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I am aware of the way Western philosophers define rationality but I do not know whether Eastern philosophers define it.
Are there any differences between the Eastern and Western philosophical traditions on what constitutes rationality? And if so, what exactly are those differences?
If it is too much to ask for what Eastern Philosophy defines as rationality, Buddhism or Hinduism would also be acceptable.
Amartya Sen addresses this question in good depth in his July 2010 essay for the New York Review of Books. Specifically, one of the notions he challenges is the insidious notion, brought up by well-meaning proponents (both Eastern and Western), that rationality is somehow a product of the West -- the corollary often being that it's not fair to impose Western standards of human rights and morality based on reason to societies where rationality is not the highest value (perhaps other values take precedence in these society, such as community, they claim). Sen shows that this conception of rationality is flawed and illustrates that rationality is a universal concept that transcends culture. He also illustrates rational traditions in non-Western cultures that are rooted in times before great strides were made in the idea during the Enlightenment.
Rationality in philosophy is the exercise of reason, reason being sound judgment (having some basis or justification for a belief). In all philosophies east or west, north or south, the basic concept of rationality is the same — everyone wants to be confident that there is some justification for their beliefs, otherwise in any society we would be cast out as madmen (insanity being exactly that - irrationality, senselessness). What truly varies from culture to culture and philosophy to philosophy is not the notion of rationality so much as the notion of what makes a belief justified.
This is a large question, and can be approached several ways.
First, at the broad level-- there is a widespread misconception that Western philosophy is rational and Eastern philosophy is mystical. This is false, but it is false in an interesting way: there is a fascinating book on the subject by Thomas McEvilley called The Shape of Ancient Thought: Comparative Studies in Greek and Indian Philosophies. For a taste of the contents, McEvilley has made a video describing his research here. Suffice it to say that both the Western and Eastern philosophical traditions contain both rational and mystical elements.
Now, more specifically: although the basic principles of Indian logic (for example) are largely identical to classical logic, the formulation of the details are somewhat different-- such as the 3-step syllogism familiar to us being replaced with a 5-step variant.
Of course, there are deviant logics in both the Western and Eastern tradition, but these are special cases.
Zhuang Zi is a chinese daoist philosopher who discussed rationality and the "adjustment of controversies" in the second part of his book of seven parts: http://ctext.org/zhuangzi/adjustment-of-controversies
Though he is not easy to understand, less yet to translate!
