Suppose that (a) memory connections are constitutive of personal identity and (b) fission of memories destroys a person. If one accepts (a), then (b) is very plausible, so (a) is the crucial assumption.
Now consider this case:
- At 4 pm, due to trauma, Sam suffers complete and irreversible amnesia with respect to events between 2 pm and 4 pm.
Then the 5 pm Sam has first-person memories of the 1 pm Sam, and it seems thus that:
- The 5 pm Sam is identical with the 1 pm Sam.
- The 3 pm Sam is identical with the 1 pm Sam.
- The 3 pm Sam is identical with the 5 pm Sam.
- At 2 pm, Sam's memories are copied into a spare brain, call it Bissam, and Bissam immediately time travels forward to 4 pm. (Forward time travel does not seem metaphysically problematic.) At 4 pm, Sam is killed.
So the memory theorist who thinks that fission kills a person should think that total amnesia with respect to a short time period also kills one.
But if that's right, then we don't survive those nights where we do not remember our dreams upon waking up. For the dreaming person has memories (skill memories at least; but also temporarily inaccessible episodic memories) of the person who went to bed. But the waking person doesn't have memories of the dreaming person, though she does have memories of the person who went to bed. So the person who went to bed fissions into the person who dreams and the person who wakes up.
This means that the memory theorist shouldn't think that fission kills. (Another standard argument for this conclusion: If fission kills and identity is constituted by memory, then you can be killed by having your brain scanned and the data put into another brain; but you can't be killed by a process that doesn't affect your body.) But if fission doesn't kill, then it seems that the best view is that in cases of fission there have always been two persons. And that leads to various absurdities, too.