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As I discuss in this question, by far the most popular school of Hindu philosophy is the Vedanta school, which bases its tenets on the doctrines laid out in the Brahma Sutras, a work by the sage Vyasa which summarizes and systematizes the philosophical teachings of the Upanishads. (You can read the Brahma Sutras here.) Now as I discuss in this answer, apart from Advaitins most members of the Vedanta school believe in three Pramanas or means of valid knowledge: Pratyaksha or perception, Anumana or inference, and Shabda or scriptural testimony.

But non-Vedantic philosophical schools believe in various different numbers of Pramanas, as described in this excerpt from B.L. Atreya's book "The Elements of Indian Logic":

The number of pramanas (indispensable causes of prama) recognized by thinkers of different schools of Indian thought goes up to ten, ... namely, Pratyaksha (Perception), Anumana (Inference), Shabda (Verbal Testimony of an authority), Upamana (Comparison), Arthapatti (Necssary Assumption)[,] ... Anupalabdhi (Non-apprehension)[,] ... Aitihya (Tradition), Chesta (Gesture), Parishesha (Elimination), and Sambhava (Inclusion).

  1. The Charvakas recognize only one pramana, namely, Pratyaksha, as the source of right knowledge.
  2. The Vaisheshikas, Jainas and the Buddhists recognize two, namely, Pratyaksha, and Anumana.
  3. The Sankhya and Yoga schools recognize only three, namely, Pratyaksha, Anumana and Shabda.
  4. The Naiyayikas recognize only four, namely, Pratyaksha, Anumana, Shabda, and Upamana.
  5. Some Mimansakas (followers of Prabhakara) recognize five, namely, Pratyaksha, Anumana, Upamana, Shabda and Arthapatti.
  6. Another group of Mimansakas (followers of Kumarila Bhatta) and the Vedantists of Shankara (Advaita) school recognize six pramanas, namely, Pratyaksha, Anumana, Shabda, Upamana, Arthapatti and Anupalabdhi.
  7. The scholars of the Puranas (Historians) recognize eight pramanas, namely, Pratyaksha, Anumana, Shabda, Upamana, Arthapatti, Anupalabdhi, Aitihya, and Sambhava.
  8. The Tantrikas (students of the Tantras) recognize nine pramanas, namely, Pratyaksha, Anumana, Shabda, Upamana, Arthapatti, Anupalabdhi, Aitihya, Sambhava, and Chesta.
  9. Some thinkers admit all the ten Pramanas, namely, Pratyaksha, Anumana, Shabda, Upamana, Arthapatti, Anupalabdhi, Aitihya, Sambhava, Chesta and Parishesha.

Thinkers who recognize a less number of indispensable means of valid knowledge (Pramanas) try to bring the rest under one or the other of those which they admit.

I'm interested in the part in bold. My question is, what philosophical school believes in these ten independent Pramanas?

This may reduce to the question of which school recognizes Parishesha as an independent Pramana, since that seems to be the distinguishing feature of this school, so it may help to give B.L. Atreya's definition of Parishesha:

Parishesha or Elimination is the process of knowing some thing by eliminating from a group of things those which it is not. For example it happens some time that we know a person to be somebody by eliminating others already known from the group where that person has been told to be present.

To my mind Parishesha clearly seems reducible to Arthapatti at least. But what philosophical school believes that Parishesha cannot be reduced to any other Pramanas?

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Jiva Goswami of Gaudiya Vaishnavism accepts all 10 pramanas.

Here is what this website has to say:

By accepting only three of the ten pramanas Jiva Gosvami does not exclude the other seven. He says that pratyaksa, anumana, and sabda includes the other seven pramanas. The breakdown is as follows: upamana, arthapatti, sambhava, and cesta are included in anumana; abhava is in pratyaksa; arsya and aitihya in sabda.

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    Everyone who believes in 3 independent Pramanas thinks the other Pramanas can be reduced to those three. My question was who believes in 10 independent Pramanas. – Keshav Srinivasan Oct 6 '17 at 14:04
  • Does that really make a difference then? You are either using the pramanas or you're not. – Ikshvaku Oct 6 '17 at 14:05
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    To give another example, the Vaisheshika school believed that Sabda Pramana was reducible to Anumana, because you could use Anumana to prove the existence of God, and then prove that God authored the Vedas, and thereby prove the authority of he Vedas. Whereas the Vedanta school says that the Vedas are Apaurusheya or authorless, and that it's only through the Vedas that we can prove the existence of Brahman, and not the other way around. – Keshav Srinivasan Oct 6 '17 at 14:15
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    What I said about everyone agreeing about the validity of Pramanas only applies to Hindus, i.e. members of the Astika schools. Nastikas disagree with Hindus on which Pramanas are valid or invalid. Buddhists and Jains think only Pratyaksha and Anumana are valid, and Sabda is invalid. Charvakas think only Pratyaksha is valid, and Anumana and Sabda are invalid. Charvakas thought that only what you can see with your own eyes and other senses is what you should accept as real. They didn't think you should draw inferences based on what you see. – Keshav Srinivasan Oct 6 '17 at 14:28
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    And yes, as a practical matter Charvakas violated their principle and used Anumana quite a bit when arguing with others. That was a point that Charvakas were frequently criticized on by other schools. – Keshav Srinivasan Oct 6 '17 at 14:32

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