Is choosing between universal and existential quantification tricky?

Question

I came across some sentences in English that are unclear about quantification: the Discrete Math textbook that I'm using says that

If you can solve any problem we come up with, then you will get A

can be interpreted either as

• If you can solve every problem we come up with...

or

• If you can solve at least one problem we come up with...

I don't understand why there is more than one interpretation: how can 'any' can be interpreted as 'at least one'?

Another example given is that

I haven't seen any of his movies

can be interpreted either as

• I haven't seen all of his movies

or

• I haven't seen at least one of his movies.

But shouldn't the above statement obviously mean 'all' rather than 'at least one'?

Answer 1

If you can solve any problem we come up with, then you will get A

how can 'any' can be interpreted as 'at least one'?

The consequent "you will get A" is too suggestive; here's a more neutral/ambiguous example, which provides a sharper contrast between the two possible meanings of 'any':

• If you can solve any problem we come up with, then we will hire you;

it is unclear whether the hiring criteria is solving at least one problem, or solving every problem.

I haven't seen any of his movies $\quad(\forall m\:\lnot Sm)\tag1$

shouldn't the above statement obviously mean 'all' rather than 'at least one'?

• It is not that I have seen any of his movies. $\quad(\lnot\exists m\:Sm)\tag2$

Well, although sentences (1) and (2) are equivalent to each other, in $(1),$ 'any' means 'all', whereas in $(2),$ 'any' means 'at least one'.

Read more here: 'Any' versus 'each/every'