With a few exceptions artificially framed (such as 'kings now reigning in Europe'), the denotation of a term is never directly and exhaustively known, but consists merely in 'all things that have the connotation.' If the value of logical training depends very much upon our habituating ourselves to construe propositions, and to realise the force of inferences from them, according to the connotation of their terms, we shall do well not to turn too hastily to the circles, but rather to regard them as means of verifying in denotation the conclusions that we have already learnt to recognise as necessary in connotation.
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.