Mr Wiener, A Simplification of the Logic of Relations. 389 
Though these definitions may seem to conflict with one 
another, they really do not conflict, for where one of them is 
applicable, the others are meaningless, since they define relations 
between objects of different types. Moreover, it is easy to see 
that our definitions are so chosen that 
Rdg (u, v) = Spy (w, p). D.C DBHd (x, v) 
=D py (a, p). UODIG (u, v) = TS py (a, p). 
This is important, as we might easily have defined relations so that 
they might have several domains or converse domains of different 
types. This is why we did not define a7(a, y) simply as 
e {(qa, y) + (4, y) kK SLU av LA) VLE Uyt, 
for this would also represent 
a8 (Ay) « (a, y)- B= Uy}. 
It will be seen that what we have done is practically to revert 
to Schréder’s treatment of a relation as a class of ordered couples. 
The complicated apparatus of ts and A‘s of which we have made 
use is simply and solely devised for the purpose of constructing 
a class which shall depend only on an ordered pair of values of « 
and y, and which shall correspond to only one such pair. The 
particular method selected of doing this is largely a matter 
of choice: for example, I might have substituted V, or any other 
constant class not a unit class, and existing in every type of classes, 
in every place I have written A. 
Our changed definition of 27p(a, y) renders it necessary to 
give new definitions of several other symbols fundamental to the 
theory of relations. I give the following table of such definitions : 
nel Kin Coy (a — x ay —y)h Df. 
aky.=.2w0{z=c2.w=y}CR.ReRel Df 
ph.=..(qa).a=h.ae Rel. ga DES 
(Rh). dR:=:a€ Rel. D,. ha Df. 
(qh). ¢6R:=.(qa).ae Rel. ha Df. 
The first two and the last two of these definitions replace *21-03°02 
and *21:07-071 respectively. From these definitions and the laws 
* We shall understand in this way any propositional functions containing 
capital letters in the positions proper to their arguments. Thus ~@R shall be 
understcod as 
(qa).a=R.ae Rel. ~ da, 
a=R.ae Rel. 3,.~ da. 
We make this definition as well as the two following ones because a propositional 
function of a class of the sort we haive defined as a relation may significantly take 
as arguments classes of the same type which are not relations, and we wish to 
define propositional functions of relations in such a manner as to require that their 
arguments be relations. 
and not as 
