18 Mr Wiener, Studies in Synthetic Logic. | 
LA | 
§ 3. There is another important sort of property which the 
ordinary serial relation, the ‘between’ relation on a given line, | 
and the separation-relation have in common. For the binary serial. 
relation, it 1s ordinary connexity; for the ‘between’ relation on a 
given line it may be expressed in symbols as 
(Wm,n):amn.V.man.Vv.mna:bmn.v.mbn.v.mnb: . 
CMN »«V«MCN«~V»MNC? Da,d,c! 
G=b).V.0=C.V.cC=Q.V.a0C. Vv. bea. Vincab. 
for the separation-relation it is 
(qm, n, 0): am||no.V.mal||no.Vv.nn||ao.Vv. mn||oa: 
bm ||no.v.mb||no.v.mn||bo.v.mn|| ob: 
cm||no.Vv.me||no.v.mn||co.v.mn||oc: 
— dm||no.v.md||no.v.mn||\do.v.mn\jod: 
Du,b,c.42¢=0.V.b=¢c.Vv.c=d.vV.d=0.V.0—¢.¥ 0 
ab ||cd.v.ac||bd.v.ad|| be. 
For the sake of brevity, let us generalize the notion of ‘field’ in | 
the following manner: , 
A 
Ges (oP @ (G15 hay acon Cpa) & 
E1201, Gz, +. Opa) nV. Ly (ay, yO; ) ao) an 
v. EF \dy, de, ..55 On 2} 
Now, I shall define a property of an n-adic relation, R, as an | 
n-connexity of that relation if it can be written in the form 
(CO) Ging Cy saen One OIE RUBS in Dip BOG Se Cin) © Bc, ° 
Uh (OOP yeeney WIAA Aa) Oh Oe oon Cn |) oi 
My iA iA 
Reidy; Ge, ..-, Om {1 Vo.. Ve (a), a) eee 
(6) C= 
where @y’... dy’, @y” ... Qn, ..., @" ... Qm'®) are each definite per | 
mutations of @...@,. It is obvious that ordinary binary connexity / 
is, by this definition, a 2-connexity, and that the properties of | 
‘between’ and separation which we have just mentioned are, | 
respectively, 3- and 4-connexities. 3 
Now, I wish to raise with regard to n-connexities the precise © 
analogue of the question which we raised with regard to n-transi- - 
tivities in the last section: is it possible, given any n-adic relation | 
and any m-connexity, to form by a perfectly definite method an - 
n-adic relation genuinely dependent on this relation, having the | 
desired sort of n-connexity ? | 
As in the former case, I shall answer this question by actually _ 
