I'm reading Exner's Logic in Elementary Mathematics, there is an exercise in which I should point all the inference schemes, if any in a bunch of statements. There is this one:
A point $P$ is called the pole of a line $p$ iff $P$ is not joined to any point on $p$ by a line.
$P$ is joined to a point on $p$ by a line.
$\therefore$ $P$ is not the pole of $p$.
I am a little confused at how should I do it. I guess that it would be the contrapositive inference:
$$\cfrac{P\rightarrow Q, ¬Q}{¬P}$$
Now:
$P'$ : A point $P$ is called the pole of a line $p$.
$Q'$ : $P$ is not joined to any point on $p$ by a line.
$$\cfrac{P'\rightarrow Q', ¬Q'}{¬P'}$$
I pointed the inference scheme for:
I guess I pointed out the scheme inference for $\rightarrow $, now should I do it for $\leftarrow$? I'm a little bit confused at what it would became, is it?
$$\cfrac{Q'\rightarrow P', ¬P'}{¬Q'}$$
Also: In my previous attempts, I tried to do the following:
$P'$ : A point $P$ is called the pole of a line $p$.
$Q'$ : $P$ is joined to any point on $p$ by a line. (Notice the absence of not)
And hence I would have something different:
$$\cfrac{P'\rightarrow ¬Q', Q'}{¬P'}$$
I would have to negate $Q'$ here but it gives a completely different thing which I'm not truly sure is the same, but I think are not, based on experimentation with truth tables. So, should I always pick "convenient" variables for the job?