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0.2. The epistemic usage and the root usage (subjective and objective)
When the speaker makes an utterance, the feeling which is epistemic or root, can be seen.
The epistemic usage means the speaker considers the event whether it is the fact or not, that is, the speaker is sure or not sure. See the following example.

She must be at the office.

This "must" expresses the speaker's assertion.

"May" expresses the speculation. The similar adverb meanings are "possibly", "probably" and "certainly" etc. (*2)

*2 These are the adverbs which mean the modality, therefore, it is called the modal adverb.
Moreover, "be possible to" and "be certain to" etc. are called the modal adjectives.

Therefore, the above can be changed into the following example.

It is certainly that she is at the office.

Now, the root usage means the speaker does not consider the event whether it is the fact or not. There are an obligation and permission etc. See the following example.

You must come by eight o'clock.

The above also uses "must", but it means the obligation.

And then, usually, the epistemic usage is parallel to the subjectiveness and the root usage is parallel to the objectiveness. This text uses subject and object instead of epistemic and root to understand the modal auxiliary easily.

For example, "can" and "may" are similar, but "can" is more root. Therefore, it is treated as the objectiveness. Moreover, the ability which is one of the meanings of "can" is the objective meaning. Thus, this means the objective auxiliary verb expresses the objective meaning.

"Can" also has the possibility, it is epistemic. It is represented in the text as "the objectiveness  the subjective meaning." It means the object auxiliary verb expresses the subjective meaning, that is to say, the object auxiliary verb also implicates the subjective meaning. It is explained in the chapter 1. (*3)

*3 "The objectiveness  the subjective meaning" is parallel to the can's possibility. In case of this, "the objectiveness" is the modal auxiliary itself and "the subjective meaning" is one of the meanings of the modal auxiliary.
These depend on the speaker's feelings, but it can be explained like this.

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