A10 The Use of a Double Negative in Russian, p. 265

An additional negative particle ничего is frequently added to expand the negative thought. In English, a double negative is normally not acceptable, except in slang expression. "He don't know nothin."

But in Russian, Он ничего не знает, though it has two negative particles, would be translated into English, "He doesn't know anything." And it is the correct way to express this thought in Russian.