[Illustration:  FIG. 1.]

An Universal Affirmative may also state a relation between two terms whose denotation is co-extensive.  A definition always does this, as Man is a rational animal; and this, of course, we cannot represent by two distinct circles, but at best by one with a thick circumference, to suggest that two coincide, thus: 

[Illustration:  FIG. 2.]

The Particular Affirmative Proposition may be represented in several ways.  In the first place, bearing in mind that ‘Some’ means ’some at least, it may be all,’ an I. proposition may be represented by Figs. 1 and 2; for it is true that Some horned animals ruminate, and that Some men are rational.  Secondly, there is the case in which the ’Some things’ of which a predication is made are, in fact, not all; whilst the predicate, though not given as distributed, yet might be so given if we wished to state the whole truth; as if we say Some men are Chinese.  This case is also represented by Fig. 1, the outside circle representing ‘Men,’ and the inside one ‘Chinese.’  Thirdly, the predicate may appertain to some only of the subject, but to a great many other things, as in Some horned beasts are domestic; for it is true that some are not, and that certain other kinds of animals are, domestic.  This case, therefore, must be illustrated by overlapping circles, thus: 

[Illustration:  FIG. 3.]

The Universal Negative is sufficiently represented by a single Fig. (4):  two circles mutually exclusive, thus: 

[Illustration:  FIG. 4.]

That is, No horned beasts are carnivorous.

Lastly, the Particular Negative may be represented by any of the Figs. 1, 3, and 4; for it is true that Some ruminants are not hollow-horned, that Some horned animals are not domestic, and that Some horned beasts are not carnivorous.

Besides their use in illustrating the denotative force of propositions, these circles may be employed to verify the results of Obversion, Conversion, and the secondary modes of Immediate Inference.  Thus the Obverse of A. is clear enough on glancing at Figs. 1 and 2; for if we agree that whatever term’s denotation is represented by a given circle, the denotation of the contradictory term shall be represented by the space outside that circle; then if it is true that All hollow horned animals are ruminants, it is at the same time true that No hollow-horned animals are not-ruminants; since none of the hollow-horned are found outside the palisade that encloses the ruminants.  The Obverse of I., E. or O. may be verified in a similar manner.

As to the Converse, a Definition is of course susceptible of Simple Conversion, and this is shown by Fig. 2:  ‘Men are rational animals’ and ‘Rational animals are men.’  But any other A. proposition is presumably convertible only by limitation, and this is shown by Fig. 1; where All hollow-horned animals are ruminants, but we can only say that Some ruminants are hollow-horned.