Which mood is created with the sentence, "Everyone should visit..." with the use of the auxiliary verb? I have decided already that it is not the imperative mood.
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It might be more helpful to consider the question in terms of modality rather than mood (with which it nevertheless overlaps). In the words of the linguist Larry Trask, modality is ‘associated with the expression of obligation, prohibition, necessity, possibility and ability’. Modality is expressed in English through the modal verbs, of which should is one, and whose main use is to express weak obligation. That is the role it performs in your example. (The expression of obligation is known to linguists as deontic modality.)
Let's take a look at what the main dictionaries say about should:
If I grasp what you mean by "mood", then the mood created by "should" is that of a suggestion: e.g. "Everyone should visit Venice. It's wonderful!"
I wouldn't say that it's imperative, since "should" is used to express something that one finds useful or desirable, not a moral obligation or an indefeasable duty.
There isn't a consensus on this matter: it really depends on your particular stance and analysis that you choose to adopt.
Usually "mood" is a label for "the grammaticalisation of modality", where 'modality' means something like 'representation of an event in a non-factual way'-- i.e. your representation is 'coloured' with various notions, typically indicating some 'scale' of notions such as necessity, possibility, volition etc.
One immediate problem is that there's no clear-cut list of concepts that definitely must or must not complete this list, and it's not clear what semantically these notions really have in common other than "the types of things usually conveyed by modal verbs"-- i.e. our definitions are a little bit circular.
But accepting that, which category of modality is represented by a particular modal verb (or other construction) is really a matter of which particular framework you want to invent or subscribe to. However, a common broad categorisation would be to talk about:
In this case, the speaker is essentially indicating their opinion, so you would probably refer to it as epistemic necessity.
There is another, broader, view of "mood" which simply looks at whether events are portrayed in a way that can be agreed with or denied. So for example, note that it is easy to use a 'tag'-based utterance to agree with the following statements:
whereas that's not possible in these cases, as there's no real fact to "agree" with and we can't use the same procedure for forming a corresponding tag:
(Of course, we could say "Yes, let it!" or "No, don't let it!" but these are different syntactic structures to the 'simple' tag, and we're not agreeing with the factual content of the utterance.)
So from this viewpoint, 'should' here doesn't indicate any special mood at all. Whereas in "let it..." and the simple imperative "rain!", we would be led to conclude that these are examples of a different mood.
So I repeat: there isn't a single answer to this question. It depends on what framework you choose to adopt and is quite a complex matter.
I leave it as an exercise to the reader to decide if this categorisation, and indeed the notion of "mood" at all as it is typically applied to English, really buys you terribly much.