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Symbollic Logic

by Lewis Carroll

“Book 4: Chapter 1”

Additional Information
  • Year Published: 1896
  • Language: English
  • Country of Origin: United States of America
  • Source: Carroll, L. (1896). Symbollic Logic. New York; Macmillan & Co.
  • Readability:
    • Flesch–Kincaid Level: 10.5
  • Word Count: 2,098
  • Genre: Informational
  • Keywords: math history, mathematics
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BOOK IV.

THE TRILITERAL DIAGRAM.

             -----------------         -----------------
            |        |        |       | xy     |    xy' |
            |        |        |       | m'     |     m' |
            |   xy   |   xy'  |       |    ----|----    |
            |        |        |       |   | xy |xy' |   |
            |        |        |       |   | m  | m  |   |
            |--------|--------|       |---|----|----|---|
            |        |        |       |   |x'y |x'y'|   |
            |        |        |       |   | m  | m  |   |
            |  x'y   |  x'y'  |       |    ----|----    |
            |        |        |       | x'y    |   x'y' |
            |        |        |       | m'     |     m' |
             -----------------         -----------------

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.]

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 Differentiae are m and m’, and that we 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 Com- partment 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.
(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’,?
A. South-East Quarter, Inner Portion.
    &c. &c.

                        TABLE IV.

      -----------------------------------------------
     | Adjunct  |                                    |
     |   of     |  Compartments, or Cells, assigned  |
     | Classes. |             to them.               |
     |----------|------------------------------------|
     | x . . . .| North Half.                        |
     | x'  . . .| South  "                           |
     | y . . . .| West   "                           |
     | y'  . . .| East   "                           |
     | m . . . .| Inner Square.                      |
     | m'  . . .| Outer Border.                      |
     |----------|------------------------------------|
     | xy  . . .| North-West Quarter.                |
     | xy' . . .|   "   East    "                    |
     | x'y . . .| South-West    "                    |
     | x'y'. . .|   "   East    "                    |
     | xm  . . .| North Half, Inner Portion.         |
     | xm' . . .|   "    "    Outer    "             |
     | x'm . . .| South  "    Inner    "             |
     | x'm'. . .|   "    "    Outer    "             |
     | ym  . . .| West   "    Inner    "             |
     | ym' . . .|  "     "    Outer    "             |
     | y'm . . .| East   "    Inner    "             |
     | y'm'  . .|   "    "    Outer    "             |
     |----------|------------------------------------|
     | xym . . .| North-West Quarter, Inner Portion. |
     | xym'  . .|   "     "     "     Outer    "     |
     | xy'm  . .|   "   East    "     Inner    "     |
     | xy'm' . .|   "     "     "     Outer    "     |
     | x'ym  . .| South-West    "     Inner    "     |
     | x'ym' . .|   "     "     "     Outer    "     |
     | x'y'm . .|   "   East    "     Inner    "     |
     | x'y'm'. .|   "     "     "     Outer    "     |
      -----------------------------------------------