It has long been settled doctrine among English grammarians that two negatives cancel each other and produce an affirmative. Just as in mathematics -(-x) equals +x, so in language "he did not say nothing" must be regarded as equivalent to "he said something".
It is going too far to say, as is sometimes said, that this proposition is self-evident. The Greeks did not think that two negatives made an affirmative. On the contrary, the more negatives they put into a sentence the more emphatically negative the sentence became. Nor did Chaucer think so, for, in a much-quoted passage, he wrote :
He never yit no vileineye ne sayde
In al his lyf, unto no maner wight
He was a verray parfit gentil knyght.
Nor did Shakespeare, who made King Claudius say:
Nor what he said, though it lacked form a little,
Was not like madness.
Nor do the many thousands of people who find it natural today to deny knowledge by saying "I don't know nothing at all about it". And the comedian who sings "I ain't going to give nobody none of mine" is not misunderstood.
Such repeated negatives, says Jespersen, are usual in a great many languages in which the negative element is comparatively small in phonetic bulk, and is easily attracted to various words. If the negation were expressed once only, it might easily be overlooked; hence the speaker, who wants the negative sense to be fully appreciated, attaches it not only to the verb, but also to other parts of the sentence: he spreads, as it were, a thin layer of negative colouring over the whole of the sentence instead of confining it to one single place. This may be called pleonastic, but is certainly not really illogical.
Still, the grammarians' rule should be observed in English today. Breaches of it are commonest with verbs of surprise or speculation ("I shouldn't wonder if there wasn't a storm ". "I shouldn't be surprised if he didn't come today"). Indeed this is so common that it is classed by Fowler among his "sturdy indefensibles". A recent speech in the House of Lords affords a typical instance of the confusion of thought bred by double negatives:
Let it not be supposed because we are building for the future rather than the present that the Bill's proposals are not devoid of significance.
What the speaker meant, of course, was "Let it not be supposed that the bill's proposals are devoid of significance".
Another example is:
There is no reason to doubt that what he says in his statement ... is not true.
Here the speaker meant, "There is no reason to doubt that his statement is true".
And another:
It must not be assumed that there are no circumstances in which a profit might not be made.
Always avoid multiple negatives when you can. Even if you dodge the traps they set and succeed in saying what you mean, you give your reader a puzzle to solve in sorting the negatives out.