What are the exact interpretations of four Mahavakyas by Dvaita, Vishishtadvaita and Advaita?

Question

Note: You can write an answer from the perspective of one or more schools, whichever you know best and also I read somewhere that some schools consider Mahavakyas as quoted out of the context, if you think so you can quote the whole verse to provide the context you intend for your interpretation.

In this question I read about much controversial alleged debate between Shri Vidyaranya and Shri Akshobhya Tirtha.

And it was really curious that the debate was over the meaning of Mahavakya Tat Tvam Asi (तत् त्वम् असि). Source

So I wanted to know what are the exact interpretations of the four main Mahavakyas from all the three schools of Vedanta.

1. Prajnanam Brahma (प्रज्ञानम् ब्रह्म)
2. Aham Brahma Asmi (अहम् ब्रह्म अस्मि)
3. Tat Tvam Asi (तत् त्वम् असि)
4. Ayam Atma Brahma (अयम् आत्मा ब्रह्म)

Answer 1

Partial Answer (for one Mahavakya Tat Tvam Asi)

Note: I recommend going through overview of the similarity and difference between Vedanta philosophies.
The debate for interpretation is due to the difference in the doctrine of Vedantic philosophies viz. Advaita, Vishithadvaita, Dvaita etc. for the relationship between Brahman and Jiva.

1. Tat Tvam Asi

   This Mahavakya occurs in Chaandogya Upanishad 6.8.7 and the text is

   स य एषोऽणिमैतदात्म्यमिदꣳ सर्वं तत्सत्यꣳ स आत्मा तत्त्वमसि श्वेतकेतो इति भूय एव मा भगवान्विज्ञापयत्विति तथा सोम्येति होवाच ॥

   6^th chapter of Upanishad contains the dialogue between Uddaklaka (Aruni) and his son Shvetaketu who has just returned after 24 years of completing his study and. His father told "did you ask for that teaching (about the Supreme Brahman) through which what is unheard becomes heard, what is un-thought becomes thought of, what is unknown becomes known?'" And Shvetaketu replied that no, he doens't know such thing. So, Uddalaka started to preach his son about Brahman by giving various examples and in the context "Tat Tvam Asi" refers toTat = That (Brahman), Tvam = You (the self of Shvetaketu) and Asi = is

   • Dvaita Interpretation:

     Dvaita considers that there is eternal difference between Jiva, Brahman and Jagat. Anand Tirtha (Madhvacharya) interpreted sa ātmā atat tvam asi (स आत्मा अतत् त्वम् असि) that means That Atman, thou art not.

     You can read Madhvacharya's commentary on Chhandogya Upanishad from here

     

     According to him Atman refers to Vishnu, the supreme being and Upanishad is preaching that Jiva is verily distinct from Vishnu.
     
     

   • Vishishtadvaita Interpretation:

     According to Vishishtadvaita, Brahman has body-soul (Sharira-Shariri) relationship with Jiva and Jagat that means Brahman is the self dwelling in Jiva and Jagat. The word Atman can refer to either Jivatma (individual self) or Paramatama (supreme self). and it needs to be interpreted properly. Brahman (as the supreme self dwells) inside of Jiva and every particles of Jagat and also outside of it.

     According to Ramanujacharya, Tat refers to the Brahman (Vishnu) and Tvam refers to the supreme inner self (Paramatma) dwelling inside individual self (Jiva). So, Tat and Tvam both refers to the Brahman (the supreme self), not Jiva (the individual self).

     Ramanujacharya has not written commentary on Upanishads, let me quote the point touching in his work Vedartha Sangraha, quoting English translation from srimatham.org: from verse 82:

     ... तथा हि-“तत् त्वम्” इति सामानाधिकरण्ये 'तत्" इत्यनेन जगत्कारणं सर्वकल्याणगुणाकरं निरवचं ब्रह्मोच्यते । “त्वम्" इति च चेतनसमानाधिकरणवृत्तेन जीवान्तर्यामिरूपि, तच्छरीरं तदात्मतया अवस्थितं, तत्प्रकारं ब्रह्मोच्यते...

     ... To explain: In the passage affirming identity ‘That thou art’, the term ‘that’ signifies Brahman, as the cause of the world, as the abode of all perfections. By the term ‘thou’ also, denotative of the individual self, Brahman itself is signified as the inner ruler of the jiva, as possessed of it as its body, as existing within the jiva as its self and as possessing the jiva as its mode...

     For further understanding, you can refer a book Ramanuja On Tat Tvam Asi & Neti Neti by Prof. Naryanacharya K. S.
     
     

   • Advaita Interpretation:

     According to Advaita, there is ultimately no difference between Jiva and Brahman as said jīvo brahmaiva nāparaḥ i. Jiva and Brahman are identical. There is only one ultimate, supreme and absolute reality/truth - Brahman. The difference between Jiva and Brahman is appearing due to ignorance (अज्ञान​), when the Brahman, the very true self is realized, no difference (द्वैत​) holds more between Jiva and Brahman.

     Adi Shankaracharya literally interpreted that verse:

     Now, that which is this Subtle Essence,- in that has all this its Self; that is the Self: that is the Truth; that thou art, O ':Shvetaketu '.

     Adi Shankaracharya explained the interpretation of Tat Tvam Asi in his commentary on the last verse of 6^th chapter. Quoting English translation from Chandogya Upanishad with Shankara Bhashya, by Ganganath Jha

     Thus liberation and bondage are due to addiction to truth and untruth (respectively) ;-and that which is the root of the universe, wherein all creatures dwell and rest,- and in which all things have their Self,- which is immortal, free from dangers, blessed, without a second,-that is true, that is thine Self-hence, that thou art, O Shvetaketu.- This sentence has been explained more than once.

     He raised one important question and answered it:

     Question—"Who is this Shvētakētu, who is spoken of as Thou?"
     Answer-I, Shvēta kētu, who know myself as the son of Uddalaka, having heard the teaching and pondered over it and learnt it, requested the Father to teach him what he had not heard, or pondered over, or known—'What, revered Sir, is that Teaching ?'—This represents the person who, being entitled to be the hearer, the ponderer and the knower,--did not, before he was taught by his father, had not reached the true nature of his own self, as Being, the Self of all, as distinct from all aggregates of causes and effects --which-as the Supreme Deity,-has entered into the aggregate of causes and effects made up of Fire, Water and Food, for the differentiating of Names and Forms,just as man enters the mirror, as his own reflection, or the Sun enters into water and other reflecting surfaces, as its own reflection ;~now however, having been enlightened by his father by the teaching 'that Thou art ', through a number of illustrations and reasons, -he understood from his father that I am Being itself ':--The repetition is meant to indicate the end of the Discourse.

     The commentary goes on and raised some possible objections with the interpretation and answered them and finally declared that "From all this it follows that what the sentence That thou art does is to set aside the idea of 'Self' in regard to that 'Living Ego' which is a product, unreal and entitled to the performance of acts."
     - - - - - - -
     
     

   Actually these interpretations from different Vedanta philosophies are dealing with the method of interpretation based on lakshaNa (implication) which are of three types 1. jahal-lakshaNa—exclusive implication. 2. ajahal-lakshaNa—inclusive implication. 3. jahad-ajahal-lakshaNa- quasi-implication or exclusive-inclusive implication. Refer this article from sanskritdocument site to get it in depth.

   Suggested paper: tat-tvam-asi from Shodhganga, a reservoir of Indian theses.

---

2. Ayam Atma Brahman

   This Mahavakya occurs in Mundaka Upanishad (which is the shortest among all Upanishads) verse 2. The Mahavakya also occurs in the Brihadaranyaka Upanishad (which is the biggest one) 4.4.5. Ayam = This, Atma = The Self
   
   

   • Adviata Interpretation:

     According to Advaita, Jiva is Atman and Atman is Brahman. In other words, the duality of selves are appearing due to ignorance and in absolute, there is only one Atman. Here Atman refers to individual self and that's why Upanishad is preaching that This self is Brahman.

     Adi Shankracharya Bhashya on Mundaka Upanishad verse 2:

     That Brahman which has been described (as existing) inferentially is now pointed out, as being directly known, by the passage, “This Self is Brahman”. The word this, meaning that which appears divided into four quarters, is pointed out as the innermost Self, with a gesture (of hand) by the passage, “This is Ātman”.

     Here one can argue that if the word Atma refers to individual self, how can it appears to be divided into four pada? Adi Shankaracharya raised this question in his commentary on next verse and relied that:

     (Objection)—while the subject-matter under discussion treats of the innermost Self (Pratyak Ātmā) as having four quarters—in the text, “This Ātman is Brahman”—how is it that (the external universe consisting of) the effulgent regions, etc., have been described as its limbs such as head, etc.?

     (Reply)—This, however, is no mistake; because the object is to describe the entire phenomena, including those of gods (Adhidaiva) as having four quarters from the standpoint of this Ātman known as the Virāt (i.e., the totality of the gross universe). And in this way alone is non-duality established by the removal of (the illusion of) the entire phenomena. Further, the one Ātman is realized as existing in all beings and all beings are seen as existing in Ātman. And, thus alone, the meaning of such Śruti passages as “Who sees all beings in the Self, etc.” can be said to be established. Otherwise, the subjective world will, verily, be, as in the case of such philosophers as the Sāmkhyas, limited by its (one’s) own body. And if that be the case, no room would be left for the Advaita which is the special feature of the Śruti. For, in the case of duality, there would be no difference between the Advaita and the Sāmkhya and other systems. The establishment of the identity of all with Ātman is sought by all the Upaniṣads. It is, therefore, quite reasonable to speak of the effulgent regions, etc., as seven limbs in connection with the subjective (individual self, Adhyātma) associated with the gross body, because of its identity with the Adhidaiva (comprising the super-physical regions) universe from the standpoint of the Virāt (the totality of the gross physical universe). This is further known from such characteristic indication (of the Śrutí), as “Thy head shall fall”, etc.....Such being the case, it is clearly established that non-duality is realized by the disappearance (of the illusion) of all duality.

     Aslo take a look at verse 196-202 of Vivekachudamani (of Adi Shankracharya) which talks about the identity as Jiva and Brahman.
     
     

   • Vishishtadvaita Interpretation

     According to Vishishtadvaita, here the word Atma refers to supreme self not individual self and that's why Upanishad is preaching that supreme self is Brahman.

     Ranga Ramanuja quoted Sri Ramanujacharya's Bhashya on Brahma Sutra verse 2.3.45 in his commentary on Mandukya Upanishad. By referring to Ramanujacharya's commetnary on BS 2.3.45, it's clear that :

     ..... Hence there is no contradiction between the individual and the highest Self--the former of which is a viseshana of the latter--standing to each other in the relation of part and whole, and their being at the same time of essentially different nature. This the Sûtra declares 'not so is the highest,' i.e. the highest Self is not of the same nature as the individual soul. For as the luminous body is of a nature different from that of its light, thus the highest Self differs from the individual soul which is a part of it. It is this difference of character--due to the individual soul being the distinguishing clement and the highest Self being the substance distinguished thereby--to which all those texts refer which declare difference. Those texts, on the other hand, which declare non-difference are based on the circumstance that attributes which are incapable of separate existence are ultimately bound to the substance which they distinguish, and hence are fundamentally valid. That in declarations such as 'Thou art that' and 'this Self is Brahman,' the words thou and Self, no less than the words that and Brahman, denote Brahman in so far as having the individual souls for its body, and that thus the two sets of words denote fundamentally one and the same thing, has been explained previously.

   • Dvaita Interpretation:

     According to Dvaita Jivatma (individual self) is totally different than Brahman. Though Brahman is the controller, initiator of every action that's done by Jiva, however Brahman doesn't dwell inside Jivatma, it dwells inside the physical heart of body. According to Madhvacharya Brahman (Vishnu) is independent and Jiva is dependent since Brahman energizes the Jivas (which is eternally different entity) and hence perceived.

     Ananda Tirtha (Madhvacharya)'s Bhashya on Mandukya Upanishad:

     

---

^{This is the partial answer, will try to add interpretation of other Mahavakyas.}