Reductio ad absurdum
Reductio ad absurdum
The method of proving a proposition by deducing a contradiction from the negation of the proposition taken together with other propositions which were previously proved or are granted. It may thus be described as the valid inference of the propositional calculus from three premisses, B and B[~A] ? C and B[~A] ? ~C, to the conclusion A (this presupposes the deduction theorem, q.v.). Such an argument may be rearranged so that the element of reductio ad absurdum appears in the inference from ~A ? A to A.
The name reductio ad absurdum is also given to the method of proving the negation of a proposition by deducing a contradiction from the proposition itself, together with other propositions which were previously proved or are granted.
The first of the two kinds of reductio ad absurdum, but not the second, is called indirect proof.
Whitehead and Russell give the name principle of reductio ad absurdum to the theorem of the propositional calculus
[p ? ~p] ? ~p.
— A.C.