If the statement “None but the brave win the race” is false, which of the…
2017
If the statement “None but the brave win the race” is false, which of the following statements must be true?
Answer: B. Some persons who win the race are not brave. — ConceptThe expression “none but A are B” means that only A can be B. It states the universal inclusion: every B is an A. The negation of a universal statement…
- A.
All brave persons win the race.
- B.
Some persons who win the race are not brave.
- C.
Some persons who win the race are brave.
- D.
No person who wins the race is brave.
Show answer & explanation
Correct answer: B
Concept
The expression “none but A are B” means that only A can be B. It states the universal inclusion: every B is an A.
The negation of a universal statement produces a particular counterexample: if “every B is A” is false, at least one B is not A.
Application
Let W denote people who win the race and B denote brave people.
“None but the brave win the race” translates to W ⊆ B: every winner is brave.
If W ⊆ B is false, there exists at least one person in W who is outside B; equivalently, some winner is not brave.
Contrast
“All brave persons win the race” reverses the direction of inclusion and asserts B ⊆ W.
“Some persons who win the race are brave” is a particular affirmative claim and can remain true even when W ⊆ B is false.
“No person who wins the race is brave” is a universal negative proposition and is stronger than the single counterexample required by the negation.
Result
Therefore, the claim that can be inferred is: some persons who win the race are not brave.