Both Nyaya and Samkhya believe in non eternal vedas. What are their arguments against eternally of the Vedas.
The validity and authority of the Vedas were acknowledged by all six Darshanas. But what is the difference?
Mimamsa and Vedanta: Eternal and authorless. They were not written by anyone, and had no beginning in time nor end and were eternal. Their authority was not derived from the authority of any trustworthy person or even Ishvara (Saguna Brahman/God).
Nyaya/Vaishesha: Non-eternal and authored by Ishvara
Yoga/Samkhya: Non-eternal and authorless.
The question asked is twofold. We cannot combine Nyaya and Yoga and answer the question because these schools are entirely different and claim authorship of vedas differently. Therefore, herein we discuss why Nyaya school claims that the Vedas are non-eternal and authored by Ishvara. Because why Yoga/Samkhya school claims the vedas are authorless is discussed elsewhere.
The Nyaya sutras attributed to Gautama and the Vaishesha sutras attributed to Kanada represent the same system for all practical purposes. These tacitly admit Vedas as a pramana. Verses VI. i. 1, and VI. i. 3, of the latter says the validity of the scriptures depends upon the trustworthy character of their author. All knowledge derived from the Vedas is valid, for the Vedas were uttered by Isvara himself. The Vedas give us right knowledge not of itself, but because they came out as the utterances of the infallible Isvara.
This cause is Isvara. He has infinite knowledge and is all merciful. At the beginning of creation, He created the Vedas. He is like our father who is always engaged in doing us good.
Why do Nyaya philosophers say this? They argue against the Mímáṃsá. The eternity of the Veda depends on this tenet of the Mímáṃsá that sound is eternal. Jaimini maintains that the vibrations of the air "manifest" the always existing sound. However, it is argued by Nyaya that Eternal things are imperceptible to senses. The views of the eternality of sound is totally different in the Mímáṃsá school compared to Nyáya school.
The purvapaksha arguments in the Jaimini Mímáṃsá sutras state the position of Nyaya philosophers in this regard
How can the Vedas be said to be underived from any personal author, when there is no evidence to establish this? Would you maintain that they have no personal author because, although there is an unbroken line of tradition, there is no remembrance of any author….
There is actual proof that the Veda had a personal author. The sentences of the Veda must have originated from a personal author, since they have the character of sentences like those of Kálidása and other writers. And, again, the sentences of the Veda have been composed by a competent person, since, while they possess authority, they have, at the same time, the character of sentences, like those of Manu and other sages.
They also refer to particular sages by name.