[The Reader should make out all these for himself. In the "books" example, these three Propositions would be "No old foreign books exist", &c.] pg030 We have seen that the Proposition "No x exist" may be represented by placing _two Grey_ Counters in the North Half, one in each Cell. ·-------· |( )|( )| |---|---| | | | ·-------· We have also seen that these two _Grey_ Counters, taken _separately_, represent the two Propositions "No xy exist" and "No xy' exist". Hence we see that the Proposition "No x exist" is a _Double_ Proposition, and is equivalent to the _two_ Propositions "No xy exist" and "No xy' exist". [In the "books" example, this Proposition would be "No old books exist". Hence this is a _Double_ Proposition, and is equivalent to the _two_ Propositions "No old _English_ books exist" and "No old _foreign_ books exist".] § 3. _Representation of Propositions of Relation._ Let us take, first, the Proposition "Some x are y". This tells us that at least _one_ Thing, in the _North_ Half, is also in the _West_ Half. Hence it must be in the space _common_ to them, that is, in the _North-West Cell_. Hence the North-West Cell is _occupied_. And this we can represent by placing a _Red_ Counter in it. ·-------· |(.)| | |---|---| | | | ·-------· [Note that the _Subject_ of the Proposition settles which _Half_ we are to use; and that the _Predicate_ settles in which _portion_ of it we are to place the Red Counter. In the "books" example, this Proposition would be "Some old books are English".] Similarly we may represent the three similar Propositions "Some x are y'", "Some x' are y", and "Some x' are y'". [The Reader should make out all these for himself. In the "books" example, these three Propositions would be "Some old books are foreign", &c.] pg031 Let us take, next, the Proposition "Some y are x". This tells us that at least _one_ Thing, in the _West_ Half, is also in the _North_ Half. Hence it must be in the space _common_ to them, that is, in the _North-West Cell_. Hence the North-West Cell is _occupied_. And this we can represent by placing a _Red_ Counter in it. ·-------· |(.)| | |---|---| | | | ·-------· [In the "books" example, this Proposition would be "Some English books are old".] Similarly we may represent the three similar Propositions "Some y are x'", "Some y' are x", and "Some y' are x'". [The Reader should make out all these for himself. In the "books" example, these three Propositions would be "Some English books are new", &c.] We see that this _one_ Diagram has now served to represent no less than _three_ Propositions, viz. (1) "Some xy exist; (2) Some x are y; (3) Some y are x".