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 example, the proposition All devils are ugly need not imply that any such things as 'devils' really exist; but it certainly does imply that Devils that are not ugly do not exist. Similarly, the proposition No angels are ugly implies that Angels that are ugly do not exist. Therefore, writing x for 'devils,' y for 'ugly,' and ȳ for 'not-ugly,' we may express A., the universal affirmative, thus:
A. xȳ = 0.
That is, x that is not y is nothing; or, Devils that are not-ugly do not exist. And, similarly, writing x for 'angels' and y for 'ugly,' we may express E., the universal negative, thus:
E. xy = 0.
That is, x that is y is nothing; or, Angels that are ugly do not exist.
On the other hand, particular propositions are regarded as implying the existence of their terms, and the corresponding equations are so framed as to express existence. With this end in view, the symbol v is adopted to represent 'something,' or indeterminate reality, or more than nothing. Then, taking any particular affirmative, such as Some metaphysicians are obscure, and writing x for 'metaphysicians,' and y for 'obscure,' we may express it thus:
I. xy = v.
That is, x that is y is something; or, Metaphysicians that are obscure do occur in experience (however few they may be, or whether they all be obscure). And, similarly, taking any particular negative, such as Some giants are not cruel, and writing x for 'giants' and y for 'not-cruel,' we may express it thus: