How does Buddhist Logic work?

In the original Pali scriptures, it is taught to not rely on logic:

So in this case, Kalamas, don't go by reports, by legends, by traditions, by scripture, by logical conjecture, by inference, by analogies, by agreement through pondering views, by probability, or by the thought, 'This contemplative is our teacher.' When you know for yourselves that, 'These qualities are unskillful; these qualities are blameworthy; these qualities are criticized by the wise; these qualities, when adopted & carried out, lead to harm & to suffering'...

Kalama Sutta

I would say Buddhist logic is no different from Aristotle's. Nagarjuna uses Aristotelian logic in Fundamental Wisdom to refute all false views. This is often misunderstood for two reasons.

First, where he addresses positive metaphysical propositions he refutes not just the proposition and its contradictory alternative (this or that) but also their combination or separation (neither or both). This is sometimes seen as a novel form of logic but if we see this as two contradictory pairs of proposition then it is just the application of Aristotle's dialectic method.

Second, Nagarjuna denies that positive metaphysical statements form legitimate contradictory pairs of the form A/not-A. Thus when we ask 'Does the world begin with something or nothing' he would deny that this is a Aristotelian contradiction and this is why he is able to propose a third alternative without breaking Aristotle's rules.

Nagarjuna's logic would not work if he rejected Aristotle's rules. Buddhist logic does not reject them but transcends them. I would say that if you understand Aristotle's logic them you understand Buddhist logic.

Many people do not understand this issue because they forget Aristotle's definition for a true contradictory pair. For a legitimate case of A/not-A for which the 'laws of thought' would apply it must be the case that one of A and not-A is true and one false. Nagarjuna shows that metaphysical dilemmas and Kant's antinomies do not take this form and this is how he is able to refute both the extreme views that create these antinomies and endorse a third alternative. He does this without any need to modify Aristotle's rules.

This issue is much misunderstood because a surprising and dismaying number of philosophers do not properly apply Aristotle's rules in their own reasoning.

In short, it is my opinion that Buddhist logic is no different from the logic we use every minute of the day in our thinking. Were it to break the 'laws of thought' we would not be able to understand Buddhist doctrine.

As a consequence, and as Nagarjuna demonstrates (also Francis Bradley, Spencer Brown and others) it is possible to logically prove that all positive metaphysical positions are logically indefensible and in this way prove by the dialectical method and 'abduction' that Buddhist non-dual metaphysics is the only global theory that works.

For this reason the dialectic is big deal in Buddhist education.

In Aggi-Vacchagotta Sutta, it has mentioned Vacchagotta used Catuskoti to ask questions from Shakyamuni Gautama Lord Buddha. But Shakyamuni Gautama Lord Buddha has rejected all the conditions in Catuskoti.

"Any consciousness by which one describing the Tathagata would describe him: That the Tathagata has abandoned, its root destroyed, made like a palmyra stump, deprived of the conditions of development, not destined for future arising. Freed from the classification of consciousness, Vaccha, the Tathagata is deep, boundless, hard to fathom, like the sea. 'Reappears' doesn't apply. 'Does not reappear' doesn't apply. 'Both does & does not reappear' doesn't apply. 'Neither reappears nor does not reappear' doesn't apply."

This is because Dhamma (doctrine) is atakkavachara. In other words, Dhamma is beyond reasoning and logic.

Buddhist logic is to acknowledge that there is suffering.

Buddhist logic is to acknowledge that there are causes to these suffering.

Buddhist logic is to acknowledge that it is possible to be free from all suffering.

Buddhist logic is to acknowledge that there are paths that lead one to be free from all suffering.

The article shown by @llya Grushevskiy is discuss very interesting logic. Some extracts from it gives bellow.

... The four original possibilities, {T}, {F}, {T, F} and {}, as t, f, b and n, respectively....And now there is a fifth possible value – none of the above, ineffable, that which lies beyond language. Call it i, ....

We started with two possible values, T and F. In order to allow things to have both of these values, we simply took value of to be a relation, not a function. Now we have five possible values, t, f, b, n and i, and we assumed that value of was a function that took exactly one of these values. Why not make it a relation instead? That would allow it to relate something to any number of those five value.....In this construction, something can relate to both t and i: and so one can say something true about something ineffable after all.

The similarities between this and our Buddhist paradox of ineffability are, you must admit, pretty unnerving ....

i like to give example to 'i'.
Take "Life" as a fact.
So t = life is,
f = life is NOT,
b = life is AND life is NOT,
n = NOT life is OR NOT life is NOT,
then what should be the i?
i = 'life is' BUT 'a dream'.

....This does not, of course, show that they are true. That’s a different matter. But it does show that these ideas can be made as logically rigorous and coherent as ideas can be. As the Buddha may or may not have said (or both, or neither): ‘There are only two mistakes one can make along the road to truth: not going all the way, and not starting.

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.