The thing with logical or philosophical arguments for reincarnation is that they either cannot be proven empirically or rely too much on semantics. The minimalist reincarnation hypothesis is, AFAIK, the only falsifiable reincarnation hypothesis.
There is something essential to some human personalities, however we
ultimately characterize it, which we cannot plausibly construe solely
in terms of either brain states, or properties of brain states, or
biological properties caused by the brain and, further, after
biological death this non-reducible essential trait sometimes persists
for some time, in some way, in some place, and for some reason or
other, existing independently of the person's former brain and body.
Moreover, after some time, some of these irreducible essential traits
of human personality, for some reason or other, and by some mechanism
or other, come to reside in other human bodies either some time during
the gestation period, at birth, or shortly after birth.
Though vague-souding, it doesn't commit the common errors of other hypotheses, i.e. it focuses explicitly on humans, it doesn't assert that all humans reincarnate and it cares not how they would reincarnate.
Ian Stevenson (and to some extent, his protégé Jim B. Tucker) was the most prolific researcher who collected data for this hypothesis. The man had sound methodology and had been to all continents of the world for more than four decades to investigate claims of reincarnation. He has a lot of critics, unsurprisingly for a researcher of the 'paranormal'. However, it's hard to dismiss many of his collected cases as purely fraudulent or confirmation bias. To summarise, the common features of his cases are a combination of:
- Violent injuries or cause of death in the purported previous life.
- Statements pertaining to events in the previous life.
- Physical anomalies and extreme phobias in the current life corresponding to the cause of death.
- Physical or behavioural resemblance between the previous and current life.
There are many great threads on BSE discussing the doctrine of reincarnation or rebirth in Buddhism, to which I have little to add. I just want to say that beliefs in reincarnation or the paranormal in general are acquired experientially. From the perspectives of many natural sciences, they are inherently unreliable and most likely unprovable. Much to my dismay, I believe it's futile to try to prove the existence of reincarnation.
Currently I'm trying to distance myself from BSE. I may or may not response to your comments.