Syllogism — Track inclusions and existence across three classes Statements: • All bricks are stones. • Some stones are rocks. • All rocks are mountains. Conclusions: I) Some mountains are stones. II) Some mountains are bricks. III) Some stones are bricks. Choose the correct set of necessary conclusions.

Introduction / Context:
We combine a universal inclusion (Bricks ⊆ Stones), an existential overlap (Some Stones are Rocks), and another universal inclusion (Rocks ⊆ Mountains). The aim is to identify which conclusions must hold in every model consistent with these premises.

Given Data / Assumptions:

• Bricks ⊆ Stones.
• ∃x: x ∈ Stones ∩ Rocks.
• Rocks ⊆ Mountains.

Concept / Approach:
From “Some Stones are Rocks” and “All Rocks are Mountains,” at least one stone is also a mountain. That directly gives “Some mountains are stones.” Claims about bricks require the existence of at least one brick, which the premises do not guarantee.

Step-by-Step Solution:

Verification / Alternative check:
Build a model with rocks and stones overlapping, zero bricks, and with all rocks inside mountains. I holds but II and III fail, confirming that only I is necessary.

Why Other Options Are Wrong:
Any option including II or III assumes existence of bricks without premise support.

Common Pitfalls:
Reading “All A are B” as implying “Some A exist;” it does not. Existence must be given separately.

Final Answer:
Only I follow.