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

Fun question. Here's how I'd break it down:

The usage is definitely incorrect in a prescriptive grammatical sense. The "or" implies "either," which means we're negating one thing or the other. These sentences appear to be trying to negate both things. The correct conjunction would be nor, because we want to negate both.

Realistically, in speech or in casual writing, one won't have an issue with people confusing the meaning of this "no...or...any" construct. It is a bit of a mess in a strict grammatical sense, but most people will understand that it means the chicken isn't sauced or seasoned, or that you aren't allergic to anything and have no medical issues.

This construct appears to be something derived from a stream of consciousness style of speech - for instance, one might say "I have no allergies," pause for a moment of thought, and then further clarify "or any medical issues, for that matter."