12-2 
MR. G. UDNY ’i^ULE OX THE THEORY OF CONSISTENCE 
X 
+ 
u 
4 
2 = 
0-25 
(a) 
X 
+ 
y 
— 
Z = 
0-25 
X 
—• 
y 
4 
Z 
0-25 
(r) 
— X 
+ 
y 
4 
Z 
0-25 
(S) 
X 
+ 
y 
4 
Z = 
0-5 
(<^) 
X 
+ 
y 
— 
2 = 
0 
(0 
X 
4- 
y 
4 
Z = 
0 
('^) 
— X 
4 
y 
4 
Z = 
0 
(^). 
There are twelve somewhat interesting infinite series of definite inferences 
corresponding to tlie twelve sides of the octahedron. They may he grouped in three 
divisions following tlie contours EBCF, EDCA, BDEA, viz.:—- 
Given. 
X — y = 0'125 1 
or y — £c = 0’125 ! 
orrc + ,y = 0-125 f 
or X y — \ 
lor the contour EBCF, and the two similar systems tliat may he written down by 
cyclical sid)stitution. These are all, it will he seen, inferences of indeijendence 
— 1 X T X i = -g), and therefore somewhat strikingly different to the usual definite 
inferences which are all inferences of complete association (all A’s are B — no A’s 
are B). F'rom the same values of x and y we could, of course, infer n' = -125, so the 
theorem may be ex})ressed in words thus— 
“ In any case where cross-equality"^' and independence both suljsist for the second- 
order frequencies, independence must subsist for two at least of tlie four positive 
third-ordei' frequencies if eithei' (1) the sum of the remaining two is equal to |^th or 
fths of the total frequency ; or (2) the difterence between tlie remaining two is equal 
to I'th of the total frequency.” 
I should, jieiTiaps, note that in tlie present case the fact that the criterion of 
independence holds for the })ositive fourth-order class necessitates its holding for all 
the remaining classes of the aggregate. This is not so in general. It may further 
be remarked that the independence of the attributes, })aii’ and pair in the second- 
order classes, does not connote independence for the groups of the third m'der. I 
hope to I'eturn to the logic of independence on a future occasion. 
§ 29. Case (2)— 
(AB)/(U) = (BC)/(U) &c. = q = 0 20. 
Tlie })airs of attributes ai’e nov/ all negatively associated. 
* I propose to use this term in lieu of the more lengthy “ equality of contraries 
Inferred. 
2 = 0-125 
