Use of can't/cannot to indicate the opposite

I cannot find any references online that would help me know what this topic in English grammar is called, but I'm trying to guide a non-American friend to understand why some would use the word "can't" or "cannot" to imply the opposite, as in

"I will see if I can't get to the bottom of this for you."

Of course it is clearer to say "I will see if I can get to the bottom of this for you," however, I suspect the former is not technically incorrect grammar, though it certainly may be a localized expression. At any rate, it is definitely a common occurrence.

However, being as that I don't know what this usage of "can't" to imply the opposite is called and that it's ridiculously hard to Google, I can't seem to find out either way if it's right or wrong.

EDIT

Thinking over this some more, I feel that the sentence using "can't" is actually clearer than the one with "can." I'm thinking that using "can't" in this case implies the difficulty/perplexity of the situation, and makes it clear that I'll be trying hard?

The idiom seems to be something like Vᵖ if/whether S/¬S, where

• Vᵖ refers to some mental Perception verb (see, discover, figure out, ...) that can take
• S is an Embedded Question complement from a Yes/No Question. These use whether or if as a complementizer (Can/Did they do it? ==> whether/if they can do it/did it).

Either S or ¬S can occur in the complement; i.e, if they can do it vs if they can't do it. This is because it eventually occurs to everybody that, if you can discover that X will occur, then you also know if X won't occur, and vice versa. Yes/No questions are binary, so an answer to either one gives you the same information. (p ⊃ q) ≡ (¬p ∨ q).

I don't know whether a modal is strictly necessary in the S, though of course modals like will and can do occur very frequently with hypotheticals like if, and interact strongly with negatives. See the many Language Log postings on "Overnegation".

I will see if I can't . . . and I will see if I can . . . are both grammatical, and both sentences can be used to make an offer. The difference lies in their use. The negative form is more colloquial and is slightly more forthcoming than the positive form, which is more likely to be followed by . . . but I can’t promise anything. I know of no name for this construction, although there might a Greek word for it somewhere in a list of tropes.