I was watching a movie the other day and one character to another said "Don't let's fight" instead of "Let's not fight." Is this the proper usage, and if so, what is the grammatical rule that applies here?
Michael Swan's book "Practical English Usage" has this to say:
There are two possible negatives, with let us not and do not let us (informal let's not and don't let's)
Forms with don't let's (and let's don't in American English) are very informal.
Your question brings to mind Alexandra Fuller's book "Don't Let's Go to the Dogs Tonight." I think the "don't let's.." construction is a plea: Please, oh please, let's not...
The long form would be do not let us fight, which is perfectly acceptable and conveys the idea that the speaker would like to avoid an argument. Your question immediately put this song in my head.
Contractions are not always grammatical just because their expanded forms are. The issue of the grammaticalness of contractions is determined by the rules of forming contractions, and their required context, and required subjects, objects and relations. These rules are not apparent just by writing down the expanded forms. The contraction "let's" is very specific, and is not just short for "let us".
The second is not a contraction of the first. Without a break between the "Mom" and the rest, it is ungrammatical and meaningless. The essential issue is that the "let" in "let's" must always be a command. But consider now the commands:
You see that the meaning is different. For non-native speakers, the second is saying that Mom should go to the movies with us, the first is (usually) commanding mom's permission.
In principle, the first can be a long form of the second, but only in certain formal contexts where contractions are disallowed. But the second form can never mean that the command is directed at mom. Let's just is not allowed to command a person like that. The "let" in "let's" always has as a subject, not a person, but a non-entity, similar to providence, or the French "on".
So you get "Mom! (Oh, providence,) let us go the movies!" as the best non-contracted approximation to "Mom! Let's go to the movies!", while the permission meaning doesn't allow the contraction, although it is the same meaning of "let", just with a different implicit subject.
In the case of "Don't let's fight", all these rules for the let's construction are fulfilled. So it is ok by these criteria.
has the same difference in meaning as the first pair. So this is an ok construction, it really is grammatical to say "Don't lets" according to the explicit let-us contraction rules.
American vs. British English
The real difference here is in the different unspoken rules of implicit subjects for "let's" in British and American English. When there is a "don't", the "don't" has a verb "do" there which is also commanding of providence. So if you can say:
When the implicit subject of the command is providence, then you can say "don't let's". If you can't say "Do" with an implicit subject providence, you're cooked.
The do business is a pure British construction, which is un-American. In American, the implicit subject providence can "let" things, but it can't "do" things.
So American English allows:
But in American, when you say
There is a real subject, not providence, not a neutral subject. If there is no one who is obviously being adressed, it actually sounds like a prayer, because you fill in the subject as God. This is different than "Let's go!" which hardly ever sounds like a prayer, becuase the subject is the implicit neutral providence thing (I am sure that linguists have a name for this). In british, "don't let us go to the movies" obviously isn't a prayer. It has the same neutral providence subject as "let's go to the movies."
So when an American says:
There is an implicit actual subject who is letting, like mom, or God, it isn't the neutral On-like providence. So the contraction is not allowed, because the meaning is never the same:
now is ungrammatical to the American, who believes that let's is sitting in a place where a subject-filled command should be.
(GEdgar's graph is explained by the rise of American)
There is, as usual, a succinct summary in ‘The Cambridge Guide to English Usage’: