Case 1. The reductio ad absurdum
The latin term, reductio ad absurdum means "to reduce (the claim) to an absurdity", where "absurdity" is meant to be taken in the logical sense of something which is impossible . This method is often used to prove a statement false. It works this way: we first assume a statement of interest is true, then by careful reasoning, where each step in the reasoning process follows validly from previous steps, if we can derive a set of inconsistent statements by this process, then the original statement must be false.
Here is an an easy example first.
Suppose you are working a crossword puzzle, and the clue for 1 across is:
Loses its cover in the fall.
You also know the letters given above for 1 across are all correct.
The clue for 1 down is:
A state of rage.
The word F R EN Z Y works.
But assuming this is true, that gives 1 across as F R E E, which is inconsistent with the clue given that 1 across is something that loses its cover in the fall.
Hence your assumption that 1 down is F R E N Z Y must be false.
Surprise! Who knew that working corssword puzzles can involve using a reductio ad absurdum!
Now let's turn to a more philosophical argument which uses the reductio ad absurdum.
First we will establish a miniature argument (which is nice in this case, but certainly not a normal part of a reductio ad absurdum).
Suppose you want to travel from here to some distant place in the universe. Suppose that your spaceship is really fast. But there is a catch, the distance to the place you want to travel is infinitely far away - will you ever get there? Clearly not, since no matter how long you travel or how far you go, the destination is no closer, since it is infinitely distant. Hence things infinitley distant (in space or time) are impossible to reach.
With this in mind we want to consider whether the following claim is true.
Any object that is not at rest was put into motion by something else.
(Think of a string of falling dominoes, if you need a picture of things put into motion by something else).
We want to use the reductio ad absurdum to examine whether the above claim is true, hence we assume it is true and follow the necessary consequences:
Claim: Anything that is not at rest was put into motion by something else.
Consequence 1: If the claim is true, then whatever it is which puts the object into motion must itself be in motion. We will call this object 'motion giver #1"
Consequence 2: But "motion giver #1" must also be itself in motion, and hence it was always in motion (which is impossible) or put into motion by some other object itself, which we will call "motion giver #2"
Consequence 3: But the same reasoning applies to "motion giver #2", it had to be put into motion by another motion giver, which we will call "motion giver #3"
Consequence 4. The above sequence goes on forever or it does not. If it goes on forever then, like never being able to arrive at a destination infinitely far away, the same is true that the chain of events that stretch infinitely into the past could reach the object just placed into motion. Hence it is not true that any object that is not at rest was put into motion by something else, or there must be something that was itself not put into motion by something before it which was the object that put everything else in motion.
While this argument is a bit more complicated that the crossword puzzle example, it uses the reductio ad absurdum in part to establish the final conclusion stated in Consequence 4. This argument is actually similar to an argument by Thomas Aquinas in an attempt to argue for the existence of God.
But we will not go further into that argument here, and instead note that this method is also known as proof by contradiction or indirect proof. It should be noted that this method can only work when the original assumption must be false - hence it will not always work since the original assumption may not be false!
The following illustrates the reductio ad absurdum graphically: