The grammatical rule against double negatives applies to sentences that combine not with no or with other negatives such as hardly, nobody, nothing, never, and nowhere:
I can’t hardly see through these glasses.
He didn’t meet nobody on the mountain.
They never lied about nothing.
On the other hand, double negatives formed with not followed by a word that begins with a negative prefix like un- or ir- are permissible in formal English.
This type of double negative is a stylistic device of understatement, a type of litotes: a figure of speech in which an affirmative is expressed by the negative of the contrary. Its use can convey a subtle difference in meaning that saying the same thing without not wouldn’t.
Some speakers who object to the “not un-” construction seem to believe that there’s a rule against it. This belief is bolstered by the often quoted example made up by George Orwell: “The not unblack dog chased the not unbrown cow across the not ungreen field.”
Orwell’s sentence is amusing, but simplistic. No English speaker is going to try to plant an un- on adjectives like black and green. Many English speakers, however, will use the “not un-” construction to achieve a nuanced meaning with adjectives like justifiable, intelligent, and convinced.
The following statements are not identical in meaning:
I am convinced by his argument.
I am not unconvinced by his argument.
A note at the online Oxford Dictionaries site points out the difference:
The use of not together with unconvinced suggests that the speaker has a few mental reservations about the argument.
Writing in 1926, H. W. Fowler (“Modern English Usage”) regarded the “not un-” usage as “a faded or jaded elegance.” He condemns the unnecessary use of the construction, but recognizes that this form of litotes is “congenial to the English temperament” and that there are contexts in which its use is suitable. He concludes, “The right principle is to acknowledge that the idiom is allowable, and then to avoid it except when it is more than allowable.”
The construction is often used unnecessarily, but sometimes it expresses a thought in a way that the positive form would not.
The following examples are probably “allowable” uses:
Arthur Ransome’s book is a not unsuccessful attempt to provide an introduction to the criticism of types of fiction.
—The reviewer does not like the book, but acknowledges that some readers may learn something from it.
Sabina Franklyn is a sweet, pretty Jane, not unintelligent but less of a presence than her lively sister.
—The character Jane is not a stupid person, but her intelligence is not a key feature of her personality the way it is for her sister.
A musty but not unpleasant odor came from inside, together with a blast of pent-up heat.
—The odor cannot be described as pleasant, but it is not repugnant either.
However, the “not un-” construction is frequently used without justification, as in the following examples:
Surprises will come to be outnumbered by cringes, as the not unappealing Patton finds herself sold short by a director.
—In the context of the review, there’s nothing to suggest that Patton is anything but appealing.
While not unentertaining, there is very little of this film that is astounding.
—In the context of the article, the reviewer seems to feel that the film is entertaining.
Some reviewers, perhaps not quite understanding the device, go all to pieces in their attempts to make use of it:
Jessica Biel is not unconvincing as a love interest, but she’s not entirely convincing.
But there’s something uniquely dis-appealing about “Don Jon.” Not unappealing, not immediately repulsive. More like simply not-appealing.
The “not un-” construction has a long history in English and remains a valid stylistic choice for writers who understand how and when to use it.