Syllogism — Be careful with shared supersets and localized existentials Statements: • All sands are beaches. • All shores are beaches. • Some beaches are trees. • All trees are hotels. Conclusions: I) Some shores are hotels. II) All beaches are shores. III) Some beaches are hotels. IV) Some sands are trees. Pick what is logically forced.

Introduction / Context:
Several sets lie under the common umbrella “Beaches.” One premise adds an existential overlap between Beaches and Trees, and another pushes Trees into Hotels. We must see which conclusions are guaranteed and which remain optional.

Given Data / Assumptions:

• Sands ⊆ Beaches.
• Shores ⊆ Beaches.
• ∃x: x ∈ Beaches ∩ Trees.
• Trees ⊆ Hotels.

Concept / Approach:
The existential at Beaches ∩ Trees, combined with Trees ⊆ Hotels, guarantees an element in Beaches ∩ Hotels — i.e., “Some beaches are hotels.” There is no information linking Shores (also a subset of Beaches) to Trees or Hotels, so we cannot guarantee that Shores overlap Hotels. Similarly, nothing forces Sands to overlap Trees.

Step-by-Step Solution:

Verification / Alternative check:
Construct a model where Beaches has disjoint regions: one region overlaps Trees (hence Hotels), other regions cover Shores and Sands without touching Trees. Then only III holds.

Why Other Options Are Wrong:
Options that include I, II, or IV assume overlaps not forced by the premises. The provided option phrased as “Only either I or III follows” effectively reduces to “Only III follows,” since I does not follow while III does.

Common Pitfalls:
Assuming that subsets of a common superset must intersect each other; they need not.

Final Answer:
Only either I or III follows (here, effectively, only III follows).