The force of the different propositional forms is to exclude one or more of these possibilities.
All S is P limits us to one of the two α, β ;
Some S is P to one of the four α, β, γ, δ ;
No S is P to ε ;
Some S is not P to one of the three γ, δ, ε.
It will be observed that there is great want of symmetry in the number of circles corresponding to the different propositional forms; also that there is an apparent inequality in the amount of information given by A and by E, and again by I and by O. We shall find that these anomalies disappear when account is taken of negative terms.
It is most misleading to attempt to represent All S is P by a single pair of circles, thus
or Some S is P by a single pair, thus
159 for in each case the proposition really leaves us with other alternatives. This method of employing the diagrams has, however, been adopted by a good many logicians who have used them, including Sir William Hamilton (Logic, I. p. 255), and Professor Jevons (Elementary Lessons in Logic, pp. 72 to 75); and the attempt at such simplification has brought their use into undeserved disrepute. Thus, Dr Venn remarks, “The common practice, adopted in so many manuals, of appealing to these diagrams—Eulerian diagrams as they are often called—seems to me very questionable. The old four propositions A, E, I, O, do not exactly correspond to the five diagrams, and consequently none of the moods in the syllogism can in strict propriety be represented by these diagrams” (Symbolic Logic, pp. 15, 16; compare also pp. 424, 425). This criticism, while perfectly sound as regards the use of Euler’s circles by Hamilton and Jevons, loses most of its force if the diagrams are employed with due precautions. It is true that the diagrams become somewhat cumbrous in relation to the syllogism; but the logical force of propositions and the logical relations between propositions can in many respects be well illustrated by their aid. Thus, they may be employed:—
(1) To illustrate the distribution of the predicate in a proposition. In the case of each of the four fundamental propositions we may shade the part of the predicate concerning which information is given us.
We then have,—