6 
Proceedings of the Royal Society 
daughters of a man , sons of a woman , and daughters of a 'woman. 
For example, sA was used to denote the sons of the man A. Thus the 
symbol sA selects a portion of the universe of mankind, and that 
portion embraces no, one, or more than one individual. We have seen 
that the symbol A is also selective, but embracing only one individual. 
The symbol s is different ; it may be considered as indicating to the 
mind to pass from A to sA. It is convenient to have words to 
express the relations of these symbols to one another. The ex- 
pression sA may be called a term , A the origin of the term, and s 
the relationship of the term. In this paper I proceed mainly by 
means of the two more general fundamental relationships c and y, 
where c denotes the children of a man , and y the children of a 
woman. 
Let cA = B 4- C+ D. This statement is a logical equation ; it 
asserts that the children of the man A are B and C and D, and only 
these. Let cA > B + C + D. This statement is a logical inequation; 
it asserts that the children of the man A include B and C and 
D. Let cA < B + C+D. This statement is a logical inequation ; it 
asserts that the children of the man A are included in B and C 
and D. 
1 1 1 
What is the proper meaning of -A and -A ? The term -A 
means the father of the person A. It is a singular term by reason 
of physiological laws which apply to mankind. There are other 
functional symbols, whose reciprocals may form plural terms ; for 
example, let eA denote the employes of A, then -B may denote 
the employers of B. 
The meaning of ccA follows from the meaning we have attached 
to cA. It means the children of the sons of the man A. Similarly 
the terms ycA^ cyB , yyB denote respectively the children of the 
daughters of the man A, the children of the sons of the woman B, 
and the children of the daughters of the woman B. The symbol c 
operating on cA, directs us first to select the male children of A, 
and then to take the children of each of these. Similarly y, opera- 
ting on cA, directs us first to select the female children of A, and 
then to take their children. Thus the symbols c and y, in so far as 
they are selective in their operation, partake of the nature of the 
quaternion symbols S and V. 
