The book "Introduction to Logic" by Irving Copi, Carl Cohen and Kenneth McMahon says the following (in Chapter "Categorical Propositions" under heading 5: Traditional Square of Opposition, part D: Subalternation):
In subalternation (in the classical analysis) the superaltern implies the truth of the subaltern. Thus, from the universal affirmative, “All birds have feathers,” the corresponding particular affirmative, “Some birds have feathers,” was held to follow.
In general this says that the truth of "All x are y" implies the truth of "Some X are Y". We write in symbols the superaltern statement followed by its subaltern:
$\forall x\in X$ we have $x\in Y$
$\exists x\in X$ such that $x\in Y$
Consider the case that $X$ is empty.
Then one argues that since the empty set is the subset of every set, we have that $X$ is contained in $Y$ which is equivalent to the superaltern statement above.
But in this case the subaltern statement is instantly falsified because there does not exist an $x\in X$ at all, let alone an $x$ which is in $X$ as well as $Y$.
Does this not contradict the quoted sentences above from the logic text? Does the logic system discussed (Aristotlean) consider vacuous cases?
It's a well-known omission from the depths of antiquity.
In the modern treatment, indeed, the vacuity of $X$ is covered as a special case.
In Copi's Symbolic Logic (I have the 4th edition of 1973) he covers this in section $4.1$:
I don't like Copi particularly, he's wordy and uses obfuscatory language.
An analysis of what's going on here can be found on ProofWiki here:
https://proofwiki.org/wiki/Universal_Affirmative_implies_Particular_Affirmative_iff_First_Predicate_is_not_Vacuous
The "prev" and "next" links will take you back and forth through the source works which have been covered on that site, illustrating the analysis that is done on this whole area of logic.