MHB Stuck on Derivation (Natural Deduction)

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The discussion focuses on deriving the conclusion R ⊃ (X v V) using natural deduction. The user outlines their progress in proving the implications step by step, ultimately reaching a contradiction by assuming R and ~(X v V). They explore the necessity of employing proof by contradiction or the law of excluded middle to advance their argument. The user expresses uncertainty about the length of their proof and mentions generating a formal proof in a proof assistant program. The conversation highlights the complexity of natural deduction and the challenges in deriving specific conclusions efficiently.
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1. ((N ⊃ P) ⊃ (I ⊃ P)) ⊃ ((P ⊃ I) ⊃ P) P

2. (I v X) ≡ (R ⊃ P) P

3. R ⊃ (I ⊃ (N ^ V)) P

4. R A

5. Reiteration of 3

6. I ⊃ (N ^ V) 3,4 ⊃E

7. I A
8. I v X 7, vI
9. Reiteration of 2
10. R ⊃ P 8,9 ≡E
11. R 4R
12. P 10,11 ⊃E
13. I ⊃ P 7-12 ⊃I

14. N ⊃ P A
15. I ⊃ P 13 R
16. (N ⊃ P) ⊃ (I ⊃ P) 14-15 ⊃I

17. Reiteration of 1

18. (P ⊃ I) ⊃ P 16, 17 ⊃E

Conclusion: R ⊃ (X v V)

Hello!

I got up to this point in my work. I realized that I have to somehow get I v V out to the scope line of R, but I am not sure how to go about this. I am thinking of (P -> I) or (R -> P), but I don't know how from there.
 
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Welcome to the forum.

Apparently, this requires a proof by contradiction or the law of excluded middle. So we assume that R and ~(X v V). The latter implies ~X and ~V. From ~V follows ~(N /\ V), and I -> N /\ V gives ~I. Together with ~X this implies ~(I \/ X), which by the second premise gives R /\ ~P. ~I implies I -> P and (N -> P) -> (I -> P), so from the first premise we have (P -> I) -> P. Then ~P proves P -> I, and we have P and ~P, a contradiction.

I am not sure if there is a shorter proof. I generated a formal proof in a proof assistant program, but it is also not very short.
 

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