11 
crystallization from man, because man is not a crystal. 
There is no such restriction in arithmetic, because every 
number is a factor of every other number. 
It is also to be observed that, though we may always 
multiply both sides of a logical equation by the same term, 
we cannot always so divide both sides. Let sr mean strati- 
fied rocks, and wr aqueous rocks, then, as a matter of fact, 
sr = wr. 
Let y mean fossiliferous, then we may multiply both sides 
by f, and write 
fsr =fvjr. 
But we cannot divide both sides by the common factor r, 
so as to unite 
s = w, 
That is to say, from the fact that stratified rocks are the 
same as aqueous rocks or rocks deposited from water, we 
cannot infer that all stratified things are the same as things 
deposited from water. 
All that has been now stated is perfectly well known and 
quite undisputed ; but I have stated it, in my own way, on 
account of what follows, and with the view of endeavouring 
to show that the laws of logical addition and subtraction 
are closely analogous to those of logical multiplication and 
division. 
Addition and subtraction have primarily the same mean- 
ing in logic as in arithmetic or common algebra. Thus if x 
means animals and y vegetables, x y means animals and 
vegetables ; and if v means vertebrates, then x — v means 
invertebrate animals. 
As we have seen, abstraction, or logical division, is uninter- 
pretable unless the quality abstracted is a quality of that 
from which it is abstracted; and similarly, subtraction in 
logic is uninterpretable unless that which is subtracted is 
part of the extension of that from which it is subtracted, 
