Objection 1: This question is especially asked in the light of David Hume's Problem of Induction, which is a well known and accepted problem in philosophy world-wide. How does an Advaiti deal with the Problem of Induction as it was already raised by the Charuvakas, and Advaitis must have had some answer to it.
There are two types of reasoning. First is inductive reasoning and second is deductive.
Inductive reasoning is a method of reasoning in which the premises are viewed as supplying some evidence for the truth of the conclusion, this is in contrast to deductive reasoning. While the conclusion of a deductive argument is certain, the truth of the conclusion of an inductive argument may be probable, based upon the evidence given.
Source - Wikipedia
Example of inductive reasoning:
Smoke always arises from fire, therefore if there is smoke, there has to be fire.
A -> B, therefore B -> A (not always true)
Example of deductive reasoning:
Smoke always arises from fire, therefore if there is no smoke there is no fire.
A -> B, therefore ~B -> ~A (always true)
(NOTE: ~B means not B)
Notice that in first example, what we arrived at is not 100 % certain but the result we arrive at second example is 100 % certain.
So, the problem you mentioned only affects the anuman praman done inductively not deductively. And as far as Advaita goes, it doesn't accept inductive anuman praman because if it did Shankracharya wouldn't have stated that Atman exist even though there is nothing like it to which we can compare it. On the other hand deductive anuman praman is used extensively in Advaita. For example,
Premise : Sat (Real object) can never give rise to Asat (Unreal object)
Conclusion : Brahman (sat) is not the cause of world (asat).
(This is clearly deductive reasoning.)
From the standpoint of the Ultimate Reality, things can, in no way, enter into causal relation. How? An unreal cannot be the cause of another unreal. An1 unreal entity such as the horns of a hare, which may be said to be the cause of another unreal entity such as a castle in the air, has no existence whatsoever. Similarly,1 an object like a jar, which is perceived and which is the effect of an unreal object like the horns of the hare, is never existent. In3 like manner, a jar which is perceived and which is the effect of another jar that also is perceived to exist, is, in itself, non-existent. And4 lastly, how is existence possible of a real object as the cause of an unreal one? No other causal relation is possible nor can be conceived of. Hence men of knowledge find that the causal relation between any objects whatsoever is not capable of being proved.
(Mandukya 4.10)
Objection2: Dvaithis and Vishishtadvaithis accept Pratyaksha, Anumana and Agama(Shabdha). But I was kind of surprised to know that Advaithis accept all the six pramanas. This is quite surprising as we consider world as maya, and everything in world is maya.
In Advaita, Atman is called aprameya (that which cannot be known by any praman). The 6 types of praman you see accepted by Advaita is only to prove that world is unreal, and Atman is attribute-less.
अन्तवन्त इमे देहा नित्यस्योक्ताः शरीरिणः।
अनाशिनोऽप्रमेयस्य तस्माद्युध्यस्व भारत।।2.18।।
(Gita 2.18)
Excerpt from Shankra's commentary on 2.18
Therefore, in order that this contingency may not arise, it is said, 'Of the everlasting, indestructible'. Aprameyasya, of the indeterminable, means 'of that which cannot be determined by such means of knowledge as direct perception etc.'
Objection: Is it not that the Self is determined by the scriptures,
and before that through direct perception etc.?
Vedantin: No, because
the Self is self-evident. For, (only) when the Self stands
predetermined as the knower, there is a search for a means of
knolwedge by the knower. Indeed, it is not that without first
determining oneself as, 'I am such', one takes up the task of
determining an object of knowledge. For what is called the 'self' does
not remain unknown to anyone. But the scripture is the final authority
[when the Vedic text establishes Brahman as the innermost Self, all
the distinctions such as knower, known and the means of knowledge
become sublated. Thus it is reasonable that the Vedic text should be
the final authority. Besides, its authority is derived from its being
faultless in as much as it has not originated from any human being.]:
By way of merely negating superimposition of alities that do not
belong to the Self, it attains authoritativeness with regard to the
Self, but not by virtue of making some unknown thing known. There is
an Upanisadic text in support of this: '৷৷.the Brahman that is
immediate and direct, the Self that is within all' (Br. 3.4.1).
