Double negatives are illogical
If double negatives were illogical then it would also be illogical that double negation is in so many world languages. Standard variety English is one of the few where double negations have fallen out of favour.
In some cases, the use of double negatives is preferred. Think of the use of not unfriendly for instance, this means the person is not friendly but he is not unfriendly either. Double negation is useful when you are not sure of something.
Baur, L., & Trudgill, P. (1998). Language myth. In Baur, & Trudgill, Language myths. London: Penguin Books.