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Buddhist logic is a branch of Indian logic (see also Nyaya), one of the three original traditions of logic, alongside the Greek and the Chinese logic. It seems Buddha himself used some of the features of such a non-standard logic in his philosophical reasoning which makes it important from a philosophical perspective. (see also Trairūpya and Hetucakra) However, in this post, I am going to tackle its mathematical aspects rather than philosophical ones.

I came across the term "Buddhist logic" in a personal discussion with an Indian fellow of analytic philosophy background who asked me whether Buddhist/Indian logic could have any applications in the modern (predominantly Greek-logic based) mathematics/physics. Eventually, we came up with the idea that in the absence of a robust formalism and a fully clarified set of axioms for any such "non-standard" logic, there is not much to say about its mathematical properties and (dis)advantages in comparison with other logics, let alone finding any actual applications in daily mathematics, computer science or theoretical physics.

My initial search revealed some intriguing pieces of information about Buddhist and Indian logic/mathematics which motivated me to go through a more thorough investigation of the topic:

  1. In the following papers, Graham Priest discusses the connection between some features of Buddhist logic such as Catuṣkoṭi and paraconsistent logic. He also provides some formalism for Jania logic a variant of Indian logic corresponding to Jainism.

    • Priest, Graham, None of the above: the Catuṣkoṭi in Indian Buddhist logic. New directions in paraconsistent logic, 517–527, Springer Proc. Math. Stat., 152, Springer, New Delhi, 2015. (MR3476967)

    • Priest, Graham, Jaina logic: a contemporary perspective. Hist. Philos. Logic 29 (2008), no. 3, 263–278. (MR2445859)

  2. Douglas Daye has published a series of papers investigating some issues with formalizing Buddhist logic. (See also his follow-up papers: MR0536102 and MR0547735)

    • Daye, Douglas Dunsmore, Metalogical incompatibilities in the formal description of Buddhist logic (Nyāya). Notre Dame J. Formal Logic 18 (1977), no. 2, 221–231. (MR0457129)
  3. Prior to Daye, there has been some effort by Staal in the direction of formalizing Buddhist logic:

    • Staal, J. F. Formal structures in Indian logic. Synthese 12 1960 279–286. (MR0131338)
  4. Also, some connections between Indian logic and more applied areas of research such as computer science has been investigated:

    • Sarma, V. V. S., A survey of Indian logic from the point of view of computer science. Sādhanā 19 (1994), no. 6, 971–983. (MR1362512)
  5. There are several references to logic in the context of Vedic mathematics including in Vyākaraṇa where a scholar named Pāṇini has developed a grammar which "makes early use of Boolean logic, of the null operator, and of context-free grammars, and includes a precursor of the Backus–Naur form (used in the description programming languages)" (cf. Wikipedia)

  6. Last but not least, it is worth mentioning that various set-theoretic concepts including the notion of infinity (see the Sanskrit terms Ananta and Purna) has a strong presence in the Buddhist/Indian logic literature where they make a significant distinction between various types (and sizes?) of infinities (i.e. Nitya, Anitya, Anadi, and Anant) in a terminology fairly similar to modern treatment of ordinals. According to Staal, "Rig Veda was familiar with the distinction between cardinal numbers and ordinal numbers". There is also a very detailed treatment of transfinite and infinitesimal numbers in Jain mathematics as well as a variant of cardinal arithmetic and Hilbert's Hotel Paradox in Isha Upanishad of Yajurveda where it starts with the following verses: "That is infinite, this is infinite; From that infinite this infinite comes. If from that infinite, this infinite is removed or added then infinite remains infinite." Along these lines is the classical book Tattvacintāmaṇi which "dealt with all the important aspects of Indian philosophy, logic, set theory, and especially epistemology, ..." (cf. Navya-Nyāya).

I am not quite sure if this short list is comprehensive and satisfactory enough to grasp the whole mathematical theory behind Buddhist logic, particularly because I am not of some Buddhist background to be fully aware of terminology (which is a crucial factor in the searching process). Also, most of the mentioned articles deal with the topic only marginally or from a slightly philosophical perspective. So I am looking for the possible help of some native Indian mathematicians or expert logicians who have worked along these lines.

Question. What are some other examples of papers on Buddhist/Indian logic in which the axioms and properties of these logics are formally investigated?

I am particularly interested in rigorous mathematical papers (rather than historical and philosophical ones) in which some theorems about their logical properties such as expression power, syntax, semantics, number of values, compactness, interpolation, etc., have been discussed (in a possibly comparative way). Any reference to possible implications of these formal systems in the foundation of mathematics especially in connection with the concept of infinity is welcome.

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    $\begingroup$ this is an intriguing question, but you give already a quite extensive list of references, including formal treatments of Indian logic, it seems you have answered the question yourself... $\endgroup$ – Carlo Beenakker Jun 5 '18 at 19:47
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    $\begingroup$ Why the close votes on this question? It is written in such a polished manner, and is easily above par, please don't close. $\endgroup$ – Suvrit Jun 6 '18 at 11:44
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    $\begingroup$ @Suvrit: I didn't vote to close, but writing in a polished manner (or in a Polish manner, for that matter) is not an argument for being on-topic. I think that this might be more relevant to Philosophy rather than to MathOverflow. $\endgroup$ – Asaf Karagila Jun 6 '18 at 14:10
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    $\begingroup$ @AsafKaragila I am not using the polishedness merely as the reason to keep it open; clearly it displays the huge amount of effort the OP put into writing the question, I learned many things from the question itself, and this question is as good as or even better than numerous other related questions on MO (many of which one would also call "more relevant to either history, or academia, or CS, or something else). Thus, I added the "and easily above par" to indicate this additional qualification of this question beyond the polishedness. $\endgroup$ – Suvrit Jun 6 '18 at 14:28
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    $\begingroup$ @Suvrit: I guess that this is where we differ. I still think that this is borderline on topic here, and do believe it would have been "more on topic" on Philosophy.SE. I agree that it is clear that a lot of work has been put into this, yes. $\endgroup$ – Asaf Karagila Jun 6 '18 at 14:31
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It seems that some aspects of Buddhist Logic are formalised using Martin-Löf Type Theory in Kuen-Bang Hou's (Favonia) PhD thesis.

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    $\begingroup$ (+1) Nice reading! Thanks for sharing, Michael! This is really one of those hard references to find, particularly because this certain connection between concepts in Buddhist logic and theorems in type theory is not mentioned in a separate article with Buddhist and type theory keywords in its title or classification, so it barely appears in the normal searches. $\endgroup$ – Morteza Azad Jun 7 '18 at 1:03
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    $\begingroup$ Let me also mention that the excerpt, "When will a banana rot enough that it is no longer a banana? Similarly, how much do I have to disassemble a chair so that it is no longer a chair? Various Buddhist schools hold the view that all conventional objects are interdependent and interconnected, and that any attempt to draw a line between bananas and no-bananas or chairs and no-chairs is misguided at best", reminds me Sorites Paradox. It is somehow suggesting that the paradox arises due to the emptiness of the notion of "heap". Interesting! $\endgroup$ – Morteza Azad Jun 7 '18 at 1:06
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    $\begingroup$ @MortezaAzad is the "banana" example arising because of a mixing of "physical" objects with "mathematical" ones? Does it not boil down to a test of equality, something that should be impossible in "physics" due to quantum stuff, uncertainty, etc.; whereas, defining equality is more reasonable in math? Just some naive thoughts... $\endgroup$ – Suvrit Jun 13 '18 at 13:33
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Perhaps this paper is of interest.

MR3620856 Bilimoria, Purushottama, Thinking negation in early Hinduism and classical Indian philosophy. Log. Univers. 11 (2017), no. 1, 13–33.

Summary: "A number of different kinds of negation and negation of negation are developed in Indian thought, from ancient religious texts to classical philosophy. The paper explores the Mīmāmsā, Nyāya, Jaina and Buddhist theorizing on the various forms and permutations of negation, denial, nullity, nothing and nothingness, or emptiness. The main thesis argued for is that in the broad Indic tradition, negation cannot be viewed as a mere classical operator turning the true into the false (and conversely), nor reduced to the mainstream Boolean dichotomy: 1 versus 0. Special attention is given to how contradiction is handled in Jaina and Buddhist logic.''

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  • $\begingroup$ (+1) This seems a very interesting article! Thanks for introducing, Gerry! It somehow reminds me the failure of double negation elimination in intuitionistic and minimal logic in contrary to their classical counterpart. $\endgroup$ – Morteza Azad Jun 6 '18 at 8:47
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Maybe useful: Indian Logic by J.N. Mohanty et alii; Ch.18 of Leila Haaparanta (editor), The Development of Modern Logic, Oxford UP (2009).

See also : Indian Logic by J.Ganeri, into : Dov Gabbay & John Woods (editors), Handbook of the History of Logic. Volume 1: Greek, Indian and Arabic Logic, Elsevier (2004).

An older one is : The Indian Variety of Logic; Part VI of Joseph Maria Bochenski, A History of Formal Logic, German ed.1956, English transl. Univ.of Notre Dame Press (1961).

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You may find this is also interesting. Syed Nizar Alam seems to be interested in the sort of logics you mention and a description of his recent research is to be found at

juniv.edu/teachers/nizaralam

I think a copy of his MSc thesis should be obtained by contacting the author directly (I was unable to find a copy myself).

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    $\begingroup$ (+1) Thank you very much! I would like to draw the audience's attention to the description of his Ph.D. project as well: "I am currently enrolled for the doctoral degree at the University of Canterbury, and investigating non-western modes of rationality, particularly the Buddhist logic system called 'Catuṣkoṭi', which offers four possibilities of a situation – it can be true, false, both true and false simultaneously, or neither- to develop a formal logical calculus." $\endgroup$ – Morteza Azad Jun 7 '18 at 13:54
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The 2015 World Congress on Logic and Religion -- which mentions the Bilimoria paper -- also lists a talk "The Logical Nature of the Nalanda Tradition of Buddhism" by Razvan Diaconescu of Simion Stoilow Institute of Mathematics of the Romanian Academy (IMAR), Romania. Abstract:

In contemporary Tibetan Buddhism, which inherits the old Indian Buddhist tradition of Nalanda in a rather complete way, logic plays an active role at various levels.

The importance of logic in the process of spiritual development of Buddhist practitioners can be traced back to the central methodological principle formulated by Buddha Shakyamuni himself, that any truth can be accepted only upon an extensive and careful analysis performed on a personal basis. In this talk we will explore the role played by logic in the Mahayana Buddhist thinking, both from a historical and methodological perspective and will discuss possible captures of Buddhist logics as modern formal logical systems.

We will also briefly look at the relationship between logic and Buddhist thinking from the other side, namely some influence of the Buddhism perspective to modern logic, especially to the universal trend in logic.

Although I'm not sure if this was ever published.


edit:

The Stanford Encyclopedia of Philosophy entry on paraconsistency says this:

[definition:] A logical consequence relation is explosive if according to it any arbitrary conclusion $B$ is entailed by any arbitrary contradiction $A$, $¬A$ (ex contradictione quodlibet (ECQ))

...

In the history of logic in Asia, there is a tendency (for example, in Jaina and Buddhist traditions) to consider the possibility of statements being both true and false. Moreover, the logics developed by the major Buddhist logicians, Dignāga (5th century) and Dharmakīrti (7th century) do not embrace ECQ. Their logical account is, in fact, based on the ‘pervasion’ (Skt: vyāpti, Tib: khyab pa) relation among the elements of an argument. Just like the containment account of Abelard, there must be a tighter connection between the premises and conclusion than the truth-preservation account allows. For the logic of Dharmakīrti and its subsequent development, see for example Dunne 2004 and Tillemans 1999.

With the SEP bibliography here for convenience:

Dunne, John D., 2004, Foundations of Dharmakīrti’s Philosophy, Boston: Wisdom Publications.

Tillemans, Tom J.F., 1999, Scripture, Logic, Language: Essays on Dharmakīrti and His Tibetan Successors, Boston: Wisdom Publications.

The Tillemans book is a collection of previously published essays. Of note is Chapter 6, which the book says

Chapter 6: originally published as "Formal and Semantic Aspects of Tibetan Buddhist Debate Logic." Journal of Indian Philosophy 17 (1989): 265-97. Some corrections have been made. The account of vyāpti has been taken up again and revised in the introduction to the present book.

It seems you can find the original paper online without too much hassle, and this paper in particular seems to be the kind of reference asked about.

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