Indirect ProofThe method of indirect proof provides a powerful technique for proving arguments to be valid. It has been employed since ancient times and is often known by its Latin name: the

reductio ad absurdumtechnique.You construct an indirect proof by assuming the

denial of the conclusion, and then,addingthis to the list of premisses, you show that it is possible to deduce anexplicit contradiction. Since an explicit contradiction cannot possibly be true, by showing thatthe denial of the conclusionleads to an explicitcontradiction, what you are in effect showing is that it isnot possible for all the premisses to be true and yet for the conclusion to be false. In other words if the premisses are all true, then the conclusion must be true, or in other words, the argument isvalid.The general form of an argument is the conjunction of the premisses implies the conclusion:

(Premiss

_{1 }Premiss^{ .}_{2}... Premiss^{ .}_{n}.) => ConciusionBy using the rules "impl." and "DeMorgan's" this statement can be shown to be equivalent to

~[ (Premiss

_{1 }Premiss^{ .}_{2}... Premiss^{ .}_{n}~Conclusion)]^{ .}which says, in effect, that "it is false that the conjunction of the premisses and the denial of the conclusion is true".

In an

indirect proof, we deny this statement, thus asserting that(Premiss

_{1 }Premiss^{ .}_{2}... Premiss^{ .}_{n}~Conclusion)^{ .}when we deduce from this a

contradiction, we show that thiscannot be true, so therefore~[ (Premiss

_{1 }Premiss^{ .}_{2}... Premiss^{ .}_{n}~Conclusion)]^{ .}is true, which is equivalent to the original argument.

An "explicit contradiction" is any statement of the form: p

~p^{ .}Once an explicit contradiction has been reached, we can in fact prove

anyconclusion, including thedesired oneby the following routine:Step m pTo construct an I.P.:~p^{ .}

Step n. p (first term of explicit contradiction) m. simp

Step n+l ~p (second term of explicit contradiction) m, simp

Step n+2 p v Desired Conclusion (any we like) n, add

Step n+3 Desired conclusion n+2, n+l, D.S.1) After listing the premisses write a line of the form

~(conclusion) I.P. (indirect proof)

2) proceed deriving lines using the 19 rules, but now your "intended conclusion" is an explicit contradiction.

3) as soon as you get a line of the form p

~p your proof is over. STOP^{ .}Remember:

Anyexplicit contradiction will do.

Itmaybe of the form (conclusion)~(conclusion) , but it does not^{ .}haveto be of this form.