The Nyaya school of philosophical speculation is based on a text called the Nyaya Sutra. It was written by Gautama (not to be confused with the founder of Buddhism), also known as Aksapada, round about the fourth or fifth century B.C. The most important contribution made by this school is its methodology. This 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, religion and philosophy can be said to be largely based on Aristotelian logic.
But Nyaya is not merely logic for its own sake. Its followers 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, and the Nyaya scholars again went to great pains to identify, in each case, what it took to make knowledge valid, in the process coming up with a number of explanatory schemes. In this sense, Nyaya is probably the closest Indian equivalent to contemporary Western analytical philosophy. But we should never lose sight of the fact that the Nyaya sages performed their labours for a specifically religious purpose.
See also Hinduism