Syllogism – Blankets, beds, and pillows (using a universal to confirm existence) Statements: • Some blankets are beds. • Some pillows are blankets. • All beds are pillows. Conclusions to test: I. Some blankets are pillows. II. Some pillows are beds. III. Some beds are blankets.

Introduction / Context:
Here, a universal statement (“all beds are pillows”) combined with an existential (“some blankets are beds”) guarantees existence inside the larger class (pillows). This allows multiple “some”-type conclusions to be derived safely.

Given Data / Assumptions:

• Bl ∩ Bd ≠ ∅ (some blankets are beds).
• Pl ∩ Bl ≠ ∅ (some pillows are blankets).
• Bd ⊆ Pl (all beds are pillows).

Concept / Approach:
Any bed is a pillow. Therefore, the specific blankets that are beds are also pillows. In addition, because there exists at least one bed (from “some blankets are beds”), there exists at least one pillow that is a bed. Finally, the original “some blankets are beds” itself ensures some beds are blankets.

Step-by-Step Solution:
I: From Bl ∩ Bd ≠ ∅ and Bd ⊆ Pl, the same elements lie in Bl ∩ Pl. Thus “some blankets are pillows.” I follows.II: Since Bd ⊆ Pl and at least one bed exists (by Bl ∩ Bd), there is at least one pillow that is a bed. “Some pillows are beds” follows.III: “Some beds are blankets” is literally the first premise (Bl ∩ Bd ≠ ∅) rewritten, so III follows.

Verification / Alternative check:
Draw three sets with beds entirely inside pillows. Place at least one element in the overlap of blankets and beds. Instantly, all three conclusions can be read off the diagram.

Why Other Options Are Wrong:

• Any option suggesting only one or two conclusions undervalues the chain from “some blankets are beds” and “all beds are pillows.”
• Exclusive “either/or” choices are inappropriate because all three hold simultaneously.

Common Pitfalls:
Overlooking that a universal joined with an existence statement yields existence in the superset; forgetting that restating a premise (“some beds are blankets”) is a valid derived conclusion.

Final Answer:
All I, II and III follow