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.
Best Answer
Do/does
A verb takes its infinitive form when combined with auxiliary do:
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:
There are/exist
In English, we don't usually use the verb exist. Usually we use the dummy subject there with be, like this:
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."
Do there exist
It is within the limits of grammaticality to combine dummy there with exist, but it sounds awkward:
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.