Bader against contingent and occasional identity
In a nice little paper, "The Non-Transitivity of the Contingent and Occasional Identity Relations", Ralf Bader argues that if identity is relative to times or worlds, then it becomes non-transitive and thus no longer qualifies as real identity.
Following Gallois, Bader assumes that a proponent of occasional identity must insist that identity statements are always relativised to a time. Now he considers a case where between times t1 and t2, two objects B and D simultaneously undergo fission in such a way that one fission product of B fuses with one fission product of D. Of the three resulting objects A, C and E, one (C) is a fission product of both B and D. Bader argues that at the initial time t1, it is then true that A=C and C=E, but not that A=E. So identity at t1 is not transitive.
According to Bader, the same problem arises in the modal domain. Here we would have two objects B and C at world w1 such that at w2, B has two counterparts A and C, and D has counterparts C and E. Then at w1, A=C and C=E but not A=E.
(I'm not quite sure why Bader uses the fission-plus-fusion setup rather than a simple case of fusion. Suppose between t1 and t2, B and D fuse into a single object C. According to Bader, it should then be true at t1 that B=C and C=D without B=D.)
In any case, I think the argument does not work. The crucial question is how to interpret the name 'C' at t1 (or w1). The name is introduced to denote an object at t2 which is a counterpart of both B and D at t1. Bader assumes that at t1, 'C' picks out both B and D. Since 'A' uniquely picks out B and 'E' picks out D, one can then truly say 'at t1, A=C and C=E'. But a friend of occasional (or contingent) identity need not agree that this is how to evaluate 'C' in the context of 'at t1'.
In fact, Gallois (who seems to be Bader's main target) disagrees. On pp.106--109 of Occasions of Identity, Gallois argues that a proper name should never pick out more than one thing at any given time. He also suggests that in the context of t1, names like 'C' should be interpreted as Russellian definite descriptions, picking out whatever object is uniquely identical, at t1, to C at t2. Since the uniqueness clause is violated, 'C' is then empty at t1. So it is not true that at t1, A=C and C=E. Similarly, Gibbard and Stalnaker would arguably say that there are several individual concepts (or individuating functions) one might associate with the individual C at w2. Relative to one of them, C satisfies A=x at w1; relative to another, C satisfies x=E at w1. But we never get both A=x and x=E.
I am actually sympathetic to the view, assumed by Bader, that 'C' is multiply referring at t1. As I explained in the previous entry, I also think we shouldn't relativise identity statements to times and worlds. So the transitivity of identity can be expressed in the standard, untensed manner:
if A=C and C=E, then A=C.
But what shall we say about this if 'C' is multiply referring, picking out both A and E? In ordinary language, such cases happen all the time, since names are rarely unique. For example, what shall we say about
if London = the capital of England and London = the 15th largest city in Canada, then the capital of England = the 15th largest city in Canada?
The consequent is clearly false. One might argue that the antecedent is also false, since it is false on every resolution of the ambiguity. This would mean that the conditional is true. But I think we should allow for "mixed resolutions" where different occurrences of an ambiguous word are resolved differently. This way, the antecedent can be true. The whole conditional comes out true on some resolutions and false on others. I think truth-on-some-resolutions is a better candidate for truth than truth-on-all-resolutions, so if I had to tick either "true" or "false", I would still say the conditional is true.
The upshot is that even if we follow Bader and assume that terms like 'C' are multiply referring at t1 or w1 -- which Gallois, Gibbard and Stalnaker would arguably reject -- we do not automatically get false instances of the transitivity principle. It depends on the general rules for interpreting statements with multiply referring terms. Even if someone were to suggest that certain instances of the principle are false -- say, because mixed resolutions should be allowed and because truth is truth-on-all-resolutions -- I don't think we can conclude that this person does not mean identity by 'identity'.