Restricted Identity II: Analyses
Now restricted identities threaten to violate Leibniz's Law: If R1 is identical with R2, then how can they differ in their courses? If AD1 is AD2, how can they differ in their history? If A1 is A2, how can they differ in their modal properties?
They can't. So either R1 and R2 (and AD1 and AD2, and A1 and A2) are not really identical, or the don't really differ. Let's look at the first option first. It says that R1 and R2 are not really identical. Hence "R1 = R2" is false, even though
1) At Cologne, R1 = R2
is true. How does the modifier "At Cologne" produce a truth when applied to a falsehood? Again, I see two possibilities.
a) "At X" restricts the denotation of terms in its scope. In (1), "R1" denotes the part of R1 that is located at Cologne; "R2" denotes the part of R2 located at Cologne, and those parts are indeed identical.
b) "At X" alters the relation expressed by "=". In (1), "=" expresses a relation that holds between Y and Z iff the parts of Y and Z located at Cologne are identical. (On this view, "=" in "At present, AD1 = AD2" expresses what Lewis calls tensed identity ("Survival and Identity").)
(Note by the way that "At X" cannot be interpreted as an intensional operator on propositions. At least I do not see how.)
In my formulation, (a) and (b) both presupppose that the relevant things have parts at the spatial, temporal oder modal location indicated by the modifier. But at least in the case of "At present, AD1 = AD2", and "At our world, A1 = A2", this is controversial. I don't think (a) allows a more neutral formulation, but (b) does. Thus:
b*) Inside the scope of "At X", "=" expresses the relation that holds between Y and Z iff Y is identical-at-X with Z
where we leave open what it is to be identical-at-X. It might mean to have appropriate parts that are identical simpliciter, or it might mean to stand in four-place relation to identity simpliciter and a place/time/world, or to instantiate-at-a-place/time/world identity simpliciter, etc. What we cannot allow is that "identical-at-X" expresses identity simpliciter, because we are currently trying to spell out the first option mentioned in the beginning, which is based on the assumption that R1 is not really identical with R1.
At least in the modal case, this might seem like an unnatural move, so we might try the other option instead, which is to grant that the relevant things are identical but to deny that they differ in any properties.
More precisely, we have to say that a river's course (or its historical course, or its shortest possible length) is not a property in the sense of Leibniz's Law. That is, statements like
R1 has such and such a courseare not referentially transparent with respect to "R1".
For modal contexts, this is precisely the view Lewis defends in "Counterparts of Persons and their Bodies". He argues that even though A1 = A2,
A2 is essentially longer than 1m
is true but
A1 is essentially longer than 1m
is false, because these sentences are not referentially transparent with respect to their subject position.
I don't know Ted Sider's position on this issue, but I believe he should say that
AD1 went to the Caspian in 1500
is not referentially transparent with respect to "AD1". That's because he holds that "AD1" refers to a temporal stage of AD1, and I guess this stage is also a temporal stage of AD2. But that last assumption could of course be rejected. Indeed, even though regarding one of the three contexts as opaque goes naturally with a corresponding view about the spatio-temporal-modal extension of ordinary things, these views are quite independent. For instance, Arthur Schopenhauer claimed that deep down in noumenal reality, all things are identical. Hence all terms denote the same thing (the Welt als Wille, which is not the mereological fusion of all ordinary things) and virtually every context is referentially opaque.