Nyaya begins from a practical problem: two philosophers disagree, and there has to be a way to settle it other than shouting. The Nyaya Sutra, attributed to Gautama Akshapada, answers with a formal structure for debate built around sixteen categories, covering everything from the topic under dispute to the fallacies that disqualify a move. Its best known device is the five membered syllogism: state the thesis, there is fire on the hill, give the reason, because there is smoke, offer an accepted example, as in a kitchen, apply the reason to the present case, and this hill has smoke of that kind, then draw the conclusion, therefore there is fire on the hill. Every step is checkable by an opponent.
Behind the method sits a theory of knowledge. Nyaya recognizes four independent sources, perception, inference, comparison and testimony, and works out exactly when each can be trusted. Nyaya used that apparatus for large claims: a creator god exists, the self is a substance distinct from the body, the Veda counts as reliable testimony, and each claim faced determined opponents ready to attack the reasoning behind it.
By the eleventh century Nyaya's logic had absorbed Vaisheshika's inventory of the world's basic categories, producing a combined Nyaya-Vaisheshika system. In the thirteenth or fourteenth century the philosopher Gangesha rebuilt the logical core in his Tattvachintamani, the Jewel of Thought, founding what later scholars call Navya-Nyaya, the New Logic, whose precise technical definitions were adopted even by the Buddhist logicians Nyaya had argued against for centuries.