# Troubles with proving the following expression $((p \implies q) \implies p) \implies p$ using Fitch System

I having heavy difficulties with this exercise beacuse it doesn't have any premise. I can't get the result I want (which is posted on the title of the question) straight to the top level

1.              p Assumption
2.                  (p => q) => p Assumption
3.                  p Reiteration: 1
4.              ((p => q) => p) => p Implication Introduction: 2, 3
5.          p => ((p => q) => p) => p Implication Introduction: 1, 4


ps: I don't want the entire proof of the expression. I would appreciate hints. If it is really complicated, then I accept the entire proof so I can study further.

ps: I know you all want to help, but I have only these operations available: -assumption -reiteration -delete -negation introduction -negation elimination -and introduction -and elimination -or introduction -or elimination -implication introduction -implication elimination -biconditional introduction -biconditional elimination It was cool to know Peirce's law, however I didn't find a way to implement it through the resources above.

• See Peirce's Law. – Mauro ALLEGRANZA Mar 18 '17 at 14:21
• Mauro, could you please post your answer so I can mark it as the solution? – Carlos Oliveira Mar 19 '17 at 14:42

## 2 Answers

For a proof of Peirce's law with Natural Deduction's based Fitch system, you can see the answer to the post: using the Fitch system how do I prove $((p \implies q) \implies p) \implies p$ ?

If you are not falimiar with the rules using $\bot$ (the falsum) you can rewrite it as follows:

1) assume $(P \to Q) \to P$

2) assume [a]: $P$ and [b]: $¬P$ and derive by $¬$E: $Q$

3) derive $P \to Q$ by $→$I, discharging [a]

4) use it with 1) to derive $P$ by $→$E.

Now we have again a contradiction with the [b] assumption: $¬P$.

The next step needs $¬$I, to derive from the said contradicition: $¬¬P$, discharging [b].

Up to now, having discharged the "temporary" assumptions [a] and [b], we have:

$(P \to Q) \to P \vdash ¬¬P$.

The following step needs $¬¬$I (Double Negation elimination) to derive $P$ from $¬¬P$ and conclude with: $(P \to Q) \to P \vdash P$.

The final step needs $\to$I to conclude with:

$\vdash ((P \to Q) \to P) \to P$.

HINT

whenever you want to prove something of the form $\phi \rightarrow \psi$, do a conditional proof where you assume $\phi$ and try to get to $\psi$.

So in your case, assume $(p \rightarrow q) \rightarrow p$, and try to get to $p$

You did do a conditional proof, but notice that the conditional you obtained was not the one you want.

HINT 2

To get to $p$ inside the conditional proof ... Do a proof by contradiction

HINT 3

If you are still stuck ... See the 9th post under 'Related' on the right.

• Thanks for the hints. I have tried do it that way, but i can't get "p" by a contradiction. I have seen this symbol ⊥, which I don't have on hands for use and my course didn't explain yet what it means. – Carlos Oliveira Mar 18 '17 at 15:19
• @CarlosOliveira If you don't have $\bot$, then you probably need to get to $p$ as well as $\neg p$. That is: if you assume $\neg p$, and can get to $p$ as well as $\neg p$ (That last one is trivial by reiteration), then you can infer $\neg neg p$, or maybe even directly $p$. – Bram28 Mar 18 '17 at 15:29