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There are puranas like Brahma purana, Skanda purana, Shiva Purana, Garuda purana. On what basis are these names given to the puranas? Is it that the puranas talk about some god or is that some god spoke particular purana or any other?

How exactly were puranas named?

The Puranas are named according to following points (these are my observations):

1. On the basis of which deity is glorified.


2. On the basis of which Kalpa events are mentioned.


3. On the basis of who is narrator of events.


4. On the basis of to whom the events are being narrated.

The complete list of Puranas and why they are named so is given in Shiva Purana, Uma Sanhita, Chapter 44.

Nandikeśvara said:—

125. O Taṇḍi, Brahma the four-faced deity is the main speaker. Hence, O sage, the first Purāṇa is called Brāhma.




126. The second Purāṇa is called Padma. It is so called because it mentions the greatness of the Padmakalpa.




127. Another Purāṇa composed by Parāśara and enlightening the details of Viṣṇu is called Vaiṣṇava Purāṇa. It is said to be composed by Vyāsa since there is no difference between father and son.




128. Those who know the Purāṇas speak that purāṇa as Śiva Purāṇa wherein there are many stories of Śiva in its earlier and later forms.




129. Where the stories of the goddess Durgā are mentioned, it is said to be Bhāgavata Purāṇa as well as Devīpurāṇa.

130-131. The Purāṇa narrated by Nārada is called Nāradīya. The seventh Purāṇa is called, O Taṇḍi, Mārkaṇḍeya because the great sage Mārkaṇḍeya is the speaker therein. Since it is related to the fire-god, the Purāṇa is called Āgneya. Since it recounts future events the Purāṇa is called the Bhaviṣya Purāṇa.

132. Since the transformation of Brahman is narrated the Purāṇa is called Brahmavaivarta. Since the story of Liṅga is mentioned it is called Liṅga Purāṇa.

133-135. O sage, the twelfth Purāṇa Varāha is so called because it contains the story of Varāha, the great Boar. In the Skanda Purāṇa the speaker is lord Śiva himself and the listener is Skanda. In the Vāmana Puraṇa the story of Vāmana (the Dwarf-god) is mentioned. The Kūrma Purāṇa contains the story of Kūrma (the tortoise-god). The Matsya Purāṇa is so called because it is expounded by Matsya. The Garuḍa Purāṇa is so called because the speaker is Garuḍa himself. Since the story of the entire cosmic egg is mentioned, the last Purāṇa is called Brahmāṇḍa Purāṇa.

But please note that there is some contradiction on Bhagavata Purana, as you can see Shiva Purana said it is Devi Bhagavata but in Vishnu Purana preface, section 5, it is said to be Shrimad Bhagavata Purana.

It is named Bhágavata from its being dedicated to the glorification of Bhagavat or Vishńu.

I read in Tattva Sandarbha by Jiva Goswami(verse 15) that:

skandam agneyam ityadi- samakhyas tu pravacananibandhanah kathakadivat
anupurvi- nirmananibandhana va / tasmat kvacid anityatvasravanam tv
virbhavatirobhavapeksaya /

And the names "Skanda","Agneya",etc. (by which the various Puranas
are known) refer either to those who first declared them, as is the
case with the Kathaka etc, or to those who rearranged them. Therefore,
if one sometimes hears (the Puranas) spoken of as non-eternal, it is
merely with reference to the fact that they are sometimes manifest and
sometimes unmanifest.

The above seems one convention in naming though not always true.

Skanda purana, Agni purana, Garuda purana,
Markendeya purana, Brahma purana, Vayu Purana, Narada purana ,
Matsya purana, Varaha purana, Kurma purana. These puranas are named after their first speakers or those who repeated it later.

Shiva Purana is spoken by Lord Shiva as well as it contains glories of Lord Shiva.

Bhavisya Purana is spoken by Lord Bhava(Shiva), but I am not sure whether it gets the name because of that or because it speaks about future.

Padma purana seems to get the name because it speaks of Padma Kalpa events.

Vishnu purana speaks about Lord Vishnu spoken by Parashara.

Linga purana speaks glories of Linga.

There are various ways people interpret Srimad Bhagavatam( The beautiful story of God, devoted to god etc.) spoken by Sukadeva

Brahmanda is that in which Lord Brahma described greatness of universe.

BrahmaVaivarta puranas is spoken by Savarni Manu to Narada.

272nd chapter of Agni Purana describes different puranas as follows:

In the KulArnava Tantra's (KT) 17th Chapter, Lord Shiva gives interesting etymological derivations (why the texts are so-named) for both the words - PurAna and ItihAsa.

ItihAsa:

IstadharmAdi kathanAt timirAjyAna bhajanAt |
HaranAt sarva
dukkhAnAm itihAsa iti smritah ||

Because it narrates of approved (Ista), Dharma etc.,breaks the
darkness (timira) of ignorance, and removes (haranAt
sarva), all miseries, it is called ItihAsa.

KT 17.42

PurAna:

For PurAna, the definition is the following:

PunyapApAdi kathanAd rAksasAdi nivAranAt |
NavabhakyAdi jananAt
purAna iti kathyate ||

Because it tells the merit of merit and demerit (punyapApa),
because it dispels evil beings like RakshAsas, and because it
generates nine-fold devotion (Navavidha Bhakti), and the like,
it is called PurAna.

KT 17.39

Navabhidha Bhakti:

These are usually spoken only in relevance to Vishnu.

NAmasravana, Kirtana, Smarana, serving the feet of deity (pAdasevA),
archanA, vandanA, dAsya, sakhya and surrender to the deity - the nine-fold
devotion.

And, that's possibly why PurAnas have (apparently) given more importance to Vishnu and Bhakti.