Despite its score, ryang's post is totally wrong. [I'm a native English speaker and logician. And the downvotes on this post demonstrate the failure of the SE voting system.]
He incorrectly claims that "the phrase ‘not any’ actually means ‘not some’ instead of ‘not every’.", and says that the following are equivalent and examples of this claim:
(1) She does not have any disease.
(2) She does not have some disease.
This is bogus, as is obvious to any native English speaker. Consider another pair:
(3) He does not understand any complicated sentence.
(4) He does not understand some complicated sentence.
This pair obviously mean completely different things, contrary to ryang's claim.
He also incorrectly attempts to connect his incorrect claim with the FOL equivalence between "∀x ( P(x) ⇒ Q )" and "∃x ( P(x) ) ⇒ Q". This is quite obvious once you consider the following:
(5) If any student fails the course, he/she must retake the course.
Obviously, the pronoun "he/she" refers to "any student" that "fails the course". So one is NOT permitted to express this in anything close to the form of "∃x ( P(x) ) ⇒ Q"!
~ ~ ~
The true answer is that ryang is NOT describing actual English, and hence is not at all an answer to Bernd's question (and one wonders how many upvoters actually know proper English).
In proper English, "any" and related determiners "anyone" and "anything" function by inducing a ∀-quantifier at the global level, but not passing any indirect-statement boundary (including modal boundary) unless forced to. The determiners "some" and "every" and "each", on the other hand, stay as local as possible. And "some" means "any" when forced to extend beyond a local region.
The correct analysis of the above key examples are as follows:
(3) He does not understand any complicated sentence.
(3') ∀X∈ComplicatedSentences ( he does not understand X )
(4) He does not understand some complicated sentence.
(4') ∃X∈ComplicatedSentences ( he does not understand X )
(5) If any student fails the course, he/she must retake the course.
(5') ∀X∈Students ( X fails the course ⇒ X must retake the course )
Here are more examples:
(6) If some student fails the course, he/she must retake the course.
(6') ∀X∈Students ( X fails the course ⇒ X must retake the course )
(7) If every student of Paul fails some course, he will be fired.
(7') ( ∀X∈StudentsOf(Paul) ∃C∈Courses ( X fails C ) ) ⇒ ( Paul will be fired )
(8) If any student of Paul fails some course, he will be fired.
(8') ∀X∈StudentsOf(Paul) ( ∃C∈Courses ( X fails C ) ⇒ ( Paul will be fired ) )
(9) If everyone agrees on any additional rule, we will adopt it.
(9') ∀R∈AdditionalRules ( ∀X∈People ( X agrees on R ) ⇒ We will adopt R )
(10) If this job can be done by anyone, we would not need to hire any expert.
(10') ∀E∈Experts ( ( ∀X∈People ( this job can be done by X ) ) ⇒ we would not need to hire E ) )
In particular take a good look at (9) and (10), which demonstrates that my explanation is correct. In (9), "any additional rule" is quantified at the global level. In contrast, in (10), "done by anyone" is quantified only within the "can ..." boundary, but "any expert" is quantified at the global level.
Finally, the use of "if" instead of "iff" (if and only if) in a definition in conventional mathematics is an abuse of terminology and once you fix that then it becomes obvious that the quantifier represented by "any" does not pass the "iff". This fact about "iff" being a hard boundary for quantification is a convention but also partially logically motivated, but it will take too long to explain why...