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4. The negative sentence and the interrogative sentence
The modal auxiliary verbs have different negative scope.
Some modal auxiliary verbs are not used in the interrogative sentence.

4.1. The negative sentence
The negative scope depends on the modal auxiliary.

a. She may not be at home now.
b. It is possible that she is not at home now.

a. It can't be true.
b. It is not possible that it is true.

The above is the possibility. (1a) denies the main verb as you see (1b). (2a) denies the modal auxiliary like (2b). These are derived from the meaning whether it is subjective or objective.
The subjective meaning denies the main verb, and the objective meaning denies the modal auxiliary.

If both of them mean objective, for example, the permission, they deny the modal auxiliaries. See the following examples.

a. You can not go there.
b. You are not permitted to go there.

a. You may not stay here.
b. I do not permit you to stay here.

But the stress is put on "not", it can deny the main verb.

a. You can NOT go there.
b. You are permitted not to go there.

a. You may NOT stay here.
b. I permit you not to stay here.

Now, the obligation of "must" and "should" denies the main verb, since it has some premise. See the following examples. (*26)

*26 The volition of "will" and "shall", and "should" deny the main verb, but they can deny the modal auxiliary.
"Must" implicates the objective meaning, but denies the main verb because of the premise.

a. You must not smoke in this room.
b. I oblige you not to smoke in this room.

To deny the modal auxiliary, "need not" is used.

a. You must not go.
(It is necessary not to go.)
b. You needn't go.
(It is not necessary to go.)

It is similar to "should."

a. You should not smoke in this room.
b. I oblige you not to smoke in this room.

(9a) can be changed into (9b). To deny the modal auxiliary "need not" is also used.

a. We should not go.
b. We needn't go.

"Have to" denies the modal auxiliary since it has the objective meaning which is the exterior factor.

He doesn't have to work hard.

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