additional 'not'

[suprunp]

Might and might not in the permission sense are rare, occurring chiefly in polite questions:
Might I (not) have a word with you?
I wonder whether I might (not) have a word with you.
A comprehensive grammar of the English language (Randolph Quirk) [1]

Unfortunately I couldn't find information (although, I am sure [1] contains it) as to when and where (and with what modal verbs) I can use an 'additional not' and what tinge it gives to a sentence. I can surmise from [1] that it can occur in polite questions (and I'm confident that sometimes I have seen it used, although I can't provide you with a decent example now).

Would you be so kind to help me with this issue?

Thanks.

[5jj]

The simple answer is - Forget 'not'. If 'not' seems illogical to you, then it probably is to most native speakers. Such utterances are more natural without 'not'.

A longer answer would involve a discussion on (pseudo-)diffidence in dated upper-middle-class English, which would be irrelevant to most learners today.

[Tdol]

I could still use not if I were asking a big favour or pushing my luck.

[suprunp]

Might I not ask you another small question?

I have recently encountered a sentence that looked like this:
Could I not justifiably claim [...].

Partly because

Such utterances are more natural without 'not'.

I had had a hard time working out what the author actually wanted to convey until I inquired what it really meant. In this case "could I not" was just another way of saying "couldn't I" and so "not" could be justifiably dispensed with.

Another reason why I struggled with this sentence was that given the examples A comprehensive grammar of the English language (Randolph Quirk) gives us:

I visit them not infrequently. = fairly frequently.
He writes not inelegantly. = rather elegantly.

I thought (somewhat unjustifiably, as turned out) that not could negate justifiably here.

Is it possible to view this kind of sentences in the latter way, or should I always view "could I not" (and such) in similar cases as "couldn't I" or simply as "could I"?

Thanks.

[5jj]

Is it possible to view this kind of sentences in the latter way, or should I always view "could I not" (and such) in similar cases as "couldn't I" or simply as "could I"?

It depends on context.

A: I think you'd better stay at home tonight.
B: Could I not go? = Couldn't I go? (Could I go?) - I want to go.

A: I really think you ought to go.
B: Could/Couldn't I not go? = Is it possible for me not to go? - I don't want to go.

A: You couldn't pass that test even if you studied for a year.
B. Could I not? = Couldn't I? Could I?- You are wrong; I could.