Last week Vicki Hartley wrote to ask me about "the use of a negative construction when a positive construction is simpler". One example is "I'll see if I can't do that". Strangely, this sentence means essentially the same thing as "I'll see if I can do that". Another example she asked about was, e.g., "I don't have but three students", to mean "I have only three students". The way I see it, these are two separate constructions, though there is an underlying theme.
In the first, you're saying that you'll try the opposite of what you intend. I told Vicki that I saw this as similar to what goes on in the scientific method -- we'll start with our hypothesis and try to disprove it; hopefully we'll fail. Logically, this seems to make sense, since either way the result is the same: you find out whether or not you're able to do something. On the other hand, as a speaker, I wonder if this "scientific method" approach is really what's going on here. To me "I'll see if I can't do that" really doesn't mean "I'm going to attempt to not do that". To me I can't find any semantic difference in "see if I can't" versus "see if I can". Pragmatically there are differences -- the redundant negative version seems to presuppose that there will be some difficulty associated with the course of action, hence the negative. On the other hand, it seems to also presuppose that the course of action will result in a positive outcome. To me, telling a sopping wet child "let's see if we can't find you some dry clothes" would be infelicitous if it were my child at my house. I could only felicitously say that to my child's friend at my house, where there would be no reason to expect to find them dry clothes that fit properly. (An aside: I picked this example because I associate "let's see if we can't..." with a parent talking to a child -- not sure if this is relevant to the discussion at hand.) However, I'll also say it implies that I'm relatively sure of a positive outcome (finding dry clothes that fit). If I say "let's see if we can find you some dry clothes" I don't get quite the same expectation; there's more tolerance for the negative outcome "oh well, I guess not".
The second construction is a slightly different version of the redundant negative. For me, "I don't have but three students" is slightly marked, but fully grammatical, whereas "I have but three students" sounds archaic almost to the point of ungrammaticality. So we have the following, which all entail having three students:
1) I have three students.
2) I have only three students.
3) I have but three students.
4) ?I have but only three students.
5) I don't have but three students.
I have a question mark by (4) because I'm not sure if I like it or not. I
think I don't, but it doesn't seem totally wrong. And then we have the
following, which all entail NOT having three students:
6) I don't have three students.
7) I don't have only three students.
8) *I don't have but only three students.
I'm pretty sure (8) is bad, but if it's not, I think it would mean the
positive, not the negative.
My guess is that we should treat this second construction as parallel to the first one. Despite my archaic interpretation of "I have but three students", this was definitely fine in earlier versions of English, and my guess is that many people would find it unremarkable even today. Thus "I don't have but three students" is essentially the same phenomenon as "Let's see if we can't do that". One avenue of research that might prove useful is semantic research on some of the North American languages. Salishan languages have suffixes that create a "managed to" reading (non-control transitivizers, for those in the know). Navajo has an adversative reading that indicates that a proposition is counter to expectation. Blackfoot (an unrelated language) has the same thing (which coincidentally is also the affix for "please"). This might be what we're seeing in English: variation based on presuppositions of the speaker's ability to bring into being some desired or discussed resultant state.