Introduction / Context:
This syllogism mixes “some” overlaps with a universal negative. We must decide which conclusions necessarily hold in every arrangement that satisfies the premises, without adding extra assumptions about further overlaps.
Given Data / Assumptions:
- Some Rats are Cats (∃ R ∩ C).
- Some Cats are Dogs (∃ C ∩ D).
- No Dog is a Cow (D ∩ Cow = ∅).
Concept / Approach:
- “Some X are Y” is symmetric, so “Some Cats are Rats” is equivalent to “Some Rats are Cats.”
- To invalidate a conclusion, it suffices to create a compliant diagram where that conclusion is false.
Step-by-Step Solution:
Conclusion III: “Some cats are rats.” This is exactly the first statement restated; it necessarily follows.Conclusion I: “No cow is a cat.” We only know Dogs are disjoint from Cows. Cats that are not Dogs could still be Cows; no statement forbids it. Hence I does not follow.Conclusion II: “No dog is a rat.” We know some Cats are Dogs and some Rats are Cats, but those two “some” groups need not be the same Cats. A model can keep R∩C and C∩D disjoint, so II does not follow.
Verification / Alternative check:
Draw Cats as a large set; place one overlap with Rats and a separate overlap with Dogs; ensure Dogs are disjoint from Cows. Premises hold; I and II can be false simultaneously while III holds.
Why Other Options Are Wrong:
Options including I or II assume extra exclusions not specified. Only III is compelled by the premises.
Common Pitfalls:
Treating two independent “some” overlaps as if they must coincide.
Final Answer:
Only III follows
Discussion & Comments