## Wednesday, 2 January 2013

### Notes on Identity (and the idea of a 'law of metaphysics' concerning it)

(I) Everything is identical with itself and with no other thing.

Consider what results when we try to express (I) in first-order logic:

(FOL I) (x) x = x ~ (y)(y = x  ~ y = x)

For a closer articulation, let us introduce a function SelfOf() - the identity function – and a distinctness relation D, yielding

(FOL I 2) (x){ x = SelfOf(x) ~ (y)(y = x y D x)}

This shows what we are doing in the case where we use something like (I) to introduce 'identity' and cognates, e.g. making sure it's not taken qualitatively. We are presupposing, using, the notions of self and distinctness/otherness, and showing how identity connects with them; if we had fixed interpretations for 'D' and 'SelfOf', we could use (FOL I 2) to define '='. And we could of course reverse this, presupposing '='. (In the first use, we would naturally emphasise 'self' and 'no other thing', or just 'other', in pronunciation. In the second, we would emphasise the word 'identical'.) We can also presuppose nothing, and take these propositions as just specifying how these notions are to relate to each other.

All these construals can be called grammatical in Wittgenstein's sense. But now, is there some other way of taking (I)? It can certainly look that way in philosophy. ('Law of Metaphysics'.)

It can look as though (I) is ruling out worlds where the identity relation looks like this:

or like this:

and ruling in this sort of picture:

But no one wants to say that these worlds are real, or even possible. So then can't we – mustn't we - take this ruling in and out as grammatical too?

Somehow, instead of seeing it that way, philosophical thinkers try to give it another, impossible sort of application – or so I want to say. But what is this other application, and how am I to describe it without falling prey to the sort of confused thinking I am trying to correct?

What makes these mistakes, or confusions, hard to correct is their slippery, elusive character. (This no doubt has to do with the way so many of our key terms slip and slide so naturally between slightly different uses.)

It is not as if my opponent thinks their Law of Metaphysics is empirical, or anything like that. Rather, they have a spurious category for it, I want to say – or, at the very least, they spuriously categorize it.

They fail to make a category at the right level, so to speak – fail to regard identity statements as being sufficiently special in their fundamental workings. Instead, they are assimilated to other relational statements, giving rise to the conception of a special sort of fact (which we have some kind of special access to, perhaps).

A purported picturing of the world as being one way rather than another. A purported division in some space of representations. Also, perhaps some vague conception of an analogue of empirical verification – as it were, rational perception (cf. Plato, Goedel, the notion of an “eye of Reason”). The sort of thing no one would ever feel like positing for 'Bachelors are unmarried' (except perhaps in a heroic effort to be consistent).

How is it that this comes about here, and not, say, with 'Bachelors are unmarried'? I believe it arises from the mixture of two modes of representation. This should become clear in a moment.

* * *

It is odd the way (I) can inspire phrases like 'Law of Metaphysics' while 'Everything exists' is much less likely to. Of course, for the latter to be false, there would have to be things which do not exist, which is obviously contradictory. OK, but isn't that also true of the following natural description of “what it would take for (I) to be false”?: there would have to be things which do not bear the identity relation to themselves, or distinct objects between which it holds.

It is as though the two conflicting things here are less similar, a bit further apart in language, than in the existence case.

What if I had said 'there would have to be things which aren't themselves, or things which are other things'? That sounds more flatly contradictory; no one, unless doubly perverse, would formulate a “Law of Metaphysics” running 'Things are themselves'. (Butler's Dictum is not generally presented as a Law of Metaphysics.) The flavour changes markedly when we bring in 'identical', talk of 'bearing the identity relation' etc. That is where we begin to get the mixture of modes of representation.

A natural picture to illustrate, or put by, 'Everything is what it is' would be:

Things are represented as dots and are shown “just being themselves”. Whereas the natural picture to put by (I) is:

And here we get the feeling of a contrast, of substance, of something being ruled out. Namely stuff like:

The mixture of two modes of representation here consists in the fact that each dot in the picture is taken to represent a different object, and yet lines are drawn indicating the identity relation – lines which could only be of use if two dots sometimes represented one object.

* * *

With identity statements, merely specifying the relation in question (identity) and the pair of objects involved in the ascription (assuming the names involved refer) fails to specify 'what is said' in any natural sense, however minimal. This problem cannot be avoided either by expelling repetitive identities from language – there can be multiple different non-repetitive identities concerning one and the same object.

If you find this strange or unacceptable, I suggest you have in the back of your mind a conception of relations which identity does not really fall under. Perhaps you are picturing something like the dots above, and imagining relations as encoding further information on top of that structure. This may be a good conception, in which case you should stop classifying identity as a relation, stop classifying identity statements along with propositions like 'John loves Mary'. This, rather than, e.g., moving to an unnatural conception of 'what is said' – which is, I think, what hard-line Millians such as Scott Soames do.

This move in semantics to an unnatural conception of what is said, then, may have an origin related to that of the conception of a Law of Metaphysics discussed critically above. Also, they can be made to support each other: if it's a 'substantial metaphysical fact' that everything is identical to itself, then a repetitive identity is an instance of this, and so perhaps it inherits some metaphysical substantiality for itself, in which case perhaps there is an informative, substantive extensional core to extensionally equivalent identity statements – something which they all say. Conversely, if we take this last thing for granted: what sort of thing does a repetitive identity say which an empirically informative counterpart also says? It had better not be something trivial, since the latter doesn't seem to say anything trivial, in any sense of 'say'. And so what sort of thing is this non-trivial thing which repetitive identities say? Perhaps we could call it 'an instance of a Law of Metaphysics'!