20 Mr Wiener, Studies in Synthetic Logic. ‘ 
have R,, {d,, ds, ...,0, ...,@n}. Weare thus led into a contradiction, 
It will be noted that this property too is characteristic of ordinary 
binary serial relations, of ternary relations such as the ‘ between’ 
relation, and although in this case not clearly stated, of Vailatis | 
separation-relations. : 
§ 4. Now two interesting questions arise: first, what hypo- 
thesis 1s necessary concerning the n-adic relation & if ins‘f is to 
have a given sort of n-transitivity ? and secondly, is it possible to | 
build a function of A which has any given sort of n-transitivity, | 
any given sort of n-connexity, and is such that if this function: 
holds between 4, K2,...,4n, the «’s are all distinct? The first: 
question is exceedingly easy to answer. Let the transitivity in} 
question be that of (1), and the connexity that of (7). Modify (1) | 
in the following manner: if in any of the products that, added, | 
make up 7’p, a term, say w, occurs as argument to several R’s, | 
replace it in all but one of its occurrences by some term, so that in: 
no two occurrences is it replaced by the same term; multiply) 
the product in which it occurs by all the expressions which can be 
formed by taking £,, [derived from the connexity expressed in) 
(7)], and giving it as arguments any n (not all necessarily distinct) 
of the terms which replace x, including # itself; and introduce | 
the terms, other than a itself, which replace #, as apparent | 
variables, in such a manner that their range is the whole left side | 
of (1), and that they are preceded by an q. If we transform (1) | 
in this way, it is easy to see, though tedious to prove, that we: 
obtain a sufficient condition for ins‘f’s possessing the sort of | 
n-transitivity indicated in (1) and the sort of n-connectedness 
indicated in (7). | 
As to the second question, it is almost self-evident that 
ins‘[(ins‘f),] possesses the sort of m-transitivity indicated in (1), | 
the sort of n-connexity indicated in (7), and that if 
{ins‘((ins‘R)p]} {«1, 2, ---, Kn}, 
and «;, «;,1=j. The two latter properties follow simply from the | 
fact that this relation is an ins of something; the fact that it | 
has the former quality follows obviously from the following con- | 
siderations. If Q has any sort of n-connexity, and Q G P, then P, | 
a fortiori, has the same sort of n-connexity, if its field is that of Q; | 
for the hypothesis of (7) (with R changed throughout to Q), | 
remains unchanged, while, if 4 
Qa) a9, a,°}, then P lai, a," enema 
so that the conclusion of (7) is true of P if it is true of Q. There- - 
fore, (ins‘f), has the desired sort of n-connexity and n-transitivity, - 
though it may be possible for us to have 7+), «,=«;, and | 
(ins‘R), {k, Ko, ..., Kn}. Since (ins‘R), is connected in the way 
determined by (7), [(ans‘R),]c, can only hold between a,, a, ..-, 
