  Along with Richard Dedekind and Georg Cantor, Peirce was  one of the first scientific thinkers to argue in favor of the existence of  actually infinite collections, and to maintain that the paradoxes that  Bernard Bolzano had associated with the idea of infinite collections were  not really contradictions at all. His criterion of the difference between finite  and infinite collections was that the so-called “syllogism  of transposed quantity,” which had been introduced  by Augustus de Morgan, constituted a deductively valid argument only when applied to finite sets; as applied to infinite sets it was invalid.  The syllogism of transposed quantity runs as follows. We have a binary relation R defined on a set S, such that the following two premises are true of the relation  (where the quantifications are taken over the set S). First, for all x there is a y such that Rxy. Second, for all x, y, z, Rxz and Ryz implies that x = y. The conclusion (of the syllogism of transposed quantity) is that for all x there exists a y such that Ryx. One of Peirce’s favorite examples helps elucidate the idea, even if it perhaps be not perfectly politically correct: Every Texan kills some Texan; no Texan is killed by more than one Texan; therefore every Texan is killed by some Texan. The argument’s conclusion follows validly only if the set of Texans is finite.