12 
For instance, x — v would not be interpretable, w^ere it not 
true that 
v = voc. 
We can subtract vertebrates from animals, because verte- 
brates are animals, but we cannot subtract vegetables from 
animals, because vegetables are not animals. Here ao-ain 
there is no similar restriction in arithmetic or common 
algebra. 
In all the foregoing there is no difficulty nor controversy. 
This begins when we come to such expressions as x-\-x, 
which involve the addition of a term to the whole or a part 
of itself. Boole combines a term with itself, and places the 
equation 
/yi/v» /v> 
i/jih t/j 
at the foundation of his system; but he rejects the ex- 
pression as uninterpretable, and admits of the addition 
of terms only when they are completely separate. Jevons, 
and most of those who have built on Boole's foundation, 
differ from him in this, and admit the formula 
x^x — x, 
I regard the introduction of this formula into the science 
as a great improvement to the symmetry of the system. It 
seems to me that Boole must have been led to reject it by 
an erroneous, or too limited, interpretation of the expression 
all, and by a needless and injudicious attempt to interpret 
his symbols in extension rather than in comprehension. 
When X, y, z, are taken to be the names of qualities, they 
are interpreted in comprehension ; when they are taken to 
be the names of the classes of things having the qualities, 
they are interpreted in extension. Boole interprets them in 
the latter way; he reads his logical equations as asserting 
the co-extension of classes, not the co-existence of qualities. 
Either interpretation gives a true meaning, but there is a 
difference between them on which I wish now to insist. 
