The three steps in analysing mahAvAkya-s
The words "that" and "thou" are used to denote different things in ordinary life. "That" would refer to something remote whereas "thou" is immediate to the person (i.e, one is not remote to oneself). So how can an equation be made identifying these two seemingly disparate things?
sureshvarAchArya explains in Ma 2.15-2.16a
sAmAnAdhikaraNyAkhyas-sambandhaH padayoH-iha |
visheshhaNa-visheshhyatvaM sambandasyAt.h padArthayoH ||
laxya-laxaNa saMyogAt.h-vAkyam-aikyaM cha bodhayet.h |
The relation of the words here is [what is] called grammatical apposition, [The understanding] of the meaning of the words is from the relation of the qualifier and qualified, And unity is taught by the sentence by equating the implied and implication.
The three steps have also been explained by sureshvarAchArya.
Step 1: relation of sAmAnAdhikaraNa:
The two words, tat.h and tvam.h, bear the relation of sAmAnAdhikaraNa (refer Ma 3.15-a above, NaiSi 2.54s and NaiSi 3.3). Grammatically, adhikaraNa means placing two substantives in apposition, which is precisely the situation in tattvamasi. Since they are placed in apposition, tat.h and tvam.h cannot really refer to two different things. Now, adhikaraNa also means substratum. sAmAna implies same-ness, and hence sAmAna-adhikaraNa means having the same substratum. Thus, sAmAnAdhikaraNa means that the two different words refer to essentially the same thing (refer also Ma 3.18). The situation is similar to interpreting the compound nilotpala -blue lotus-. Since the two words nIla and utpala are placed in apposition, it means blue-ness and lotus-ness inhere in the same substratum, namely the blue-lotus.
Some advaitins, like AchArya maNDana mishra, say that the knowledge of unity produced by the mahAvAkya-s are not immediate and cannot produce liberation. Immediate knowledge is achieved only through prasa.nkhyAna, i.e., repeated meditation on such texts. But, sureshvarAchArya points out that such a reasoning is incorrect. The fallacy of the prasa.nkhyAnavAdin-s is clearly seen when the other two steps are understood.
Step 2: relation of visheshhaNa-visheshhya
visheshhaNa is the qualifier and visheshhya is the qualified. In the sentence the two words tat.h and tvam.h mutually qualify each other (NaiSi 3.10). The individual self (tvam.h) is qualified by brahman (tat.h), thus saying that the individual self is actually free from suffering (nirduHkhitvam.h). Conversely, brahman is qualified by the individual self, which shows that brahman is immediate and the innermost self, and not something remote.
Here an objection may arise: the relation of qualifier and qualified can exist only if brahman has attributes. For example, a lotus can be red or blue. Further, the color blue could belong to some object other than a lotus. But brahman is known to be only one, ekameva-advitIyam; and without attributes, nirguNam. Moreover, the jIva is the sufferer -duHkhin- and goes through painful transmigratory existence. How is this resolved?
Step 3: relation of laxya-laxaNa:
Although there is the relation of visheshhNa and visheshhya, the words is not used in the sense of an object having different attributes, as in a lot s being blue or red. They are used in the sense of laxaNa-laxya. laxaNa is the implication and laxya is what is implied. To give an example: when the ether in the pot and the ether outside it are equated, there is the relation of visheshhaNa-visheshhya. But, there is really one ether which has no attributes. The relationship is understood by laxya-laxaNa-artha. I.e, the incompatible elements between the pot ether and the other ether, namely the idea that ether is constrained within and outside the pot, is discarded - ghaTetarakhyariva (see NaiSi 3.9). Similarly, the wrong notions that the individual self is the duHkhin, limited by the body, senses, etc, and that brahman is remote, not the immediate self, etc, are both sublated by the sentence.