Vagueness and Identity

Many metaphysicians are interested in a question which can be put roughly in the following way: can it be vague whether A and B are identical?

Sometimes sloppy metaphysicians ask whether "identity can be vague." But this sloppiness borders on being a category mistake. Vagueness most naturally, and in the first instance, applies to questions, not objects or relations. It can vague as to whether Harry is bald, or vague as to how many grains of sand are in the Sahara Desert. But the notion of a vague object or a vague relation is confused. The best we can say of a "vague object" is that there are objects such that it is vague as to where their borders are. And the best we can say of "vague relations" is that there are objects and relations such that it is vague as to whether those objects bear the relation in question, or that it is vague how many pairs of objects bear that relation. If we want to investigate vagueness and its relation to identity, we must investigate it via a question, such as: can it be vague as to whether A and B are identical.

Here is an argument that it can be vague as to whether A and B are identical.

N.B., I will use 'clearly' in the following way: it is clearly the case that P iff it is true that P and it is not vague as to whether P.

Consider the Sahara Desert. There are grains of sand such that it is vague as to whether those grains of sand are parts of the Sahara Desert.

Now imagine that we discover a heretofore unknown tribe which uses the name 'Schmahara Desert' in the following two ways. First, the Schmahara Desert, like the Sahara Desert, is a desert. Thus (we may pretend) both the Sahara and the Schmahara are entirely composed of grains of sand. Second, the following bi-conditional holds: a grain of sand G is clearly part of the Schmahara Desert iff G is clearly part of the Sahara Desert.

There are three options:

(i) It is clearly the case that the Sahara Desert and the Schmahara Desert are identical.
(ii) It is vague as to whether the Sahara Desert and the Schmahara Desert are identical.
(iii) It is clearly not the case that the Sahara Desert and the Schmahara Desert are identical.

We can rule out (iii) straight away. For it to be clearly not the case that the Sahara Desert and the Schmahara Desert are identical, there would have to be some grain of sand such that it is clearly a part of one but clearly not a part of the other.

Here is a way to rule out (i). It might be that for every grain of sand G except for G*, it is vague as to whether G is part of the Sahara Desert iff it is vague as to whether G is part of the Schmahara Desert. But, we may imagine, G* is such that it is vague as to whether G* is part of the Sahara Desert but it is clearly not the case that G* is part of the Schmahara Desert. In other words there are more grains of sand by one for which it is vague whether they are part of the Sahara Desert than there are grains of sand for which it is vague whether they are part of the Schmahara Desert.

Because it is vague as to whether G* is part of the Sahara Desert, it is vague whether the Sahara Desert is bigger than the Schmahara Desert.

If it were clearly the case that the Sahara Desert and the Schmahara Desert were identical, then it could not be vague as to whether the Sahara Desert is bigger than the Schmahara Desert.

So not (i).

So, (ii), it is vague whether the Sahara Desert is identical to the Schmahara Desert.

So, in general, it can be vague as to whether A and B are identical.