[The Reader should make out all these for himself. In the "books" example, these three Propositions would be "No English books are new", &c.] pg033 ·-------· |( )| | |---|---| | | | ·-------· We see that this _one_ Diagram has now served to present no less than _three_ Propositions, viz. (1) "No xy exist; (2) No x are y; (3) No y are x." Hence these three Propositions are equivalent. [In the "books" example, these Propositions would be (1) "No old English books exist; (2) No old books are English; (3) No English books are old".] The two equivalent Propositions, "No x are y" and "No y are x", are said to be 'Converse' to each other. [For example, if we were told to convert the Proposition "No porcupines are talkative", we should first choose our Univ. (say "animals"), and then complete the Proposition, by supplying the Substantive "animals" in the Predicate, so that it would be "No porcupines are talkative animals", and we should then convert it, by interchanging its Terms, so that it would be "No talkative animals are porcupines".] Similarly we may represent the three similar Trios of equivalent Propositions; the whole Set of _four_ Trios being as follows:-- (1) "No xy exist" = "No x are y" = "No y are x". (2) "No xy' exist" = "No x are y'" = "No y' are x". (3) "No x'y exist" = "No x' are y" = "No y are x'". (4) "No x'y' exist" = "No x' are y'" = "No y' are x'". Let us take, next, the Proposition "All x are y". We know (see p. 17) that this is a _Double_ Proposition, and equivalent to the _two_ Propositions "Some x are y" and "No x are y'", each of which we already know how to represent. ·-------· |(.)|( )| |---|---| | | | ·-------· [Note that the _Subject_ of the given Proposition settles which _Half_ we are to use; and that its _Predicate_ settles in which _portion_ of that Half we are to place the Red Counter.] pg034 TABLE II.