(1) You cannot use can not.
(2) You can not use cannot.
Is one or the other of these sentences more grammatically correct? If so, does that make it also more semantically correct?
First off, here’s what I think about them. Up until some time last year, I was convinced that can not was the only acceptable way to write it, and cannot was an error, a neologism, or at least something a bit informal (when it actually has been in use for six centuries). Then for some reason I became convinced that both were acceptable and began using cannot because it is less ambiguous:
(3) I (cannot/can not) eat the cake, because I am too full.
(4) I (*cannot/can not) eat the cake if you want to save it for later.
See, the negation in can not could either negate the modal can (i.e., I am unable to do something) or the predicate (i.e, I am able to not do something), whereas the negation in cannot can only negate the modal. So I personally try to use cannot when I want to negate the modal and can not when I want to negate the predicate. This distinction is relevant to me because I actually do intend to negate the predicates of such sentences sometimes. Most reasonable people do not. If you are one of the people who don’t do this, then there is no reason for the choice of cannot/can not to matter to you. That’s not entirely true; some people argue that can not must be used when you want to emphasize the not, and I’ll drink to that.
Other people’s opinions vary. Neither Strunk nor Fowler seem to explicitly state a preference between the forms, although I can’t find an instance of either of them using can not. The OED, MWDEU, AskOxford, the Columbia Guide to Standard Modern English, and Paul Brians (and me) all agree that while cannot is the more common modern usage, both are acceptable. But of course there are also dissenters, and let me attempt to counter one argument against can not.
This argument is that of Language Hat, (the author of this brilliant diatribe against language “snoot” David Foster Wallace) who is justly irate that the definition of cannot in one dictionary is “can not”. We agree that these two forms are not equivalent (see sentence (4)), but I disagree with his secondary argument that cannot is properly one word because it is pronounced as such. Many phrases are properly written out as multiple words even though they are pronounced as single words. These include should have, going to, want to, go to, out of. And English orthography is hardly beholden to pronunciation (tho it mite be nice if it were).
Summary: Both cannot or can not are generally fine. The one-word form is preferred in contemporary English, but the two-word form must be used when you want to emphasize the negation or you want to negate the predicate. And as for the questions at the beginning, both sentences are grammatically correct, but only (2) is semantically correct: you are allowed to not use cannot.
***
The Inner Spaces series so far:
I: A lot about alot (10/24/07)
II: All right (10/26/07)
III: Can not be split? (10/27/07)
IV: It’s Been a While (01/14/08)
Motivated Grammar 
5 comments
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October 27, 2007 at 9:10 pm
renaissanceguy
I agree with your rationale and your conclusion.
June 9, 2009 at 2:58 pm
jettaodernie
Hi there!
Thank you very much for that splendid explanation.
Although I found the answer to my question here already, I will definitely have a closer look at some of the other articles. I am sure there is so much more on this site that is really interesting and useful, not only for me as a ESL teacher to-be.
Would that be alright if I liked your page on my blog?
Thanks again,
take care, M.
June 9, 2009 at 6:57 pm
jettaodernie
That’s a typo, sorry. Of course, I do not only like your page but I would like to link it as well.
September 22, 2010 at 4:43 am
Chantel
Would it be proper English to say, “Cannot you do this for me?” since it is okay to say, “Can’t you do this for me?”
September 22, 2010 at 12:54 pm
Gabe
Chantel: No, come to think of it, you can’t. The problem has to do with subject-auxiliary inversion. To form a question in English, you have to invert the subject and auxiliary, so “You can do this.” becomes “Can you do this?” So the question here is what else inverts with the auxiliary. If you’re not using a contraction, a negative won’t follow the auxiliary, so “You are not coming.” becomes “Are you not coming?” If you do use a contraction, the negative is joined to the auxiliary in a way that allows for both of to be inverted: “You aren’t coming.” -> “Aren’t you coming?” But this auxiliary+negative inversion requires the two to be joined in a contraction, so you can’t say “Are not you coming?” (Or at least I can’t in Pittsburgh English; some poking around on Google suggests this may be possible in Indian English and some 19th century British English.)
“Cannot you” doesn’t work for the same reason, which means that you really can’t say “Can’t you do this for me?” without a contraction, because “Can you not do this for me?” is a significantly different question. Thanks for making me think about this!