Let's suppose that personal identity over time is secured by continuation of psychological states. Now imagine Jim and Sally are robots who are persons (if you don't think robots could be persons, just suspend disbelief for a while until I get back to the issue) and have almost all of their psychological states on their hard drives. According to the psychological theory, if you swap Jim and Sally's hard drives, Jim and Sally will go with the hard drives, rather than with the rest of their hardware. But here is something odd. When you unplug Jim and Sally's hard drives during the swap, either Jim and Sally continue existing or they don't. If they do continue existing, then by the psychological theory, they are surely located where the hard drives are, since that's where the memories are. They are basically reduced to hard drives.
There is a case to be made that they do continue existing, at least given the psychological theory of personal identity. First: To kill an innocent person, even temporarily (after all, many people, including me, believe that all our deaths are temporary!), is seriously wrong. But swapping hard drives doesn't seem problematic in this way. Second: There is some reason to think temporally gappy existence is impossible, and if gappy existence is impossible, then if Jim and Sally exist before and after the swap, they exist during it. Third (and specifically to the psychological theory): It is plausible that if the identity of a person across time is secured by a part of the person, then the person can exist reduced to that part. Thus, if the identity of a person comes from the soul, then the person can survive reduced to a soul.
So we have this: Given the psychological theory, Jim and Sally exist reduced to hard drives. But that's absurd! For we can replace hard drives by cruder mechanisms. We can suppose a computer where memory is constituted by writing in a large book. It is absurd to think a person can exist reduced to a book. So we should reject the psychological theory.
Well, that assumed that robots could be persons. Maybe they can't. And our memories do not sit on a convenient isolated piece of hardware in the brain. Indeed, that is true. But surely agents could have evolved whose memories are stored on a convenient isolated piece of hardware, and such agents could be persons. And the argument could be run for them.