My textbook states that for the conditional statement "p implies q",
"p is a sufficient condition for q and q is a necessary condition for p."
How is this so? One might be lead to believe that p is independent of q and that it is a necessary and sufficient condition for q, no?
Additionally, what is the correct way to interpret the truth table of this conditional statement given below:
Edit: Please keep in mind that I have only just graduated high school and am not taking any advanced courses in logic.
Best Answer
It's important to realize that $p \implies q$ is not any kind of causative relationship - it's not saying $p$ causes $q$, or anything like that. It's saying "if $p$ happens to be true, then $q$ also happens to be true, possibly by pure coincidence".
Likewise, "necessary condition" and "sufficient condition" don't say anything about causation. $q$ is a necessary condition for $p$ exactly if $p$ can only be true if $q$ is; $p$ is a sufficient condition for $q$ exactly if whenever $p$ is true, so is $q$ - possibly coincidentally.
So suppose we know that the sentence $p \implies q$ is true. Then we know that if $p$ happens to be true, so does $q$; that's what the $\implies$ symbol means. So $p$ can't be true if $q$ isn't - if it were the case that $p$ is true with $q$ false, our sentence $p \implies q$ would be false. But that's the definition of $q$ being a necessary condition for $p$! Likewise, $q$ has to be true whenever $p$ is, which means $p$ is a sufficient condition for $q$.
Notice that it doesn't work the other way around - $p \implies q$ doesn't mean $p$ is a necessary condition for $q$, because $q$ might be implied by other, unrelated things as well.
We remind that if $p \implies q$, then: - $q \implies - p$