I once asked here a question about how I can get from a false premise to a true premise, and people noted that it is called "Principle of Explosion" – meaning that from a false premise you can get to whatever you want. Link is can be found here: Using deductive reasoning to prove a true statement with false premises?.
But, now came to my mind the proof that $\sqrt{2}$ isn't rational, which is a proof by contradiction (the famous one) and I thought to myself, assuming $\sqrt{2}$ is rational, is as we now know a false premise as it is not rational, meaning that we have again the principle of explosion! (Or in Latin "Ex falso quodlibet").
What is going on here? Why can we assume $\sqrt{2}$ is rational and say that we come to a contradiction when in the first place we assumed a false statement!
For me, it is like saying "Assuming $1=2$, now I can prove that $1=3$" (Hey that is a contradiction no? we said $1=2$ ) thus $1 \ne 3$… but as we know $1=2$ isn't true also!.
Maybe the example of $\sqrt{2}$ is pretty obvious that there is a dichotomy between those relations, either it is rational or not. But there are proofs which do no deal with a dichotomic "property" of something… or proof by contradiction is always for dichotomic properties? (like, being even or odd, rational and irrational, etc…)
Thank you!
Best Answer
A proof by contradiction has an important step that you omitted: you start from a false premise, reason to a contradiction, and then discard the false premise as incorrect. It is in fact exactly the avoidance of "proof by explosion."
In other words, we assume various proven facts, call them $A,B,C...$, as well as facing a claim that $P \implies Q$. If you start from an assumption of $\neg (P \implies Q)$, and use it to derive $\neg C$, you have shown that $\neg (P \implies Q) \implies \neg C$, and the contrapositive of that is $C \implies (P \implies Q)$. Since you know $C$ is true, you can conclude that $P \implies Q$.