Let us take, first, the Proposition "Some xm exist". [Note that the _full_ meaning of this Proposition is (as explained at p. 12) "Some existing Things are xm-Things".] This tells us that there is at least _one_ Thing in the Inner portion of the North Half; that is, that this Compartment is _occupied_. And this we can evidently represent by placing a _Red_ Counter on the partition which divides it. ·---------------· | | | | ·---|---· | | | (.) | | |---|---|---|---| | | | | | | ·---|---· | | | | ·---------------· [In the "books" example, this Proposition would mean "Some old bound books exist" (or "There are some old bound books").] Similarly we may represent the seven similar Propositions, "Some xm' exist", "Some x'm exist", "Some x'm' exist", "Some ym exist", "Some ym' exist", "Some y'm exist", and "Some y'm' exist". pg044 Let us take, next, the Proposition "No xm exist". This tells us that there is _nothing_ in the Inner portion of the North Half; that is, that this Compartment is _empty_. And this we can represent by placing _two Grey_ Counters in it, one in each Cell. ·---------------· | | | | ·---|---· | | |( )|( )| | |---|---|---|---| | | | | | | ·---|---· | | | | ·---------------· Similarly we may represent the seven similar Propositions, in terms of x and m, or of y and m, viz. "No xm' exist", "No x'm exist", &c. * * * * * These sixteen Propositions of Existence are the only ones that we shall have to represent on this Diagram. § 2. _Representation of Propositions of Relation in terms of x and m, or of y and m._ Let us take, first, the Pair of Converse Propositions "Some x are m" = "Some m are x." We know that each of these is equivalent to the Proposition of Existence "Some xm exist", which we already know how to represent. ·---------------· | | | | ·---|---· | | | (.) | | |---|---|---|---| | | | | | | ·---|---· | | | | ·---------------· Similarly for the seven similar Pairs, in terms of x and m, or of y and m. Let us take, next, the Pair of Converse Propositions "No x are m" = "No m are x." We know that each of these is equivalent to the Proposition of Existence "No xm exist", which we already know how to represent. ·---------------· | | | | ·---|---· | | |( )|( )| | |---|---|---|---| | | | | | | ·---|---· | | | | ·---------------· Similarly for the seven similar Pairs, in terms of x and m, or of y and m. pg045 Let us take, next, the Proposition "All x are m."