Unusual change of meaning of word "any" in negative sentences form "for all" to "there exists". Predicate logic

Question

Question. Why does the word "any" in negative sentences changes its meaning from "for all" to "there exists"?

Origin of the question. I have a question about translating English to predicate logic. I don't understand well enough how the word "any" should be translated in predicate logic in general. I know that this word has principal meaning "for all", but I have found one example where it, in my opinion, means "there is at least one".

None of Occam's followers likes any realists. This example is from Suppss's "introduction to logic"

(x)(Ox->(y)(Ry->Lxy))

In my opinion this is equivalent to: "There is not at least one follower of Occam who likes at least one realist".

My attempt to solve this issue. So let us consider simpler example. Say, the sentence: No dogs likes any cat. (x)(Dx->(y)(Cy->Lxy)) In my opinion it is equivalent to: "There is no single dog who likes at least one cat" and not equivalent to: "There is no single dog who likes all the cats".

First one. There is a set of all the dogs. There is a set of things that like at least one of the cats. So intersection of these sets is empty.

Second one. There is a set of all the dogs. There is a set of all the things that like all of the cats (even mad and ill ones). So intersection of these sets is empty.

From second one we can infer that there might be a dog who likes some cats, but not all of them (It might like only fluffy ones). From first one it follows that there can't be any cat that is liked by any dog.

So I concluded that "any" changes it's meaning from "for all" to "there exists" in some sentences, possibly in all negative sentences.

My research. There is a similar question: Reference request for the logical analysis of the word "any" In my opinion it doesn't fully answer the question, they say that it is some artifact of English language. So there is grammar book, that tells that in these cases we use "any" as existential and in other cases as universal quantifier.

I think that it deeper phenomenon than just some grammatical artifact of English. I know that there is similar issue with translation of "any" in other different languages. So in these languages there is the same issue.

P.S.: I have read 6 books in logic, I know what a square of opposition is. I am a chemist who has logic as a hobby.

Answer 1

Conifold's comment is incorrect. See this post for a more accurate representation of the meaning of "any" in English. That post also gives a sufficient counter-example:

(1) She does not have any disease.
(2) She does not have some disease.
(3) For any disease D, she does not have D.

(1) does not mean (2), but rather (3). Applying the same explanation to your example:

(4) No dog likes any cat.
  ≡ ∀A∈Cats ¬∃B∈Dogs ( B likes A ).

Your idea (i.e. that "any" changes it's meaning from "for all" to "there exists" in some sentences, possibly in all negative sentences.) is therefore wrong when implemented in the natural way (i.e. changing (1) to (2)). Here are yet more counter-example of different kinds:

(5) If any person does not understand English, he/she would answer your question wrongly.
  ≡ ∀R∈People ( ¬( R understands English ) ⇒ R would answer your question wrongly ).

(6) If this job can be done by anyone, we would not need to hire any expert.
  ≡ ∀Y∈Experts ( ( ∀X∈People ( this job can be done by X ) ) ⇒ we would not need to hire Y ) ).

(7) If this job had been done by anyone, we would not have any problem now.
  ≡ ∀X∈People ∀Y∈Problems ( this job had been done by X ⇒ we would not have Y now ).

As explained at the linked post, in (6) the "anyone" is stuck under the modal "can", whereas the "any expert" is not under any modal boundary so it goes all the way to the global level. And (7) looks superficially similar to (6) but lacks the modal "can" and so the "anyone" does go all the way to the global level!