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Would it be possible in a statement not a question?
Good point. It brings to mind . . ., well, another "rare" usage. Consider this,
Here's a riddle. I have two coins in my pocket. One is not a nickle (= 5 pennies) and one is not a quarter (= 25 pennies). Both are not nickles, and both are not quarters.
[1] Both are not secretaries.
=> Of the two one is (not) a secretary
It's not the default reading, agreed, but it's possible, at least in my dialect. "Both" refers to a set, and that set is made up of two individual items. Negate the set and we get reading [a]; negate the individual items within that set and we get reading [b].
[a] neither is a secretary
[b] one is (not) a secretary
Re: Both + negative sentence 
It brings to mind . . ., well, another "rare" usage. Consider this,
Linear Mode
