I can imagine splitting my brain in two and placing the individual hemispheres in individual bodies. This affects the psychological continuity view of personal identity because then I would not be numerically identifiable to either of the individual bodies.
There is a distinction between what is physically possible and what is logically possible. I appreciate the significance of the thought experiment if it is physically possible for me to split my brain in two. And I appreciate that when some people say that brain splitting is logically possible, they may just mean we have not ruled it out as a physical possibility. Therefore, it is logically possible that sometime in the future we may have discovered that it is physically possible.
However, if we discovered (somehow) that it was not physically possible to split the brain, then would the thought experiment would still be relevant? It seems implied that the mere logical possibility of brain splitting is enough for it to be relevant. If this is the case can you explain why it is relevant to me?