Let p,q, and r be the propositions,
p: Grizzly bears have been seen in the area.
q: Hiking is safe on the trail.
r: Berries are ripe along the trail.
Then in the book Discrete Mathematics And Its Applications by Kenneth H Rosen the following sentence,
For hiking on the trail to be safe, it is necessary but not sufficient that berries not be ripe along the trail and for grizzly bears not to have been seen in the area
is translated into propositional logic is (q -> (~r ^ ~p)) ^ ~((~r ^ ~p) -> q).
But, in my logical thinking it has to be only the first part of the above expression (q -> (~r ^ ~p)).
So, what's wrong with my thinking?
Best Answer
The first part is the necessary condition. The second part is the "but not sufficient" condition
The claim is that $(1)$ if the berries are ripe or bears have been seen, then it's not safe and $(2)$ even if the berries are not ripe and bears have not been seen, it still might not be safe (for example, there might be a hurricane approaching). In other words, condition $(1)$ is necessary for safety, but not sufficient to guarantee it.
The statement "$B$ is necessary for $A$" can be expressed as $A\rightarrow B$. The statement "$B$ is sufficient for $A$" can be expressed as $B \rightarrow A$. The statement "$B$ is necessary but not sufficient for $A$" can be expressed as $(A \rightarrow B) \land \lnot(B \rightarrow A)$.
Thus, in your proposed version, you've only indicated the necessary condition. I agree, your version doesn't claim it's sufficient. But the statement in the book says more: It explicitly says that the necessary condition is not sufficient.