The two words, anvaya and vyatireka, are frequently used in Indian works of philosophy and logic to prove a point. They invariably go together. Anvaya is a positive statement indicating a universal agreement whereas vyatireka is the opposite of it, proving the same point by indicating the universal absence of the contrary.
To illustrate: Smoke is seen on the yonder hill. We conclude that there is fire on that hill. This conclusion is based on two premises from our previous experience:
Of these, the first one is a positive statement, asserting the presence of fire wherever there is smoke. Hence it is called anvaya-vyāpti, or simply anvaya. The second statement asserts a negative factor, the universal absence of smoke when there is no fire. This is called vyatireka-vyāpti or vyatireka (negation or contrariety). To prove the existence of fire on the hill on the evidence of smoke, both these methods of argument are necessary.