  By way of explanation, consider the case Parfit uses in support of  his claim that identity is not what matters: fission (Ibid., 254–255). Suppose both of my brain hemispheres are functional  duplicates of the other, and that each of my other two triplet  brothers has suffered irreversible brain damage. A brilliant  neurosurgeon can transplant one of my brain hemispheres into each  brother, and so each survivor (we will stipulate) will be fully  psychologically continuous with me upon waking up. What has happened  to me? If we lack the “no branching” clause, we are  forced to say that, because both brothers are psychologically  continuous with me, they are both me. But then (given the  transitivity of identity) both survivors would also have to be  identical to each other, which seems obviously false  (although see Belzer 2005 for doubts about this assertion). So to  avoid violating this transitivity requirement, we simply have to  stipulate in our criterion of personal identity that, if the  relations in which identity consists may hold one-many, they must  obtain uniquely for identity itself to obtain.