Propositional Logic: ¬p → ¬q, q V ¬r ⊢ r → p

Question

Is this proof correct?

¬p → ¬q, q V ¬r ⊢ r → p

1. ¬p → ¬q                P
2.  q V ¬r                P
3.         r              A
4.           q            A
5.               ¬p       A
6.               ¬q       E→(1,5)
7.                p       E¬(4,6)
8.                r → p   I→(3,7)
9.           q → (r → p)  I→(4,8)
10.              ¬r       A
11.               p       E¬(3,10)
12.               r → p   I→(3,11)
13.         ¬r → (r → p)  I→(10,12)
14.          r →  p       Ev(2,9,13)
15.       p               E→(14,3)
16. r → p                 I→(3,15)

Answer 1

No. It is, I'm afraid, quite hopeless, even with the probably intended (3) inserted.

So I suspect it won't help much just for someone to plonk down here a correct proof; your effort suggests that you are really struggling with the basics here and correcting one proof won't sort things out. Sorry!

So read a good intro or two to Fitch-style natural deduction for advice about proof-strategies. There are a number of decent ones.

For example there's mine, originally CUP but now freely downloadable from https://logicmatters.net/ifl. The book has lots of examples, and even better there are lots of worked examples to the exercises online, with explanations of proof strategies.

Answer 2

I disagree with @PeterSmith’s answer and think that you’re on the right track.

Without seeing the definitions of your rules, the instance of $\to$ intro on line 8 is incorrect. Either way, assuming you have Indirect Proof available, on line 7 you should have $\bot$ (or something functionally equivalent like $p \land \neg p$) so that you can discharge the assumption made on line 5 and derive $p$ on line 8. Then, you can derive $q \to p$ on line 9, followed by a derivation of $\neg r \to p$ so that you can use $\lor$ elim to derive $p$.