gatives, the equational interpretation is certainly less obvious. The
proposition _No X is Y_ (E.) cannot be said in any sense to equate X and
Y; though, if we obvert it into _All X is some not-Y_, we have (in the
same sense, of course, as in the above affirmative forms) X equated with
part at least of 'not-Y.'

But what is that sense? Clearly not the same as that in which
mathematical terms are equated, namely, in respect of some mode of
quantity. For if we may say _Some X is some Y_, these Xs that are also
Ys are not merely the same in number, or mass, or figure; they are the
same in every respect, both quantitative and qualitative, have the same
positions in time and place, are in fact identical. The proposition
2+2=4 means that any two things added to any other two are, _in respect
of number_, equal to any three things added to one other thing; and this
is true of all things that can be counted, however much they may differ
in other ways. But _All X is all Y_ means that Xs and Ys are the same
things, although they have different names when viewed in different
aspects or relations. Thus all equilateral triangles are equiangular
triangles; but in one case they are named from the equality of their
angles, and in the other from the equality of their sides. Similarly,
'British subjects' and 'subjects of King George V' are the same people,
named in one case from the person of the Crown, and in the other from
the Imperial Government. These logical equations, then, are in truth
identities of denotation; and they are fully illustrated by the
relations of circles described in the previous section.

When we are told that logical propositions are to be considered as
equations, we naturally expect to be shown some interesting developments
of method in analogy with the equations of Mathematics; but from
Hamilton's innovations no such thing results. This cannot be said,
however, of the equations of Symbolic Logic; which are the
starting-point of very remarkable processes of ratiocination. As the
subject of Symbolic Logic, as a whole, lies beyond the compass of this
work, it will be enough to give Dr. Venn's equations corresponding with
the four propositional forms of common Logic.

According to this system, universal propositions are to be regarded as
not necessarily implying the existence of their terms; and therefore,
instead of giving them a positive form, they are translated into symbols
that express what they deny. For exam