(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.