What do the Nyaya Sutras say about the existence of God? Does it believe in the existence of God or deny it?
The Nyaya Sutras accept the existence of God.. Here is what Gautama says in Adhyaya 4 Pada 1 of the Nyaya Sutras (from Ganganatha Jha's translation):
- īśvaraḥ kāraṇaṁ puruṣakarmāphalyadarśanāt
God is the cause; because we find fruitlessness in the actions of Men.
- na puruṣakarmābhāve phalāniṣpatteḥ
It is not so; because as a matter of fact, no fruit appears without man's action.
Inasmuch as it is influenced by Him, there is no force in the reason (put forward).
Now this translation is written from the viewpoint of Vatsyayana's Nyaya Sutra Bhashya, but all commentators on the Nyaya Sutras agree that these Sutras assert the existence of God. But different commentators differ on the precise interpretation of these Sutras, as the translator says in a footnote:
In regard to this section there is difference among Commentators. According to the Bhashya, the Vartika and Vishwanatha, it is meant to propound the Naiyayika Siddhanta that the Universe has been created by God; and in accordance with this view, Su. 19 puts forward the final Siddhanta. Su. 20 puts forward an objection against the Siddhanta and Su. 21 answers that objection from the stand-point of the Siddhanta. It is this interpretation that we have adopted in the translation.... The Tatparya, followed by the Parishudhhi and Prahasha, has taken it as representing the criticism of the Vedanta doctrine that "God is the constituent cause of the Universe." By this interpretation Su. 19 represents the Vedanta view, Su. 20 shows the untenability of that view, and Su. 21 puts forward the final Nyaya-Siddhanta that God is the creator, the operative cause, not the constituent cause, of the Universe.
So as you can see, some commentators think that in these Sutras Gautama discusses the question of whether God exists, and he concludes that a supreme being does exists. Other commentators think that in these Sutras Gautama discusses the question of whether God is both the efficient cause and the material cause of the Universe, or just the efficient cause, and he concludes that the supreme being is just the efficient cause. But either way, all commentators are agreed that Gautama's Nyaya Sutras endorse the existence of God.
On a side note, out of the six Astika schools, only Kapila's Samkhya school and Jaimini's Purva Mimamsa school don't unequivocally accept the existence of God. The Samkhya school rejects God outright, whereas in the case of the Purva Mimamsa school some of its adherents accepted the existence of God and others held agnostic or atheistic views. On the other hand, Patanjali's Yoga school, Gautama's Nyaya school, Kanada's Vaisheshika school, and Vyasa's Vedanta school all accept the existence of God.
It may also be worth noting the differences between the Nyaya school and the Vedanta school on the nature of God, since most Hindus today belong to the Vedanta school. As discussed above, Nyaya school believed that God is only the efficient cause of the Universe, whereas most members of the Vedanta school (other than Madhwas) believes that God is both the efficient cause and the material cause of the Universe. Also, the Nyaya school believed that the existence of God can be proven through logical arguments, whereas the Vedanta school believes that the existence of a supreme being requires scripture to establish, as I discuss here. These two differences are not unconnected; if you try to prove the existence of God through logic, then the kind of God that you'll produce is one who is only the efficient cause of the Universe. That is why most religious people outside the Vedanta school, both inside and outside of Hinduism, whether Naiyayikas, Vaisheshikas, Shaiva Siddhantins, Christians, Muslims, etc. all believe that God is only the efficient cause of the Universe. Whereas the Vedanta school, since it says that Brahman is beyond logical argument and the other non-scriptural means of knowledge, arrives at a very different conception.