One of the most striking aspects of the Nyaya School (and thus Indian logic in general) is its different formalism for deductive reasoning. Many subtle issues in logic are common to East and West (for example, the issue of dealing with fictional entities), but this is not common. It seems very strange: Aristotelian deductive logic appears so simple and straightforward, so obvious, that even if one personally could not have invented it, one feels that surely enough mathematicians and logicians put together will – that is universal and pre-determined.
[I don’t think Indian logic is that different - it does the same work, and generally provides the same result, as Aristotelian logic.]
It is almost as if one were to travel to a land with many able geometers, who had developed out it its furthest ramifications a complete Euclidean-style plane geometry – except they all assume parallel lines diverge and so their geometry is just plain different. One says to oneself, Riemannian geometry certainly has many uses (such as in Relativity), and is quite as internally consistent and as valid as our system which assumes parallels are always equally distant from each other – but how on earth did their geometry happen to develop in that way? One feels an explanation would be nice.
One starts the beginning. Scholars discuss and debate topics. Sometimes they even change their minds. There is some set of properties that makes a given set of statements by a scholar convincing, or not convincing, or perhaps even obviously unacceptable and ‘wrong’. They eventually begin meta-discussions analyzing and classifying arguments. This tradition of vadavidya begins around 0 AD . We can find a few different classifications. Drawing on the earliest classifications is the tripartite division found in the Nyaya Sutra, where
- one can seek to discover truth and establish some thesis
- engage in disputation where one seeks merely to establish a thesis (truth be hanged)
- “destructive criticism or ‘wrangling’” , where one may no longer seek to establish something at all by any means, but rather seeks to tear down the opponent’s establishment.
This division is clearly normative: —–
[ re ‘never’: no, actually - in certain contexts, like the brahmodaya (competitive debate), you are supposed to engage in #2 - because if you don’t & your opponent does, you will lose. #1 is generally for teacher-student interacting.]
—- we are supposed to engage in #1, and never 2 or 3. But the reason for this set of splittings is unclear. The summaries mention several distinguishing factors such as one’s emotional state or intention, one’s methods, and one’s goals – 3 binary variables mean 9 possible divisions, so why these particular 3? Well, the split between 1 and 2 indicates that some methods were seen as good and some as bad, but the last one is somewhat odd. It would seem one cannot be purely skeptical or critical in a ‘good’ debate. But does this make sense? An example: perhaps I have no good suggestion as to why the current heir to the throne of Bhutan is , but I could surely argue in all honesty and good faith and in a valid way against a proposal that the heir is Abraham Lincoln (who is dead, for starters).
In line with this intuition, there is another categorization with 4 sets. The 4th is “(iv) the type modeled after the honest one where only the refutation of a thesis is needed.”  (From Gauda Sanatani’s division; Matilal mentions this position was supported by philosophers like Nagarjuna and Sriharsa.)
By this point, we’ve settled on an idea that the goodness or badness of a debate depends not on the goal of a debate (now that we’re allowed to seek a proof or a refutation of a proposition), but rather on how it is conducted – it is the tactics and arguments (tarka) that divide a good ‘honest’ type of debate from the bad, sophistic, dishonest ‘tricky type’.
What tactics are acceptable? We return back to the kind of discussions these debates were: attempts to persuade oneself or others  of something. What sorts of things persuade others falls under epistemology, and authors have rightly identified this: “Logic developed in India out of two slightly distinct traditions: 1. vada tradition ie. Tradition of debate which was concerned with dialectic tricks, eristic arguments, and sophistry, and 2. Pramana tradition which was concerned with the criteria of empirical knowledge. On account of this genesis, Indian logic imbibed an epistemological character which was never removed throughout its history.” 
Well, we already know informal discussions can persuade people. So the obvious starting place is to try to abstract and formalize what’s going on in a discussion. Just list out the steps of someone getting interested in a topic, laying out a specific proposition, being convinced an answer is findable, laying out his premises, examples, and concluding with satisfaction at the answer.
Barlingay tells us that this is what happened: “The history of Indian logic reveals that originally the syllogistic argument consisted of 10 parts or ‘avayanas’.”  (Matilal 1985 concurs.) The specific 10 do not matter (the Jain logicians and Vatsyayana who are Barlingay’s sources here differ on them anyway). They were eventually ‘streamlined’ down into a 5-part argument.
The five are, per King :
1.“Pratijna – The Statement, Premise, or Position that is to be established (sadhya) 2. Hetu – The Cause or Reason for the statement 3. Udaharana – The Example 4. Upanaya – The Application of that example 5. Nigamana – The Conclusion”
With this schema, it’s easy enough to substitute in the standard example:
What? There’s fire on that there hill
Why? Because there’s smoke rising from it.
So? Well, wherever there is smoke there is fire.
[ [strikethrough of #3] example: ‘As in the kitchen’]
There’s smoke on that hill, and thus there is also fire.
[strikethrough of #4 ‘This is such a case (that example applies to this case, because the same inferential mark is present in both - ie. smoke’)]
There’s fire on that there hill.
There have been attempts in both Indian’s past and modern times to formalize this further and shrink it: “This kind of argument with five parts was subjected to severe criticism at the hands of the Buddhist logicians like Dignaga, Dharmkirti, and Dharmottara. They reduced the syllogistic argument to the first 3 or the last 3 and in some cases even to two in the fashion of Aristotelian syllogism and enthymeme  . They argued that either the first 2 or the last 2 parts were redundant.”
This certainly seems fair to me. The last/first ones are clearly redundant – there’s no need to repeat the goal/proved-theorem, which brings us down to 4 lines. Line 4 merely substitutes
[no, it refers to the previous example]
in premises to a basic law of inference, and thus is also somewhat redundant. And with 3 lines, we seem to have arrived at an Aristotelian syllogism in everything but name.
What sort of argument can one muster against viewing the 5-part syllogism as just a variant on the Aristotelian one? Well, “Vatsyayana the commentator of Nyaya Sutras has tried to defend the 5 parts by arguing that each of these parts stands, as it were, for each Pramana; since Pramanas or the sources of knowledge, are four the syllogistic argument should consist of 4 parts and a conclusion bringing the total to 5.” 
[ no - there is no example in Aristotle. The example is why Matilal calls the Indian syllogism ‘epistemological’. It is about truth, not just validity ]
If this is Vatsyayana’s apology accurately presented, then it seems very poor indeed to me – mere analogical reasoning with little more philosophical foundation than arguing that the syllogism should have 5 parts because hands have 5 fingers and thus it is more convenient to count.
Matilal writes, “The model of reasoning with which the Indian logicians were chiefly concerned was not PURELY [sic] deductive…there have been some confusing and futile attempts to reduce the arguments studied by the Indian logicians to [the] Aristotelian syllogistic model.”  Why these attempts are confusing and futile does not seem to be well-explained by Matilal. Fortunately, King addresses the issue at length . He writes of the classic ‘Socrates is mortal’ syllogism that “The Naiyayika and other Indian logicians would argue that this argument is acceptable only because one knows from experience (ie. by induction) that men are in fact mortal. Indian logic then is both formal and empirical and does not allow for a purely deductive logic.”
I believe that in attempting to avoid the Scylla of parodying Indian logic as some purblind groping toward Western sophistication and correctness, we have plummeted into the Charbydis of unfairly characterizing Greek logic. Aristotle’s logic was one of the great edifices of Greek thought; we should be wary of assuming it was purely formal with no epistemological connections in it. Formalism as a philosophy of math and logic would not take shape for many centuries, and attributing it to the Greeks is anachronistic. And by considering the history of Greek thought, we can see that their arguments appealed to reality in the same way as the Naiyayika arguments did.
[ ! See Kreale and Kreale, The Development of Logic. p.z. Aristotle distinguishes between demonstrative and dialectical argument. The first is concerned w/ truth - the second, only w/ validity. ]
Consider Parmenides, famous for his argument that the changing world is an illusion and that true reality is a vast unchanging block of something. His argument proceeds quite deductively, starting from the 2 premises that things that exist, exist; and that there is no such thing as nothingness, the void, a vacuum. With no empty space, things cannot be divisible; with no divisibility, there cannot be any changes in something itself. Further, anywhere a thing might try to move to would already be occupied (since an unoccupied space would be a vacuum, and thus non-existent). And this can be put easily in syllogistic form, and the syllogisms all proves our everyday world is a lie.
Were the Greek logics purely deductive, Parmenides’s argument would rest there and
[ You misunderstand Matilal’s distinction. Of course the Greeks used logic because they wanted to reach true conclusions! But, unlike the Indians, they also addressed the strictly formal aspects of proof & falsity (as in Aristotle’s ‘dialectical’ argument, which may or may not be based on true premises). ]
presumably we would all have to believe it on pain of being irrational. But the Presocratic Atomists did not accept Parmenides’s view, and instead appealed to sense-data: we observe change and motion, therefore there must be vacuums for things to move into, therefore there must be no-things, therefore one of Parmenides’s two premises is false.  A fine bit of reasoning from contradiction. And this is not a theoretical example; this actually happened.
Presumably, the way in which the five-part syllogism operates is that where the Atomists pointed that one did in fact observe change, the Indian will point out that smoke can be created by machines, that smoke is not always accompanied by fire. A premise is contradicted, and one no longer needs to accept the argument. This is the same exact process. How then can we claim the two traditions are completely incomparable?
[ We aren’t claiming that. Nor is Matilal. ]
Matilal writes: “The general form of the arguments studied by the Indian logician is: A is B because of C. The middle term or the ‘reason’ hetu C, can be either adequate or inadequate (instead of being strictly valid or invalid). An adequate middle term or ‘reason’ will establish the conclusion and the argument will be sound. If the middle term is not adequate, the conclusion will not be established and the argument will be unsound. We may thus be tempted to introduce an additional premise so that these arguments will be deductively valid. We can resort to the theory of ‘suppressed premises’ and decide that we are dealing in fact with deductive arguments in all these cases. But this seems at best to be a distortion, and at worst the demolition of the original structure of the actual argument.” 
I must confess that I don’t see any distortion or demolition when we think about suppressed premises (cf. quote in note 9). Indeed, suppressed premises make the 5-part syllogism much clearer to me. We have a premise ‘There is smoke on the hill.’ We add the general rule, ‘Smoke at a place implies fire at a place’, and by Modus Ponens, we obtain ‘fire on the hill’.
Our argument is now valid, and we can begin investigating the soundness. This business of adequacy and inadequacy mixed with sound and unsound confuses. There is absolutely nothing at all wrong with saying that a counterexample of smoke-with-no-fire renders an argument valid but unsound. To say that ‘the middle term is not adequate’ obscures what is going on.
[ re ‘relation is between’: ‘it is vyapt, ’pervasion’: look it up in Matilal’s index. ]
Further, the 5-part form can lead one into error. It isn’t clear what the relation is between smoke and fire. Is it a mere logical conjunction we can derive from, or are we speaking of causality? If it is just logical conjunction, another way of saying
[ re ‘causality’: It has no causal implication. Think of another syllogism: 1. John possesses mortality. 2. Because he posses human-ness. 3. As in the case of Socrates. 4. This is such a case. 5. therefore it is so. where is the causal implication here? Does human-ness cause mortality? No, they are co-incident. This co-incidence is called vyapt, ‘persuasion’, in Indian logic. No causal link is implied. ]
- (p^q) or ~(p or q)
- QED; q
Then our 5-part syllogism is alright. It is correct and not fallacious. But if it is causality, then it is another way of saying
- For Every p, there is q
[ for every human-ness, there is mortality. Is this causal reasoning? It isn’t doing this! ]
And our p is fire, and our q is smoke, then we are reasoning abductively, from effect to cause. And this is the fallacy of affirming the antecedent:
- p -> q
- QED: p
The logical expressions are of course quite different, but it isn’t so clear in the informalities of the 5-part. If we wish to reason abductively and also validly, then we need a suppressed premise. We need to affirm that smoke is always caused by fire (and not alternative smoke-source B), otherwise we could erroneously think there is fire on a hill when it was just smoke-source B. So then:
- fire -> smoke [unnecessary]
- smoke -> fire
- QED: fire
An alternative formulation might be ‘~smoke -> ~fire; smoke; fire’, which is an flip of Modus Tollens, with strong support from Dignaga, who has three conditions for a ‘logical sign’:
- “It should be present in the case (object) under consideration.
- It should be present in similar case or a homologue.
- It should not be present in any dissimilar case, any heterologue.” 
The first criterion clearly is a question of soundness. Do our premises correspond to reality? The second two are in a sense the same, per above. If smoke is only present when there is fire, and non-smoke is only present with non-fire things (this sounds odd, but Hempel’s Raven paradox with its ‘all non-black things are non-ravens’ is much the same), then it’s the same.
Thus: Indian and Western logic share many properties, and can profitably be analyzed in terms of each other. Contrary to what I thought earlier, the 5-part syllogism is valid insofar as the two signs are specified to be correlated, but interpreting implication as causality gets us into trouble. We can extricate ourselves and retrieve a valid (but not necessarily sound) argument by adding an additional restriction, and it turns out to hew to Dignaga’s triple nature of signs.
References 1. Pg 2, Matilal 1998 2. Pg 130, King 1999 3. Prince Jigyel Ugyen Wangchuck, as it happens 4. Pg 3, Matilal 1998 5. “It is ‘Anumiti’ which is inferential knowledge, ‘Anumana’ is a means to that knowledge. For the Nyaya logician Anumana is the most important means of knowledge. The later Nyaya, following the Buddhist logicians, accepted two kinds of Anumana. One is the Anumana for oneself, the other is ‘Pararthanumana’ or inference for others. It is the latter form which takes explicitly the linguistic form…” pg 107, Barlingay 1965 6. Pg 127, Matilal 1971 7. Pg 108, Barlingay 1965 8. Pg 131, King 1999 9. ‘enthymeme’ here being used in the logical sense of an argument with hidden premises; Barlingay clarifies this later on pg 109 when he writes: “It is not difficult to see that the Pancavayavi sentence does not have a syllogistic form. There is a suppressed major premiss [sic] which is contained in the first two propositions. The third is a particular example of the universal major premiss. It can be observed without much difficulty that the arrangement of the Pancavayavi vakya emanates from the procedure of controversy. The lacuna that Vyapti or the major premiss is suppressed must have been observed by Indian logicians. So in the course of time with the instance or the third “premiss”, they also supplied the universal concomitance.” 10. Pg 131-132, King 1999 11. This summary of the Atomists’ logical process follows Karl Popper’s The World of Parmenides, ISBN 9780415173018 12. Pg 128, Matilal 1971 13. Pg 6, Matilal 1998
Bibliography - Barlingay, S.S. A Modern Introduction to Indian Logic 1965 - King, Richard. Indian Philosophy: An Introduction to Hindu and Buddhist Thought, 1999, ISBN 0-87840-756-1 - Matilal, Bimal Krishna. Epistemology, Logic, and Grammar in Indian Philosophical Analysis 1971 - Matilal, Bimal Krishna. The Character of Logic in India, 1998, ISBN 0-7914-3739-6
[ Causality is not a necessary feature of the 5-step proof. You are getting distracted because in the one example you cite, there happens to be a causal relation as well. The only necessary relation in question is _vyapti. (Acc. to Gaingesu, there are 29 different kinds! See Matilal pp 164-168). I really think that Matilal is correct in distinguishing the 5-step Indian & 3-step Aristotelian syllogism. I suggest you pick up Kreale & Kreale, The Development of Logic, which is the standard work on the history of logic. ]