Abstract

1. Despite the plain fact that there is nothing in this world that can be proved without reliance on some assumption or another, there is an inalienable difference between an argument that begins by assuming what it is designed to establish and one that begins by assuming the contradictory of what it is designed to establish. Arguments of the first kind are uncontroversially acknowledged to be circular, or question-begging; though valid they achieve nothing. Those of the second kind conform to the classical pattern of reductio ad absurdum or indirect proof. A typical example is the proof of the irrationality of √2, which begins by assuming that for some integral m, n the identity m2/n2 = 2, and arrives at a contradiction. Nothing whatever is established or justified by arguments of the first kind, even granted the truth of all the other assumptions, and the validity of the rules of inference; while the second may establish that, if those assumptions are true, then the additional assumption is false