We have the following argument:
$P \implies Q$ (Premise 1)
$P$ (Premise 2)
The accompanying truth table is:
I don't understand this section of my textbook:
The premises are both true only in line four of the table, and in this line the conclusion is true as well.
You can also see from the truth table that both premises are needed to make this argument valid. But if we were to change the truth table for the conditional statement to make P → Q false in the first line of the table, then the second premise of this argument would no longer be needed.
We would end up with the conclusion that, just from the single premise P → Q, we could infer that Q must be true, since in the two lines of the truth table in which the premise P → Q would still be true, lines two and four, the conclusion Q is true too.
If we would change the $P → Q$ to be false in the first line of the truth table. Then why couldn't the conclusion be false as well? I don't understand what line $2$ and $4$ have anything to do with line $1$?