Let me explain. Suppose someone says "We must play a game now". I disagree, but only on that we have to do it.
I shouldn't answer "We must not" because I would be saying that the game is not to be allowed to happen.
"We don't must" feels incorrect, I'm not sure if it is.
So, other than completely changing the phrase answering things like "I don't feel like playing" or "I disagree", how do I express that idea that I don't think that we must, using the verb must? Is it possible?
The key issue is the ordering of auxiliaries in English.
(Subject) + Modal + PerfectMarker + ProgressiveMarker + PassiveMarker + (MainVerb) + (...)
An example of all the auxiliary positions being used is:
We must have been being taught English grammar.
Negatives occur after the element they are most relevant to (often the first). Since must is a modal, it occurs in the first slot. But this blocks any other modals from occurring there. Adding not runs into the conundrum of meaning you already mentioned. You could attempt to insert another expression of necessity later in the sentence, but having multiple such words often changes the semantics in odd ways. For example:
We must not have to play a game.
(Italics to indicate that most speakers would emphasize "not") It seems right at first, but this expresses that you believe (or you've come to the conclusion) that we don't have to play. This is not precisely the same thing, and the sentence is a little odd as well.
Also you are right: you cannot generally say "We don't must" as the dummy and emphatic forms of "do" cannot occur with any auxiliaries, and even if they could, they could not occur before must. (The 'lexical' do still can, but this occupies the main verb slot, as in "We must not do that.")
Thus our only real choice is to reword the sentence and lose must. Generally the most native-sounding choice is to use a periphrastic modal like "have to". (Note that periphrastic modals occur in the main verb slot, as they have not fully grammaticalized into real modals. Yet.) I think the choice the other respondents both used sounds the best:
We don't have to play a game.
To get the logical negation of "must", switch must with a different verb.
A: We must study English grammar.
B: No, we need not study English grammar.
or B: No, we don't need to study English grammar.
or B: No, we don't have to study English grammar.
or B: No, we are not required to study English grammar.
Remember that one of the defining characteristics of modals is that they can't function as infinitives. If you bear that in mind, it helps you predict the impossibility of what you are suggesting and similar sentences.
So, the solution is that you need to find another periphrastic construct (usually "need to" or "have to" etc) with a similar meaning.
Normally, a native English speaker would say something like:
We don't have to play a game.
They'd probably say it with an accent on "have" to denote that their intention is not the fact that they aren't playing the game, it's just that playing the game isn't an absolute necessary.
Both must and must not express obligations, in fact opposite ones. To express there is no obligation you can use do not have to. Or you can use noun must, e.g:
There is no such must.
Negated form, must not, has different meaning than do not have to. Consider modal verbs: can, may, have to etc. Negative forms of these modals have meaning that there is no obligation. In this sense, the meaning of must't seems odd, counterintuitive, esp. for a new learner. However, in some contexts, using must / mustn't is more useful than have to / do not have to and vice versa.
For example, must / mustn't is useful in exact/provable contexts like mathematics, logic. E.g., given the statement A is B, i.e. A must be B, if one disproves it then the disproval implies the opposite: A isn't B, i.e. A mustn't be B. In fact, in contemporary use: Must and must not are often used in deductions, e.g. for must see (merriam-webster.com/dictionary/must), points 4,5 or (collinsdictionary.com/dictionary/english/must_1), point 4.
Another example where the meaning of modal mustn't is useful is in detailed instructions, manuals, highway code, legal regulations (https://www.merriam-webster.com/dictionary/must), points 1,2,3. In such cases there is important to list obligations, steps how to proceed, and then do no have to is not useful. E.g. You must proceed when you are given green light but you mustn't when there is red light.
This is not possible. In most dialects, English auxiliary verbs take only plain-form verbs as their complements, not tensed verbs.
The modal auxiliary verbs are defective, in that they do not have the full complement of forms. In particular, they do not have a plain form. Most have a present-tense form and a past-tense form, which would disallow them as complements to any auxiliary verb. Incidentally, must is more defective than most modal auxiliary verbs in not even having a past-tense form.
