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_.