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Mr Wrener, Studies in Synthetic Logie. 19 
constructing such a relation. I shall define the relation A,, as 
the relation such that R {a,, dz, ...,a@,} when, and only when, 
Ay, Ag, +++, An € CR, 
and the conclusion of (7) is false*. 
I shall define the class, wp, as follows: 
A 
(8) @p = a {a, | € a . Ay ? As, eee, An—2 (Si GSR . 2 i, a1, MQ, «+5 In—2 a 
LCN CA Osy 6-25 On al « Len (0) Gy, Ys Og, 10, Opie) vem 
Ee ENON Oss 222) naa, Yl » Lign Ys 2) Oi, Os, «5 Ono « 
in [Cay Bh Cay ee Cia toe tien (in ees Ch once Cran UB nee 
MC rs. Oreo hi = ce Luan (Ci, Cg, <5 Enon, ©, Yh os 
te ety (0, Dis Da, 2 On_at «Va Loa (Ou, ©, Oa 5---)0n a} » Vi .-- 
Vv, IRAN {b,, b., ADA: c} te Dye xp cconnan © b,, OS sisiery (Drees ea} Df. 
Next, I define ins as follows: 
5 AA 
emis 2 fore Oona) On| ay ay a, ON, Ons) ln EQ 
(Gaia Ga, <2.) Bp) «0 € Oy « Gy EAs... Un EA G [0y, >, ---, Gn} DE. 
Now, I claim, ins‘R possesses the desired sort of n-connexity, 
whatever A may be. 
For did it not, by (7), it would be possible to find n distinct 
Q's, SAY M4, @2,---, %, Such that none of those relations hold between 
them which can be made from those in the conclusion of (7) by sub- 
stituting ins‘R for R, and each a for the a with the same number ; 
while, as we learn from (9), each ais a member of w,. ‘That is to 
say, if we pick out one member from @, say 2, one from a, 
say z,,and so on till we come to a,, from which we pick out w,, then 
2, Up, ..., Lp» Will stand to one another in none of the relations men- 
tioned in the conclusion of (7), and hence will stand to one another 
in the relation R,,. This will be true whatever the values that x, 
takes in a, 2, in 4, etc. It is easy to see that from this and the 
second half of the proposition in the brackets in (8), we can 
conclude that a,=a,=...=4,, which contradicts our hypothesis. 
Hence, ins‘R always possesses the n-connexity expressed in (7). 
Another and equally important property possessed by ins‘R 
is that, if (ins‘R) {a4 %,..., An}, 1, M,---,% are all distinct. For 
suppose that (ins‘R) {a,, 4, ...,%,-..,4n}. Then we shall have to 
have, by the definition of ins, RF {a, de, sreaiOs = -- Onion WHEL, By 
belongs to %, a, to %, etc, b to a, and so on till we get to dp, 
which belongs to an; %,%,+--,4 are all, by the definition of ins, 
members of wz. Therefore, by the definition of w,, we shall 
* Tt will be seen, of course, that Roa, wa, and ins are essentially functions of 
the particular sort of n-connexity asserted in (7). 
2—2 
