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_. § 6. _Venn's Method of Diagrams._ Let us represent "not-x" by "x'". 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:-- _____ _____ _/ \_/ \_ / / \ \ | | + | | \_ \_/ _/ \_____/ \_____/ _____ _____ _/ \_/ \_ / /#\ \ | |###| | \_ \#/ _/ \_____/ \_____/