including &, forming a well-defined function of R, having the 
desired sort of transitivity. 
_ This is proved as follows: let the n-transitivity in question be 
the one given in (1). Decompose 1p {ay, ..., dn, by, +0) Op}, Os 
indicated, into a sum of expressions of the form 
IRAs Crp bok, Colt a 18 1h Gyn 55040, |b 55 0 (6,0, C30 ee ene 
Let there be, say, f such expressions, the pth one always with 
l, R's. Replace each of these R’s by one and one only of the 
variable relations X,, X.,..., Xm, with the same arguments as. 
Daf 
the & it replaces, and let m= > 1,. We shall thus transform 
p=l 
1’, into a relation which is a function of the m variable relations 
: T 
X,, Xo,..., Xm. - Let us call this relation Xs ae Now, let | 
us define the relation X,X,... Xi» as follows: 
i ——____f 
(2) XsKye Xm (eas Bay Oa} «= « (0s ay nna Bi) 
T | 
Ae ae (G1, An, +045 Mn, Dy, by, .., bx} Df. | 
| 
| 
| 
16 Mr Wiener, Studies in Synthetic Logic. 
Like rx. ace XC INOS ye a function of Age AG. ene 
where the latter may assume any values which are n-adic re- 
lations. Now, I define the class of T-powers of R, or, as I write it, — 
—>- 
TR, as follows : 
CB) LEE Gs XG 005 hyn 6 [oc Dy, es 
XOX 6 Min € po eon ae Sep} Df. 
sae aa 
I make the further definition, 
— 
(4) Bp=sTSR Def. 
Now, Ry includes R and is a function of it, and has the desired 
sort of n-transitivity. 
First, Rp includes R. For, since, as may be seen on inspection, — 
. — | 
RT,,R, Re T,,.“R. Since every member of a class is included in | 
= | 
the sum of the class, RE&T.“RGER,. Secondly, as Rp is 
derived from R by a process which is really perfectly definite — 
(though I admit that some of the stages of the process by which — 
I have derived Ry from R are not uniquely determined, ’a little 
reflection will convince one that all the possible determinations of 
a yield the same value of R,), it is a function of R, and 
1 94 30! ™ 
