This arrangement would involve us in very serious trouble, if we ever attempted to represent “No x′ are y′.” Mr. Venn once (at p. 281) encounters this awful task; but evades it, in a quite masterly fashion, by the simple foot-note “We have not troubled to shade the outside of this diagram”!
To represent two Propositions (containing a common Term) together, a three-letter Diagram is needed. This is the one used by Mr. Venn.
Here, again, we have only seven closed Compartments, to accommodate the eight Classes whose peculiar Sets of Attributes are xym, xym′, &c.
“With four terms in request,” Mr. Venn says, “the most simple and symmetrical diagram seems to me that produced by making four ellipses intersect one another in the desired manner”. This, however, provides only fifteen closed compartments.
For five letters, “the simplest diagram I can suggest,” Mr. Venn says, “is one like this (the small ellipse in the centre is to be regarded as a portion of the outside of c; i.e. its four component portions are inside b and d but are no part of c). It must be admitted that such a diagram is not quite so simple to draw as one might wish it to be; but then consider what the alternative is of one undertakes to deal with five terms and all their combinations—nothing short of the disagreeable task of writing out, or in some way putting before us, all the 32 combinations involved.”
[pg176]This Diagram gives us 31 closed compartments.
For six letters, Mr. Venn suggests that we might use two Diagrams, like the above, one for the f-part, and the other for the not-f-part, of all the other combinations. “This”, he says, “would give the desired 64 subdivisions.” This, however, would only give 62 closed Compartments, and one infinite area, which the two Classes, a′b′c′d′e′f and a′b′c′d′e′f′, would have to share between them.