Logic: Proof?

Question

I'm given:

P → (Q → R)

P → (R → S)

And I must show that:

P → (Q → S)

This is what I've done so far:

1) Show P → (Q → S)

2) P............................... Assumption (CD)

3) P → (Q → R)........... Premise 1

4) P → (R → S)........... Premise 2

5) (Q → R).................... 2, 3 MP

6) (R → S) .................... 2, 4 MP

Now here is where I get stuck I know that logiclaly Q → S from what I have above. But I don't know how to actually derive it.

I though of continuing on like this:

1) Show P → (Q → S)

2) P............................... Assumption (CD)

3) P → (Q → R)........... Premise 1

4) P → (R → S)........... Premise 2

5) (Q → R).................... 2, 3 MP

6) (R → S) .................... 2, 4 MP

7) Q................................ Assumption

8) R................................. 5, 7 MP

9) S.................................. 6, 8 MP

But I'm not sure if with what I've done I've actually shown that P → (Q → S).

Thanks!

Answer 1

That's how I would do it.

You have not "actually" shown it, but the logic is sound and valid so it is reasonable to form that conclusion.

This is the point is it not?

Answer 2

Well solve it like you would do a math problem.

If a=b and b=c then a=c.

I think you need to continue how you were doing it the first time. The second one is a little bit more winded.

Can't you also do

R=R

right before R=S

And then put R=Q and next put R=S

And then put Q=S

Or something like that.