It appears in ancient India there was a style of debate that became associated with Buddhism and is still a part of at least Gulugpa Tibetan Buddhism.
What is it and how would I get started?
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In the suttas a debate usually takes the form:
I hold such and such to be the case.
I hold such and such a different case to be the case.
There is no confrontation. No dealing with the other persons argument. You accept or do not accept. You are held to be capable of understanding both arguments and hold the wrong one only because you have not heard the correct one before. Argument concerning cases is held to be 'thinking out loud' and is not done. Again, you are held to be capable of reasoning through all the arguments yourself. The original statement can be made up to three times with the same response being made up to three times. Then a deadlock is declared and the issue is submitted to an authority both can trust.
Here is one example: http://obo.genaud.net/dhamma-vinaya/pts/an/05_fives/an05.166.hare.pts.htm
A later form of debate, not found in the suttas, but found in the controversies that arose around the second Council, involved directly dealing with the substance of the opponant's argument.
You say such and such, but here it is understood that this is the case. Do you agree that here it is understood that this is the case? If so you must agree that your argument is defeated.
I agree that such and such is understood to be the case, but it does not apply in the case I bring up for such and such reasons.
More or less the way a reasonable dialogue would be conducted today, but at an essentially lower level than the earlier form.
There is never any reference to the individual involved, only the issues are discussed.