bigideas

“Identity” is one of those ideas that causes trouble almost as soon as you start thinking about it. By “identity” philosophers mean “the relationship something has with itself and nothing else”. You need some concept of identity to be able to say things like “I ate the plumbs you bought”, because I mean “the plums I ate and the plums you bought are identical”.

If you want to speak rigorously, you might think all you need is a suitably rigorous version of this identity relationship. The standard account of identity is based on Leibniz’s Law, which states that x is identical to y if and only if everything that’s true of x is also true of y, and vice versa.

The first thing this suggests is that everything is identical with itself. The second is that if there’s no difference between two things then they’re the same thing (this is sometimes called the principle of “identity of indiscernables”). Both of these are reassuringly close to our intuitions about what “identity” means.

This idea is useful, but under certain circumstances it admits paradoxes, the most obvious of which involve identity across time. Let’s say you saw a dog yesterday, and you see it again today. In the intervening time it lost one of its hairs in what must have been a very minor incident. Now, the dog you saw yesterday had one more hair than the dog you saw today, so they’re obviously not identical. Yet surely they’re the same dog?

It seems that Leibniz’s Law is far too strong to capture the idea of identity across time. What’s worse, there are other identity paradoxes of the same kind that don’t involve time. Think about making a statue out of clay (and nothing else) — are the clay and the statue identical? The linked posting contains various other examples of identity problems). Most troublesome of all may be Church’s Paradox, which suggests that thinking two things are different makes them so even if you’re “wrong” (which, obviously, you can’t be).

Enter “relative identity” (“RI”). Under RI we’re never allowed to make a statement like “x is identical to y“. Why not? because we just don’t allow ourselves that “is identical to” construction that causes all the problems. Instead we say “x is an identical F to y“, where F stands for a category or class of things.

The difference is huge. We can say that the dog you saw yesterday is not the same lump of physical matter as the one you’re seeing today, but that it’s the same dog. The paradoxes of identity seem to melt away. And we can make this new relative identity relation as rigorous as we like; all we need is a bit of set theory. There’s a nice defence of RI, with more examples, here.

But RI is still controversial; many writers feel such a radical a revision of informal logic isn’t warranted. The Stanford Dictionary has a good section about objections to RI near the end of the article. I have to say I find it attractive because it reduces a big, numinous, metaphysical idea to something concrete that we can use. And I think — although I accept this is controversial, too — that that’s one of the things philosophy is for.