Can you give an example in the Pali canon where the Buddha uses LEM?

Here is your problem, the statement itself. The statement isn’t even true…. truth itself isn’t found per that statement nor statements. The statement is a mundane one and therefore has limitations.

Very simple said, words are merely a raft, that leads or points to the truth itself…. which is directly experienced for oneself. Proof here isn’t the mundane words but rather that direct experience itself. It’s beyond words!

I believe it is not correct to think that the sutta’s are literally the words of the Buddha. Probably the logic you see, the reasonings, the line of thinking, is more those of the composers then the Buddha.

Maybe you seek argumentum ab auctoritate?

I’m aware of this contention, but this isn’t the subject of this particular question nor does it seem obviously related. That question is completely orthogonal here so I think it better to leave it to other discussions.

Fascinating thread. Thank you.

As I think through possible counter-examples, I keep finding they’re RAA not LEM.

An off the top of my head pre-morning-coffee thought. It makes sense in terms of his overall teaching. One of the poisons is delusion. And he uses RAA as a tool to demonstrate things we believe are in fact false, are delusions. But knowing what is true isn’t a logic task. That requires the Noble Eightfold Path.

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I’ve thought a lot about how to make this more intuitive and this is what I’ve come up with:

  • Let ⊥ be “An illogical conclusion.”
  • Let ¬P <=> ⊥ be “Assuming P leads to an illogical conclusion.”
  • Let ¬¬P be “Assuming that, assuming P leads to an illogical conclusion, itself leads to an illogical conclusion.”
  • Let ¬¬¬P be “Assuming that, assuming that, assuming P leads to an illogical conclusion, itself leads to leads to an illogical conclusion, itself leads to leads to an illogical conclusion.”

Triple negation elimination:

  • Let ¬¬¬P → ¬P be If “Assuming that, assuming that, assuming P leads to an illogical conclusion, itself leads to an illogical conclusion, itself leads to an illogical conclusion’, then 'Assuming P leads to an illogical conclusion.”

You can understand this in a constructive way as follows:

  1. ¬P <=> ⊥: This means that assuming P leads to an illogical conclusion, or a contradiction. In constructive logic, this doesn’t imply falsehood in the classical sense, as constructive logic focuses on what can be constructively proven. Thus, ¬P doesn’t mean P is false, but that assuming P leads to an unacceptable conclusion. This approach is more about the process of knowledge acquisition than assigning a binary truth value.

  2. ¬¬P: This does not mean P is true or P is known. It simply means that when we assume ¬P we arrive at an illogical conclusion. ¬¬P is weaker than P in this context, as it does not constructively assert the truth of P.

  3. ¬¬¬P: This statement asserts that assuming ¬¬P (i.e., assuming that the negation of P leads to an illogical conclusion) itself leads to an illogical conclusion. This triple negation doesn’t directly comment on the truth or falsehood of P but rather on the logical structure surrounding the proof or disproof of P.

  4. ¬¬¬P → ¬P: This is interesting because it allows us to collapse the layered assumptions into the original negation without invoking any affirmative proof of P. Essentially, ¬¬¬P → ¬P means that if a triple negation of the assumption leads to an illogical conclusion, then the single negation must also lead to an illogical conclusion. It’s like peeling back the layers of negation to return to the initial negation, but without making a direct statement about P’s truth.

We can get away with the reducing the triple negation but not the double negation precisely because the former does not involve any positive affirmation of the truth value of P, but reducing the latter does.

Given the above semantics I’m analyzing the graft of this logical structure on:

“There is, mendicants, an unborn, unproduced, unmade, and unconditioned. If there were no unborn, unproduced, unmade, and unconditioned, then you would find no escape here from the born, produced, made, and conditioned. But since there is an unborn, unproduced, unmade, and unconditioned, an escape is found from the born, produced, made, and conditioned.” Iti 43 Ajātasutta

My initial hypothesis is I think it fits, but rendering it into english how you graft is not completely straightforward. The biggest problem is finding convincing evidence that the statements are one of a logical proof rather than assertions of logical inference. Probably there is a lot to say about this, but I think I can offer that if it is treated as a logical proof, then it does not necessarily involve LEM, but can be interpreted constructively as I spelled out above.

Hope this helps to explain the triple negation elimination :pray:

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No, but I have read his papers about Buddhism and logic with great interest and enthusiasm! I think he was well motivated to study the topic and it makes sense given the prolific usage of the catuskoti tetralemma in the canon to ascribe a paraconsistent or many-valued structure.

What’s interesting about paraconsistent logics is they are in some ways the antithesis of constructive logic. It looks like this:

  • Constructive logic - one truth-value (focuses on proof of construction/existence)
  • Classical logic - two truth values and every proposition must be one or the other
  • Paraconsistent/Many-Valued logic(s) - N truth values where the catoskiti is the specific four-valued logic.

The classical logic has the LEM, but the catuskoti has the 4-valued equivalent. They say that every proposition must be assigned one of the 4-valued values.

Well, the Buddha rather famously refused to acknowledge any of the four-values of his Indian interlocutors thus violating the 4-valued LEM. The way that Graham Priest handles this is to assume that perhaps the Buddha ascribed to some 4+N logic. However, there is another possibility - maybe he ascribed to only constructive (or possibly even only !minimal!) logic?

As a student of Nagarjuna, this interests me as he also rather famously flouts the 4-valued equivalent of LEM for the catuskoti.

I’ve always been drawn to seeing if I can understand the logic of Nagarjuna’s MMK through the Pali canon as these scriptures were my first encounter with Buddhism. As I indicated elsewhere I’m working on a paper describing a particular chapter of Nagarjuna’s MMK as understood through the lens of Pali suttas and this project of evaluating the logical rules the Buddha employed in the Pali canon is so motivated.

I assume that you’ve then read Graham Priest’s numerous articles on Buddhist logic? :pray:


I appreciate your effort to explain the triple negation elimination in intuitionistic logic. I don’t think it’s “intuitive” enough as an explanation though.

However, for me, personally, because I know that’s it’s true, I agree with the triple negation elimination, both in classical logic and constructive (intuitionistic) logic: ¬¬¬P → ¬P

Now, let’s look at your attempt to apply the triple negation elimination to Iti 43:

By saying “unborn would be ¬P”, in intuitionistic logic - BHK interpretation, it means “born would be absurdity”. Also in the language of intuitionistic logic: “a proof of born is the same as proof of 1=0”

The problem does not lies in the “triple negation elimination” when you try to attempt the conversion “proof of contradiction” (LEM) in Iti 43 into “refutation by contradiction” (RAA).

The problem lies instead in the claim “a proof of born is the same as proof of 1=0”

When you do this at the last step, you are effectively conclude that the Buddha said “the born can not be proven”. I don’t think it’s worth the price.

There is more to say here, but I’ll leave that to another time and place. For now I’ll just note this part of the BHK Interpretation that you point out:

It is not, in general, possible for a logical system to have a formal negation operator such that there is a proof of “not” P exactly when there isn’t a proof of P; see Gödel’s incompleteness theorems.

Is in some tension with your last statement that the constructive reading of sutta in question has the Buddha affirming that, "the born can not be proven.” Rather, I think it has the Buddha saying something like, “Assuming the born leads to an illogical conclusion” if we interpret the sutta in question as containing an attempt at a logical proof of the unborn in constructive logic. As I said before I think that isn’t the only possible interpretation.

However, I do think there is an unambiguous case where the Buddha makes a logical proof - not just stating inferences - in the Pali canon where if you take make a constructive reading something like the above is the result, but again another time and place. :pray:

I have not, I’m afraid! My studies in logic were long ago and I haven’t kept up. I should put them on my to-do list.

By the way, Finnigan’s article, which I posted yesterday cites Priest and deals with the logical structures of MN 60.

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Take note that the BHK interpretation is the standard explanation of intuitionistic logic.

Even when you interpret “Assuming the born leads to an illogical conclusion”, it is still a huge price to pay for the 1st Noble Truth about dukkha.

Well, there is another example here: AN 10.76 Tayodhammasutta

“Mendicants, if three things were not found, the Realized One, the perfected one, the fully awakened Buddha would not arise in the world, and the teaching and training proclaimed by the Realized One would not shine in the world. What three? Rebirth, old age, and death. If these three things were not found, the Realized One, the perfected one, the fully awakened Buddha would not arise in the world, and the teaching and training proclaimed by the Realized One would not shine in the world. But since these three things are found, the Realized One, the perfected one, the fully awakened Buddha arises in the world, and the teaching and training proclaimed by the Realized One shines in the world.

  1. The proposition to be proved is P. (which is: “these three things are found”)
  2. Assume ¬P. (which is: “if three things were not found”)
  3. This consequence would be evident. (which is: “the Realized One, the perfected one, the fully awakened Buddha would not arise in the world, and the teaching and training proclaimed by the Realized One would not shine in the world”)
  4. However, this consequence is not evident. (which is: “the Realized One, the perfected one, the fully awakened Buddha arises in the world, and the teaching and training proclaimed by the Realized One shines in the world.”)
  5. Conclude P. (which is: “these three things are found”)

Just wanted to mention, although you appear to have read my undeclared points thread so maybe it’s redundant, but I have found Priest’s and Westerhoffs analysis of the tetralemma very implausible. Take what I think is the simplest example, given already at DN1, of an infinite cosmos:

Q: is the cosmos infinite?:

4 A:
the cosmos is infinite
the cosmos is finite
the cosmos is both finite and infinite (i.e has both finite and infinite parts/characteristics)
the cosmos is neither finite nor infinite (i.e has neither finite nor infinite parts/characteristics)

So, firstly, it appears that interpreted as above, there is really nothing logical involved here, certainly nothing requiring “paraconsistency” or truth values beyond T and F. What is involved is a question about a property in relation to an object, i.e does the object “the cosmos” have the property “infinite” and in the first and third case this question would evaluate as T and in the second and the 4th case it would evaluate as F, and one can evaluate it with any logic one likes, nothing essentially logical is at stake.

I mean, the above question is still a live question in contemporary cosmology, and there is nothing logical about rejecting the third option, for example, on the basis that it somehow violates the law of excluded middle, which in the form above, it absolutely does not.

I am fascinated by the potential of making logic and mathematics compatible with buddhism (as opposed to either nominalism or platonism, the 2 mainstays of philosophy of math currently) and think that type theory has lots of quite intense resonances with buddhist thought.

for example, in type theory types and tier terms must be introduced (arise), must have an elimination rule (cease), and must (at least in homotopy type theory, have a path! very exciting

anyway, very much hope your research arrives on here one day soon, I will be fascinated to check it out!!

Yes, and I don’t object to it at all, rather what I put above with the triple negation elimination depends upon it! I never intended to give the impression that I object to the BHK interpretation.

I’ll leave that to another time and place, but I think at least I understand your objection.

I agree that if this is indeed a logical proof, then when rendered into constructive logic it would fit the same structure as the previous example. However, I do think there is some ambiguity as it can be read as other than a logical proof.

For the purposes of this discussion, I wonder if you would object to either of the two following statements:

  • None of the examples provided involve the use of LEM. That is, if we stipulate that the examples above are logical proofs, and we try to map them onto classical logic instead, they still do not involve proof using LEM.

  • Mapping them onto classical logic instead would still entail the undesirable outcome you see above to the 1st Noble Truth

I don’t think I have! Can you link it? I’d be interested for sure. I’ll wait to read your thread to try and understand your point, but at the outset I’ll say that I found Priest’s work very illuminating and logically consistent. Where I part with Priest is I don’t think the Buddha was adopting the 4-valued logic himself - or a 4+N valued logic as Priest would have it - but rather only using the 4-valued logical premise to reject his interlocutors arguments.

Yes, this is echoing some reminiscent thoughts that I’ve have that led me here to this discussion :slight_smile: I’m sure you are aware there is a very deep connection between constructive logic and everything from computer aided mathematical proof checking, mathematical foundations theory, the Church-Turing thesis, the Godel incompleteness theorems, category theory, the lambda cube, and so much more of modern mathematical/logical machinery. I think it is possible there is a very deep connection as well with sunyata.

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I think Priest is interesting, and first degre entailment is interesting, I just think theres no reason to believe that anything other than “common sense logic” was at play in the undeclared points, they seem to be pre-buddhist, in DN2 they are associated with Sañjaya Belaṭṭhiputta and cover a range of different questions, none of which seem to me to require any other truth values than true or flase, at least if the 3rd and 4th options, where they occur, are treated merologically, which seems to me to agin be perfectly commonsensical.

As I see now, I object to both statements.

As I see, they do involve the use of LEM for “proof by contradiction”. Both “proof by contradiction” and “refutation by contradiction” use “law of non-contradiction”.

In the highlight part below, it’s the “double elimination negation” that makes the distinction between “proof by contradiction” and “refutation by contradiction”.

Intuitionistic logic excludes both “double elimination negation” and “LEM”, that’s why it’s compatible only with “refutation by contradiction” through “law of noncontradiction” that still holds in intuitionistic logic.

Those examples in Iti 43, Iti 42, , AN 3.105, AN 10.76 are “proof by contradiction”. They are not “refutation by contradiction”. As highlighted in the image: “Proof by contradiction is equivalent to the law of the excluded middle”. So, they do involve LEM.

As I see, it is instead: “Mapping them onto intuitionistic logic would entail the undesirable outcome to the 1st Noble Truth by claiming "to prove dukkha is the same as to prove 1=0"

Thank you, that is very clarifying. I appreciate the time you took to explain as well. What is remarkable to me is it seems we are echoing a various famous debate in Madhyamaka buddhism: the Svātantrika–Prāsaṅgika distinction. Huge compendiums of commentary in the Tibetan Buddhist tradition are devoted to this very topic. Many modern western books have been written about it. In this famous debate, the jargon goes by slightly different technical names. The proof by contradiction is called “affirming negation” or “implicative negation.” The refutation by contradiction is called “non-affirming negation” or “non-implicative negation.”

A simple way to understand the Prasaṅgika argument is that by using proof by contradiction aka indirect proof this is what was meant by the Buddha of “holding a view.” On the other hand the Svatantrika believed that by denying proof by contradiction the Prasaṅgika was annihilitionist. This is a simplistic or very brief summary, but I think it gets to the heart of the matter.

What’s amazing is we seem to be echoing that debate based soley on sutta from the Pali canon and using analysis from modern mathematical/logical formalisms. What’s more the controversy of whether sunyata represents just the selflessness of persons or both the selflessness of persons and selflessness of phenomena is orthogonal to this debate.

You seem to be taking the Svatantrika side of the argument while I am taking the Prasaṅgika. Another striking feature here is a meta-level debate that occurred in this famous debate: the two sides argued over whether the distinction was meaningful. The Prasaṅgika side said the distinction was meaningful and that the Svatantrika side’s error was enough to destroy “the Middle Way.” The Svatantrika side claimed that the Prasaṅgika understanding was tantamount to annihilationism. Perhaps I was too quick to dismiss the connection between Madhyamaka and LEM as just a mere slogan.


Hi. Are you suggesting the OP’s definition of LEM was inaccurate, similar to how I also questioned it? It can be difficult for some to comprehend these linguistics.

Hi. How is this related to the Pali Suttas and other Early Buddhist texts?

In reading some Pali Suttas, it was clear to me the Buddha held views, particularly, ‘Right View’, despite the Buddha did not clinging to any views.

In reading the the Pali Suttas, I found the word ‘annihilitionist’ used in ways about views about self. Such as in Brahmajāla Sutta, ‘annihilitionist’ was talked about as the view a self is annihilated at death. And in Acela Sutta, it read as though ‘annihilitionist’ is attributing suffering to another self, possibly such as blaming others for one’s own self-made suffering.

Hi. I do not wish to sound rude but to me it looks like most people here are struggling to follow this topic. I cannot sense any “we” interacting in any debate, at this stage.

Hi. I currently cannot see this is occurring.

In reading Pali Suttas, it reads like there is no “persons” that are selfless. The Pali Suttas say all phenomena are selfless, therefore there is no “person” to be selfless, because what only exists is selfless phenomena. The five aggregates are selfless phenomena ignorantly taken to be a “person” but there is no person. There is only the five aggregates. I did read the word “person” used frequently in the Pali Suttas I read however this word “person” read as though it is just conventional language. As an example, I remember reading one sutta that said of the Buddha: there is one person that arises in the world for the benefit of the many. Obviously, the Buddha is not a real person. The word “person” must just be used here in a very worldly or conventional way. It would probably sound very strange to say: there is one set of five aggregates that arises in the world for the benefit of the many.

I now read Wikipedia: According to Tsongkhapa, Prāsaṅgika asserts that all phenomena are empty of inherent existence or essence, because they are dependently co-arisen with mental imputation. All phenomenon in all possible worlds lack inherent existence and come into existence relative to a designating consciousness which co-arises with that phenomena.

These ideas sound unfamiliar to the Pali Suttas I have read, where the primary subject matter was how suffering originates and how suffering ends. The ending of suffering is described as the ending of craving and not the ending of mental imputation. Setting the Wheel of Dhamma in Motion: And this, monks, is the noble truth of the cessation of stress: the remainderless fading & cessation, renunciation, relinquishment, release, & letting go of that very craving.

If all phenomena are empty of inherent existence or essence it must be the case the assertions of Tsongkhapa & Prāsaṅgika are also empty of inherent existence or essence therefore those empty assertions without essence of Tsongkhapa & Prāsaṅgika mean nothing. This being the case, it is easy to empathize with how the Svatantrika side claimed that the Prasaṅgika understanding was tantamount to annihilationism.

The Middle Way is the Noble Eightfold Path. Setting the Wheel of Dhamma in Motion: Precisely this Noble Eightfold Path: right view, right resolve, right speech, right action, right livelihood, right effort, right mindfulness, right concentration. This is the middle way realized by the Tathagata.

What could be a good subject for debate is if the Indian logistician philosopher Nagarjuna understood or misunderstood the Kaccayanagotta Sutta?

I’ll let Clarity answer that, but in the meantime I wonder if this video might be helpful. There are other videos that also describe intuitionistic logic on youtube, but the one above seems to do a good job IMHO. Just a note: “intuitionistic” is a really unfortunate name that has to do with the particular historical way that this logic came about and much less to do with the actual content. That’s why I prefer to call it constructive logic. :pray:

In so far as the disagreement between Clarity and myself about the specific interpretation of the Pali Suttas discussed above echoes a previous debate, but I wish this Q&A to stay focused on the Pali canon suttas and LEM. Thanks for pointing out the digression.

Not surprising given the formalisms of logic discussed. They are deep subjects in themselves full of unknown jargon to those who have not studied modern logic. I posted a link to a video above hoping it might help. There are a lot of good resources on the web if you wish to learn more about the modern logical formalisms mentioned in this thread. When I said “we” were echoing a debate in a different context in the previous comment it was specifically meant to refer to Clarity and myself. :pray:

The cleansed one has no formulated view
Dhonassa hi natthi kuhiñci loke,
at all in the world about the different realms.
Pakappitā diṭṭhi bhavābhavesu;
Having given up illusion and conceit,
Māyañca mānañca pahāya dhono,
by what path would they go? They are not involved.
Sa kena gaccheyya anūpayo so.

One here who has no wish for either end—
Yassūbhayante paṇidhīdha natthi,
for any form of existence in this life or the next—
Bhavābhavāya idha vā huraṁ vā;
has adopted no dogma at all
Nivesanā tassa na santi keci,
after judging among the teachings.
Dhammesu niccheyya samuggahītaṁ.

For them not even the tiniest idea is formulated here
Tassīdha diṭṭhe va sute mute vā,
regarding what is seen, heard, or thought.
Pakappitā natthi aṇūpi saññā;
That brahmin does not grasp any view—
Taṁ brāhmaṇaṁ diṭṭhimanādiyānaṁ,
how could anyone in this world judge them?

“But does Master Gotama have any convictions at all?”
“Atthi pana bhoto gotamassa kiñci diṭṭhigatan”ti?

“The Realized One has done away with convictions.
“Diṭṭhigatanti kho, vaccha, apanītametaṁ tathāgatassa.

Did you just miss those in your sutta study @Dunlop?

Thank you. I may not be as well read as you. I have not read this Snp.

The link provided is amazing. It includes Pali that says: pakappita past participle 1. considered; designed; arranged; thought over. This does not negate the Right View of the Buddha. Pakappita reads as though it is thought created invented view, including views using worldly logic, such as LEM. It does not read it is Right View according to reality. Maha-cattarisaka Sutta Thus the learner is endowed with eight factors, and the arahant with ten. A Buddha, an Arahant, has Right View.

This apparent error attempting to use Snp 4.3 to say the Buddha had no Right View is sufficient to support the Right View a Buddha has Right View. Snp 4.3 says That brahmin does not grasp any view. A Buddha has Right View, which is not a formulated view and also not grasped. :slightly_smiling_face:

Does diṭṭhigatan mean “convinctions”? This reads very strange. I am sure a Buddha has convictions. The link says diṭṭhigatan means a (false) view, a theory. I might stick to my Access To Insight Suttas.

The verse is relevant to the topic. This verse indicates LEM is an example of formulated view. This verse shows the Buddha did not & could not rely on LEM. The Kalama Sutta says to not rely on logic.