Logical Labyrinths

12. All and Some 105
doesn’t smoke, but that is not possible by our assumption that all of
them are knaves! So our assumption that they are all knaves leads
to a contradiction, and hence they are all knights. It then further
follows that all the inhabitants are knights and all of them smoke.
12.5. Again, all the natives must be of the same type. If they were knights,
they certainly wouldn’t say that some knaves on the island smoke
(since this would imply that some of them are knaves). Thus they
are all knaves, and since their statements are therefore false, it follows
that none of them smoke.
12.6. We are given that they are all of the same type. Now, consider
any one of the natives. He says that if he smokes, then all of them
smoke. The only way that it could be false is if he smokes but not
all of them smoke. But since they all said that, the only way the
statements could be false is if each one of the inhabitants smokes,
yet not all of them smoke, which is obviously absurd. Therefore the
statements cannot be false, and so the inhabitants are all knights.
Since their statements are all true, there are two possibilities: (1)
None of them smoke (in which case their statements are all true,
since a false proposition implies any proposition); (2) All of them
smoke. And so all the inhabitants are knights, and all that can be
deduced about their smoking is that either none of them smoke or
all of them smoke, but there is no way to tell which.
12.7. Again, all the inhabitants are of the same type. Each inhabitant
claims that if any inhabitant smokes, then he does; and if that
were false, then some inhabitant smokes, but the given inhabitant
doesn’t. Since they all say that, it follows that if the statements
were false, we would have the contradiction that some inhabitant
smokes, but each one doesn’t. Thus the natives are all knights, and
again either all of them smoke or none of them smoke, and again
there is no way to tell which.
12.8. Again, they are all of the same type. They couldn’t be knights,
because if their statements were true, then some of them smoke, yet
each one doesn’t, which is absurd. Hence they are all knaves. Since
their statements are false, it follows that for each inhabitant x, either
it is false that some inhabitants smoke, or it is false that x doesn’t
smoke—in other words, either none of them smoke or x smokes. It
could be that none of them smoke. If that alternative doesn’t hold,
then for each inhabitant x, x smokes, which means that all of them
smoke. So the solution is that all of them are knaves, and either all
of them smoke or none of them smoke, and there is no way to tell
which.