IL.—CONSISTENCE. 21
possible. The conditions (1) and (2) therefore give all the con-
ditions of consistence for the case of two attributes, conditions of
an extremely simple and obvious kind.
11. Now consider the case of three attributes. There are
eight ultimate frequencies. Expanding the ultimate in terms of
the positive frequencies, and expressing the condition that each
expansion is not less than zero, we have—
or the frequency given below
will Le negative.
(a) (4BC)<0 4B0C))
L(4R) + (40) - (4) (4/3)
FHS 2
(In) + = a
(45) (4.3) | 4)
C30 (480)
a) FO) (aBC)
B) P(AB)+(4C)+ (BC) - (4) - (B)- (C) +N (afy)
These, again, are not conditions of a new form. We leave it
as an exercise for the student to show that they may be derived
from (1) (a) and (1) (4) by specifying the universe in turn as
BC, By, 3C, and By. The two conditions holding in four universes
give the eight inequalities above.
12. As in the last case, however, these conditions will be im-
possible to fulfil if any one of the major limits (¢)—(%) be less than
any one of the minor limits (a)-(d). The values on the right
must be such as to make no major limit less than a minor.
There are four major and four minor limits, or sixteen compari-
sons in all to be made. But twelve of these, the student will
find, only lead back to conditions of the form (2) for (4B), (40),
and (BC) respectively. The four comparisons of expansions due
to contrary frequencies ( (a) and (&), (6) and (g), (¢) and (f), (d)
and (e) ) alone lead to new conditions, viz.—
(a) (4B) + gio +(BC) 4(4) +(B) +(C) - N)
(6) (4B)+(-.)=(LC)}(4) 4)
(e) (AB)—(A0)+(LC)3(B)
(d) - (4B) + (40) + (BC) » (C)
13. These are conditions of a wholly new type, not derivable
in any way from those given under (1) and (2). They are con-
ditions for the consistence of the second-order frequencies with
each other, whilst the inequalities of the form (2) are only conditions
for the consistence of the second-order frequencies with those of
lower orders. Given any two of the second-order frequencies, e.g.
Q.
