A double-negative sentence is one that has two "negatives," and as we all know, two negatives make a positive. This is also known as a "lelote," but I am just going to say "double negative," because, duh, there are two negatives.
I don't not know what's going on.
I know what's going on.
Now there are more subtle double-negatives that don't begin with "N". For example:
I doubted that she didn't know me.
"Doubted" is a negative (synonym is "didn’t think").
So: I think she knew me.
There are a few of these subtle negatives:
In fact, some of the double-negative constructions below don't use "not" terms. But they're still presenting two negatives in the sentence.
What's the problem with that? This construction forces the reader to re-read and recalculate what the sentence really means. Once in a while you might want to confuse the reader, but if you don't, recast the sentence clearly, with positive terms that make clear what you mean. Here are some examples, where you end up with a single negative perhaps:
There is no reason to believe that this primary campaign won't be not only effective but also responsive.
This primary campaign will probably be both effective and responsive.
Few people outside of the United States disbelieve in the power of American commercialism.
Most people outside the United States believe in the power of American commercialism.
The point is not that the decline in income has not been arrested.
The point is that the decline in income has been arrested.
In our inability to write with emotion, we are not so different from those alienated suburban teens who not only don't have the vocabulary to express feelings but also don't have the freedom to invent a new vocabulary.
In our inability to write with emotion, we are like those alienated surburban teens who have neither the vocabulary of feeling nor the freedom to invent it.
No less wise a person than Charles Buckley was incapable of facing age gracefully.
Even as wise a person as Charles Buckley was incapable of facing age gracefully.
So what's the issue?
You usually want the reader to understand your meaning and not have to "translate" the sentence to make sense. So you might try translating it yourself first! Try to recast it as a positive sentence so that the reader can understand it immediately.
Now if most of your sentences are easily understandable, the infrequent double-negative construction will stand out as particularly meaningful. You've gained the trust of the reader that you do know how to convey information concisely in clear sentences. So that trust will have the reader approaching your few over-complicated sentences and trying to figure out why you made these more difficult.
Every sentence construction has its purpose. So there are reasons you might want to use the double-negative and other complicated constructions. Why would you choose to make a sentence more difficult to read? Well, you might want the reader to slow down and read slowly, maybe even read it twice or three times. You might want to convey a hedge or irony:
Bill is not unworthy of promotion.
Bill is more or less worthy of promotion, but I am not enthusiastic enough to recommend him.
Often double negatives are followed by a "but" or "however" which states something clearer:
Hemingway is not an unaccomplished stylist; however, his terse prose is an acquired taste.
Finally, though, an important purpose of the double negative is to allow for a second meaning, a deeper meaning, a subtext or ambiguity. Here's an example, from that master of subtext, Shakespeare:
What wouldst thou write of me, if thou shouldst praise me?
O gentle lady, do not put me to't;
For I am nothing, if not critical.
Here is why double negatives shouldn't be wasted just to over-complicate simple thoughts. The double negative here conveys the most subtle of character revelation.
Iago is a complicated man who lives within deception, and here, for a moment, he is being honest, if only in his sentence formulation. He is nothing except his contrariness, his resentment, his envy, and the very construction of the sentence ("I am nothing") presents his sense of his own emptiness, and then, the sad but honest corollary there-- If I am not critical, I am nothing. That is, as with enjambment in poetry, the double-negative sentence lets the reader get two simultaneous and perhaps contrasting or complementary meanings.
I can't stress this enough, however: LESS IS MORE. The fewer times you use over-complicated sentences, the stronger will be the effect. The reader will assume that it there is some reason you broke your usual comprehensible pattern, and look for the deeper meaning.
(Oh, if you're wondering if two positives can make a negative, let me introduce you to my teenagers. Their favorite term is, "Yeah, right." Now you have to say that aloud, with full adolescent sarcasm, to get that they mean, "Mom, you are so totally wrong.")