Gautama, also known as Aksapada and Dirghatapas, is the author of the Nyayasutra and the founder of the Nyaya school. He is credited with formulating the principles of Hindu logic and establishing it on a firm foundation. In fact, the methods delineated by him are used in all schools of philosophy including Vedanta.
Gautama’s ideas were explained and systematized by Vatsyayana in his Nyayasutrabhashya. Uddyotakara’s (c. 500—600 CE) Nyayavarttika is devoted almost exclusively to logic and epistemology in an attempt to refute the Buddhist Dignaga’s logic. The Nyaya School, almost from its inception, was in conflict with Buddhist logicians and their epistemological theories, and Uddyotakara was greatly influenced by and incorporated Dignaga’s theories into the Nyaya system.
A subsequent Buddhist attack, this time from Dharmakirti (600 CE), arguing that the Nyaya and Vaishesika proofs for the existence of god (Ishvara) were logically fallacious, resulted in Udayana (900—1000 CE) formulating the first systematic account of Nyaya theism, where he gave a rational enquiry into the existence of God .
The Nyaya system is an elaboration of sixteen philosophical topics (padartha) as elucidated in the opening verse of the Nyayasutra which states that supreme bliss is attained by knowledge of the true nature of the sixteen categories: means of correct knowledge (pramana), objects of correct knowledge (prameya), doubt (sanshaya), purpose (prayojana), familiar example (drishtanta), established conclusion (siddhanta), members of a syllogism (avayava), hypothetical argument (tarka), decisive knowledge (nirnaya), discussion for truth (vada), controversy (jalpa), destructive arguments (vitanda), fallacies (hetvabhasa), quibbling (chala), specious objections (jati), and vulnerable points (nigrahasthana). Nyaya teaches that ignorance is the root of all suffering and rebirth and that only complete knowledge of the true nature of things will bring deliverance (apavarga). Salvation is the supreme goal of life, and Nyaya employed logic and epistemology to not only to know reality but also to correct false, fallacious, and sentimental fallacies.
A new school of Nyaya known as Navya Nyaya (new logic) developed in the twelfth century. Unlike the earlier works which had concentrated on an elucidation of the categories as enumerated in the Nyayasutra, this school put an emphasis on the valid means of knowledge (pramana) and did not concern itself with metaphysics. The Tattvacintamani of Gangesa is the major work of this school and the basis upon which all later developments derived. With it, the Navya Nyaya School developed not only a highly complex epistemology with its own technical language but also initiated a unique style of philosophical writing in India that is noted not only for its brevity but also for its precision.