"Wherein does personal identity consist, then? [...]
My overall conclusion then, is that persons in all probability to comprise in what I would call a basic sort, for which no adequate criterion of identity can be formulated"
PP. 135-6 EG Lowe - Kinds of Being, A Study in Individuation, Identity and the Logic of Sortal Terms
Personally I adore this question; it's what draws me into Philosophy at the moment. How do we identify anything that changes over time?
What technique we usually use for identity is Leibniz's Identity of Indiscernibles, which says that if A is identical to B then any property of A must** also be a property of B, and vice-versa. It makes a lot of sense; if Object A is orange, then we'd expect that if Object B was identical, Object B would also be orange.
So what happens when I do take away a plank of wood from Theseus' ship? Does the ship lose the property of having _that_ plank of wood and so stop being Theseus' ship he sailed in on?
Loux's reply is my personal favourite. Loux claims that the moment the piece of wood was removed from the ship; it stopped being a property of that ship. If you took away all the bits of wood from the ship, there would no longer be** a ship to take objects off, but if you put a new piece of wood onto the ship, it would be adopted into the ship as a part of the ship. It solves the problem nicely, but like all problems in Philosophy, it doesn't matter if you have a solution, it matters if the solution can be applied to learn something.
The issue is time, or Diachronic[Dia=Through / Chronos=Time] Identity if you want the technical term. If time had stopped, a river could be compared to be identical to another river and we would have no issue. The problem has also been stated as Heraclitus' "You cannot step in the same river twice"
Quine answers this question in From a Logical Point of View, he says that we should consider time in slices, like 2D parts of a 3D object. So when you take time slice A of a river, you can see all the water molecules that are in it at that time, and if you drive, faster than the river flows, down to a further part of the river and take time slice B, you could see all the same molecules of water in the river again. Would that be sufficient to step in the same river twice? Well that depends how we identify the river, so Quine looks at how we teach someone to identify a river. We point to many spots of say, the Tiber, and say “That’s the Tiber”, and we do it again later, at another point in the Tiber, until the person understands all the points in that river are the same river. We accept the flow of water, the process of water flowing, as what makes up that river, not what’s in the water or when we looked at it. Heraclitus’ example plays on the fact that river is a name for a temporal group of objects, not a concrete one, and so we become confused. We can** step in the same river twice, and should not worry ourselves over that.
But how does this apply to humans?
Let’s take Henry, an example from Loux.
"Henry acquires a tan. We would say that a property of Henry is the fact that he is tanned. In winter, Henry loses his tan. Henry-in-summer does not have the same property, as Henry-in-winter does"
I'm gonna borrow heavily** from Loux's Metaphysics here, using the 2nd Edition if anyone desperately wants to check my sources.
So what do we do? Obviously Leibniz's Law says they are NOT identical, but we find this very distasteful. I'd want to say I was identical to me before I went on holiday. The reason we find these views to conflicting is that we are usually acting under an Endurantist viewpoint; our parts are persisting three-dimensional individuals wholly present at every moment of their existence
An Endurantist would reply that we should describe Henry according to him at certain times. This is Henry at 13:47 Friday 20th Feb 2012 when he had a tan, so he is still identical to Henry, but not by time-specific measurements. A Perdurantist would reply to this that this measure is too steep; we wouldn’t want to have to do that to describe identity, and so the answer is one we can accept, but only if there isn’t a better one. The Perdurantist thinks there definitely is a better example; we are time segments of a single identity. We should not worry about whether we lose something, or gain something in our spatial being, only that we are linked over time to our spatial being before that.
So you would be a Perdurantist; you believe that we persist because of our change over time, not that we are one concrete entity.
You might ask who could believe the opposite. Well, anyone who believes in a soul, or anything that’s non-physical could easily conjure up a few examples of a concrete entity that happens to exist within a physical manifestation. They would say that you are not your physical entity, because you can lose your arm and not seem to be a different person, and they’d be right in that regard. To answer your question of what actually is identity, I’d refer you to the quote at the beginning; there isn’t a way to define a person’s identity. I believe this is because our descriptive terms don’t play nicely with time, but that’s my own opinion. Some people would believe that if you have enough matching properties, or matching essential properties, then you’re identical, but I feel this is too weak and likely to falter.
I swear I write all these things at 2 or 3am. I apologise for any mistakes, and I’ll clear this up tomorrow to make it more legible, but I hope that it might give you a better understanding of some of the viewpoints floating around at the moment.