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Consider this scenarios:

A: Can I do X?
A: Can't I do X?

In both the cases, the B replies with "Yes" to indicate A can do X and with "No" to indicate he cannot.

The 1st one seems to ask for permission while the 2nd one seems to imply a tone of indignation.

Consider another scenario:

A: Can I not do X?
A: Can't I not do X?

Here, B replies with "Yes" to indicate A doesn't have to do X and with "No" to indicate A has to.

Here also, the difference is subtle: A seems to just ask for clarification or permission in the 1st and seems frustrated or even resigned in the 2nd.

My question is, what are the differences between the 2 then?

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You've really hit the nail on the head with the overtones of indignation often present when the second variant is used. Intonation is controlling; a different tone with the second variant indicates the suggesting of a hitherto unstated and perhaps ingenious course of action - one that the addressee probably hasn't thought of; another tone, perhaps with an 'Oh!', indicates that the speaker has just had a eureka moment. The 'Can't' suggests asking for clarification of rules (rather than for overt permission), or feedback on the drawbacks about the suggested course of action X. – Edwin Ashworth Apr 5 '13 at 7:59

The difference is that, when you add a negation to the verb Can, you don't really change the meaning of the question, which is to simply ask the permission.

So for instance, "Can I do X" is not different from "Can't I do X", so the same answer to either of these questions would have the same meaning.

However, when you add a negation to the verb Do, you change the meaning of the question, because now you're changing the action to which you wanted permission.

So "Can I do X" means something entirely different than "Can I not do X" and the same answer to these questions would therefore have entirely different meanings.

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in simple terms, open and closed answers, Can I do X? implies you want a quick yes or no. Whereas Can't I do X? implies you need further explanation on why you can't. The others are just double negatives as Henrique said.

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Ah, but if you put a exclamation mark at the end like "Can't i do X!?" (although not thought of as 'good English') would imply you needed a quick yes or no almost overriding your passive stance! – Edward Apr 8 '13 at 19:53

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