I am currently watching the Yale open course "Philosophy of Death" with Shelly Kagan. So far he has made the case that there is not good enough reason to believe in the existence of an immaterial soul. Thus everything is now being argued mostly from a physicalist perspective.
He is now discussing personal identity, specifically the perspective of personality theory of identity which says
Your personality (beliefs, desires, memories etc etc) is what makes you you. Although your personality may change across space-time, the entire continuously connected space-time worm of beliefs, memories, desires, etc etc is your personality and forms your identity.
We are presented with roughly the following science-fiction thought-experiment:
Imagine I die and a scientist is able to copy the information from my brain that composes my personality and then is able to load (implement) my personality onto the brain of some other dead body, John Smith from Iowa. Certainly "I" would awake with my same (space-time worm of) desires, beliefs and memories but in the body John Smith. Based on the personality theory of identity, this person would be me. Now imagine the scientist loads my personality onto another dead body, Kris Kringle from Florida. Are the person in Iowa and the person Florida both me?
We only have three options:
Only one of the two are me.
Both of them are me. Meaning the space-time worm that was just one me has now split into a two me's, one in Iowa and one in Florida.
Neither is me.
As there is no reason to logically accept one as me and the other as not, option 1. cannot be the case.
Option 2. directly follows from the personality theory of identity as stated above. Kagan immediately dismisses this as absurd and therefore appeals to option 3. (requiring a revision of the personality theory by adding a no-splitting clause). However Kagan does not elaborate on this dismissal.
Why dismiss option 2?
Edit for further simplification:
1st premise (soundness aside)
Personality theory of personal identity:
You := your personality (whatever that is)
Your personality is transferable and copiable to different objects in such a way that preserves it.
Your personality has been copied to two different objects A and B.
You are A and you are B. (Option 2 and not option 1 nor option 3)
What is wrong with this conclusion?
Furthermore, what is so repulsive about the idea in general?