belamo,
While it is true that Either/Or are coupled when implying choice is available, OR being in a sentence does not mandate choice.
From Boolean logic:
If a then b MAY require that
If Not a then Not b,
but it is a fallacy to imply
If b then a from the above.
So, too, with la lingua propria.
As Crio mentioned, earlier, a more archaic form of the negative was
Neither/Nor.
The antecedent "n" in Nor has been lopped off in most modern English, but it being gone does not mean it is forgotten in the Rules of the language.
A suggestion, any time you have a sentence using Neither, or where a correct substitution would be Neither, silently append the "n" from Neither onto the subsequent

or, and you will see that no choice is left to you. The lazy shortening of the Nor to Or does not negate the lack of choice in the sentence structure.
Ni uno, dos, o tres =
Ni uno, ni dos, y tambien no tres.
(I hope my Spanish is not so rusty as to have made an abomination of that example.)
As for Crio's example
Crio said:
If I say "I do not have apples, grapes, and pears"
Going back to the Mathematical Logic you love so well -
Boolean Logic
Not A, G and P =
Not [A, G, ans P] as far as Boolean Logic is concerned.
It is only true if one has NONE of the three.
Missing one or more of the set makes the whole statement false.
It is awkward in English words in a grammatically correct sentence, but clear if you set it up the way it would be in math.
The solution to some of the confusion is to recast some of the examples to make them less ambiguous for all. It does not negate their actual meaning, if argued in a literary "court of law", however.
I see Crio has posted just ahead of me!