Statements: All cups are books. All books are shirts. Conclusions: Some cups…

Concept:

Two categorical rules govern this pair of statements. First, the chain (transitivity) rule: when two universal affirmative statements ("All A are B", "All B are C") share a common middle term, they combine into a single definite universal statement between the outer terms — "All A are C". Second, the conversion rule: a universal affirmative "All A are B" always yields the valid particular converse "Some B are A", but it can never yield a negative particular such as "Some A are not B", since that would directly contradict the universal itself.

Application:

1. The two statements — "All cups are books" and "All books are shirts" — share the middle term "books", so the chain rule combines them into one definite statement between the outer terms: all cups fall inside shirts.

2. The first conclusion, "Some cups are not shirts", is a negative particular between cups and shirts. Since a chain of two universal affirmatives can never produce a negative particular between its outer terms, this conclusion does not follow.

3. The second conclusion, "Some shirts are cups", is the converse of the definite statement from Step 1. By the conversion rule, a universal affirmative's converse is always a valid particular statement, so this conclusion does follow.

Cross-check:

A quick concrete check confirms this: let the only cup, c, also be a book and a shirt (cup within book within shirt). Every cup (c) is indeed a shirt, so "some cups are not shirts" is false. And c itself shows a shirt that is a cup, so "some shirts are cups" is true — matching Step 2 and Step 3 exactly.

Only the second conclusion follows.