How does Parfit's "brain division" thought experiment refute the brain theory of personal identity?

Question

Shoemaker proposes that identical persons are individuals who are psychologically continuous and share the same brain. This is the brain theory of personal identity. However, Parfit takes objection to this with his brain-division thought experiment. Suppose you severed person A's corpus callosum, and put one hemisphere into person B, and another hemisphere into person C. What, then is the identity relationship between A, B and C?

This can be thought to be a valid objection to brain theory because:

1. The identity of person A cannot be definitely associated with person B or C
2. Person A is psychologically continuous with B and C,
3. A, B, and C all share the same brain.

Ok, maybe I'm misunderstanding Parfit's thought experiment, but I'm having trouble rectifying premise 3 (that A B and C share the same brain). If you sever the corpus callosum, and put fragments of A's brain into other bodies, how could you possibly claim that those fragments are numerically identical to A's original, unified brain?

Of course I understand it's a thought experiment, and that this doesn't have to be practically possible, but really, I'm finding this a bit of a stretch--which I suspect is because I'm misunderstanding the thought experiment.

Answer 1

It's an incredibly unhelpful thought experiment because it is far from clear that someone with a severed corpus callosum is actually "one person" in the cognitive sense that we normally mean. (Also, nobody has actually severed all the basal connections. Also, there is no reason to believe two half-brains of different people would be compatible in any meaningful sense.)

So the premise is to do something that may be impossible (not just practically but logically, given how the individuality of brains are implemented) and lands us in a philosophically perplexing state at the best of times. Needless to say, this doesn't bring clarity to anything, nor should it be used to tell us anything about identity. (Even if it did, it doesn't tell us anything that the Ship of Theseus does not.)