  If by Rxy in the syllogism of transposed quantity we take  f(x) = y, where the function is defined on and has values in the set S, then the second premise of the  syllogism of transposed quantity says that f is a one-one function.  The conclusion says that every member of S is the image  under f of some member  of S. Thus the syllogism of transposed quantity says that no one-one  function can map the set S to a proper subset of itself. This  assertion holds, of course, only if S is a finite set. So, as it turns  out, Peirce’s definition of the difference between finite and infinite sets is  virtually equivalent to the standard one, which is found in Section 5 of Richard  Dedekind’s Was Sind und Was Sollen die Zahlen?, to the effect that an  infinite set is one that can be placed into a one-to-one correspondence with a  proper subset of itself. Peirce claimed on various occasions to have  reached his definition of the difference between finite and infinite collections  at least six years before Dedekind reached his own definition.