On the expression "no [noun 1] or any [noun 2]"?

Question

I have often seen the following expressions: [ex.] 1. I have **no** allergies or **any** medical issues. 2. John serves a chicken with **no** sauce or **any** kind of seasoning. I suspect that such a use is wrong from a prescriptive grammatical point of view, because "no" modifies "allergies" in (ex.1) and the sentence is split as follows: "I have no allergies" or "I have any medical issues". But this is not what sentence (ex.1) means. (ex.1) means that I have no allergies or (and?) no medical issues. Probably, I think that (ex.1) is reanalyzed as "I don't have any allergies or any medical issues". My question is: (1) Is the use like (ex.) correct (in prescriptive grammatical terms)? (2) If not, what is the correct expression to tell the expressions like (ex.1) and (ex.2)? (3) Or is there some grammatical rule where when "no" is used, the word that comes after "or" may be "any". (4) Given that I am wrong and the use like (ex.) is correct, why is this so?

Answer 1

It seems that this colloquial usage is conflating two types of negation. For the first, one might say, "I have no allergies or medical issues," in which case "no" would apply to the group of "allergies or medical issues" as a unit; for the other, one might say, "I have neither allergies nor medical issues," in which case it would be acceptable to include "any" before "medical issues". However, as you've noticed, the sentence "I have no allergies or any medical issues" confuses the extent and application of the negation; thus, this form should be avoided.

Answer 2

Technically, it should be "nor" any medical issues, which makes both sides of the sentence negative. However, nor is one of those words that has fallen out of use in United States English, probably because we are lazy.