Syllogism — Non-forced intersections vs. guaranteed propagation Statements: • All desks are rooms. • Some rooms are halls. • All halls are leaves. Conclusions: I) Some leaves are desks. II) Some halls are desks. III) Some leaves are rooms. Determine which conclusion(s) necessarily follow.

Introduction / Context:
This problem contrasts conclusions that require a specific intersection (not guaranteed) with a conclusion that follows from pushing an existential through a universal inclusion. The key is to see which sets are forced to overlap by the premises.

Given Data / Assumptions:

• Desks ⊆ Rooms.
• ∃x: x ∈ Rooms ∩ Halls.
• Halls ⊆ Leaves.

Concept / Approach:
From “Some rooms are halls” and “All halls are leaves,” the same witness element is both a room and a leaf, which guarantees “Some leaves are rooms.” However, nothing forces desks to be among those rooms that are halls, so claims about leaves∩desks or halls∩desks are not necessary.

Step-by-Step Solution:

Verification / Alternative check:
In a Venn diagram, place a dot in Rooms ∩ Halls (which lies within Leaves). No dots are needed in Desks anywhere. The premises hold and only III is forced.

Why Other Options Are Wrong:
Any option including I or II assumes extra overlap not stated.

Common Pitfalls:
Assuming that because Desks are Rooms, and some Rooms are Halls, therefore some Desks are Halls; this is a classic error (illicit conversion of “some”).

Final Answer:
Only III follows.