Chapter 41

_REPRESENTATION OF PROPOSITIONS IN TERMS OF x AND m, OR OF y AND m._

§ 1.

_Representation of Propositions of Existence in terms of x and m, or of y and m._

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."

·---------------· |( ) | ( )| | ·---|---· | | | (.) | | |---|---|---|---| | | | | | | ·---|---· | | | | ·---------------·

We know (see p. 18) that this is a _Double_ Proposition, and equivalent to the _two_ Propositions "Some x are m" and "No x are m' ", each of which we already know how to represent.

Similarly for the fifteen similar Propositions, in terms of x and m, or of y and m.

These thirty-two Propositions of Relation are the only ones that we shall have to represent on this Diagram.

The Reader should now get his genial friend to question him on the following four Tables.

The Victim should have nothing before him but a blank Triliteral Diagram, a Red Counter, and 2 Grey ones, with which he is to represent the various Propositions named by the Inquisitor, _e.g._ "No y' are m", "Some xm' exist", &c., &c. pg046 TABLE V.

·---------------------------------------------------------· | ·---------------· Some xm exist | ·---------------· | | | | | = Some x are m | | | | | | | ·---|---· | = Some m are x | | ·---|---· | | | | | (.) | | | | |( )|( )| | | | |---|---|---|---| ·-----------------· |---|---|---|---| | | | | | | | | No xm exist | | | | | | | | ·---|---· | | = No x are m | ·---|---· | | | | | | | = No m are x | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· Some xm' exist | ·---------------· | | | (.) | = Some x are m' | |( ) | ( )| | | | ·---|---· | = Some m' are x | | ·---|---· | | | | | | | | | | | | | | | | |---|---|---|---| ·-----------------· |---|---|---|---| | | | | | | | | No xm' exist | | | | | | | | ·---|---· | | = No x are m' | ·---|---· | | | | | | | = No m' are x | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· Some x'm exist | ·---------------· | | | | | = Some x' are m | | | | | | | ·---|---· | = Some m are x' | | ·---|---· | | | | | | | | | | | | | | | | |---|---|---|---| ·-----------------· |---|---|---|---| | | | | (.) | | | No x'm exist | |( )|( )| | | | | ·---|---· | | = No x' are m | ·---|---· | | | | | | | = No m are x' | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· Some x'm' exist | ·---------------· | | | | | = Some x' are m'| | | | | | | ·---|---· | = Some m' are x'| | ·---|---· | | | | | | | | | | | | | | | | |---|---|---|---| ·-----------------· |---|---|---|---| | | | | | | | | No x'm' exist | | | | | | | | ·---|---· | | = No x' are m' | ·---|---· | | | | (.) | | = No m' are x' |( ) | ( )| | | ·---------------· | ·---------------· | ·---------------------------------------------------------· pg047 TABLE VI.

·---------------------------------------------------------· | ·---------------· Some ym exist | ·---------------· | | | | | = Some y are m | | | | | | | ·---|---· | = Some m are y | | ·---|---· | | | | | | | | | | |( )| | | | | |---|(.)|---|---| ·-----------------· |---|---|---|---| | | | | | | | | No ym exist | |( )| | | | | | ·---|---· | | = No y are m | ·---|---· | | | | | | | = No m are y | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· Some ym' exist | ·---------------· | | | | | = Some y are m' | |( ) | | | | | ·---|---· | = Some m' are y | | ·---|---· | | | | | | | | | | | | | | | | |(.)|---|---|---| ·-----------------· |---|---|---|---| | | | | | | | | No ym' exist | | | | | | | | ·---|---· | | = No y are m' | ·---|---· | | | | | | | = No m' are y |( ) | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· Some y'm exist | ·---------------· | | | | | = Some y' are m | | | | | | | ·---|---· | = Some m are y' | | ·---|---· | | | | | | | | | | | |( )| | | | |---|---|(.)|---| ·-----------------· |---|---|---|---| | | | | | | | | No y'm exist | | |( )| | | | | ·---|---· | | = No y' are m | ·---|---· | | | | | | | = No m are y' | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· Some y'm' exist | ·---------------· | | | | | = Some y' are m'| | | ( )| | | | ·---|---· | = Some m' are y'| | ·---|---· | | | | | | | | | | | | | | | | |---|---|---|(.)| ·-----------------· |---|---|---|---| | | | | | | | | No y'm' exist | | | | | | | | ·---|---· | | = No y' are m' | ·---|---· | | | | | | | = No m' are y' | | ( )| | | ·---------------· | ·---------------· | ·---------------------------------------------------------· pg048 TABLE VII.

·---------------------------------------------------------· | ·---------------· | ·---------------· | | |( ) | ( )| All x are m | | (.) | | | | ·---|---· | | | ·---|---· | | | | | (.) | | | | |( )|( )| | | | |---|---|---|---| ·-----------------· |---|---|---|---| | | | | | | | | | | | | | | | | ·---|---· | | All x are m' | ·---|---· | | | | | | | | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· | ·---------------· | | | | | All x' are m | | | | | | | ·---|---· | | | ·---|---· | | | | | | | | | | | | | | | | |---|---|---|---| ·-----------------· |---|---|---|---| | | | | (.) | | | | |( )|( )| | | | | ·---|---· | | All x' are m' | ·---|---· | | | |( ) | ( )| | | (.) | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· | ·---------------· | | | | | All m are x | | | | | | | ·---|---· | | | ·---|---· | | | | | (.) | | | | |( )|( )| | | | |---|---|---|---| ·-----------------· |---|---|---|---| | | | |( )|( )| | | | | (.) | | | | | ·---|---· | | All m are x' | ·---|---· | | | | | | | | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· | ·---------------· | | | (.) | All m' are x | |( ) | ( )| | | | ·---|---· | | | ·---|---· | | | | | | | | | | | | | | | | |---|---|---|---| ·-----------------· |---|---|---|---| | | | | | | | | | | | | | | | | ·---|---· | | All m' are x' | ·---|---· | | | |( ) | ( )| | | (.) | | | ·---------------· | ·---------------· | ·---------------------------------------------------------· pg049 TABLE VIII.

·---------------------------------------------------------· | ·---------------· | ·---------------· | | |( ) | | All y are m | | | | | | | ·---|---· | | | ·---|---· | | | | | | | | | | |( )| | | | | |---|(.)|---|---| ·-----------------· |(.)|---|---|---| | | | | | | | | | |( )| | | | | | ·---|---· | | All y are m' | ·---|---· | | | |( ) | | | | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· | ·---------------· | | | | ( )| All y' are m | | | | | | | ·---|---· | | | ·---|---· | | | | | | | | | | | |( )| | | | |---|---|(.)|---| ·-----------------· |---|---|---|(.)| | | | | | | | | | | |( )| | | | | ·---|---· | | All y' are m' | ·---|---· | | | | | ( )| | | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· | ·---------------· | | | | | All m are y | | | | | | | ·---|---· | | | ·---|---· | | | | | |( )| | | | |( )| | | | | |---|(.)|---|---| ·-----------------· |---|---|(.)|---| | | | | |( )| | | | |( )| | | | | | ·---|---· | | All m are y' | ·---|---· | | | | | | | | | | | | ·---------------· | ·---------------· | |---------------------------------------------------------| | ·---------------· | ·---------------· | | | | ( )| All m' are y | |( ) | | | | | ·---|---· | | | ·---|---· | | | | | | | | | | | | | | | | |(.)|---|---|---| ·-----------------· |---|---|---|(.)| | | | | | | | | | | | | | | | | ·---|---· | | All m' are y' | ·---|---· | | | | | ( )| | |( ) | | | | ·---------------· | ·---------------· | ·---------------------------------------------------------·

pg050