388 Mr Wiener, A Simplification of the Logic of Relations. 
is determined uniquely by that of $, and vice versa—if, to put it : 
in symbols, when w’ bears to ¢’ the same relation that ~ bears 
to d, k:.d' (4, y)- =zy+ 9 (@ y)i =: Wa.=, . ya—, we can entirely ~ 
avoid the use of *12°11, and interpret any proposition concerning 
the extension of @ as if it concerned the extension of yw; for the 
existence of the extension of a propositional function of one 
variable is assured to us by *12:1, quite as that of one of two 
variables is by «12:11. Now, is such a w the propositional 
function 
(qa, y)» O(a, y) a= UCU Vv UA) VLU. > 
For it is clear that for each ordered pair of values of # and y there — 
is one and only one value of a, and vice versa. On the one hand, 
as U(u't'a U UTA) is determined uniquely by «, and 1‘0‘t‘y is deter- 
mined uniquely by y, u“(e“esa v LA) v LUu’y is determined uniquely 
by # and y. On the other hand, if 
(Uefa V UA) vu UUsity = UU Sz U UA) Uw, 
either Ufufy = Uz UV UEA or Uuty=U'tfw. The former supposition 
is clearly impossible, for, as efz + A, life v USA is not a unit class. 
From the latter alternative we conclude immediately that y = w. 
Similarly, # =z. 
Therefore, when x and y are of the same type, we can make 
the following definition : 
296 (0, y) = BMG, y)-$(w, y).0= Kua viEA) veey} DEF 
It will be seen that in this definition of 27¢ (a, y) it is essential 
that the x and the y should be of the same type, for if they are 
not U(Uuseut*A) and 6c%t%y will not be, and U(iftfe vu UA) vu UU ty © 
will be meaningless. To overcome this limitation, and secure 
typical ambiguity for domain and converse domain of 29 (a, y) — 
separately, we make the following definitions : | 
an (4, y) =k (qa, y) x (a, y) p 
= (Ua UA) OU OE) ee, 
KYp (K, y) a (AK, y) p(k, y) c 
p= UU UA) UV LU U(Utty UA) UL AT} 6 Df. 
ete. 
Bo (a, B)=R (an, B)- (a, B)- 
w= [eG v fA) uv fA] Utes) DE 
aRp (@, %) = A ((aqm, »)- 
p= Ue [essa v A) vu tcAlulAturttA}l Df 
ae ter 1-2". Ae are ON ae 
ete. 
is q 
* This may seem circular as vis a relation, defined in the Principia as I, but it 
really is not circular, for “x may be defined directly as the class, 7 (y=). 
