There is a lot of confusion surrounding the concepts of Reincarnation and Rebirth. At first glance it seems like they are synonyms, but there is a big difference between these two concepts.
The doctrine of reincarnation, also referred to as transmigration, is taught by Hinduism, Jainism and Gnostic Christianity. It's main postulate is that there is an indestructible, eternal, personal element that travels from one life to the next. This element is called atman in the Hindu religion, jiva in the Jain faith and soul by Gnostic Christianity. Reincarnation assumes identity between the occupant of this body and, when this body dies, the occupant of the following body.
In contrast, Buddhism teaches the doctrine of rebirth. Between a series of lifetimes there is a relationship of causality, not of identity. To make this more clear, let's use the example of the falling dominos. If I place a series of dominos standing up in line next to each other, and I strike the first one to make it fall down, this causes the second one to fall down, and the next, until the last in line falls down. The fall of the first domino is the cause of the fall of the last one, but there is not a shared identity between the first and the last domino.
When somebody dies, rebirth is caused by the continuation of the mental processes of the dead person in a new body. The new person however is not identical to the one that left the previous body (a lot of causes and conditions come into play to shape this new being), nor a completely different person (because of causality relation).
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