Consciousness as the ground (adhiṣṭhānam) of everything we affirm by being the logical connection (anvaya) of all its fickle, dual, contrasting (vyatireka) appearances, which we rightly deny.

That is why James Swartz calls this means of knowledge (pramāṇa) 'the elimination of non-essential variables', whereby the joke is of course that every variable is non-essential.

Anvaya comes from anu plus √i, go. That which follows each other literally.
Anvaya therefore means, for example, word order in grammar. It refers to the syntax, the arrangement, ordering in a sentence. This inspires the following: The message, the knowledge of a sentence, is that which connects the different (vyatireka) words of a sentence (anvaya). In the same way: Pure knowledge is what connects the apparent difference from reality for the jñānī (the knower of himself as truth).

It is the distinction between what is always true (anvaya), whatever appears, consciousness must always be present for something to be there. Everything that is something is capricious and different from each other. Sometimes it is there, sometimes it is not, and sometimes it seems true and is irrelevant or absent (vyatireka).

Vyatireka comes from vi-, distinction, setting apart + ati, over, beyond, intensifying + √ric or √rañj → reka, abandoning, separating. So: vi + ati + reka = separating, splitting off, exclusion

Vyatireka literally means: seclusion, distinction, or negative distraction. Consciousness shows itself as a contrasting (vyatireka) plurality, which is logically connected by consciousness itself.

The comparison with space is useful here again. Outer things can only be there, by the grace of space. In the same way, all differences such as different feelings, thoughts, dreams, physical objects in the waking state, can only exist by the grace of sat cit, existence-awareness.

Anvaya vyatieka is the logical procedure for determining truth based on what does and does not always match. Anvaya focuses on agreement, no difference/no change, vyatireka on non-agreement, difference/change.

Someone once said it like this: The presence of the effect when the cause is present is anvaya. The absence of the effect when the cause is absent is vyatireka. Thus, for a cause-and-effect relationship to exist, anvaya and vyatireka must match. So Anvaya vyatireka is not a dual relationship, they are totally reconciled. We know this from the satya mithyā teaching. Satya is the only real reality, mithya is the relative reality which is in reality the real reality.

For example, if we accept that happiness arises from an object or event, it must be shown that happiness is always present with the object or event (anvaya), and is always absent in its absence (vyatireka).

If a candy sometimes gives a child happiness and other times it doesn't, there is no agreement: the candy is not the cause of happiness. Happiness may or may not be blocked by ignorance, because of apparitions. The child can be completely happy without sweets. So what Anvaya is must be permanent existence: consciousness.

Determining what is true by seeing what is changeable and unchanging in situations and phenomena helps in understanding the true nature of the self.