## Friday, October 20, 2006

### Lowe on necessity of identity

The standard proof of the necessity of identity runs as follows:

1) For all x, necessarily x=x (Premise)
2) a=b (Assumption)
3) Necessarily, a=a (From 1)
4) a has the property of being necessarily identical to a (From 3)
5)
b has the property of being necessarily identical to a (From 2,4, and Leibniz’ law)
6) Necessarily, a=b (From 5)

I used to be concerned about Lowe’s objection to this proof. Lowe says (rightly) that we must be careful to distinguish between two properties: the property of being necessarily self-identical, as had by a, and the property of being necessarily identical by a, as had by a. The properties are clearly distinct: everything has the former, but only a has the latter. (3), Lowe said, is ambiguous: it could mean that everything has the property of being necessarily self-identical or it could mean that everything has the property of being necessarily identical to itself (i.e. that for all x, x has the property ‘being necessarily identical to x’). Read in the first way, (3) is uncontroversial, but then (4) doesn’t follow: all that follows is that a is necessarily self-identical and, hence, that b is necessarily self-identical. That is also uncontroversial, and a far cry from the necessity of identity. To get the claim that b is necessarily identical to a we need to get that a is necessarily identical to a, which requires the second reading of (3). But this, says Lowe, is not uncontroversial: to rule out contingent identity, we need to be given an argument for it.

Lowe is definitely right that there are two properties here. But is there any case to be made for the claim that a is necessarily self-identical but is not necessarily identical to a? I don’t think so. We can argue very simply as follows:

1) a is necessarily self-identical (Premise)
2) If
is self identical then a is identical to a (Tautology)
3)
Necessarily (If a is self identical then a is identical to a) (From (2))
4)
Necessarily (a is self-identical) (From 1)
5) If
Necessarily (a is self-identical), then Necessarily (a is identical to a) (From (3)
6) Necessarily (a is identical to a) (from (4) and (5))
7) a
is necessarily identical to a (From (6))

Where could one resist that? Lowe definitely agrees with (2) because he is happy with the standard proof of the symmetry of identity, which relies on this (this is a point that Bob Hale has emphasised in his discussion of the necessity of identity). And surely (2) is no contingent truth – how could it fail? (Putting aside worries about the contingent existence of a.) (5) follows from (3) in any normal modal logic. Maybe one could resist the moves from (1) to (4) and (6) to (7) if one played silly-buggers with necessity as a predicate modifier – but that looks a bit desperate. The only other step is modus ponens. So the prospects don’t look bright. If you’re going to resist the standard argument, it shouldn’t be on Lowe’s grounds.