Thus, the Diagram, here given, exhibits the two Classes, whose respective Attributes are x and y, as so related to each other that the following Propositions are all simultaneously true:—“All x are y”, “No x are not-y”, “Some x are y”, “Some y are not-x”, “Some not-y are not-x”, and, of course, the Converses of the last four.
Similarly, with this Diagram, the following Propositions are true:—“All y are x”, “No y are not-x”, “Some y are x”, “Some x are not-y”, “Some not-x are not-y”, and, of course, the Converses of the last four.
Similarly, with this Diagram, the following are true:—“All x are not-y”, “All y are not-x”, “No x are y”, “Some x are not-y”, “Some y are not-x”, “Some not-x are not-y”, and the Converses of the last four.
Similarly, with this Diagram, the following are true:—“Some x are y”, “Some x are not-y”, “Some not-x are y”, “Some not-x are not-y”, and of course, their four Converses.
Note that all Euler’s Diagrams assert “Some not-x are not-y.” Apparently it never occured to him that it might sometimes fail to be true!
Now, to represent “All x are y”, the first of these Diagrams would suffice. Similarly, to represent “No x are y”, the third would suffice. But to represent any Particular Proposition, at least three Diagrams would be needed (in order to include all the possible cases), and, for “Some not-x are not-y”, all the four.