Indian logic

The development of Indian logic can be said to date 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. 2nd century BCE), the tetralemma of Nagarjuna (c. 2nd century CE), and the analysis of inference by Gotama (c. 2nd–3rd century CE), founder of the Nyaya school of Hindu philosophy. Indian logic stands as one of the three original traditions of logic, alongside the Greek and Chinese traditions.

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 5th century BCE, refers to the anviksiki and tarka schools of logic. Pāṇini (c. 5th century BCE) developed a form of logic which had some similarities to Boolean logic 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.

Vaisheshika
Vaisheshika, also Vaisesika, (Sanskrit: वैशॆषिक)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.

Tetralemma
In the 2nd century, the Buddhist philosopher Nagarjuna developed the tetralemma form of logic, also known as catuskoti, though an early precursor to this can be traced back earlier to the Rig-Veda.

Nyaya
Nyaya (pronounced "nyα:yə") is the name given to one of the six orthodox or astika schools of Hindu philosophy &mdash; 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 science and 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.

Navya-Nyaya
The Navya-Nyāya or Neo-Logical darśana (school) of Indian philosophy was founded in the 13th century CE by the philosopher Gangeśa Upādhyāya of Mithila. It was a development of the classical Nyāya darśana. Other influences on Navya-Nyāya were the work of earlier philosophers Vācaspati Miśra (900–980 CE) and Udayana (late 10th century).

Gangeśa's book Tattvacintāmani ("Thought-Jewel of Reality") was written partly in response to Śrīharśa's Khandanakhandakhādya, a defence of Advaita Vedānta, which had offered a set of thorough criticisms of Nyāya theories of thought and language. In his book, Gangeśa both addressed some of those criticisms and – more importantly – critically examined the Nyāya darśana himself. He held that, while Śrīharśa had failed successfully to challenge the Nyāya realist ontology, his and Gangeśa's own criticisms brought out a need to improve and refine the logical and linguistic tools of Nyāya thought, to make them more rigorous and precise.

Tattvacintāmani dealt with all the important aspects of Indian philosophy, logic, set theory, and especially epistemology, which Gangeśa examined rigorously, developing and improving the Nyāya scheme, and offering examples. The results, especially his analysis of cognition, were taken up and used by other darśanas.

Navya-Nyāya developed a sophisticated language and conceptual scheme that allowed it to raise, analyse, and solve problems in logic and epistemology. It systematised all the Nyāya concepts into four main categories: sense or perception (pratyakşa), inference (anumāna), comparison or similarity (upamāna), and testimony (sound or word; śabda).

Influence of Indian logic on modern logic
In the late 18th century, British scholars began to take an interest in Indian philosophy and discovered the sophistication of the Indian study of inference, culminating in Henry T. Colebrooke's The Philosophy of the Hindus: On the Nyaya and Vaisesika Systems in 1824 (in Ganeri, 2001), which provided an analysis of inference and comparison to the received Aristotelian logic, resulting in the observation that the Aristotelian syllogism could not account for the Indian syllogism. Jonardon Ganeri observed that this period was the period in which George Boole and Augustus De Morgan were making their pioneering applications of algebraic ideas to the formulation of logic (such as Algebraic logic and Boolean logic), and suggested that these figures were likely to be aware of these studies in xeno-logic, and further that their acquired awareness of the shortcomings of traditional logic are likely to have stimulated their willingness to look outside the system.

As a parallel, mathematicians are now aware of the influence of Indian mathematics on the European. For example, Hermann Weyl wrote:
 * "Occidental mathematics has in past centuries broken away from the Greek view and followed a course which seems to have originated in India and which has been transmitted, with additions, to us by the Arabs; in it the concept of number appears as logically prior to the concepts of geometry." (Weyl, 1929)