Advaita — Śaṃkara (8th century)

Brahman alone is ultimately real (pāramārthikasatya) — pure undifferentiated consciousness-bliss-being (saccidānanda). The individual self (jīva) is Brahman under the limiting adjunct (upādhi) of buddhi; on liberation the upādhi dissolves and identity with Brahman is recognised. The world (jagat) has empirical (vyāvahārika) reality — not hallucination — but is ultimately indeterminate (anirvacanīya). Māyā conceals and projects multiplicity from the one Brahman. Liberation = direct non-dual cognition (aparokṣānubhūti) of tat tvam asi. Jīvanmukti possible.

Rāmānuja's objection: if knower, act of knowing, and object known are ultimately identical as one Brahman, knowledge itself becomes impossible. "Indeterminate being" (anirvacanīya) is incoherent — a thing is either real (sat) or unreal (asat); there is no coherent third category.

Viśiṣṭādvaita — Rāmānuja (11th–12th century)

Brahman is one but qualified (viśiṣṭa) by real internal attributes — including jīvas and jagat as Brahman's body (śarīra). Three mutually dependent real tattvas: Brahman/Īśvara (Viṣṇu), cit (jīvas — conscious, atomic, real and eternally distinct), acit (matter). Māyā is Brahman's real creative power. Liberation = sāyujya with Viṣṇu, retaining full individual identity. Degrees: sālokya → sāmīpya → sārūpya → sāyujya.

Viśiṣṭādvaita navigates between Advaita's identity (incoherent) and Dvaita's pure difference (theologically unsatisfying): the qualified whole is one — non-dualism preserved — but the modes are real and distinct — the devotee-Bhagavān relation is not dissolved at liberation.

Dvaita — Madhva (13th century)

Five eternal real differences (pañcabheda): (1) Īśvara ≠ jīva, (2) Īśvara ≠ jagat, (3) jīva ≠ jagat, (4) jīva ≠ jīva, (5) jagat-entities ≠ each other. All five are real and eternal. Viṣṇu alone is absolutely independent (svatantra); all jīvas and matter are absolutely dependent (paratantra). Even in liberation, ānanda has gradations (tāratamya) — the highest liberated being (Lakṣmī) still falls infinitely short of Viṣṇu's ānanda.

Madhva posits three classes of jīvas: mukti-yogya (capable of liberation), nitya-saṃsārin (condemned to eternal saṃsāra), tamo-yogya (condemned to eternal tamas). No other Hindu school posits preestablished condemnation — a position that has no parallel in the tradition and is closer to Calvinist double predestination.

Tat tvam asi — the three-school fault-line

All three schools interpret Chāndogya 6.8.7 ("That thou art") differently, mutually exclusively. Advaita: strict identity — both "tat" and "tvam" refer to the same pure consciousness after limiting adjuncts are stripped via lakṣaṇā. Viśiṣṭādvaita: identity-in-difference — "tvam" refers to the jīva as a mode (prakāra) of Brahman; identity holds at the level of the qualified whole. Dvaita: tadātmya = similarity, not identity — the jīva is Brahman's image (pratibimba).

The three readings are logically exhaustive (identity, qualified identity, difference) and mutually exclusive. No synthesis has been achieved despite centuries of argument. This is arguably the deepest philosophical fault-line in Indian thought — generating completely different soteriologies, ritual logics, and theologies. The debate continues in living form in the Vedānta traditions of South India.