Identity Is Not A Relation

In the Tractatus Logico-Philosophicus, Wittgenstein makes many sweeping, cryptic claims, sometimes quite philosophically radical, with little to no argument or explanation. These are some of the most frustrating parts of the book, but also the funniest. (“Logic must take care of itself.” “Superstition is the belief in the causal nexus.”) One of these claims is as follows:

That identity is not a relation between objects is self-evident.

For as long as I can remember, that is, for as long as I have been thinking about these sorts of issues, I have always agreed with Wittgenstein on this point. I agreed with him when I first read the Tractatus many years ago, and I agree with him now having just read it again. Identity is not a relation between entities (an entity and itself).

What is going on here may be unclear to the metaphysically uninitiated, so let me explain. It is philosophical commonplace to say that every entity is related to itself by a relation of identity. (I use ‘entity’ in a neutral way to refer to any particular thing, object, property, event etc. of any kind.) Further, it is often thought that this is something that we can be as sure about as anything. This may seem like an odd claim, but may make more sense when you see how it can be written:

A=A

Or put in other, semantically equivalent ways: A is A, A equals A, what is A is A, whatever A is, A is too, A is identical with A. What could be more obviously true than this? Plug in whatever entity you want for A – Rowan, my desk, my mug, the pattern on my mug, God, Pegasus – and you have a series of truisms like Rowan=Rowan, my desk is my desk, and what is God is God. These surely cannot be denied. Here is David Lewis from On The Plurality of Worlds:

…we should not suppose that we have here any problem about identity. We never have. Identity is utterly simple and unproblematic. Everything is identical to itself; nothing is ever identical to anything except itself. There is never any problem about what makes something identical to itself; nothing can ever fail to be (1986, p. 192).

My problem with this philosophical commonplace is not with the truth or obviousness of the claim that A=A, or cognates of this claim. Obviously statements like “Rowan=Rowan” are obvious and true. My problem (and, I think, Wittgenstein’s problem) is with the metaphysical interpretation of the claim. As I said above, it is philosophical commonplace to say not only that A=A is true, and obviously so, but also that it expresses a relation of identity, a relation between an entity and itself, a relationship that an entity stands in to itself.

Here are some unproblematic examples of claims that expresses relationships between entities: “A is taller than B”, “A loves B”, “A is the parent of B”, “B is the child of A.” The relationships expressed are “taller than”, “loves”, “is the parent of”, and “is the child of”, and the entities that stand in these relationships are A and B. Claims that express such relations are true when the entities actually stand in that relation. So, if A really is taller than B, if A really stands in the relation of being taller than B, then the claim that “A is taller than B” is true. If not, if A is not taller than B, and therefore does not stand in the relation of being taller than B, then it is false.

There are two things that can, in general, be said about relations that I take to be truisms. (This is not to say that they are not controversial; truisms often are in metaphysics.) First, the relata of a relation – the things which it relates – must exist for the relation to exist. In order for A to stand in the relation R to B, A and B must actually be there for each other to be related to each other. If they were not out there to be related, then there can be no relation between them. Second, for a relation to exist, it must be something over and above the relata of the relation. If A could exist and B could exist without therefore standing in some relation R, then R, if it exists, must be something over and above A and B when they do not stand in that relation. Even if the existence of A and B guarantees that they stand in the relation R, then R, if it exists, must be something over and above A and B. Otherwise the relation expresses nothing. There would not really be A and B related in some way R, there would just be A and B, and their existence does not guarantee the existence of R, after all.

Do these two truisms hold of the putative relation of identity between an entity and itself, supposing there is such a relation? Suppose that A stands in the relation R to B, where A designates Rowan, R designates the relation of identity, and B designates Rowan. Call this supposition ARB. The relata of this relation are Rowan and Rowan, the very same entity. So, assuming I exist, then the relata of the relation ARB – Rowan and Rowan – exist, even though Rowan and Rowan are the same entity. The first truism seems, therefore, to hold. Is the relation something over and above the relata? Suppose that the relation R in ARB is not something over and above the relata. If this were the case, there would not really be A (Rowan) and B (Rowan) related in some way R, there would just be A and B. The relation R in ARB must therefore be something over and above the relata. And there is no principled reason this cannot be the case. The second truism seems to hold as well. This shows that there is no conceptual barrier to an entity standing in a relation of identity to itself; the identity relation is not self-contradictory.

What I want to argue is that, while we can make sense of the identity relation, there is no good argument for its existence. In the following I will assume that if we do not have to postulate an entity for some explanatory reason (i.e. to explain some apparent phenomenon or other), then we should not do so. If we do not assume this standard, then metaphysics becomes impossible, as any entity for any reason whatsoever can be postulated. (I take it that this is the import of Ockham’s razor.) Given this, our default position should be: we should not think that the identity relation exists between an entity and itself unless there is some explanatory reason to do so, i.e. unless it explains some phenomenon or other. (This is the least plausible place to quibble with what I have to say.)

What conceivable reason could there be to think that the identity relation explains some phenomenon or other? Consider the sorts of situations where we would posit a relation to explain some phenomenon or other. Generally, you have two entities, A and B, and there is something going on between them. I mean this in the broadest possible and utterly ordinary sense. A may love, be the parent of, be the child of, be to the left of, be to the right of, be above, be below, be to the past of, be to the future of, or be thinking of B. In these cases, we want to explain each situation by saying that A stands in a specific relation (whatever it is in each case) to B. It seems that something can even be going on, in the above sense, between A and B, even where A and B are the same entity. For example, suppose that I love myself. “Myself” and “I” both designate Rowan, so the phenomena to be explained is that Rowan loves Rowan. In this case, we would say that there is a relation (the relation of love) between I and myself. So far so good.

(Or perhaps not. I suspect that the entity designated by the word “Rowan”, Rowan, would already include whatever constitutes my loving myself, so it is unclear what the putative relation is supposed to add. Given this, I am tentatively sceptical even of relations between an entity and itself for basically the same reasons as those set out below. However, I put aside this thought here.)

The question now is: can the relation of identity be something that is “going on” – again, in this broadest sense – between an entity and itself. We can make sense of the idea that something is related to something else or itself when the relation in question characterises that going on. We know what it is to love (or so we hope we do), and so when we say that A loves B, whether B is other than A or just is A, we can understand what the relation of loving amounts to, we have an idea of the sorts of things we may observe when we observe A and B. Furthermore, the loving relation is something more than merely A and B (otherwise, as above, it amounts to nothing). You could have A and B without A loving B. So, the loving in question is something that is going on between them. But what is going between A and A when A stands in the relation of identity to A?

It is not clear to me that we know what it is for something to stand in the relation of identity to itself, that is, it is not clear how this characterises the goings on of an entity. Nor is it clear to me that it adds to our understanding of A, our understanding of what it is and the things going on between it and itself. And it is not clear to me what it could possibly be, if it existed, over and above A itself. After all, it necessarily rises and falls with A. If A exists, then the relation of identity it stands in to itself does. If A does not exist, then the relation does not either. And there are no other possibilities. This is not dispositive, but it suggests not just the relation of identity is nothing more than A (i.e. that it is identical with A), but that the relation does not exist, as it in no way characterises A in a way not already characterised by itself. There is just A. This suggests that, following Wittgenstein, “to say of one thing that it is identical with itself says nothing at all.” But if saying that something stands in the relation of identity to itself says nothing at all, then it cannot do any explanatory work, and we have no possible reason to say that it exists. And if we have no possible reason to say that something exists, then our best bet is that it does not. Identity is not a relation.

What, then, should we say identity is, if it is not a relation between an entity and itself? We certainly speak of identity, and it certainly seems to have some utility, sometimes. But if identity is not a relation, then what is it? Here is a proposal. To say that A and B are identical (or that A=B, or that A is B, etc.) is just to say that “A” and “B” refer to the (exact) same entity. It is informative to say that “A is identical to B” only when those to whom such a claim is addressed are not aware “A” and “B” refer to the (exact) same entity. Otherwise, to those who already know this, it says nothing at all.