Indirect derivations depend on the concept

that with true premises we could not have a false conclusion.  Suppose we assume that the original conclusion is wrong.  If that happens then we should end up with a situation in which there is a contradiction--an impossible scenario.  We thus reason backward and say that for there to be a clearly false inference (like saying P&~P) we had to have a false hypothesis, since for there to be a valid argument with a false conclusion there would have to be a false premise.  Of course, the same reasoning applies when the premises are already inconsistent, but then this only shows that anything can be made to follow from a false premise.  (Remember saying that something follows only means it is true when indeed the premises are true.)

Let's take an argument form for which we could also do a direct derivation, then one for which a direct derivation would not be possible.

P -> (Q & R), ~R |- ~P

1.  P -> (Q & R)    Ass
2.  ~R                    Ass
3.         P                Hyp     Here we are indenting to show the beginning of a subordinate derivation.  Anything that             follows from this hypothesis should not be seen as "true" otherwise.
4.        Q & R      1,3  -> elim
5.        R              4  & elim
6.        R & ~R    2,6  & int    This is clearly an impossibility, so we know that the imagined possibility of P can be ruled out--and then the opposite is shown to be true
7.  ~P                   3-6  hyp elim   notice that now we cite the entire set of lines
 

P -> Q, Q -> R |- P -> R

1.  P -> Q    Ass
2.  Q -> R    Ass
3.      ~(P -> R)     Hyp
4.     ~ (~P v R)     3  MI
5.      P & ~R         4  DeM
6.      P                   5  & elim
7.      Q                  1,6  -> elim
8.      R                  2,7  -> elim
9.      ~R                 5  & elim
10.     R & ~R         8,9  & int
11.  P -> R              5-10 hyp elim

The advantage of indirect proofs is that they can be used with any valid argument form.  The disadvantage is that often direct proofs and, as we shall see next, conditional proofs are shorter and sometimes more interesting.
 

Practice Exercises


Go back to the derivation problems in earlier modules and rework them with indirect proofs.