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.

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Solve this multiple-choice question and choose the correct option.

Accepted Answer

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: Derive I: pick r ∈ Stones ∩ Rocks (from the existential). Since Rocks ⊆ Mountains, r ∈ Mountains. Hence Mountains ∩ Stones ≠ ∅ ⇒ I follows.Test II: “Some mountains are bricks” needs at least one brick; the premises never state that any brick exists. II is not forced.Test III: “Some stones are bricks” would also require at least one brick. From Bricks ⊆ Stones alone, existence does not follow. III is not forced. 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.

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