Scope

When translating sentences in accordance with Russell's theory of descriptions, one needs sometimes to pay attention to scope. Indeed it is one of the virtues of the theory that it allows one to make distinctions of scope.

Consider

            Buttercup is not Farmer Giles's cow.

One could translate this either as:

            $x[x is Farmer Giles's cow Ù ¬b=x],

which says that there is such a thing as Farmer Giles's cow, but that Buttercup is not it. Or,

            ¬$x[x is Farmer Giles's cow Ù b=x],

which is compatible with there not being any such thing as Farmer Giles's cow.

Actually one might take the original to mean simply that Buttercup is not a cow belonging to Farmer Giles; in which case one would not make use of the theory of descriptions at all.

Consider,

            Buttercup wants to be the first cow to jump over the moon.

We certainly don't want to translate this as,

            $x[x is the first cow to jump over the moon Ù Buttercup wants to be it].

It is Buttercup who is deranged, we imagine, and not the speaker. So we should prefer:

Buttercup wants it to be the case that she is the first cow to jump over the moon,

Alas, we have not got the tools to deal with "Buttercup wants it to be the case that"; but we might (perhaps) translate "she is the first cow to jump over the moon" in accordance with the theory of descriptions.

Again consider:

            The first cow to jump over the moon does not exist,

i.e.

            There is no such thing as the first cow to jump over the moon.

This should not be translated as,

            $x[x is the first cow to jump over the moon Ù x does not exist],

but rather as,

            ¬$x x is the first cow to jump over the moon.

Notice, by the way, that we rendered "exist" by using the existential quantifier rather than a predicate "x exists". And that is the normal way of translating "exists".

Thus,

            Something F exists,

Is normally formalised as,

            $xFx.

How about,

            Something exists.

We can formalise that as:

            $xx=x,

which tells us merely that there is something which is the same thing as itself, which effectively tells just that something exists.

Notice that it is not that we cannot treat "exists" as a predicate. We can perfectly well translate "x exists", as we did just now, as "x=x". Or (equivalently) we could instead translate it as "$yx=y" (i.e. "there is something which x is".) 

So we could have formalised something F exists as:

            $x[FxÙx=x]

But that is a long-winded way of saying something equivalent to "$xFx".

We could indeed have formalised it as,

            $x$y[x=yÙFx].

But that is an even longer-winded way of saying the same thing.

Notice here in particular a virtue of Russell's way of treating definite descriptions compared with translating them with individual constants such as "a".

We have seen that we can render "There is no such thing as the first cow to jump over the moon" as,

            ¬$x x is the first cow to jump over the moon.

And then if, not trying to be too penetrating, we use "Mx" to mean "x is a first cow to jump over the moon", and using Russell's theory, we get,

            ¬$x[MxÙ"y[My®y=x]],

which has the virtue of being true.

But if were were to use "c", say, to mean "the first cow to jump over the moon", the best we could do would be.

            ¬$xx=c

or, equivalently,

            ¬c=c.

But these would say something false, thanks to the fact that our convention for using individual contants requires that they name something. (What if we dropped that convention? Well, that is another story. We would certainly have to change our tableau rules, since $x¬Fx would no longer follow from ¬Fa.)