Mr Wiener, Studies in Synthetic Logic. 17 
of R alone, once J is determined. Thirdly, Rp has the desired 
sort of n-transitivity. For we can write 
Pp, {Ay , Me, see, n, b,, bs, S05 Op 
as a sum of products of the form 
CH Cr 2 Cp} o Pepi (Cy 5 Cary on On} «2» Lop {C1 Cg y.5205 One 
Now to say Rp {d,, dz, ...,dn} is, by the definition of Ry, the 
same as to say that there is some S such that ST,,R, and 
iS1d,,d>,.-.,d,}. Therefore 
| Dp Gop ono Ups Ons Usp soe Oi 
is equivalent to 
iy, 
di 
(qs: So, 00.0 5) Sim) 0 SE Carne Gis |@y, Os, ore, An; Oy, Os, 0005 by} o 
: SR Gene STs 1h, 1 Sy Toke 
| Therefore 
foe eie ra (0), 0s,---, 0;)- T I Chisf Oy ova Cunt Os Oy be anh oo 
i 
= ce (qiSte say coon ye S182 .-+ Sim (as Ms step Giphc 
SOP 1B 0 Spelt 0 SM IE soe Sap ll aadas be 
28 (GiShip/Shp cose rao Rens one Sy Chip Chis onan Cin to 
Si 
Ky Xo Xn Me Ix, y,, x, AAs Xm eH 
JSG IB Dyno Sap (Stn coon 73/2 OB 
5) 55 (ap Stin(Ses abbas) easier Boe Sip) Oks Chip sob yamie ce 
ar 
X,, X55... XmepeDy x x sBidiges thin B/E 
(ll 
Qo +889 “hon 
JRRETTECD Vis Sui Sieeas (ies (Vibe 
aaa T 
5) 82 (Gp Shy Sen onns (Spy orsnishoos(Sip. (in Capinen Cpl o 
SS 
[Si iSts\see (Stn Jlidas He 
eer 
pr 
>> 
2) 88 (Gl) e OG Cee anon tna UG It ee 
= 
=) #8 (GUO) Gig Ginn can Cal) Oe 
S) 8B Ltn by soon ial 
This is what we wished to prove, for, if we compare this with (1), 
it shows that R,, has the desired sort of transitivity. 
_ When the transitivity in question is ordinary binary transi- 
tivity R, becomes R,,.. In general, the appropriate form of Rp 
performs the function of R,, in systems whose order is given by a 
triadic or tetradic or other polyadic relation. 
VOL. XVIII. PT. I. 2 
