Variously formulated as saying that no proposition can be both true and not true; or that nothing can be - without qualification - the case and not the case at the same time; or that nothing can -without qualification - both have and lack a given property at the same time.
The law cannot be logically proved without begging the question, though arguments of a different kind (among those called transcendental arguments) have been offered in its defence since Aristotle (384-322 BC) in his Metaphysics (book 4, chapter 4).
However, recently a notion of dialetheism has been defended which allows breaches of the law in certain cases.
Also see: paraconsistency
G Priest, 'Contradiction, Belief and Rationality', Proceedings of the Aristotelian Society (1985-86)