false.

2. The translations of "some" and "all" are not quite parallel. The one uses "Ù" where the other uses "®". So what does ""x[x is a cow Ù x is brown]" mean? It means everything is a brown cow. To see what "$x[x is a cow ® x is brown]", notice that it is equivalent to "$x[¬x is a cow Ú x is brown]", which means something very peculiar: that there is something which is either not a cow or is brown. So it will be true if, for example, there is a green field, or a brown goat, as well as if there is a brown cow.