II.—CONSISTENCE. 22 
Example iii.,—In a certain set of 1000 observations (4)=45, 
(B)=23, (C)=14. Show that whatever the percentages of B’s 
that are 4 and of (’s that are 4, it cannot be inferred that any B’s 
are C. 
The conditions (a) and (%) give the lower limit of (BC), which 
is required. We find— 
(BC), (4B) _(40) _. 
(a) 7 < WV V 918. 
(BO), (4B), (40) _. 
(5) 7 < ry 045. 
The first limit is clearly negative. The second must also be 
negative, since (4B8)/N cannot exceed ‘023 nor (4C)/N -014. 
Hence we cannot conclude that there is any limit to (BC) greater 
than 0. This result is indeed immediately obvious when we 
consider that, even if all the B’s were 4, and of the remaining 
22 A’s 14 were (’s, there would still be 8 A4’s that were neither 
B nor C. 
14. The student should note the result of the last example, as it 
illustrates the sort of result at which one may often arrive by 
applying the conditions (4) to practical statistics. For given 
values of &, (4), (B), (C), (AB), and (4C), it will often happen 
that any value of (BC) not less than zero (or, more generally, not 
less than either of the lower limits (2) (a) and (2) (8) ) will satisfy 
the conditions (4), and hence no true inference of a lower limit is 
possible. The argument of the type ‘So many 4’s are B and 
so many B’s are C' that we must expect some 4’s to be C'” must 
be used with caution, 
REFERENCES. 
(1) MorcAN, A. DE, Formal Logic, 1847 (chapter viii, ‘On the Numerically 
Definite Syllogism ”). 
(2) Boog, G., Laws of Thought, 1854 (chapter xix., ‘‘ Of Statistical Condi- 
tions”). 
The iors are the classical works with respect to the general theory 
of numerical consistence. The student will tind both difficult to follow 
on account of their special notation, and, in the case of Boole’s work, 
the special method employed. 
(3) YuLe, G. U., “On the Theory of Consistence of Logical Class-frequencies 
and its Geometrical Representation,” Phil. Trans., A, vol. excvii. 
(1901), p. 91. (Deals at length with the theory of consistence for 
any number of attributes, using the notation of the present chapters.) 
Ler