OF LOGICAL CLASS-FREQUENCIES, ETC. 
93 
Boole, while exhibiting in a clearer light the data that are necessary for any 
inference, and the limitations of inference caused by assigned limitations of data. 
The whole of the conditions of consistence are derived from one source, or one 
condition only, viz., that no frequency can he less than zero. But as it is evident that 
if aU the frequencies of any order be greater than zero, the frequencies of all lower 
orders must, d fortiori, be greater than zero, we may limit the above statement by 
saying that all the conditions of consistence are covered by the dictum that no 
ultimate freeiuencjf can he less than zero. If, however, we are dealing with groups of 
the nth and lower orders only in m specified attributes, it is convenient to divide 
the conditions into two classes—(l) the “ inferior conditions of consistence,” which may 
be derived from the fact that no nth order frequency can be less than zero; (2) the 
“ superior conditions of consistence ” which can only be derived from the consideration 
that frequencies of order greater than n cannot be less than zero. 
A distinction of this sort is, it may be noted, made by De Morgan. Inferences 
drawn from the inferior conditions of consistence for second order groups he terms 
spurious inferences;! they do not really follow from given premises (?.(?., given values 
of (AB) and (AC) or (AB) and (BC), &c.), but are “ true by the constitution of the 
universe. 
§ 3. It will be convenient to use the following terms in addition to those defined in 
my previous memoir. 
A set of frequencies formed by taking the frequency of any positive group 
ABCD . . . N, together with the frequencies of all possible groups of the same 
order formed by substituting the contraries a /5 y 8 . . . v for one or more of the 
attributes ABCD . . . N, will be called an “ aggregate ” of frequencies. Any one 
aggregate contains only one positive group which may be used to denote the 
aggregate, so that one may speak of the AB aggregate or the ABCD aggregate. 
The order of an aggregate may be defined as the order of the groups contained in 
it. An aggregate of order n contains 2” groups. The sum of the frequencies of 
these 2" groups of the aggregate, is evidently equal to the total frequency or number 
of observations (U). 
If m attributes be specified, the number of positive groups of the ti-th order that 
can be formed from them is 
m{iii — 1) {m — 2) . . . {vh — n + V) 
DB 
The complete set of consistent aggregates corresponding to these positive groups 
will be termed a congruence of aggregates or simply a congruence. Tlie number 
* The frequency of a group specified by all the attributes noted. “ On the Association of Attributes,” 
&c., loc. cit., p. 2.59. 
t ‘Formal Logic,’ p. 153. | Note on same page. 
