People who believe there is only the physical aspect to life will say reincarnation is illogical, or they'll come up with the silly idea that our atoms get reincarnated into other life by consumption.
However, if you've ever witnessed someone die (unfortunately, I have, more than once) you can see that the life force leaves the entire body all at once, in a way that can't be explained by physical means. Atoms do not produce life; there is a life force, a non-physical thing, which is commonly called a soul.
Since this soul entered one body to give it life, and left it at the time of death, it seems logical that it could go on to give life to another body.
It seems logical enough that an American university's school of medicine has a department devoted to studying reincarnation.