How does Buddhist Logic work? In the Wikipedia article Buddhist Logic, it is mentioned that Buddhist Logic is different from classical western logic.


  ‘Indian Logic’ should be understood as being a different system of logic than modern classical logic (e.g. modern predicate calculus), but as anumāna-theory, a system in its own right. ‘Indian Logic’ was influenced by the study of grammar, whereas Classical Logic which principally informed modern Western Logic was influenced by the study of mathematics.


How does Buddhist logic work then? And how is it used in the exegesis of Buddhist scriptures?
On the one hand, there is logic which is a set of rules for making deductions, inductions, etc. Even in the West, there are various. For instance, (1) classical logic and (2) intuitionistic logic. I believe that they have different ways of dealing with the law of excluded middle. I used to study more philosophy of logic than logic itself, and there are great differences in that respect between formal and intuitionistic.

In Buddhism, when logic is applied, there is no "set of rules for making deductions" that is different from those of classical formal logic.

Ilya Grushevskiy gave an explanation of the law of identity and law of non-contradiction that makes you think that Indian logic is different from classical logic. However, Tibetans give another explanation. We say: X does not exist inherently. Therefore, X is not inherently one with X, but it is conventionally so. It is not inherently different from not-X, but it is conventionally so. And so forth.

In the same way, the first verse of Nagarjuna's Karika says "Neither from itself nor from another, Nor from both, Nor without a cause, Does anything whatever, anywhere arise." We explain: of course, a tree arise from another conventionally, since a sprout, or a seed, is not a tree. But since it is not inherently produced, it does not inherently arise from another. We call that "refuting the four extremes of production."

When you say "Indian Logic should be understood as being a different system of logic than modern classical logic[...]" it could refer, not to logic itself but to:


Dialectic, as for instance that of Nagarjuna. Western scholars tend to describe it as negative, but that is a category that is applied as well to that of Western philosophers (such as Kierkegaard, for instance)
A set of propositions.
Epistemology.


Indian and Indo-Tibetan epistemology mainly describes types of consciousnesses. Consciousness is qualified as an object-posessor since it knows its object, and so we speak of seven types (or more or less, depending on the text). For instance, a "wrong consciousness" that engages its object erroneously, an "inferential cognizer", etc. Consciousness is always described in relation to the causes of its arising (a sense power, an object, a immediately preceding condition) and is thus not taken independently.
In the Indo-Tibetan traditions, one studies epistemology so as to know how mind works. It is all aimed at mind training and is not merely "philosophical" in that sense.