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
