14 
Term/' I have made a somewhat similar distinction, thus : — 
meaning that every x is in the relation R to a 2 / ; — and 
lir<]Ely, 
meaning that every x is in the relation R to every y. In 
these expressions, the 1 before the x is superfluous, and is 
introduced merely for visible symmetry. 
We have next to find an interpretation for x — x, and 
after what we have seen above this will be easy. If we 
take water from water, there is left an indefinite quantity 
of water, varying from none up to a quantity indefinitely 
near to the original quantity. Let us as before use - as the 
symbol of indefinite quantity ; then 
x-x = 0 
so that there is a perfect symmetry between the following 
two pairs of equations : — 
0 
X X = X, X — X = 
X 0 . 
XX = X, -= X - x). 
X U 
To complete the parallel between multiplication and divi- 
sion on the one hand, and addition and subtraction on the 
other; — we have seen that in logic, though from the truth of 
x = y, 
we can infer the truth of 
zx = zy, 
yet from the truth of 
zx = zy, 
we cannot inversely infer the truth of 
x^y. 
