92 
yin. G. UDXY YULE OX THE THEORY OF COXSISTEXCE 
we deal a certain number (viz., (AB)) of cards marked (B), one into each box, leaving 
only (A) — (AB) nnoccupied boxes, and then proceed to deal into the remaining boxes 
cards marked C, one into each ; some of these must fall into boxes already occupied by 
a B if their number exceed (A) — (AB). From similar simple reasoning Be Moegax 
derived the complete conditions of consistence for (AB), (AC), and (BC)."^ He does 
not, however, consider the case of more than three attributes, and the whole 
discussion is rendered very lengthy owing to his standpoint being still that of the 
older logicians. 
The theory of numerical logic was carried somewhat further by Boole in 
Chapter 19 of the ‘Laws of Thought’ (1854), entitled “Of Statistical Conditions.” 
After taking a series of projiositions for finding the major and minor limits to class- 
frequencies or sums of frequencies of any order, in terms of the first-order frequencies 
and the total frequency only, he proceeds to the general problem “ given the 
respective numbers of individuals compiised in any classes, s, f, &c., logically defined, 
to deduce a system of numerical limits of any other class 'iv also logically defined.” I 
must confess myself unalfie to follow the physical meaning of the processes symboli¬ 
cally developed in Boole’s general theorem, and this chapter has not, to my knowledge, 
been discussed by subsequent logicians. One naturally turns to the “ Symbolic 
Logic ” of Dr. Venn, whose lucid treatment clears up many difficulties of the “ Laws 
of Thouglit,” but he does not appear to deal with the problems “ of statistical 
conditions.” 
In the following; memoir I have endeavoured to deal with the general theorv of 
logical consistence, as I prefer to term it, from a standpoint slightly different to 
that of Boole, t 
§ 2. Let (U) be the total frequency in some defined universe, and let 
( A) (B) (C) . . . (AB) (AC) . . . (ABC) . . . &c., 
lie the frequencies of the positive groups (classes) up to, say, groups of the nth order, 
the number of attributes specified being on [on > n). It will be remembered that all 
other group-frequencies can be expressed in terms of those of the positive groups, so 
that no others need be considered. Boole, in his general theorem, quoted above, 
supposes certain of these frequencies to be known, and requires to find the resulting 
limits to some one other. I propose instead to make, at the outset, no supposition as 
to the frequencies that are knovai, but simply to discuss what conditions must hold 
if the whole set of frequencies is to be self-consistent. By proceeding in this way 
symmetrical systems of conditions are obtained of great interest and generality. 
They may be applied at once to such cases of limit-inference as are dealt Avitb by 
* See § 5 helow. 
t I nia}^ pei'haps state that this work was not directly suggested by De Morgan’s or Iw Rooi.e’s 
writings. Difficulties had arisen in the invention of numerical examples to illustrate certain points of 
theiwy, and I was driven to working out the theory of consistence in order to clear up these difficulties. 
