The paradox exists because of the modal verb expressing a possibility. It is explained how:
I could care means there is a possibility of caring and the possibility may vary in degree from a degree zero to infinitive. If one cares to any degree at all, it is possible to care a lot or you could care less which may even amount to zero degree. In other words: You could not care less because it is impossible to care less than that amount(zero degree).
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