In some other languages, eg. Romance languages (Italian, Spanish, French, Romanian etc.) the double negative reinforces the negativity.

eg. (Italian):

Italian = non ho niente
gloss = * not / I have / nothing
ie. 'I have nothing' or 'I do not have anything'

ObJoke (or is it a true story/urban legend?):

A linguistics professor is giving a lecture. He says: "In English, a double negative like 'I can't not go' is a positive. But in some other languages, like Russian, a double negative is an emphatic negative. However, there is no language where a double positive is a negative."

"Yeah, sure," someone shouts from the back row.

Not to be picky, but I don't see a good clear statement of the problem of double negatives here.

The way English grammar is set up, using a double negative should make a positive: "I can't not go" = "I have to go"; "Well, it doesn't not work" = "It works! (albeit not as we planned)".

But many English users treat double negatives as if they are negative: "I'm not never going"" = "I'm never going"; "There ain't nothing wrong" = "Nothing's wrong".

This is annoying to people who like to keep the language neat. Double negatives are often considered to indicative of a lower class upbringing and a poor education. There is some truth to this, but not as much as the grammar police would have you believe.

In Spanish, and many other European languages, double negatives are fine. They are neither positive nor used for emphasis, they are simply a way of speaking.

We have 18th century mathematician and grammarian Robert Lowth to thank for the received wisdom that two negatives are ungrammatical. In his 1762 book A Short Introduction to English Grammar, Lowth, who was enamoured of Latin-derived logical models, decreed that two negatives cancel each other out and create a positive; until that time, two negatives were taken to reinforce each other. Lowth's rule, though arbitrary and invented, has become "truth", as has his insistence that sentences should not end in prepositions, that split infinitives are ungrammatical, and that "they" cannot be used as a gender-neutral pronoun. Not all languages consider a double negative to be not correct.