negating nested generalized quantifiers

From the definition of the square of opposition, I know that the sentence “All the apples are red.” has a same meaning as “It’s not the case that some apples are not red. ” and “No apples are not red.”

But I’m not sure how to deal with sentences contains two quantifiers.

E.g. Assuming there are balls which have two sizes (small and big) and two colors (red and blue)

Is the sentence “All the red balls are the same size as most blue balls. ” has the same meaning as “It’s not the case that some red balls are not the same size as most blue balls.”

And is the sentence “All the red balls are larger than some blue balls” equal to “It’s not the case that some red balls are not larger than some blue balls.” and “No red balls are not larger than some blue balls.”?

Could anyone please tell me whether to also change the second quantifier?

Note that the existential quantifier means at least one, i.e., “some [thing] is” rather than “some [things] are”.

You're not negating statements, just using double negation to equivalently re-express them.

And is the sentence “All the red balls are larger than some blue balls” equal to “It’s not the case that some red balls are not larger than some blue balls.” and “No red balls are not larger than some blue balls.”?

Assuming that the choice of the blue ball(s) depends on the choice of red ball, instead of vice versa:

$$\forall x\,(Rx\to \exists y \,(By \land Lxy))$$ is equivalent to $$\lnot\,\exists x\,(Rx\land\forall y \,(By \to \lnot\,Lxy)),$$ so, corrections:

• “It’s not the case that some red ball is not larger than every blue ball.”
• “No red ball is not larger than every blue ball.”