I'm still astounded when I hear claims from reasonably intelligent people that are along the lines of:
In a past life, I was an Egyptian Queen
Leaving aside that Egyptian leaders were called Pharohs, how likely is that we can believe such claims? It's not enough to say my uncle Billy went under hypnosis and was able to tell the story of [insert name here]. How could he have done that if he'd not been [insert name here] in a past life? He didn't even study History at school. Why? Consider this: Billy just happen to be a rubbish storyteller. He gets flustered, stammers becuase of nerves, and, ultimately, becomes tongue-tied. Yet, under hypnosis Billy is able to remember his past life as an Egypitan Queen (presumably of the female, rather than male, kind). One only has to watch the likes of a Derren Brown (or, numerous others) show to know that people's behaviour can be markedly different when under hypnosis than not. In part, this is down to suggestibility, but in part it's down to the ability for us to lose our inhibitions when under hypnosis. So, it's quite possible that Billy has been able to cast off his stammer and his nerves and become a fantastic storyteller when under the influence of hypnosis. Indeed, maybe he is accessing areas of the brain normally dormant and is simply retelling a repressed memory[^1] of a TV show or film he saw years ago.
Hypnosis aside (many such claims are made without its use), what is the logical position and how do we arrive at it?
The most famouse story of reinacarnation that I know of is the story of Jesus' reincarnation. Like most stories, it requires one to believe the testimonies and of others. There is no physical proof, that I am aware of.
[^1] The memory may have been repressed for the simple reason that it held no intrinsic value. The human mind sifts through millions of pieces of information every moment of conciousness and discards anything it deems unimportant (which is most of it).