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I'm pretty new to the topic of natural deduction using the Fitch method. I found a very helpful site (http://proofs.openlogicproject.org/) in which you can construct your proofs, but I'm having a lot of trouble with the following:

What I have so far

I get as far as the proof for P but I'm not sure how to then use P to further the proof. I've done a lot of scouring on the net but I just cannot figure out how to continue. Any help would be appreciated.

Thanks

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Hint

From $P \land Q$ you have correctly derived both $P$ and $Q$.

Now use $Q$ with $Q \to R$ to derive $R$ and conclude with $P \to R$ by $\to$-intro.

The same wit the other disjunct, in order to use $\lor$-elim to conclude with $P \to R$.

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  • $\begingroup$ Thanks so much for your help - I've managed to prove P → R for the first disjunct as I have proven both P and Q, and by then using Q → R. However, I'm still struggling with the other disjunct. Having only proven P and R, I have no idea how to get from that to P → R as I do not have anything on Q for that one. $\endgroup$ – Gerhardus Carinus Feb 9 at 19:47
  • $\begingroup$ @GerhardusCarinus - you have $P$ and $R$ and thus use $\to$-intro to get $P \to R$. $\endgroup$ – Mauro ALLEGRANZA Feb 9 at 19:51
  • $\begingroup$ Thank you so much Mauro! That did the trick. I didn't know that you can infer P → R from P ∧ R. Yes, I am very new to this topic! But thanks to your help I have a much better understanding. $\endgroup$ – Gerhardus Carinus Feb 9 at 20:02
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Instead of getting to $P$, try to get to $P \rightarrow R$ in both subproofs, so that you can pull that out using $\lor \ Elim$

To get $P \rightarrow R$, do a subproof within the subproof, where you assume $P$, and get to $R$. You can then end the subproof, and conclude $P \rightarrow R$. So you need to do this within each subproof.

Another option is to immediately start your proof by assuming $P$, and then within that subproof do your two subproofs, trying to get to $R$ in both cases.

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  • $\begingroup$ Thanks Bram28! After a lot of back and forth (and help from Mauro above) it ended up exactly as you described it. I appreciate your effort! $\endgroup$ – Gerhardus Carinus Feb 9 at 20:10
  • $\begingroup$ @GerhardusCarinus Glad you figured it out! :) $\endgroup$ – Bram28 Feb 9 at 20:17

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