Syllogism — Judge the validity of conclusions from the premises: Statements: • No pencil is an eraser. • All erasers are sharpeners. Conclusions: I. All sharpeners are pencils. II. All sharpeners are erasers.

Introduction / Context:
The premises constrain relationships among three sets: pencils, erasers, and sharpeners. We must determine whether either of two strong universal conclusions is forced.

Given Data / Assumptions:

• No Pencil is an Eraser (Pencil ∩ Eraser = ∅).
• All Erasers are Sharpeners (Eraser ⊆ Sharpener).

Concept / Approach:
From Eraser ⊆ Sharpener, we only know something about the eraser portion of Sharpeners. We do not know how large the set of Sharpeners is, nor whether Sharpeners include Pencils. Therefore, universal claims about all Sharpeners cannot be justified from the given information alone.

Step-by-Step Solution:
1) C1: “All sharpeners are pencils.” This would make Sharpeners ⊆ Pencils. Nothing in the premises supports that; in fact, since Erasers ⊆ Sharpeners and Pencils are disjoint from Erasers, if Sharpeners were all Pencils, Erasers would also need to be Pencils, contradicting “No Pencil is an Eraser.” Hence C1 is false.2) C2: “All sharpeners are erasers.” This would make Sharpeners = Erasers (or at least Sharpeners ⊆ Erasers). The premise only gives Erasers ⊆ Sharpeners, not the converse. Hence C2 is not forced and can be false in a model where some Sharpeners are not Erasers.

Verification / Alternative check:
Construct a model where there are many Sharpeners, only a small subset are Erasers, and Pencils are disjoint from Erasers (but may or may not overlap Sharpeners elsewhere). Both C1 and C2 fail generally.

Why Other Options Are Wrong:
Any option that asserts one or both conclusions follow claims more than the premises guarantee.

Common Pitfalls:
Confusing a subset statement with a biconditional; assuming the converse.

Final Answer:
Neither conclusion I nor conclusion II follows.