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.
- A.
None follows
- B.
Only either I or II follows
- C.
Only II follows
- D.
Only I and III follow
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Show answer & explanation
Correct answer: A
Correct answer: "None follows".
Premises:
Some towels are brushes.
No brush is soap.
All soaps are rats.
Reasoning and counterexamples for each conclusion:
Conclusion I — "Some rats are brushes": Does not follow. Counterexample model that satisfies all premises but makes this false: towels = {b}, brushes = {b}, soaps = {s}, rats = {s, r}. Here b (a towel and brush) is not a soap and not a rat, so no rat is a brush; all premises hold but there are no rats that are brushes.
Conclusion II — "No rat is brush": Does not follow. Counterexample model that satisfies all premises but makes this false: towels = {b}, brushes = {b}, soaps = {s}, rats = {s, b}. Here b is both a brush and a rat but not a soap; all premises hold (no brush is a soap, all soaps are rats, some towels are brushes) while the statement 'No rat is brush' is false.
Conclusion III — "Some towels are soaps": Does not follow. From 'Some towels are brushes' together with 'No brush is soap' we know the towels that are brushes are not soaps, and there is no information guaranteeing any towel is a soap. A model such as towels = {b}, brushes = {b}, soaps = {s} (with b not equal to s) satisfies the premises while no towel is a soap.
Because each conclusion can be made false while all premises remain true, none of the three conclusions follows from the premises.