Generally, Buddhists believe that there is no beginning to birth and that once we achieve liberation from the cycle of existence by overcoming our karma and destructive emotions, we will not be reborn under the sway of these conditions. Therefore, Buddhists believe that there is an end to being reborn as a result of karma and destructive emotions, but most Buddhist philosophical schools do not accept that the mind-stream comes to an end. To reject past and future rebirth would contradict the Buddhist concept of the ground, path and result, which must be explained on the basis of the disciplined or undisciplined mind. If we accept this argument, logically, we would also have to accept that the world and its inhabitants come about without causes and conditions. Therefore, as long as you are a Buddhist, it is necessary to accept past and future rebirth.



For those who remember their past lives, rebirth is a clear experience. However, most ordinary beings forget their past lives as they go through the process of death, intermediate state and rebirth. As past and future rebirths are slightly obscure to them, we need to use evidence-based logic to prove past and future rebirths to them.
 


http://www.dalailama.com/messages/statement-of-his-holiness-the-fourteenth-dalai-lama-tenzin-gyatso-on-the-issue-of-his-reincarnation

To reject past and future rebirth would contradict the Buddhist concept of the ground, path and result, which must be explained on the basis of the disciplined or undisciplined mind. If we accept this argument, logically, we would also have to accept that the world and its inhabitants come about without causes and conditions. Therefore, as long as you are a Buddhist, it is necessary to accept past and future rebirth.



There are many different logical arguments given in the words of the Buddha and subsequent commentaries to prove the existence of past and future lives. In brief, they come down to four points: the logic that things are preceded by things of a similar type, the logic that things are preceded by a substantial cause, the logic that the mind has gained familiarity with things in the past, and the logic of having gained experience of things in the past.



"Of all those things that from a cause arise,
Tathagata the cause thereof has told;

'Apart from a requisite condition, there is no coming-into-play of consciousness' (aññatra paccayā natthi viññāṇassa sambhavo)

(AN 10.61 ) "A first beginning of ignorance cannot be conceived, (of which it can be said), 'Before that, there was no ignorance and it came to be after that.' Though this is so, monks, yet a specific condition of ignorance can be conceived. Ignorance, too, has its nutriment, I declare; and it is not without a nutriment. And what is the nutriment of ignorance? 'The five hindrances,' should be the answer.

13. (4) "Again, the Tathagata understands as it actually is the world with its many and different elements. That too is a Tathagata's power...

14. (5) "Again, the Tathagata understands as it actually is how beings have different inclinations. That too is a Tathagata's power..

Ultimately all these arguments are based on the idea that the nature of the mind, its clarity and awareness, must have clarity and awareness as its substantial cause. It cannot have any other entity such as an inanimate object as its substantial cause. This is self-evident. Through logical analysis we infer that a new stream of clarity and awareness cannot come about without causes or from unrelated causes. While we observe that mind cannot be produced in a laboratory, we also infer that nothing can eliminate the continuity of subtle clarity and awareness