... so we come up with different answers depending on which part -- the source term itself or the explanation -- gets our attention. The thing is, they do not match. "Teoría del absurdo" is almost a straight synonym of "Reducción al absurdo", neither more, nor less formal; simply expressing a slightly different point of view by placing emphasis on the proposition, rather than on the method of proof. Witness how both terms are explained in Spanish:
http://es.answers.yahoo.com/question/index?qid=2010050619110...
http://www.cidse.itcr.ac.cr/revistamate/propuestas-didactica...
It is incorrect to say of any such technique that it "takes reasoning beyond logics". It has been one of the most used methods in logics for millenia, and it predates English, which is probably why we don't have a "plain English" version of it.
It would be fun to see an interpretation of it that does not use Latin or Greek words, if such is possible.
The literal translation of the source term should be: "demonstrate by reductio ad absurdum".