An Identity Theory of Identity
The identity theory I have in mind says simply this: identity is identical to indiscernibility. That is, the identity relation reduces to the indiscernibility relation. Why would anyone endorse this theory? First, there is a very clear connection between identity and indiscernibility, enshrined in Leibniz’s Law: x is identical to y if and only if x and yare indiscernible, i.e. have all properties in common. Second, if the two are not identical (identity and indiscernibility), then we cannot establish identity by ascertaining indiscernibility, since the identity relation transcends the indiscernibility relation. We could simply assert an entailment from the latter to the former, but this has the look of a stipulation absent any recognition of identity. Thus there is a danger that identities will turn out to be unknowable if not reducible to indiscernibility. Third, identity would be a sort of metaphysical dangler if not reducible to indiscernibility: it would stand apart from indiscernibility in a weird and gratuitous way—better then to trim it back to indiscernibility using Occam’s razor. Reality contains no identity relation over and above the indiscernibility relation. So there are reasons to hope that identity is in fact identical to indiscernibility.
But there is a well-known obstacle: while the indiscernibility of identical things seems self-evident, the identity of indiscernible things seems not to be. The classic example is qualitatively identical spheres at different points in a symmetrical universe. The example seems exceptional and contrived—we don’t normally encounter such potential counterexamples to Leibniz’s law—and not surprisingly there are counter-replies. First, we could simply stipulate that identity-with-x should be counted among the properties of x, in which case the qualitative counterpart y will not have this property (having instead the property of being identical-with-y). Second, we can include spatial location among the properties of each sphere, so that the two are not spatially indiscernible. Third, we could bite the bullet and declare the two identical, dismissing the thought experiment as fantasy: for if nothing distinguishes them they cannot be distinct. It is important here to denude the concept of discernibility of any epistemic connotation: the point is not that our inability to discern the difference between two objects establishes that they are identical; it is that the objective fact of indiscernibility, i.e. complete sharing of properties, entails identity (because it is identity, according to the identity theory of identity). Thus we can reply to the standard objection to the right-to-left reading of Leibniz’s Law: there is no proof that indiscernibility fails to entail identity.
The theory might be compared with a similar identity theory of numbers, viz. that numbers are sets. Once we have sets, it may be said, we don’t need a separate ontological realm of numbers—we can shave off the redundant ontology. Whether that is a good argument is not to the point; the point is that the analogous identity theory of identity can claim that there is no ground to distinguish identity from indiscernibility once we have a suitably relaxed notion of indiscernibility to work with. Identity just is complete objective indiscernibility, neither more nor less. It adds nothing to the basic fact of absolute and total coincidence of properties. It is possible to be an anti-reductionist about identity, holding it to be a separate relation altogether, but one can appreciate the position of someone who can’t stomach that kind of metaphysical multiplication. The reduction is not empirical, to be sure, and it qualifies as knowable a priori, and it may even be analytic: but it is informative in some way, and hence rationally disputable—it is not an empty tautology. It is piece of substantive metaphysics. An interesting feature of it is that it applies to itself: it says that identity is indiscernible from indiscernibility. For that is what identity is, according to the theory; therefore the identity of identity with indiscernibility is the same as the indiscernibility of identity and indiscernibility (and that “same” is also equivalent to indiscernibility). The word “identical” or “same” always expresses indiscernibility. Thus Leibniz’s Law is not just a stipulated biconditional in need of a rationale; it is tantamount to the proposition that identity is identical to indiscernibility. Leibniz has discovered a true identity statement connecting two expressions of the language: “identical” and “indiscernible” both denote the same relation, variously described as identity or indiscernibility. The case is no doubt special, but in broad outline it is an instance of a Fregean informative identity statement: one denotation with two names for it. Ontologically, the world contains indiscernibility facts, and these facts constitute identity facts. Logically, the case is just like the identity theory of mind and brain: the world contains brain facts, and these facts (allegedly) constitute mental facts. There is something pleasing in the result that identity itself might be subject to reduction via identity (i.e. indiscernibility).