Vedānta Deśika (13th c. South India) stages a discussion between thinkers of the Mīmāṃsā and Nyāya schools on the topic of the validity of cognitions. The first school thinks that validity is intrinsic, the latter thinks it is extrinsic. The Naiyāyika starts by stating “Valid cognitions are produced by the cause producing cognitions plus an additional element (producing their validity), because, while being an effect, they are specified by such an additional element, like invalid cognition are specified by an additional element distinguishing them from valid ones” (vigitā pramā samyaṅmithyāvabodhasādhāraṇakāraṇāt atiriktasahitāj jāyate, kāryatve sati tadviśeṣatvāt apramāvat).
But this does not hold in the case of the Lord’s cognition, which is permanent and uncaused (the Lord has no new cognitions, but perpetually knows everything).

And, if you admit that the Lord’s cognition is intrinsically valid, why should human cognitions be different?

Nor does the syllogism hold in the case of the cognitions of liberated souls, which are also uncaused and still valid.
If the inference were construed as not including the cognitions of liberated people, then these would belong to the contrary case (vipakṣa). But one should not find the probandum in the vipakṣa, whereas this would be the case in a possible contrary inference along the lines of “The validity of the cognitions of ordinary people is unproduced, because it is a valid cognition, like the valid cognition of liberated souls”.

After this long discussion, Vedānta Deśika adds a puzzling statement:

cārvākādeḥ prakṛtaprayoge kā vārtteti cet eṣaiva vipakṣe bādhakābhāvādimayī.

Which relevance/role is there for the Cārvākas and similar [thinkers] in the syllogism under examination (“Valid cognitions are produced by the cause producing cognitions plus an additional element (producing their validity), because they are specified by such additional element, like invalid cognition are specified by an additional element distinguishing them from valid ones”)?

That only, that there are no subsequent invalidating cognitions and other invalidating factors in the vipakṣa.

How do you understand the last sentence? The following is my interpretation, but I am not sure:

The vipakṣa of the given syllogism should be “cognitions that are not effects”. For Naiyāyikas, this is not a good vipakṣa, because there are cognitions that are not effects and are valid, like in the case of the Lord and of liberated souls. For Cārvākas, the problem is that there are no uncaused cognitions, but a valid syllogism needs a vipakṣa.