From David Denison:
Not sure where this fits in the misnegation scheme of things. On Jazz Record Requests (BBC Radio 3, 19 Jan 17:00) the presenter quoted a listener's request as follows:
I think that not many listeners will be familiar with this track - and should be.
As David observes, "The intended meaning is obvious" — and the number of negatives is appropriate — but something has gone wrong. Two things combine to create the problem: (1) The placement of the negative (in the subject "not many listeners"); and (2) the syntactic level of the and coordination (partnered with the verb phrase "will be familiar"). In addition, the intended meaning is more consistent with but than with and.
If we ignore the initial qualifier "I think …", the structure looks something like this:
(I've used "***" to mark the verb phrase ellipsis "familiar with this track".)
This means that there are not many listeners for whom it's both true that they are familiar with this track and that they should be familiar with this track. And that, unfortunately, is more or less the opposite of what the presenter clearly meant, which is that (most or all) listeners ought to be familiar with the track, even though they mostly now are not.
The presenter could have fixed the problem by changing the level of coordination to sentences instead of verb phrases:
Or by putting the negation in the first conjoined verb phrase rather than in the subject:
Other solutions are obviously possible as well.
This is the sort of thing that often goes wrong in extemporaneous speech, where we're somehow managing to cast complex hierarchies and conjunctions of concepts into word-sequences in real time. It's especially common when an additional thought is added to the ingredients list after the cake is partly baked, as seems to have happened here.
Not many of us are capable of getting it right all the time — nor should be.