15 
Similarly as to addition and subtraction ; — if 
x = y, 
we can infer tliat 
x^-z = y 
But we cannot conversely from the truth of 
x-\-z = y + 
infer the truth of 
x = y. 
We can make such inference — in other words, we can prac- 
tice subtraction freely — only when we are certain that the 
terms added together — 0 -h x and 0 -f y — have no common 
part — not when one is in whole or in part identical with 
the other. Let I for instance mean living beings or orga- 
nisms, a animals and v vegetables. Then a I will mean 
animals and organisms generally, and the equation 
a + ^ = (X + 'y, 
will be true, but 
will not be true. 
By thus admitting of the addition of all terms to each 
other whether they have any common part or not, we do 
not, as a matter of fact, interfere with the facility of multi- 
plying them, and we make it practicable to use the following 
rule for finding the logical negative of any complex term: — 
Change every simple term into its negative, and for the sign 
of addition substitute the sign of multiplication, and con- 
versely. For symmetry and clearness let us unite the 
negative of x, or what is not x, in the abbreviated form x. 
The negative of 
xy + z, 
will thus be 
{x^y)z, 
{xy-^z) + {x + y)z=l, 
and 
