22 Mr Wiener, Studies in Synthetic Logie. 
‘ 
Similarly, if we understand the separation-relation to hold 
only between four distinct terms, the general separation-relation is 
completely determined by the three following propositions : 
(qe): ab ||dce.v.cd|jab.v.abljec. ae||cd: Da,o,c,4 = ab || cd, 
(qm, n, 0) :am||no.v.ma||no.v.mn||ao.Vv.mn||oa: 
bm||no.v.mb||no.v.mn||bo.v.mmn|| ob: 
em ||no.Vv.me||no.v.mn||co.Vv.mn|loc: 
dm||no.v.md||no.v.mn||do.v.mn|lod: 
De cd20—0.V.0—C.V.¢—0 «V0 —@ V0) — Cn 
ab||cd.v .ac||bd.v.ad|| be, 
QDi|cd 1 Dinca = @ 40s @ (Cle ini0) cl ODE tee 
That is, from any triadic or tetradic relation, we are able to 
construct a between-relation or a separation-relation, respectively. 
This fact should play much the same part in explaining how the — 
regular relations of space may be derived from the irregular 
relations to be found in our experience that the analogous fact 
concerning dyadic relations plays in showing how the serial | 
relation of the instants of time may be derived from the non- | 
serial relation of complete succession between events*. Logically 
too this fact has a considerable interest, for it gives a hint of | 
another method of defining mathematical systems than by the 
use of postulates; given our fundamental logical postulates to 
start with, we may be able to select the fundamental ‘ indefinables 
of a mathematical system in such a manner that whatever values | 
they may assume within their range of significance, the funda- | 
mental formal properties of the system will remain invariant. | 
§ 6. Of course, all the formal properties of a triadic or tetradic¢ © 
relation are not determined when the relation is completely deter- — 
mined as a between or separation relation. Hence there remain 
interesting and important questions yet as to whether simple » 
properties of A may be given which will give ins‘f or Rp or ° 
ins‘[(ins‘f)7] properties analogous to density or ‘ Dedekindianness, — 
etc. If density with respect to a given transitivity, say that of (1), | 
be the property of a relation R which holds when the implication — 
in (1) is converted, then it requires little proof to see that if the — 
converse of (1), modified in the manner that (1) is modified im the © 
first paragraph of § 4, is true of R, and if C‘RCs‘op, then 
ins‘R will have the required sort of density. 1 know of no simpler — 
property of R, however, by which we can replace C‘R Cs‘ap, 
and, at any rate, if R is a between or separation relation, this | 
sort of density will not be the property which we would naturally 
call by that name. If R {a,b,c} means ‘b is between a and ¢, 
| 
| 
| 
Vv 
Vv 
Vv 
Vv 
Se = — +e 
* See Proc. Camb. Phil. Soc., vol. xv11, Part 5, pp. 4419. 
