Russell's theory of descriptions is a theory about how to understand definite descriptions: that is, descriptions that are particularly suited to referring to single things, such as:
the president of the USA,
the nicest cow in the world,
John's best friend.
Russell's proposal is that we understand "The cow is brown" as saying that there is exactly one cow, and it is brown. (Or, which comes to the same thing, there is exactly one cow, and all cows are brown.). Thus:
As it stands this may strike one as obviously wrong: surely someone who says that the cow is brown, does not mean that there is exactly one cow in the whole wide world, and that it is brown. However, we may easily suppose that the domain in question is implicitly a good deal more limited than everything there is; plausibly it will be restricted (rather vaguely, perhaps), to things in the vicinity, or some such thing. In fact the context will help to determine the domain; it may be things in the picture, or, things on Farmer Giles's farm (if we happen to be talking about that), or…
One might also object to the proposal on the grounds that someone who says that the cow is brown will not actually be saying that there is exactly one cow etc. But, by way of defence, one might say that at any rate, if there is not exactly one cow, what the person says will not be true. So, supposing that the sentence is either true of false, the truth-value of the proposed paraphrase will always agree with that of the original; and that will be enough for our purposes.
Because it will be useful for further formalisations, consider how we might formalise
Using "b" for "Buttercup", we could say,
(i.e. "The cow is Buttercup".)
More compactly we can say:
So now, if we want to formalise
The farmer bought the cow,
We can say, as a first step,
$x[x is the farmer Ù $y[y is the cow Ù x bought y]].
(Notice that we have to make sure that "x bought y" is in the scope of both the quantifiers.) Or, if you prefer:
$x$y[x is the farmer Ù [y is the cow Ù x bought y]].
And we now replace "x is the farmer" and "y is the cow" with expressions which follow the pattern of the formalisation of "Buttercup is the cow". Thus:
(where, of course, "Fx" means "x is a farmer", and "Bxy" means "x bought y".)
Notice, by the way, that we used ""z" twice. This was all right because the second occurrence was not in the scope of the first. But we could, of course, have used ""w" instead.
Another example: as a first step we can translate,
Farmer Giles's cow is brown
$x[x is the cow which Farmer Giles owns Ù x is brown]
and so arrive at: