Identity

Identity is the relation everything bears to itself and to nothing else. It looks too simple to be a topic — and yet nearly every puzzle in this section is generated by it, because the innocuous-seeming principles governing identity have consequences that collide with equally innocuous claims elsewhere.

Numerical and qualitative

Two coins off the same press are qualitatively identical — exactly alike — and numerically distinct: they are two. Only numerical identity is identity strictly so called; qualitative identity is exact similarity, which is a different relation and one the problem of universals is largely about.

English encourages the confusion ("they have the same car" is ambiguous), and a surprising number of apparent paradoxes trade on it.

Leibniz's Law

The governing principle, in the direction almost nobody disputes:

$$x=y\to \forall F(Fx\leftrightarrow Fy)$$

The indiscernibility of identicals: if $x$ and $y$ are the same thing, then whatever is true of $x$ is true of $y$, because there is only one thing for the predications to be about. The converse — the identity of indiscernibles — is a substantial and contested thesis, and running the two together as "Leibniz's Law" is a standing source of confusion.

The law's importance to ontology is that it is the standard tool for proving things distinct. The whole statue-and-clay problem is an application: the statue and the lump differ in a modal property, therefore by Leibniz's Law they are two, therefore coincident objects exist.

Apparent counterexamples all involve intensional contexts. Giorgione was so called because of his size; Barbarelli was Giorgione; but Barbarelli was not so called because of his size. Lois Lane believes Superman can fly and does not believe Clark Kent can. These do not refute the law. The predicates involved — is so called, is believed by Lois to fly — are not straightforwardly about the object; the contexts are referentially opaque, and substitution fails because the term's position is not purely referential. Diagnosing which contexts are opaque is a task for the philosophy of language, not a repair to the metaphysics.

The necessity of identity

A short argument with large consequences, due to Barcan Marcus and made famous by Kripke. Assume Leibniz's Law and the triviality that everything is necessarily self-identical:

$$\square (x=x)$$

$$x=y\to (\square (x=x)\to \square (x=y))$$

Substituting $y$ for $x$ inside the modal context using Leibniz's Law yields:

$$x=y\to \square (x=y)$$

All identities are necessary. If Hesperus is Phosphorus, it is necessarily so; if water is ${\mathrm{H}}_2\mathrm{O}$, necessarily so.

This is counterintuitive because such identities are discovered empirically and might, for all we knew, have turned out otherwise. Kripke's resolution is to separate the metaphysical from the epistemic: the statement is necessary but a posteriori. What could have turned out otherwise is not the identity but our evidence — there is a possible world in which the evening star is not the morning star, but in it the object we would be pointing at is not Venus.

The argument's force depends on names being rigid designators, picking out the same object in every world. Definite descriptions are not rigid, which is why "the number of planets is 8" is contingent while "8 is 8" is not.

Contingent and relative identity

Two attempts to escape the consequences, both employed against the constitution puzzles.

Contingent identity (Gibbard): the statue is the lump, but only contingently — they could have been distinct. This requires denying the argument above, typically by denying that the modal predicate expresses a genuine property, or by treating modal claims as relative to how the object is described. The cost is that objects no longer have their modal properties absolutely.

Relative identity (Geach): there is no relation of identity simpliciter, only identity-relative-to-a-sortal. Two things can be the same lump without being the same statue, and there is no further question of whether they are the same thing. This dissolves the constitution puzzle at a stroke and requires rebuilding logic, since standard first-order logic with identity assumes an absolute relation. Few follow him, but the position is hard to refute outright rather than merely resist.

Vague identity

Can it be indeterminate whether $a=b$? The ordinary cases suggest yes: as a cloud disperses, or as parts are replaced, there seem to be moments where no answer is right.

Evans' argument says no. Suppose it is indeterminate whether $a=b$. Then $b$ has the property being such that it is indeterminate whether $a$ is identical with it. But $a$ lacks that property, since it is determinate that $a=a$. By Leibniz's Law, $a\ne b$ — and determinately so, contradicting the supposition.

The argument is three lines long and has generated a large literature. The disputed steps are whether the indeterminacy operator can figure in a genuine property, and whether the final contraposition is valid in a logic with indeterminate sentences. Its significance for this section is via the argument from vagueness: if there cannot be vague identity, and if indeterminate composition would require it, then composition is never indeterminate — which forces the choice between universalism and nihilism.

Identity over time

Persistence — what makes the person now the same as the child then — is a further question, and one this section deliberately does not treat, because it has a home in the persistence page of the space-and-time section. The two options, endurance (wholly present at each moment) and perdurance (having temporal parts), bear directly on the constitution puzzles, and the perdurantist solution to those puzzles is only as good as the general case for temporal parts.

What belongs here is the observation that "identity over time" is not, strictly, a further kind of identity. Either the thing now and the thing then are numerically the same thing, in which case ordinary identity applies, or they are different things standing in some other relation — and the perdurantist takes the second option, replacing identity with a relation among temporal parts.

Where identity sits

Identity is not itself an ontological thesis but the logical apparatus every thesis is stated with, which is why it belongs on the category question side of the ledger: criteria of identity are what distinguish genuine categories from arbitrary groupings, and "no entity without identity" is the standard test for admitting something to the inventory.

The next page turns from what makes a thing that thing to what makes it the kind of thing it is: essence and accident.