Evidently, Buddhist logic is based on grammar, whereas Greek/Western logic is based on mathematical understanding. So I checked out a book they recommended to me from my library to get started: Master of Wisdom: Writings of the Buddhist Master Nāgārjuna. From what I can tell, Nāgārjuna is sort of the Plato of Buddhist logic. He's early second-century A.D.
Here's my favorite verse so far:
'Is' and 'is not' and also 'is-is not' have been stated by the Buddhas for a purpose. It is not easy to understand!
Of course, I don't understand it in one day. Nevertheless, I am reminded of what I believe to be the most overlooked section of Plato's Republic, a section I have quoted more than once:
And so, Glaucon, I said, we have at last arrived at the hymn of dialectic. This is that strain which is of the intellect only, but which the faculty of sight will nevertheless be found to imitate; for sight, as you may remember, was imagined by us after a while to behold the real animals and stars, and last of all the sun himself. And so with dialectic; when a person starts on the discovery of the absolute by the light of reason only, and without any assistance of sense, and perseveres until by pure intelligence he arrives at the perception of the absolute good, he at last finds himself at the end of the intellectual world, as in the case of sight at the end of the visible.
The point here is that the dialectic -- the process known as "the dialectic" -- is not an end in itself. One who practices that way is destined to become, well, an "insignificant wrangler." The dialectic is designed to bring practitioners to a very particular moment of... whatever: realization, or enlightenment, or awakening, whatever. The master dialectician simply encourages his students to continue practicing dialectic until that insight comes, until the student arrives at the hymn.
I wonder if Nāgārjuna's teachings are similarly directed. If they are, the ends might be similar or even the same... apparently both disciplines have a very distinct purpose, however difficult (even impossible) it is to describe that purpose to the novice.
Follow Up: So, how does the rhetorician of contemporary academia decide that he or she believes there is no end to rhetoric? If I say "practice lifting weights until you can bench press three hundred pounds," and you say, "I could never do that, so I will not practice," aren't you missing something? Especially if I tell you that I thought I couldn't do it when I started, but now I can?