Sec. 3.  The equational treatment of propositions is closely connected with the diagrammatic.  Hamilton thought it a great merit of his plan of quantifying the predicate, that thereby every proposition is reduced to its true form—­an equation.  According to this doctrine, the proposition All X is all Y (U.) equates X and Y; the proposition All X is some Y (A.) equates X with some part of Y; and similarly with the other affirmatives (Y. and I.).  And so far it is easy to follow his meaning:  the Xs are identical with some or all the Ys.  But, coming to the negatives, 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.