# How to derive P > (Q > R) from (P > Q) > R in Fitch?

I am having a little bit of difficulty coming up with a Fitch-style natural deduction proof.

Presumably, I need to use a few conditional introduction rules, but I am not sure what I can get out of the first premise. I tried to assume P (as the outer-most layer of subproof), but I wasn't sure where I could go from there. Thanks!

You are given the premise (P > Q) > R. Since you want to show a conditional, P > (Q > R), assume as you attempted the antecedent, P. Then you need to derive the consequent, Q > R.

However, that consequent is also a conditional. So start another subproof within the first subproof by assuming the antecedent Q. Now the goal is to derive R.

At this point you have two subproofs one inside the other and you need to derive R. If you had P > Q, you could derive R from the premise. So, first you have to derive P > Q.

Since that is a conditional make a third subproof inside the second one by assuming P. You already have Q as an assumption so you can use reiteration to derive it. That gives you P > Q. From there you can derive R and from that result Q > R. From that the final goal P > (Q > R) can be derived.

Here is a proof using a Fitch-style proof checker:

Kevin Klement's JavaScript/PHP Fitch-style natural deduction proof editor and checker http://proofs.openlogicproject.org/

• Please try not to do coursework for users in full, but rather help them solving it themselves. This way, they do not really learn anything and that's what coursework is meant for. Oct 9, 2019 at 16:08
• @PhilipKlöcking I don't consider this doing someone's homework. What it does offer is a reference to a tool the OP, or someone else, may not be aware of with an example of its usefulness. No one can use this answer without understanding it. Oct 9, 2019 at 16:43
• Sounds like a good plan. I do not have the time to look for the quotes atm,, but generally, questions like these should show explicitly that there was some effort solving the problem themselves, e.g. the ways one had tried. Bits like "having a bit of difficulty" sound a bit like "(but not caring enough to try harder)". Oct 9, 2019 at 17:33
• @FrankHubeny Thanks for the help - this was just a practice problem for an exam (not homework). I was trying to actually prove it on proofs.openlogicproject.org, but only got to the second assumption - I was a bit stumped after that. Thanks again. Oct 9, 2019 at 23:33
• @Vikram It's best if you add details like that to the body of the question, this way you show your own effort and the help can hook in where you are stuck Oct 10, 2019 at 8:28