Help, have to prove the tautology (p ∨ ¬p) from nothing using fitch system.

Question

I'm new using fitch so this is all i have and don't know how to get it done. (https://i.paste.pics/d0746642a4513e1ca1799b3e92e2ace2.png)

Here's the link of the exercise: http://intrologic.stanford.edu/problems/exercise_04_14.html

Answer 1

Although there may well be a shorter way to reach the goal, you're nevertheless halfway there already. You can use the same sequence of steps you used to get $\ \text{~}(p|\text{~}p)\Rightarrow p \ $, but replacing assumptions $2$ and $5$ with $\ \text{~}p\ $, to get $\ \text{~}(p|\text{~}p)\Rightarrow \text{~~}p \ $. Then use negation introduction on this and your conclusion $9$ to get $\ \text{~~} (p|\text{~}p)\ $, then negation elimination to get the goal. I have checked that this works by entering all the steps into the Fitch checker at your link to the exercise.