Your Pain Is Our Pleasure
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July 18, 2013
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I find the distinction quite useful. Cannot means the action is impossible; can not means there is the option to not take the action. Thus "I was going to X, but if you don't want me to I can not" means I would be happy to change my plans if you want me to"; on the other hand, "I was going to X, but if you do't want me to I cannot" means that you have the power to veto my action.
@e2e4: Actually, the second was in common currency in centuries past--from Jane Austen's Emma: "'Cannot you, my dear Emma--cannot you form a guess as to what you are to hear?'" Cf. will not/do not etc. Austen. It would still be grammatically correct, but modern tongues would automatically contract it: "Can't you go there?" or "Can't you form a guess?"
Indeed, I think that today "Can you not" would be used to mean "Can't you" only by people trying to sound extra-formal, such as authors writing historical fiction (which is often wrong, since it should be "Cannot you" to avoid anachronism). Much more often I would expect a spoken "Can you not?" to have the emphasis on the 'can' (as others have noted), meaning "Are you allowed to not?" or else "Would you refrain?" rather than "Is it impossible?": "Can you not buy homeowner's insurance?" or "Can you not keep tapping your foot?"
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