Whether double negatives are allowed is a common source of disagreement with regard to both allowability and meaning. This article discusses some of the involved issues, and touches on the hypothetical use of a double positive as a negation.
Note that I differ strictly between “double” (“triple”, ...) and “repeated” negatives/positives. The former is used to indicate a traditional double, e.g. “I do not want nothing”; while the latter is a repetition, e.g. “No; I do not want anything.”, and logically something very different. This terminology need not coincide with that of other authors, and corresponding care should be taken when comparing opinions; however, the distinction is of utmost importance, and I suspect that the failure to make it is one of the leading causes of confusion.
I also have some views on the related concept of litotes.
(The Wikipedia page double negativew, in version 340180653, has been used as a source of examples, some of which are criticized below.)
Most examples of double negatives (in languages that I am familiar with) appear to be the result either of inability to think through the effects of two negatives or of following a traditional (in a certain region, group, culture, ...) use. The traditional cases arise mostly through unthinking repetition and imitation of phrases used by e.g. parents and age peers. These, in turn, either have ultimately originated through incompetence or go back so far that the origins are unclear (and may or may not have been correct in the language of that faraway time).
An often given explanation, that the double negative would be a means of emphasis, has rarely been plausible to me (at least not when looking at modern English): Someone who claims “I can’t get no satisfaction.” will typically use a double negative as a matter of course (either in this type of sentence or in all negated sentences), which implies that its use in this particular statement brings no added emphasis. It can, however, well be that some of these traditional doubles originally arose from a combination of an attempt to add emphasis by repetition and a lack of thought or understanding.
I would speculate that some claimed examples of multiple negatives from other languages are a misinterpretation of congruency, in the same way as “we are” is not a double plural, but a plural pronoun followed by a congruent verb. If a certain hypothetical language expresses “We are not at home.” by “We-no are-no at home-no.”, we just have three congruent sentence elements, not a triple negative. Of course, this could also be a reason for phrases like “can’t get no”—the problem being that such non-standard use is incorrect and potentially confusing for those who use the standard rules.
Why is a double negative (typically) incorrect? Simply because its literal interpretation is usually something very different from what the speaker actually intends. Consider “We don’t need no education.”: Here we have two negatives, “don’t” and “no”. The former negates the entire sentence (arguably, it is more correct to say that “n’t” negates “do”), giving “It is not so that we need no education.”. The latter is a part of the object (“no education”). In other words, there is something that we do not need; this something is no education, or an absence of education. In effect, we do not need education to be absent—which implies that the presence of education is acceptable or, possibly, even welcome. This, however, is in grave contrast to the actual intention of unwelcome. (Notably, depending on the exact intentions, education may or may not have been acceptable; however, it is certainly not something wished for.) All in all, the statement is unintentionally humorous, because it is proof in itself that the speaker does need education...
Note that it is not strictly correct to equate a negative with a logical negation, as in “P = not not P”, because the many negatives have a slightly different character. In the above example “not” (as part of “don’t”) is a negation; however, “no” is more akin to an empty-set qualifier. Thus, we do not have “We don’t need no education.” = “We need education.”, but = “We do not need a set of education that is empty.”. Other variations exist, notably complementary and “opposite” sets. For instance, a non-negative number is not the same as a positive number, and a negative number is not the same as a non-positive one (0 is simultaneously non-negative and non-positive).
It should further be born in mind that a triple (“not not not”) will be logically close (but rarely identical) to a single not, while a double typically takes a near opposite meaning.
The French “pas” is often cited as a legitimate use of double negative. Arguably, however, it is not a negative at all, being originally used to add emphasis (compare e.g. “no” and “no way”). Over time the use of “pas” has changed to become a fix part in compound negatives, and ultimately taking on a negative value on its own. This then is not strictly an example of a legitimate double negative, but a mixture of traditional use (as above) and a drift in meaning. The confusion is likely increased by the different word order, which means that a French speaker would not say e.g. “is no way”, but “no is way”.
There are cases of apparently double (or more negatives) that are not what they appear to be. Wikipedia gives the following examples of a triple and a quadruple negative from Chaucer (I give the modern English only):
There was not no man nowhere so virtuous
He never yet no vileness not said / In all his life to no sort of man.
These, however, do not live up to the claim:
The first amounts to “There was not no man, nowhere, so virtuous”, which nowhere has more than a double negative. Yes, the sentence contains three negatives, but (at least in the terminology used on this page) they are only doubled. Remove the one “not” (“There was no man, nowhere, so virtuous.”), and the sentence is a perfectly lucid and logically consistent statement using a repeated negative (notwithstanding that the typical English formulation would be “no man, anywhere”). In fact, we may even stretch this to “There was not [left-out object], no man, nowhere, so virtuous.”, which is a three-fold repetition of a negative without any doubling at all. (This would imply another error of some kind. Notably, however, the old English version appears to use “nas” in the sense “not was”/“was not”, which could be a further indication of repetition rather than doubling. Judging this in detail is unfortunately beyond my current knowledge.)
The second can be paraphrased as “In all his life, to no sort of man, he never yet no vileness not said.” where at least the first “no” is independent of a (probably) actual triple negative—we thus have a triple, not a quadruple. This triple could possibly be reduced further with a deeper analysis.
In both cases, the error of the Wikipedia article lies in confounding a “not x, not x” with a “not not x”: “I dislike Brussels sprouts; I dislike Brussels sprouts.” with “I do not dislike Brussels sprouts.”—repeated negative and double negative.
The same section gives a third potentially misleading example, by attributing to “a well-educated man” (Oliver Cromwell) what may well be the literal preservation of someone elses double negative.
While the existence of double negatives in the accepted language of yore is well-established, Wikipedia gives the impression of partiality here.
A well-known anecdote goes (in one variation):
A linguistics professor was lecturing to his class one day. “In English,” he said, “A double negative forms a positive. In some languages, though, such as Russian, a double negative is still a negative. However, there is no language wherein a double positive can form a negative.”
A voice from the back of the room piped up, “Yeah, right.”
(http://funpeople.org/1997/1997BIJ.htmle)
While both funny and a good example of creative thinking, this retort is specious:
Firstly, this is not a double positive, but a repeated positive. It is more akin to “No, no, no!” or “No; you must not.” than “We don’t need no education.”.
It is not the compound, but each individual statement, that causes a negative, through use of irony, likely even sarcasm. That the combination is used, instead of one of the individual words, is a matter of clarity: The combined phrase has become strongly associated with sarcasm, and it is harder to unambiguously bring over the proper sarcastic pronunciation with just one syllable to play with. In principle, however, either of “yeah” and “right” could be used to bring over the same sentiment. (Obviously, using just one word would remove the proposed counter-example and, thereby, make the statement a mere disagreement.)
In effect, the proposed statement is not a “Yes yes.” (double positive) equaling a “No.”, but a “No, no.” (repeated negative) with that meaning.
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