Both + negative sentence

[Tdol]

Woulkd it be possible in a statement not a question?

[Casiopea]

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.

Answer: One is a nickle and one is a quarter.

[Tdol]

Other than riddles?????

[Casiopea]

Other than riddles?????

I believe context is the determinant.

[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

[Tdol]

Interesting- there's no way I'd get the 'one is not' meaning from that.