d are ruminants_, namely, all the
animals between the two ring-fences. Similar inferences may be
illustrated from Figs. 3 and 4. And the Contraposition of A. may be
verified by Figs. 1 and 2, and the Contraposition of E. by Fig. 4.

Lastly, the Inverse of A. is plain from Fig. 1--_Some things that are
not hollow-horned are not ruminants_, namely, things that lie outside
the outer circle and are neither 'ruminants' nor 'hollow-horned.' And
the Inverse of E may be studied in Fig. 4--_Some things that are
not-horned beasts are carnivorous_.

Notwithstanding the facility and clearness of the demonstrations thus
obtained, it may be said that a diagrammatic method, representing
denotations, is not properly logical. Fundamentally, the relation
asserted (or denied) to exist between the terms of a proposition, is a
relation between the terms as determined by their attributes or
connotation; whether we take Mill's view, that a proposition asserts
that the connotation of the subject is a mark of the connotation of the
predicate; or Dr. Venn's view, that things denoted by the subject (as
having its connotation) have (or have not) the attribute connoted by the
predicate; or, the Conceptualist view, that a judgment is a relation of
concepts (that is, of connotations). 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
ne