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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)
jcemp
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  • LIne 3 is missing: maybe r P. – Mauro ALLEGRANZA Oct 23 '23 at 09:53
  • I do not understand 14. – Mauro ALLEGRANZA Oct 23 '23 at 09:54
  • Basically, you have to assume r and then apply (Ev) to 2nd premise to derive p in both cases. – Mauro ALLEGRANZA Oct 23 '23 at 09:56
  • What do the letters represent on the right hand side? Is 8) being used on the way to determine 14)? Looks circular at 1st glance. Do you know Modus Ponens and Modus Tollens? I think they can cut your proof by more than half. By Modus Tollens 1) is equivalent to $q\implies p$ 2) is logically equivalent to $r\implies q$ since you get a contradiction otherwise. Then I think you can deduce a contradiction from $r \land \lnot q$ proving the result. – TurlocTheRed Oct 28 '23 at 02:15

2 Answers2

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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 $p$.

PW_246
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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.

Peter Smith
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