3 
with zero index is equal to its own second power, and also 
to its own reciprocal. This is also true in arithmetic; but 
it is moreover generally true in arithmetic that every such 
term has the value of unity, which is not true in logic. 
The four terms L, and (L~^y when combined 
give sixteen equations, whereof each is the canon of a 
syllogism. Two of these have been stated above as the 
direct and the inverse expression of the canon of Barbara.” 
Of the sixteen syllogisms, fourteen a.re conclusive ; that is 
to say, in fourteen cases, whereof the two just mentioned 
are examples, the two relations combine into one, which is 
expressed by one of our four terms. Another example is 
that given above where the middle term is Irishman”; — 
this in the present notation is expressed by A = AM, 
Mr=A“^B, therefore A=A®B ; or, simply as a canonical equa- 
tion and without using any but relative terms, by L.L~^=L^. 
The forms of the two inconclusive syllogisms are L^.(L~'^y 
and 
The truth of the equations of this system does not depend 
on the interpretation of L as inclusion, but solely on the 
transitiveness of the relation symbolized by L ; and they 
consequently remain true if L means, for instance, superior, 
or cause ; postulating in the latter case that the cause of a 
cause is a cause. 
For the purpose of the present essay, relation is regarded 
as analogous, though not closely so, to ratio, and relative 
terms to numerical coefficients. In logic, as in arithmetic, 
if one of the following four equations is true, the rest are 
necessarily so. 
A 
A = ZB B = A-A 
A 
where ^ is taken to mean, in logic, any relation whatever 
of A to B. 
