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:
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)
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)
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)
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)
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)
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.