I don't understand how to deal with this question:
Show that the argument form with premises p1, p2,...,pn and conclusion q → r is valid if the argument form with premises p1, p2,...,pn, q, and conclusion r is valid.
Assuming r is valid. where will i put the r in the solution?
I tried:
1. p (premise)
2. q (premise)
3. p ^ q (simplification, conclusion of 1,2)
4. p ^ q -> r
5. r = T
since r is valid, q must be valid based on line 4. So q -> r is T?
Instead of a formal proof, you can think about this question purely in terms of the definitions of the concepts involved.
We know that the argument form with premises $p_1,...p_n, q$ and conclusion $r$ is valid. This means (by definition of validity) that it is impossible for all of $p_1,...p_n, q$ to be true and $r$ to be false all at the same time. So, if we assume that all of $p_1,...p_n$ are true, then it is impossible to have $q$ true and $r$ false at the same time as well. But by the truth-table of the $\rightarrow$, that means that it is impossible for $q \rightarrow r$ to be false, still under the assumption that all of $p_1,...p_n$ are true. Hence, by definition of validity, any argument form with premises $p_1,...p_n$ and conclusion $q \rightarrow r$ is valid.
If you insist on a formal proof, first of all please know that there are many different formal proof systems with many different rules sets. Also, we can only really sketch such a formal proof, since we are talking about some unknown number of premises. Anyway, here is a pretty generic formal proof.
OK, so we know that the argument form with premises $p_1,...p_n, q$ and conclusion $r$ is valid. That is, we can infer $r$ once we have $p_1,...p_n, q$, i.e. we have a formal proof to the effect of:
$1. p_1$ premise
$2. p_2$ premise
...
$n. p_n$ premise
$n+1. q$ premise
...
$n+m. r$ conclusion
Now, what we need to show is that we can get $q \rightarrow r$ once we have $p_1,...p_n$. OK, so:
$1. p_1$ premise
$2. p_2$ premise
...
$n. p_n$ premise
$n+1. q$ assumption
... (insert above proof of deriving $r$ from $p_1,...p_n, q$ here)
$n+m. r$
$n+m+1. \ q \rightarrow r \ \rightarrow $ Intro