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addition, so far as I am aware, is now stated for the first 
time. 
The problem of the addition of relatives is : — Given any 
two relations subsisting between the same terms, to find the 
esultant relation. 
Let us, for the sake of simplification, adopt Be Morgan’s 
method of working in an arbitrarily limited universe. The 
simplest possible case is that of a universe of only two 
individuals, exactly alike except in name, and each having 
an indefinite number of names. Let the relation between 
the names of the same individual be symbolised by 1, and 
the relation between the names of the different individuals 
by -1. Four syllogisms arise by the multiplication of 
these relations, and may be expressed by the following 
“ canonical equations,” all of which are true also in common 
algebra. 
1x1 = 1 
1 x (-1)= -1 
(-l)xl 1 
(-l)x(-l) = l 
Which are thus expressed in language : 
Identical if identical is identical. 
Identical if opposite is opposite. 
Opposite if identical is opposite. 
Opposite if opposite is identical. 
By the addition of the same relatives we get the following 
canonical equation, 
i + (-i)=o, 
which is also true in common algebra, and is the expression 
for the universe under consideration, of the truth that con- 
tradictory relations cannot coexist. 
If, as before, we express the relation of inclusion by L , 
and the converse relation by L~\ then 
L + L~' = 1. 
That is to say, if A is included in B and also includes B, 
then A is identical or coextensive with B, 
