Is there any argumentative way to establish that rebirth exists? In shatdarshanas, it seems like a lot of them - samkhya, yoga, etc don't believe in rebirth. Although Yoga Sutras of Patanjali talk about siddhis attained from efforts from previous lives, it doesn't explicitly discuss or argue the 'how' part.
Gautama Buddha
Since there actually is another world (any world other than the present human one, i.e. different rebirth realms), one who holds the view 'there is no other world' has the wrong view...
— Buddha, Majjhima Nikaya i.402, Apannaka Sutta, translated by Peter Harvey
seems to have argued the closest concept of consciousness. It goes like this:
If Vijnana (Chit/Consciousness) is independent of the body and it isn't created by itself out of thin air (so it cannot be annihilated either), then it must have appeared out of some other world (different realms) and after the death of the body can also move to different realms after death, is how re-birth is established from best of my understanding here (I could be wrong, in that case, please correct).
Is there any other philosophy that discusses rebirth through inference or any logical system?
Please Note: - The question is NOT about which of them do believe in the concept of re-birth, but how do they establish them in different schools through logical tools like inference, deduction, etc. similar to what Buddha did. Proof of re-birth can be done using scientific studies, anecdotes, etc. but that's of the least interest in philosophy.
Note: If no argument can explain rebirth, then how they can argue liberation as stopping birth and rebirth? (I know certain schools like - Yoga don't prescribe liberation as a cessation of re-birth probably.