"What's" in indirect questions

Solution 1:

Grammatically speaking, the last two examples are quite different. In the first example, what substitutes the predicative

The wisdom behind education is something.

What is the wisdom behind education? (question is about the predicative)

I don't know what the wisdom behind education is. <-- correct

I don't know what is the wisdom behind education. <-- too colloquial, unacceptable in formal writing.

Now, in the second example, what substitutes the subject.

Something is on her mind.

What is on her mind? (question is about the subject)

I don't know what is on her mind. <-- correct

I don't know what on her mind is <-- incorrect

Solution 2:

The difference between embedded and non-embedded questions is that, while normal Wh-questions must undergo Subject-Auxiliary Inversion, in certain cases,

-- though, note, only in those cases; we'll return to this point later --

• What did the man eat?
• *What the man ate?

embedded Wh-questions normally don't in these cases.

• I don't know what the man ate.
• *I don't know what did the man eat.

However, this distinction is moot in

• I wonder what's on her mind.

because this is not one of those "certain cases" in which Subject-Auxiliary Inversion is required. The case where it is required in normal Wh-questions is the case in which the Wh-questioned word is the subject of the question.

Swapping the verb with the subject is designed to put something other than a normal NP at the beginning of a question to show that it's a question. In a Yes/No-question, that's the auxiliary verb. But in a Wh-question, there's also a Wh-word, which is a question marker all by itself, and when it's the subject, it's already at the beginning of the question, marking it as a question.

So, in that case, as Armen points out, no inversion is necessary in a normal Wh-question

• What's on her mind?

and therefore it's not necessary either in an embedded Wh-question

• I don't know what's on her mind.

Solution 3:

There's an inconsistency in the preamble. One has to be very careful when converting a quote structure to a report structure:

The book doesn't say, "What's the wisdom behind education?"

Changing this to an indirect question becomes the following:

The book doesn't ask what the wisdom behind education is.

This conversion largely preserves sense, but even now, the reported version may be an inaccurate statement while the quote version may be an accurate if pedantic one - the actual wording in the book might be "What is the wisdom behind education?", for instance.

Addressing the problem about the grammatical (rather than semantic) correctness of various report structures, the tendency is for much less rigid rules to be deemed necessary for both the wording and the punctuation, with the proviso that accuracy be maintained and ambiguity be avoided.

For instance, I've come across (on the web) examples of the use of bid and wish both as reporting verbs proper and quote verbs:

He bade us a fond farewell. He bade us welcome.

... bade us "Goodbye" [unusual]

She wished us a merry Christmas. She wished us Merry Christmas.

She wished us "Merry Christmas."

There may be other verbs used in both capacities. (Say appears to be, of course:

He said {that} she was going shopping. He said "She was going shopping."

but the meaning changes slightly.)

With the examples given, I'd say that

The book doesn't say what's the wisdom behind education. or The book doesn't ask what's the wisdom behind education. are clumsy elisions, better expressed as The book doesn't say what the wisdom behind education is. or The book doesn't ask what the wisdom behind education is. (the difference is now obvious).

However, "[She] doesn't say what's really on her mind." is quite acceptable grammatically, in fact the accepted idiomatic expression. Though I'm struggling to think of the related quote version.