# Natural deduction - formal proof troubles

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

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 $$P \to R$$.

• 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. – Gerhardus Carinus Feb 9 at 19:47
• @GerhardusCarinus - you have $P$ and $R$ and thus use $\to$-intro to get $P \to R$. – Mauro ALLEGRANZA Feb 9 at 19:51
• 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. – Gerhardus Carinus Feb 9 at 20:02

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.

• 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! – Gerhardus Carinus Feb 9 at 20:10
• @GerhardusCarinus Glad you figured it out! :) – Bram28 Feb 9 at 20:17