Necessarily, for any x and any y, x is identical to y only if for any P, x has P if and only if y has P. (Compare, e.g. Colin McGinn, Logical Properties (New York: Oxford University Press, 2000), 4-7.
I agree with other philosophers that this has apparent counterexamples, if “P” can stand either for any predicate or for any property.
But in my view, the underlying metaphysical intuition – that a thing can’t at one time be and not be a certain way – is undeniable. (And of course, it has important theological implications.)
I would restrict the “P” above to intrinsic properties, if I believed in properties. But I don’t. So I’ve been putting the principle like this:
The Indiscernibility of Identicals: Necessarily, any A and B are identical only if they (1) never have differed, (2) don’t differ, (3) will not ever differ, and (4) could not differ.
This formula doesn’t import any assumptions about property-theory, either for or against. Rather, it uses only a primitive concept of differing, or being different – as to qualitative aspect or way of being, not as to number. I think this well captures the intuition and fundamental conviction noted above.
But it now strikes me that the formula is needlessly complicated. Why not just this?
Necessarily, any A and B are identical only if they could not simultaneously differ.
As with any version of Leibniz’s Law, this is supposed to give us a necessary condition for identity, and a sufficient condition for non-identity. If A and B really are numerically the same, they of course “they” (really: it) can’t at any time differ from how it is at that time. There’s the necessary condition. How about the sufficient condition? If A and B have, do, or will differ, then of course they could differ. Actuality implies possibility. So actually differing at some time is sufficient for non-identity. But of course, merely possibly differing at a time – being such that they could simultaneously differ – is enough to prove non-identity too. This is just what the prior formulation said, then, in fewer words.
To spell it all out, the claim is: Necessarily, for any x and any y, x and y are identical only if it is not possible that there is a z such that z is a time or a point in timeless eternity at which x and y differ.
Philosopher-friends: what say you?
Update, in response to Anthony’s perceptive comment – I meant this:
Necessarily, any A and B are identical only if they could not simultaneously or timelessly differ.