Syllogism — Chaining several “some” statements across different sets Statements: • Some pens are pencils. • Some pencils are erasers. • Some erasers are sharpeners. Conclusions: I) Some sharpeners are pens. II) Some pencils are sharpeners. III) Some erasers are pens. Pick the conclusion(s) that necessarily follow.

Introduction / Context:
This is a classic trap: several existential (“some”) premises are given, each possibly about different elements of the sets. The test is whether you can resist “connecting the dots” without a premise that forces the same individuals to serve in each overlap.

Given Data / Assumptions:

• Some Pens ∩ Pencils ≠ ∅.
• Some Pencils ∩ Erasers ≠ ∅.
• Some Erasers ∩ Sharpeners ≠ ∅.

Concept / Approach:
From “some A are B” and “some B are C,” you cannot infer “some A are C” unless there is information that ties the same members across the overlaps. Each “some” could refer to a different subset of B. Thus, none of the cross-bridge conclusions is forced.

Step-by-Step Solution:

Verification / Alternative check:
Create a model where the overlaps are pairwise disjoint: one item is a pen-pencil only, a second is a pencil-eraser only, and a third is an eraser-sharpener only. All premises hold, but none of I–III is satisfied, proving that no listed conclusion is necessary.

Why Other Options Are Wrong:

• Only II or only III follow: neither is forced, because the critical overlaps can be on different members.
• All follow: far too strong; there is no transitive law for “some.”

Common Pitfalls:
Illicitly chaining existential statements as if they were universal inclusions; forgetting that “some” can refer to different elements in each premise.

Final Answer:
None follows.