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We didn’t see nothing. [ = We saw nothing.]
She never danced with nobody. [ = She didn’t dance with anybody.]
The rules dictate that the two negative elements cancel each other out to give a positive statement instead, so that the sentence ‘I don’t know nothing’ could literally be interpreted as ‘I do know something’.
Double negatives are standard in many other languages and they were also a normal part of English usage until some time after the 16th century. They’re still widely used in English dialects where they don’t seem to cause any confusion as to the intended meaning. Nevertheless, they aren’t considered acceptable in current standard English and you should avoid them in all but very informal situations. Just use a single negative instead:
We didn’t see anything.
She never danced with anyone.
There is one type of double negative that is considered grammatically correct and which people use to make a statement more subtle. Take a look at the following sentence:
I am not unconvinced by his argument.
The use of not together with unconvinced suggests that the speaker has a few mental reservations about the argument. The double negative creates a nuance of meaning that would not be present had the speaker just said:
I am convinced by his argument.