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.