When is it best to write "can't" versus writing "cannot"? Are they interchangeable in every situation?
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Grammatically, you can use can't instead of can not or cannot in the majority of circumstances. There is an exception. In wh-movement, the contraction should not be expanded unless you also change the word order:
Why can't I have some bacon? //OK
Why cannot I have some bacon? //not OK, archaic
Why can I not have some bacon? //OK again, although formal
Stylistically, the choice between can't and cannot is more complex. Generally, people use can't in speech and informal writing, and cannot or can not in formal writing or very formal speech. Also (as @Kris points out in a comment), cannot might be used when you need to carefully distinguish it from can't in speech.
Mark Beadles
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BTW, the OED says that the one-word construct cannot is "the ordinary modern way of writing can not" with a space between it, which apparently is archaic, or non-modern, or some such. – tchrist Aug 20 '12 at 19:22
Does "Why can I not have some bacon" sound awkward, as it is, in formal circumstances, only in fringe cases applicable? For example: "You can not have bacon because it is out of our reach." While for a denied request one would use "Why may I not have some bacon?" The use of 'can' instead of 'may' is already an informal speech pattern. Moreover, a denied request is unusual in formal settings, questioning it would be quite awkward per se.
– gschenk Aug 09 '16 at 04:10