Statements: Some towels are brushes. No brush is soap. All soaps are rats.…
2025
Statements: Some towels are brushes. No brush is soap. All soaps are rats.
Conclusions:
I. Some rats are brushes.
II. No rat is brush.
III. Some towels are soaps.
Answer: B. Only either I or II follows — CONCEPTA conclusion follows only when it is true in every model that satisfies the premises. Two conclusions form an either-or pair when one is the exact…
- A.
None follows
- B.
Only either I or II follows
- C.
Only II follows
- D.
Only I and III follow
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Correct answer: B
CONCEPT
A conclusion follows only when it is true in every model that satisfies the premises. Two conclusions form an either-or pair when one is the exact logical negation of the other: one of them must be true, although the premises may not determine which one individually.
For an existence statement “Some R are B,” the exact negation is the universal exclusion “No R is B.” They cannot both be true and cannot both be false.
APPLICATION
Let R denote rats and B denote brushes. Conclusion I says that R ∩ B contains at least one member. Conclusion II says that R ∩ B is empty. These are exact logical complements, so exactly one of I or II must hold.
The premises do not determine which member of the pair holds. A non-soap brush may also be a rat, making I true; alternatively, every brush may lie outside the set of rats, making II true. Both arrangements preserve “No brush is soap” and “All soaps are rats.”
Conclusion III needs at least one towel that is also a soap. The known towel–brush member cannot be a soap because no brush is soap, and the premises supply no other towel–soap witness. Therefore III is not forced.
CROSS-CHECK
If a rat–brush overlap exists, I is true and II is false.
If no rat–brush overlap exists, I is false and II is true.
In either case, the premises can still be satisfied with no towel–soap overlap, so III remains unproved.
Therefore the accepted response is “Only either I or II follows.”