It is easy to define identity at a particular moment, if two things have the same properties then they are identical, and really one thing. Defining identity over time, what it means for two objects at different periods of time to be the same object, is a bit more difficult. If two objects at different times are the same we can say that IOT(a, b) holds, meaning that a and b are the same object at different moments in time (this is simply a formalization, not a new claim). It is generally assumed that the IOT relation has the following properties: IOT(a, b) -> IOT(b, a) (reflexivity), IOT(a, b) & IOT(b, c) -> IOT(a, c) (transitivity), and IOT(a, b) -> a and b are identical (in the standard sense) or they don’t exist at the same time (uniqueness in time). However, no matter how you define the IOT relation in terms of properties of the world, the problem of branching arises, where IOT(a, b) and IOT(a, c) hold, and b and c exist at the same time (for example, when an amoeba divides into two, duplicates created by a matter replicator, ect). Given the fact that the IOT relation is reflexive, transitive, and unique in time this is a contradiction, meaning that there is no IOT relation, and hence no identity over time. So, can the IOT relation, and thus identity over time, be fixed to address the problems created by branching?
The “obvious” way to avoid the problem of branching would be to argue that after a branching event that produces both b and c such that IOT(a, b) and IOT(a, c) either: a) branching invalidates the IOT relation, it is not the case that IOT(a, b) and IOT(a, c), even though they meet all the other criteria for participating in the IOT relation, b) after a branching event it is true that IOT(a, b) and IOT(a, c), but it is not the case that either IOT(b, a) and IOT (c, a), a denial of reflexivity, or c) that after a branching event IOT(b, c) doesn’t follow from IOT(a, b) and IOT(a, c), a denial of transitivity. We can label these three responses as “no branching conditions”, since they state that after a branching event the rules for identity over time change. Even though there are essentially three ways to define a no branching condition they all lead to another problem for identity over time, which is that if identity over time is defined with a no branching condition then it is impossible to tell given two object at two different moments if those objects participate in an IOT relation. This in turn means that the IOT relation, thus defined, is epiphenomenal, it has no effect on the causal powers of the objects. And if it is epiphenomenal it can never be an explanation of anything non-epiphenomenal, and anything else that is defined in terms of it must be epiphenomenal as well. I admit, then, that adding a no branching condition is consistent, but it renders identity over time useless, meaning that we could do without it equally well, and for reasons of ontological simplicity probably should. Thus adding a no branching condition is a very poor defense of identity over time indeed.
There is another possibility for fixing the IOT relation, which is to discard the requirement that at any given moment there can’t be two objects that participate in the IOT relation with each other (rejecting the uniqueness in time requirement). Again, such revisions allow the IOT relation to avoid contradictions when branching occurs, but the IOT relation, so defined, seems less like one of identity, and more like a relation that picks out objects with a shared history. Allowing there to be more than one of the same individual at a time would require us to revise most theories about obligations, relationships, ect. So while the IOT relation, so revised, isn’t epiphenomenal, it doesn’t seem to be the kind of relation that can play the role that identity over time is supposed to.
So where does this leave us? We have exhausted the options for preserving identity over time as a simple relation between objects which either holds or doesn’t. Of course, we can define the IOT relation as a function that maps pairs of objects to values, basically a measure of their similarity (or some other appropriate notion). This relation can play the role that we require of identity over time (for example, defining what it means to be a certain person), but to be honest it isn’t really a kind of relation that can be properly called identity, since at best objects at different times are simply very similar to each other. Thus it is my claim that there is no such thing as identity over time if we require it to be some form of identity (as the name seems to imply).
Of course it is impossible to deny that we act as though there were such a thing as identity over time. I naively think of my computer now as the same computer that I had yesterday. If there is no such thing as identity over time where does this idea come from? It seems likely that it is an abstraction we develop to explain why objects at one point in time are very similar to objects at subsequent times, with allowances made for some degree of change. Obviously this idea isn’t robust enough to deal with branching cases, but in our everyday, macroscopic, life branching cases happen very rarely. We might be tempted then to base our philosophical theories upon this naïve understanding of identity over time, since it seems to work well enough in everyday life. We really shouldn’t though, since as I have pointed out there really is no such thing, and our naïve conception of it is simply a rough, and sometime inaccurate description of the way things work, and thus philosophical theories based upon it will be equally rough, equally inaccurate, and equally unable to deal with branching cases.