corrected 12/3/99
A valid argument, as has been pointed out repeatedly, must have a true conclusion if it has true premises. Thus it is impossible for a valid argument to have true premises and a false conclusion. So, if in doing a derivation, one finds that one has true premises and a false conclusion then one knows that the argument is not valid. This is the rationale for the indirect proof (IP), or as it is sometimes known, reductio ad absurdum, which is Latin for "reduction to absurdity." If one can from a set of premises generate a conclusion that is contradictory, that is, p & ~p, then one knows that the argument is invalid.
This technique, which, like the conditional proof, makes use of assumptions, is useful when one is having difficulty working out the derivation but can assume the contradictory of the conclusion and reduce the argument to an absurdity, or contradiction. Here is how it works.
If we have a set of premises that we think to be true, we add a false premise (the negation of the conclusion) and we get a contradiction--as a result of adding the false premise--then we know where the problem is. The argument would have been valid if we had not added in the negation of the conclusion!
The following example is adapted from David Kelley, The Art of Reasoning (W.W. Norton Co., 1990):
(A É B) É CHere is the derivation:(A & ~B) É C /\ C
1. (A É B) É C2. (A & ~B) É C /\ C
3. ~C Assume
4. ~(A É B) 1,3 MT
5. ~(A & ~B) 2,3 MT
6. ~A v ~~B 5 DeM
7. ~A v B 6 DN
8. A É B 7 Cond
9. (A É B) & ~(A É B) 8,4 Conj
10. C 3-9 IP
The indirect proof (IP) shows that the conclusion--C--follows from the premises--lines 1 and 2.
Exercise 15 provides practice opportunities.
Copyright © 1999, Michael Eldridge