In Kenneth Rosen’s book example number 13 of Chapter 1.5 says,
There is a woman who has taken a flight on every airlines
In the example the solution is like this:
Let $P(w,f)$ $:$ $w$ has taken $f$
$Q(f,a)$ $:$ $f$ is a flight in $a$
And the answer is $∃w∀a∃f(P(w,f)∧Q(f,a))$
My question is why the answer is not $∃w∀a∃f(Q(f,a)\to P(w,f))$?
What is the difference between these two answers? Please explain.
The issue is that $Q(f,a) \rightarrow P(w,f)$ is a weaker statement than $Q(f,a) \land P(w,f)$; in particular, the first statement is true if $Q(f,a)$ is false regardless of whether $P(w,f)$ is true or not. This prevents it from matching up with the verbal statement.
For example, you could have a woman $w$ who has never taken a flight by some particular airline $a$, but since there exists a flight $f$ that is on some other airline, $Q(f,a) \rightarrow P(w,f)$ is true, so this woman could satisfy your logical statement but not Rosen's, and not the verbal statement.
In fact, as long as there are flights on at least two airlines, every woman would satisfy $\forall a \exists f (Q(f,a) \rightarrow P(w,f))$.