When should I use RAA in natural deduction proofs? I can't understand exactly when should I use RAA (reductio ad absurdum) rule in natural deduction proofs? What situation should "trigger" me to think "Now it's time to use RAA"?
 A: 
What situation should "trigger" me to think "Now it's time to use RAA"?

I can think of $4$ explicit situations:


*

*Your goal is the negation of something. If you goal is $\neg \varphi$, assume $\varphi$ and see if you can get a contradiction 

*Your goal is some atomic statement $P$: Assume $\neg P$, get a contradiction, get $\neg \neg P$ using RAA, and finally derive $P$ from $\neg \neg P$ (classical logics typically have a $\neg \ Elim$ rule for this)

*Your goal is a disjunction $\varphi \lor \psi$ ... and you don't have any disjunction that you can work with (if you do have a disjunction, set up a $\lor \ Elim$ on that one.). The nice thing about doing an RAA here is that once you assume $\neg ( \varphi \lor \psi)$, you should be able to derive both $\neg \varphi$ and $\neg \psi$ (using two RAA proofs themselves!), and now you have some useful stuff to work with on your way to a contradiction.

*Your goal is an existential $\exists x \ \varphi(x)$ ... and you don't have some other existential to work with (if you do have another existential, set up a $\lor \ Elim$ on that one.). So here the assumption will be $\neg \exists x \ \varphi(x)$ which, with a bit of work (and probably another RAA inside) should allow you to prove $\forall x \ \neg \varphi(x)$ ... and that will be useful to try and get to a contradiction given whatever else you have.
