The development of Indian logic dates back to the anviksiki of Medhatithi Gautama (c. 6th century BCE); the Sanskrit grammar rules of Pāṇini (c. 5th century BCE); the Vaisheshika school's analysis of atomism (c. 6th century BCE to 2nd century BCE); the analysis of inference by Gotama (c. 6th century BC to 2nd century CE), founder of the Nyaya school of Hindu philosophy; and the tetralemma of Nagarjuna (c. 2nd century CE).
Indian logic stands as one of the three original traditions of logic, alongside the Greek and the Chinese logic. The Indian tradition continued to develop through early to modern times, in the form of the Navya-Nyāya school of logic.
Origins
The Nasadiya Sukta of the Rigveda (RV 10.129) contains ontological speculation in terms of various logical divisions that were later recast formally as the four circles of catuskoti: "A", "not A", "A and 'not A'", and "not A and not not A".
Medhatithi Gautama (c. 6th century BCE) founded the anviksiki school of logic. The Mahabharata (12.173.45), around the 4th century BCE to 4th century CE, refers to the anviksiki and tarka schools of logic. (c. 5th century BCE) developed a form of logic (to which Boolean logic has some similarities) for his formulation of Sanskrit grammar. Logic is described by Chanakya (c. 350-283 BCE) in his Arthashastra as an independent field of inquiry anviksiki.
The schools
Vaisheshika
Vaisheshika, also Vaisesika, (Sanskrit: वैशेषिक) is one of the six Hindu schools of Indian philosophy. It came to be closely associated with the Hindu school of logic, Nyaya. Vaisheshika espouses a form of atomism and postulates that all objects in the physical universe are reducible to a finite number of atoms. Originally proposed by Kanāda (or Kana-bhuk, literally, atom-eater) from around the 2nd century BCE.
Catuskoti
In the 2nd century, the Buddhist philosopher Nagarjuna refined the Catuskoti form of logic. The Catuskoti is also often glossed Tetralemma (Greek) which is the name for a largely comparable, but not equatable, 'four corner argument' within the tradition of Classical logic.
Nyaya
Nyāya (ni-āyá, literally "recursion", used in the sense of "syllogism, inference") is the name given to one of the six orthodox or astika schools of Hindu philosophy — specifically the school of logic.
The Nyaya school of philosophical speculation is based on texts known as the Nyaya Sutras, which were written by Gotama in around the 2nd century CE. The most important contribution made by the Nyaya school to modern Hindu thought is its methodology. This methodology is based on a system of logic that has subsequently been adopted by most of the other Indian schools (orthodox or not), much in the same way that Western philosophy can be said to be largely based on Aristotelian logic.
Followers of Nyaya believed that obtaining valid knowledge was the only way to obtain release from suffering. They therefore took great pains to identify valid sources of knowledge and to distinguish these from mere false opinions. According to the Nyaya school, there are exactly four sources of knowledge (pramanas): perception, inference, comparison and testimony. Knowledge obtained through each of these can, of course, still be either valid or invalid. As a result, Nyaya scholars again went to great pains to identify, in each case, what it took to make knowledge valid, in the process creating a number of explanatory schemes. In this sense, Nyaya is probably the closest Indian equivalent to contemporary analytic philosophy.
Jain logic
Jainism made its own unique contribution to this mainstream development of logic by also occupying itself with the basic epistemological issues, namely, with those concerning the nature of knowledge, how knowledge is derived, and in what way knowledge can be said to be reliable. Jain logic developed and flourished from 6th century BCE to 17th century CE. According to Jains, the ultimate principle should always be logical and no principle can be devoid of logic or reason. Thus one finds in the Jain texts, deliberative exhortations on any subject in all its facts, may they be constructive or obstructive, inferential or analytical, enlightening or destructive. The Jains have doctrines of relativity used for logic and reasoning: Anekāntavāda – the theory of relative pluralism or manifoldness; Syādvāda – the theory of conditioned predication and; Nayavāda – The theory of partial standpoints.
These Jain philosophical concepts made most important contributions to the ancient Indian philosophy, especially in the areas of skepticism and relativity.