I wrote in a forum:
Does exists any other approaches?
Someone suggests me to write:
Does there exist any other approaches?
I made some researches on the web and I found that the latter expression is the common, and maybe right, form.
I'm looking for a deep understanding of such expression.
A verb takes its infinitive form when combined with auxiliary do:
Do ghosts exist?
Does God exist?
Notice that exist does not agree with the subject. The choice of do or does agrees with the subject. Do and ghosts are both plural; does and God are both singular.
Since this is a question, you need the inverted form, where auxiliary do precedes the subject and the main verb follows the subject:
Do any other approaches exist?
In English, we don't usually use the verb exist. Usually we use the dummy subject there with be, like this:
Are there any other approaches?
Yes, there are other approaches.
That's probably how you should say it. Usually we say exist to talk about whether something is real or a superstition, for emphasis, or to speak precisely and technically, as in mathematics. By using exist when searching for a possible method ("approach") for doing something, you make the listener wonder if you mean something like what people ordinarily use exist to mean.
However, for that reason, saying exist where be is customary makes your point more forcefully. In an appropriate context, if you say "Do any other approaches exist?", people will understand this as a rhetorical question: that is, you mean "Stop searching for something that might not even exist! Let's choose the approach that we already know."
It is within the limits of grammaticality to combine dummy there with exist, but it sounds awkward:
Do there exist any other approaches?
Most people perceive this as clumsy English. Often in English, we do get clarity by combining multiple ways of suggesting the same thing, but combining dummy there with exist sounds overly wordy. I don't know if there is a principled explanation for why. English works more by customary phrases than by rules, so there might not be a rule here. In mathematics, does there exist does not sound as clumsy, because there exists is a very common phrase in mathematics.
Does there exist a greatest prime number? [Ordinary and clear in mathematics.]
Does a greatest prime number exist? [A little strange-sounding in mathematics.]
Does there exist a credible presidential candidate? [A little strange-sounding in everyday English.]
Does a credible presidential candidate exist? [Emphatic everyday English, suggesting doubt about something ordinarily taken for granted.]
It's not really too complicated. Firstly, think of the statement you're asking about:
There exist other approaches.
There in this case is a dummy subject; in English, you can't just have a verb with no subject, and we usually use it or there. For example, you have to say
There are six possibilities.
and not
Are six possibilities.
When you make a statement about a verb into a question in English, you use do and change the word order. For example.
He eats fish. / Does he eat fish?
They live in Hoboken. / Do they live in Hoboken?
You're asking a question about the statement "There exist other approaches", so it follows the exact same pattern:
There exist other approaches. / Do there exist (any) other approaches?
We have to have there in the sentence because it's still the dummy subject, which we need; you can't say "Does exist X?" or "Do exist X?" because there's no subject in it. It should also be Do if you're talking about approaches, because approaches is plural: it does, but they do.
When the subject and the verb are reversed to make a question, in modern English it is only the auxiliary and the subject that are inverted, the main verb still follows the subject. (Perhaps that is why we use "do" to form questions, so that there is an auxiliary to be inverted).
So the question corresponding to
Other approaches exist.
would be:
Do other approaches exist?
and that is perfectly grammatical.
But "exist" has an alternative sentence frame with a dummy subject "there", and the logical subject in the complement position:
There exist[s] other approaches.
and the question form of this is
Do[es] there exist any other approaches?
In contrast,
*Does exist any other approaches?
is not grammatical, because only the auxilliary is inverted with the subject, not the whole verb phrase (except occasionally in poetry).
(I put the endings in brackets because there is some disagreement about whether to use a singular or plural verb in this construction. The grammatical subject is "there", which is singular, and the traditional rule says that the verb should be singlar "exists" and "does"; but many people in sentences like this make the verb agree with the logical subject, and say "exist" and "do". I would say that at present, both options are grammatical).
Other verbs which can take this sentence frame include "appear", and "follow". It is something of a literary construction, and "there exists" is particularly used in academic writing.
There appeared a small man from behind the counter.
There followed a series of bangs and crashes.
There is nothing deep to understand really, this is basic question phrasing:
Auxiliary + Subject + Verb (+ complement).
In your first proposition, you completely forgot about the subject. Hence, the second proposition :
Do there exist any other approaches?
Is the correct one.
To stick to yours a bit more, you could also use:
Do any other approaches exist ?
Still following the "Auxiliary + Subject + Verb" syntax. To be frank, this second proposition sounds a lot better than the first one, to me.
P.s: If you're having any more trouble with question phrasing in English, keep in mind that figuring out the answer syntax is always the first, and may be most important, step of the process.
