130. Euler’s diagrams and the class relations between S, not-S, P, not-P.—In Euler s diagrams, as ordinarily given, there is no explicit recognition of not-S and not-P; but it is of course understood that whatever part of the universe lies outside S is not-S, and similarly for P, and it may be thought that no further account of negative terms need be taken. Further consideration, however, will shew that this is not the case; and, assuming that S, not-S, P, not-P all represent existing classes, we shall find that seven, not five, determinate class relations between them are possible.
Taking the diagrams given in section [126], the above assumption clearly requires that in the cases of α, β, and γ, there should be some part of the universe lying outside both the circles, since otherwise either not-S or not-P or both of them would no longer be contained in the universe. But in the cases of δ and ε it is different. S, not-S, P, not-P are now all of them represented within the circles; and in each of these cases, therefore, the pair of circles may or may not between them exhaust the universe.
Our results may also be expressed by saying that in the cases of α, β, and γ, there must be something which is both not-S and not-P; whereas in the cases of δ and ε, there may or may not be something which is both not-S and not-P. Euler’s circles, as ordinarily used, are no doubt a little apt to lead us to overlook the latter of these alternatives. If, indeed, there were always part of the universe outside the circles, every proposition, whether its form were A, E, I, or O, would have an inverse and the same inverse, namely, Some not-S is not-P ; also, every proposition, including I, would have a contrapositive. These are erroneous results against which we have to be on our guard in the use of Euler’s fivefold scheme.
We find then that the explicit recognition of not-S and not-P practically leaves α, β, and γ unaffected, but causes δ and ε each to subdivide into two cases according as there is or is not anything that is both not-S and not-P; and the 171 Eulerian fivefold division has accordingly to give place to a sevenfold division.
In the diagrammatic representation of these seven relations, the entire universe of discourse may be indicated by a larger circle in which the ordinary Eulerian diagrams (with some slight necessary modifications) are included. We shall then have the following scheme:—
172 It may be useful to repeat these diagrams with an explicit indication in regard to each subdivision of the universe as to whether it is S or not-S, P or not-P.[175] The scheme will then appear as follows:—
[175] We might also represent the universe of discourse by a long rectangle divided into compartments, shewing which of the four possible combinations SP, SPʹ, SʹP, SʹPʹ are to be found. This plan will give the following which precisely correspond, as numbered, with those in the text:—
| (i) | SP | SʹPʹ |