grammar - Grammatical function of "at best" idiom

Dictionaries state that "at best" is an idiom. But, what is the grammatical function of "at best" (for example, in the below sentences?)

• Their response to the proposal was, at best, cool.


• The government's response seems to have been at best confused and at worst dishonest.


• If he drops the course now, at best he’ll get an incomplete, and he could fail.

Thanks

Answer

In order to analyze this, it would be easier to start with one of the more standard definitions:

• taking the most optimistic view


• in the most favourable interpretation


• under the most favourable condition

Then substitute the phrase for the definition:

• Their response to the proposal was, in the most favorable interpretation, cool.


• The government's response seems to have been in the most favorable view confused and in the least favorable view dishonest.


• If he drops the course now, under the most optimistic outcome he’ll get an incomplete, and he could fail.

Now you can see it as a prepositional phrase. I think (but am not 100% sure unless someone supports this) that it would function as a disjunct adverbial. See http://www.linguisticsgirl.com/using-prepositional-phrases-disjunct-adverbials/

In the last sentence, there seems to be some element of uncertainty. Suppose it was a teacher who caught a student cheating. The teacher could give an incomplete, but the administration might fail him. In that case the teacher would say 'but':

• If he drops the course now, under the most optimistic outcome he’ll get an incomplete, but he could fail.