Syllogism — Nested inclusions and an external existential Statements: • All chairs are tables. • All tables are cushions. • Some cushions are trolleys. • All trolleys are lamps. Conclusions: I) Some lamps are tables. II) Some trolleys are chairs. III) Some cushions are lamps. IV) All chairs are cushions. Choose the correct set.
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AOnly I follows
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BOnly III and IV follow
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COnly either II or II follows
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DAll follows
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ENone of these
Answer
Correct Answer: Only III and IV follow
Explanation
Introduction / Context:Nesting inclusions allow some conclusions to be read off immediately, while an existential about another branch creates a separate chain. The trick is to avoid assuming intersections that are not enforced.
Given Data / Assumptions:
- Chairs ⊆ Tables.
- Tables ⊆ Cushions.
- ∃t: t ∈ Cushions ∩ Trolleys.
- Trolleys ⊆ Lamps.
Concept / Approach:From Chairs ⊆ Tables ⊆ Cushions, it follows that All chairs are cushions (IV). The existential witness t in Cushions ∩ Trolleys, combined with Trolleys ⊆ Lamps, implies that Some cushions are lamps (III). However, to claim “Some lamps are tables” or “Some trolleys are chairs” would require the trolley witness to lie inside Tables/Chairs, which is not guaranteed.
Step-by-Step Solution:
IV: Chain inclusions directly: Chairs ⊆ Tables ⊆ Cushions ⇒ Chairs ⊆ Cushions.III: From t ∈ Cushions ∩ Trolleys and Trolleys ⊆ Lamps, t ∈ Lamps; hence Cushions ∩ Lamps ≠ ∅.I: For “Some lamps are tables,” we would need a lamp that is also a table; nothing links the trolley-lamp witness back to Tables.II: For “Some trolleys are chairs,” we would need Trolleys ∩ Chairs ≠ ∅; again, not enforced.Verification / Alternative check:Diagram two branches: (Chairs → Tables → Cushions) and (Cushions ← Trolleys → Lamps). The trolley branch meets Cushions but need not pass through Tables/Chairs.
Why Other Options Are Wrong:They assume overlaps between Trolleys and Tables/Chairs that the premises do not create.
Common Pitfalls:Overextending an existential witness into unrelated subsets merely because they share a common superset.
Final Answer:Only III and IV follow.