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. pg174 _____ _/ ___ \_ / / y \ \ | \___/ | \_ x _/ \_____/
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.
_____ _____ _/ \_ _/ \_ / \ / \ | x | | y | \_ _/ \_ _/ \_____/ \_____/
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.
_____ _____ _/ \_/ \_ / / \ \ | x | | y | \_ \_/ _/ \_____/ \_____/
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_.
Mr. Venn's Method of Diagrams is a great advance on the above Method.
He uses the last of the above Diagrams to represent _any_ desired relation between x and y, by simply shading a Compartment known to be _empty_, and placing a + in one known to be _occupied_.
Thus, he would represent the three Propositions "Some x are y", "No x are y", and "All x are y", as follows:--
_____ _____ _/ \_/ \_ / / \ \ | | + | | \_ \_/ _/ \_____/ \_____/
_____ _____ _/ \_/ \_ / /#\ \ | |###| | \_ \#/ _/ \_____/ \_____/
_____ _____ _/#####\_/ \_ /#######/ \ \ |#######| + | | \#######\_/ _/ \#####/ \_____/ pg175 It will be seen that, of the _four_ Classes, whose peculiar Sets of Attributes are xy, xy', x'y, and x'y', only _three_ are here provided with closed Compartments, while the _fourth_ is allowed the rest of the Infinite Plane to range about in!