In the Land of Liars , “all politicians have a spine and a head” is false. Then which of the following is true in the Land of Liars?
(A) There is at least one politician who has no spine and no head.
(B) All politicians have no spine and no head.
(C) There is at least one politician who has no spine.
(D) There is at least one politician who has no head.
(E) If all politicians have a spine, then there is at least one who has no head.
On
The answer is definitely not $(A)$ or $(B)$, and I hope you can see that. $(C)$ is false because all politicians can have a spine but not a head. Then the initial statement still holds. By the same reasoning, $(D)$ is not correct either.
This leaves the answer as $(E)$.
I'm curious, what level olympiad is this question. For a senior paper it seems pretty elementary.
"All politicians have a spine and a head" is FALSE.
This implies that not all politicians must have a spine and a head, which means there might be some politicians who don't have either of those or have only one of those.
However:
Only one politician doesn't have a spine is enough to make the statement false.
Only one politician doesn't have a head is enough to make the statement false.
Only one politician has nothing is enough to make the statement false.
This means that either of the above will make the statement false, and because the statement is FALSE, we can't know if any of them is definitely correct (we can only know that at least one of them is correct).
This means neither of $(A),(C),(D)$ are definite true statements.
$(B)$ is incorrect, also because as stated above, "Only one politician has nothing is enough to make the statement false."
For $(E)$ however, if all politicians have a spine AND no one has no head then the statement "All politicians have a spine and a head is TRUE". By contradiction, $(E)$ is correct.
Note: At first, I assume that it is possible that this exercise may have multiple statements correct instead of one, that's why I'm considering all cases.