Mr Wiener, Studies in Synthetic Logic. 21 
| when %=a,=...=. Therefore, o(insR),, 18 made up exclusively 
‘of unit-classes. Now, we can write the condition for the n-transi- 
(10) (WM; re; sary rx) T ins(Gns‘R) 7] {K1, Ko, +-+,Kn, Ma Ae; seep aval . 
/ >) aren EADS (HOSTED al) sin Haan conn rahe 
| The expression in the form 7'nsqiins‘gy is here the sum of pro- 
ducts of terms of the form ins‘[(ins‘R),| {u1, Mo, .-., ln}, where the 
psare to be found among the «’s and 2’s, and all the X’s appear 
somewhere as arguments to ins‘[(ins‘h),]. Therefore, since 
| CAns‘[(ins‘R)_| C @ ins“) . 
| all the «’s and ’s are unit classes. Therefore, since 
| 
K1, Ka, .. 
fins‘[(ins‘ 2) |} {74, Ye, ----, Yn} 
holds when and only when », ... vy, are members of wins‘Ry,» and 
‘there is an a, belonging to »,, an a belonging to m,..., an & 
belonging to y,, we may write (10) as follows: 
| (11) (qi, Bo, 0905) Px) . Bi, Bo, Bx; 1, Az, +++, An € UO lins’R) mn * 
Piins'R) 7» (%, Zl coos Chae [Sing Bs, cory Bu} ee eaOS@hu), (Oy, Qo, se ral 
From (5) it follows that (11) is identically satisfied, and hence 
that ins‘[(ins‘2);| has the desired sorts of n-connexity and n-transi- 
tivity, and never, to put it roughly, relates a member of its field 
to itself, whatever R may be. Hence, if we have a system whose 
‘postulates can be put in the form of three propositions, one 
asserting a certain n-transitivity, another a certain n-connexity of 
a given n-adic relation, P, and the third asserting that P never 
relates a member of its field to itself, then, given any n-adic 
relation, R, we can construct a function of R having the desired 
properties of P. Moreover, it is easy to see that if F itself has the 
desired properties, the constructed relation will be, so we may 
put it, of the same formal properties as h, but two types 
higher. 
§ 5. Now, there are very important sorts of relations whose 
definitions may be put in the above form. The general ‘between’ 
relation between members of a series is, it 1s easy to see, com- 
pletely determined as to its formal properties by the three pro- 
positions 
(qd): abd .bde.v.abd.bed.v.adc.dbe. 
v.abd.acd.bac.v.dab.dac.bac.v. bea: Day,¢ « abe, 
(qm, n):amn.v.man.Vv.mna:bmn.v.mbn.v.mmnb: 
CNUs VINCI = Va TNC Dy 5.0% 
a OL IG Ni — CaN OU C Wan OCU ta Vien. a 
Mee 56.0). C.a + 0: 
