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

I’m looking for the best examples in the Pali canon where it can be conjectured that the Buddha uses a logical proof that relies upon the Law of the Excluded Middle to reach some conclusion.

Specifically, I’m looking for any/all examples in the canon where the Buddha draws some conclusion about the truth value of some proposition based on logically analyzing the proposition while employing LEM in the proof.

EDIT:

To clear up a misunderstanding I’m not looking for examples where the Buddha used Reductio Ad Absurdum (RAA) nor am I looking for examples where the Buddha made the proposition (P ^ ~P), but rather where the Buddha specifically used LEM as in classical logic as proof of some positive truth.

I’m looking for examples in the Pali canon where the Buddha is specifically using examples of classical logic (which uses LEM) in a way that would run afoul of constructivist logic which does not allow indirect proof, but does allow RAA.

Some terms:

• Reductio Ad Absurdum (RAA): Involves assuming a proposition P, and from this assumption, deriving a contradiction (typically of the form Q∧¬Q). The presence of this contradiction allows us to conclude ¬P. This is RAA and does not rely upon LEM as an axiom.

• Indirect Proof which implies LEM: This method starts by assuming ¬P and then deriving a contradiction (again, often of the form (Q∧¬Q). The contradiction leads to the conclusion that P must be true.

To make this very simple:

1. Assume P, find a contradiction or absurdity and conclude not PRAA
2. Assume not P, find a contradiction or absurdity and conclude PLEM

I’m looking for cases of #2 and not for #1.

Thank you !

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It can be seen in MN 117 Mahācattārīsakasutta, I highlighted the 1 part out of 10 parts that he used LEM:

So there are twenty on the side of the skillful, and twenty on the side of the unskillful. This exposition of the teaching on the Great Forty has been rolled forth. And it cannot be rolled back by any ascetic or brahmin or god or Māra or Brahmā or by anyone in the world.

If any ascetic or brahmin imagines they can criticize and reject the exposition of the teaching on the Great Forty, they deserve rebuke and criticism on ten legitimate grounds in the present life. If such a gentleman criticizes right view, they praise and honor the ascetics and brahmins who have wrong view. If they criticize right thought … right speech … right action … right livelihood … right effort … right mindfulness … right immersion … right knowledge … right freedom, they praise and honor the ascetics and brahmins who have wrong freedom. If any ascetic or brahmin imagines they can criticize and reject the exposition of the teaching on the Great Forty, they deserve rebuke and criticism on these ten legitimate grounds in the present life.

Even those wanderers of the past, Vassa and Bhañña of Ukkalā, who taught the doctrines of no-cause, inaction, and nihilism, didn’t imagine that the Great Forty should be criticized or rejected. Why is that? For fear of blame, attack, and condemnation.

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Namo Buddhaya!

Maybe this will do

https://www.accesstoinsight.org/tipitaka/mn/mn.035.than.html

Saccaka makes a proposition and contradicts it when cross-questioned and thus the proposition is established as false by contradiction.

I think that is indeed a use of Reductio Ad Absurdum (RAA) which is strictly speaking not the same as LEM. I’ll edit my question to be more precise. Thank you

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Hmm, I don’t think this qualifies. Going to edit my post to hopefully make it clear. Thanks

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Alright. How about another try at AN 3.61 Titthāyatanasutta and also MN 76 Sandakasutta?

Those are all straightforward cases of RAA.

Consider:

1. Assume that “everything that is experienced is because of past deeds” … find a problem or contradiction “In that case, you might kill living creatures, steal, be unchaste; use speech that’s false, divisive, harsh, or nonsensical; be covetous, malicious, or have wrong view, all because of past deeds.” and conclude it isn’t the case that “everything that is experienced is because of past deeds” → straightforward application of RAA and not LEM

2. Assume that “everything that is experienced is because of God Almighty’s creation” … find a problem or contradiction “In that case, you might kill living creatures, steal, be unchaste; use speech that’s false, divisive, harsh, or nonsensical; be covetous, malicious, or have wrong view, all because of God Almighty’s creation.” and conclude it isn’t the case that “everything that is experienced is because of God Almighty’s creation” → again, that is RAA

3. Assume that “everything that is experienced has no cause or reason” … → follows the same pattern of RAA.

I’m looking for Indirect Proof or LEM which would look like:

Assume “not A” … find a problem or contradiction and then conclude “therefore A!”

That’s a positive affirmation of the truth of some proposition based on finding a problem or contradiction with its opposite → indirect proof based on LEM

This would be surprising since the logic used at the time and place of the Buddha seems to be paraconsistent. I.e., they seem to have considered statements like “The Tathagata both exists and does not exist after death” as statements that could be true or false, meaning their formal logic didn’t have the law of excluded middle as in classical logic (AFAIK)

On the other hand, when the Buddha uses similes like the simile of the raft, he doesn’t say stuff like “a raft is for crossing over, and not for crossing and not crossing” – so IMO maybe the law of excluded middle is implied as a part of common-sense reasoning.

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Oh well, you can refer to the famous Anattalakkhaṇasutta SN 22.59 then.

Also, you can refer to RAA as “Proof by contradiction” while LEM as “refutation by contradiction”. There is a well explained wiki article here.

Are you trying to prove that the Buddha didn’t use “Intuitionistic logic”? It seems he didn’t go into that direction of logic.

Indeed. An excellent overview of the Buddha’s logical system can be found here (The Journal of Comparative Philosophy, 2010)

No, this is not the Buddha’s logical system. The Buddha had never presented this system at all.

While the tetralemma as found in the Indian logics at the time of the Buddha would undoubtedly seem to be paraconsistent, I am not aware of any place in the Pali canon where the Buddha actually concluded a proposition was in a paraconsistent truth class. That is, I think he accepted the tetralemma as a basis for discussion, but did not affirmatively explain anything in a paraconsistent way making use of a ‘true contradiction’ etc at least not in the Pali canon. I’d be very interested to find examples refuting this.

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I fail to see how this is indirect proof. It looks to me a straightforward use of RAA or ‘refutation by contradiction’ in your nomenclature.

Actually, my current hypothesis is that he only employed constructive logical proofs. That is, he did not affirm anything via indirect proof and would have been quite comfortable fitting into intuitionistic logic (which I call constructive logic). That is why I’m looking for examples in the Pali canon that challenge this hypothesis.

NOTE: This is a much narrower hypothesis than Graham Priest’s contention about the Buddha employing full on paraconsistent logic. Constructive logic is a strict subset of classical logic not a superset like the paraconsistent logics.

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Yeah, as I’m writing I can’t remember any use of an explicit logical argument, paraconsistent or otherwise.

There’s stuff like in MN 76:

But when a teacher relies on logic and inquiry, some of that is well reasoned, some poorly reasoned, some true, and some otherwise.

This seems at least informally to be “P or not P”, P being ‘well-reasoned’.

I can’t recall any instant of the Buddha saying something like “well reasoned and not well reasoned, neither well reasoned nor not well reasoned” in any other example. He doesn’t seem too keen on making formal logical arguments, AFAIK.

Edit: As in the Buddha using paraconsistent logic to make an argument.

I think the canon is replete with instances of the Buddha refuting others using logical arguments of the ‘refutation by contradiction’ kind, but no not a lot of him affirmatively giving a logical proof of the truth of P. Usually his way of knowing is direct experience.

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Alright. Let’s look at the wiki article that I have given:

Now compare with what you explained in the 1st post:

Sorry for the confusion, we should have now.:

• RAA is “refutation by contradiction”
• LEM is “proof by contradiction”

So, now we look at Iti 43 Ajātasutta:

“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.”

The proposition to be proved is P. (which is: “There is, mendicants, an unborn, unproduced, unmade, and unconditioned”)
Assume ¬P. (which is: “If there were no unborn, unproduced, unmade, and unconditioned”)
Derive falsehood (which is: “then you would find no escape here from the born, produced, made, and conditioned”)
Conclude P. (which is: “There is, mendicants, an unborn, unproduced, unmade, and unconditioned”)

Hopefully, this is the one you are looking for.

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Why are you seeking this knowledge and what purpose is this seeking of such serving?

Hmm, this is very interesting case! Thank you for bringing it to attention. Upon consideration though I’m thinking this is just RAA and not Indirect proof. Here’s why:

We can think of ‘born’ and ‘unborn’ as forming a disjoint set and hence mutually exclusive. In this case we can say that ‘born’ is P and hence unborn would be ¬P where (P∧¬P) is logical inconsistency.

Then we have:

1. The proposition to be proved is ¬P.
2. Assume ¬¬P.
3. Derive falsehood.
4. Conclude ¬¬¬P.
5. ¬¬¬P → ¬P
6. Conclude ¬P

None of the above used LEM and did not involve indirect proof. I’m checking to make absolutely sure, but this post gives me some confidence that the above is correct.

But your example was really good and I think is the strongest case yet, but fortunately I think this still can be interpreted to be constructive logic not a case of the Buddha using indirect proof.

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This is a fascinating topic and one worthy of much further study! Good on you for doing this!! Please write up your research for us !!!

Metta

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May i ask what purpose your gatekeeping of other peoples research serves?

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