This one is nice and interesting:

If after supposing , you achieved the conclusion that both and are true, you're not lost, since you just discovered another truth: that it's not possible for to be true, that's it, it's true.

For instance, I confess that if I use Windows, I don't profit the time I am with my computer. Since some years, I do profit it, so the conclusion is that I don't use Windows. To achieve that conclusion, the path that you would follow (maybe without thinking) is precisely the one that this rule needs: suppose that I do use Windows, in that case I wouldn't profit my computer. But I said that I do profit it, so that supposition must be wrong.

This procedure is called reduction to the absurd (reductio ad absurdum): suppose something to achieve a contradiction and be able to assert that what we supposed is false. It's specially useful if you start supposing the contrary of what you want to prove: if any contradiction can be discovered, then it's almost all done.

I should note that this is an abuse of notation: following all the laws of logic, it happens that each subdemonstration needs one conclusion (not two); and at the above hypothesis, it's not clear which one is the conclusion ( or ?). The correct way to write it would be using conjunction introduction to say that , and this one is the conclusion which shows the wrongness of the initial hypothesis. But my teachers didn't write that line.

Daniel Clemente Laboreo 2005-05-17