# Why do we need to show $\lnot \lnot P \implies P$ when justifying proof by contradiction here? Does my proof work without doing this?

I am confused of the justifications I hear for why proof by contradiction works, including the one on wikipedia. I am confused why people say we need the law of the excluded middle, and why Wikipedia says we need to show that $$\lnot \lnot P \implies P$$- I am confused by this latter thing because the same Wikipedia page I cited says we can replace an if-then with its contrapositive due to looking at the truth table of the contrapositive and seeing how the contrapositive is logically equivalent- so why can't we just replace $$\lnot \lnot P$$ with $$P$$ by seeing how the truth tables of each of these are equivalent?

I tried to write a proof justifying proof by contradiction which avoids the things which seemed to me superfluous, such as showing the statement $$\lnot \lnot P \implies P$$ and using the law of excluded middle, which tries to use the same general idea used by one of the proofs on Wikipedia. I would be very appreciative of any feedback on my proof as to why its wrong, and why we need to do things like explicitly show the statement $$\lnot \lnot P \implies P$$ which I didn't do. Here is my proof:

To clarify, what I will show is that $$(\lnot P \implies R \land \lnot R) \implies P$$ for any propositions $$P$$ and $$R$$. We will do a direct proof by first assuming that $$\lnot P \implies R \land \lnot R$$. We deduce that $$R \land \lnot R$$ is false using the law of non-contradiction. We then conclude the contrapositive $$[\lnot (R \land \lnot R)] \implies \lnot \lnot P$$ is true, due to looking at the truth table of the contrapositive statement and seeing that the contrapositive is equivalent for all assignments of $$R$$ and $$P$$ to boolean values. Then, we use modus ponens to conclude $$\lnot \lnot P$$ is true. Due to looking at the truth table of $$\lnot \lnot P$$ (that is, a table which shows the values this statement has when $$P$$ is false and $$P$$ is true) we see that $$\lnot \lnot P$$ is equivalent to $$P$$, so we replace the former with the latter to conclude $$P$$.

• Using truth tables amounts to assuming the law of the excluded middle (since you assume each proposition is either true or false), which also justifies proof by contradiction and double negation elimination. These are considered valid in classical logic but not in intuitionistic logic.
– Karl
Commented Aug 2 at 1:26