Book IV

THE TRILITERAL DIAGRAM.

CHAPTER I.

SYMBOLS AND CELLS.

First, let us suppose that the above left-hand Diagram is the Biliteral Diagram that we have been using in Book III., and that we change it into a Triliteral Diagram by drawing an Inner Square, so as to divide each of its 4 Cells into 2 portions, thus making 8 Cells altogether. The right-hand Diagram shows the result.

[The Reader is strongly advised, in reading this Chapter, not to refer to the above Diagrams, but to make a large copy of the right-hand one for himself, without any letters, and to have it by him while he reads, and keep his finger on that particular part of it, about which he is reading.] pg040 Secondly, let us suppose that we have selected a certain Adjunct, which we may call "m", and have subdivided the xy-Class into the two Classes whose Differentiæ are m and m', and that we have assigned the N.W. Inner Cell to the one (which we may call "the Class of xym-Things", or "the xym-Class"), and the N.W. Outer Cell to the other (which we may call "the Class of xym'-Things", or "the xym'-Class").

[Thus, in the "books" example, we might say "Let m mean 'bound', so that m' will mean 'unbound'", and we might suppose that we had subdivided the Class "old English books" into the two Classes, "old English bound books" and "old English unbound books", and had assigned the N.W. Inner Cell to the one, and the N.W. Outer Cell to the other.]

Thirdly, let us suppose that we have subdivided the xy'-Class, the x'y-Class, and the x'y'-Class in the same manner, and have, in each case, assigned the Inner Cell to the Class possessing the Attribute m, and the Outer Cell to the Class possessing the Attribute m'.

[Thus, in the "books" example, we might suppose that we had subdivided the "new English books" into the two Classes, "new English bound books" and "new English unbound books", and had assigned the S.W. Inner Cell to the one, and the S.W. Outer Cell to the other.]

It is evident that we have now assigned the Inner Square to the m-Class, and the Outer Border to the m'-Class.

[Thus, in the "books" example, we have assigned the Inner Square* to "bound books" and the *Outer Border to "unbound books".]

When the Reader has made himself familiar with this Diagram, he ought to be able to find, in a moment, the Compartment assigned to a particular pair of Attributes, or the Cell assigned to a particular trio of Attributes. The following Rules will help him in doing this:--

(1) Arrange the Attributes in the order x, y, m. pg041 (2) Take the first of them and find the Compartment assigned to it.

(3) Then take the second, and find what portion of that compartment is assigned to it.

(4) Treat the third, if there is one, in the same way.

[For example, suppose we have to find the Compartment assigned to ym. We say to ourselves "y has the West Half; and m has the Inner portion of that West Half."

Again, suppose we have to find the Cell assigned to x'ym'. We say to ourselves "x' has the South Half; y has the West portion of that South Half, i.e. has the South-West Quarter; and m' has the Outer portion of that South-West Quarter."]

The Reader should now get his genial friend to question him on the Table given on the next page, in the style of the following specimen-Dialogue.

Q. Adjunct for South Half, Inner Portion? A. x'm. Q. Compartment for m'? A. The Outer Border. Q. Adjunct for North-East Quarter, Outer Portion? A. xy'm'. Q. Compartment for ym? A. West Half, Inner Portion. Q. Adjunct for South Half? A. x'. Q. Compartment for x'y'm? A. South-East Quarter, Inner Portion. &c. &c.

pg042 TABLE IV.

CHAPTER II.

REPRESENTATION OF PROPOSITIONS IN TERMS OF x AND m, OR OF y AND m.

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.

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

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.

CHAPTER III.

REPRESENTATION OF TWO PROPOSITIONS OF RELATION, ONE IN TERMS OF x AND m, AND THE OTHER IN TERMS OF y AND m, ON THE SAME DIAGRAM.

The Reader had better now begin to draw little Diagrams for himself, and to mark them with the Digits "I" and "O", instead of using the Board and Counters: he may put a "I" to represent a Red Counter (this may be interpreted to mean "There is at least one Thing here"), and a "O" to represent a Grey Counter (this may be interpreted to mean "There is nothing here").

The Pair of Propositions, that we shall have to represent, will always be, one in terms of x and m, and the other in terms of y and m.

When we have to represent a Proposition beginning with "All", we break it up into the two Propositions to which it is equivalent.

When we have to represent, on the same Diagram, Propositions, of which some begin with "Some" and others with "No", we represent the negative ones first. This will sometimes save us from having to put a "I" "on a fence" and afterwards having to shift it into a Cell.

[Let us work a few examples.

Let us first represent "No x are m'". This gives us Diagram a.

"Some m are x; No m are y".

If, neglecting the Rule, we were begin with "Some m are x", we should get Diagram a.

And if we were then to take "No m are y", which tells us that the Inner N.W. Cell is empty, we should be obliged to take the "I" off the fence (as it no longer has the choice of two Cells), and to put it into the Inner N.E. Cell, as in Diagram c.

This trouble may be saved by beginning with "No m are y", as in Diagram b.

And now, when we take "Some m are x", there is no fence to sit on! The "I" has to go, at once, into the N.E. Cell, as in Diagram c.

Here we begin by breaking up the Second into the two Propositions to which it is equivalent. Thus we have three Propositions to represent, viz.--

These we will take in the order 1, 3, 2.

First we take No. (1), viz. "No x' are m'". This gives us Diagram a. pg052 Adding to this, No. (3), viz. "No m are y'", we get Diagram b.

This time the "I", representing No. (2), viz. "Some m are y," has to sit on the fence, as there is no "O" to order it off! This gives us Diagram c.

"All m are x; All y are m".

Here we break up both Propositions, and thus get four to represent, viz.--

These we will take in the order 2, 4, 1, 3.

First we take No. (2), viz. "No m are x'". This gives us Diagram a.

To this we add No. (4), viz. "No y are m'", and thus get Diagram b.

If we were to add to this No. (1), viz. "Some m are x", we should have to put the "I" on a fence: so let us try No. (3) instead, viz. "Some y are m". This gives us Diagram c.

And now there is no need to trouble about No. (1), as it would not add anything to our information to put a "I" on the fence. The Diagram already tells us that "Some m are x".]