Selected Mathematical Works: Symbolic Logic + The Game of Logic + Feeding the Mind: by Charles Lutwidge Dodgson, alias Lewis Carroll. Lewis Carroll
to begin with, that the phrase “none but the brave” is equivalent to “no not-brave men.” We then arrange thus:—
“No | existing Things | are | not-brave men deserving of the fair.”
(6)
“All bankers are rich men.”
This is equivalent to the two Propositions “Some bankers are rich men” and “No bankers are poor men.”
Here we arrange thus:—
“Some | existing Things | are | rich bankers”; and “No | existing Things | are | poor bankers.”]
[Work Examples § 1, 1–4 (p. 97).]
BOOK III.
THE BILITERAL DIAGRAM.
CHAPTER I.
SYMBOLS AND CELLS.
First, let us suppose that the above Diagram is an enclosure assigned to a certain Class of Things, which we have selected as our ‘Universe of Discourse.’ or, more briefly, as our ‘Univ’.
[For example, we might say “Let Univ. be ‘books’”; and we might imagine the Diagram to be a large table, assigned to all “books.”]
[The Reader is strongly advised, in reading this Chapter, not to refer to the above Diagram, but to draw a large 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 “x,” and have divided the large Class, to which we have assigned the whole Diagram, into the two smaller Classes whose Differentiæ are “x” and “not-x” (which we may call “x′”), and that we have assigned the North Half of the Diagram to the one (which we may call “the Class of x-Things,” or “the x-Class”), and the South Half to the other (which we may call “the Class of x′-Things,” or “the x′-Class”).
[For example, we might say “Let x mean ‘old,’ so that x′ will mean ‘new’,” and we might suppose that we had divided books into the two Classes whose Differentiæ are “old” and “new,” and had assigned the North Half of the table to “old books” and the South Half to “new books.”]
Thirdly, let us suppose that we have selected another Adjunct, which we may call “y”, and have subdivided the x-Class into the two Classes whose Differentiæ are “y” and “y′”, and that we have assigned the North-West Cell to the one (which we may call “the xy-Class”), and the North-East Cell to the other (which we may call “the xy′-Class”).
[For example, we might say “Let y mean ‘English,’ so that y′ will mean ‘foreign’”, and we might suppose that we had subdivided “old books” into the two Classes whose Differentiæ are “English” and “foreign”, and had assigned the North-West Cell to “old English books”, and the North-East Cell to “old foreign books.”]
Fourthly, let us suppose that we have subdivided the x′-Class in the same manner, and have assigned the South-West Cell to the x′y-Class, and the South-East Cell to the x′y′-Class.
[For example, we might suppose that we had subdivided “new books” into the two Classes “new English books” and “new foreign books”, and had assigned the South-West Cell to the one, and the South-East Cell to the other.]
It is evident that, if we had begun by dividing for y and y′, and had then subdivided for x and x′, we should have got the same four Classes. Hence we see that we have assigned the West Half to the y-Class, and the East Half to the y′-Class.
[Thus, in the above Example, we should find that we had assigned the West Half of the table to “English books” and the East Half to “foreign books.”
We have, in fact, assigned the four Quarters of the table to four different Classes of books, as here shown.]
The Reader should carefully remember that, in such a phrase as “the x-Things,” the word “Things” means that particular kind of Things, to which the whole Diagram has been assigned.
[Thus, if we say “Let Univ. be ‘books’,” we mean that we have assigned the whole Diagram to “books.” In that case, if we took “x” to mean “old”, the phrase “the x-Things” would mean “the old books.”]
The Reader should not go on to the next Chapter until he is quite familiar with the blank Diagram I have advised him to draw.
He ought to be able to name, instantly, the Adjunct assigned to any Compartment named in the right-hand column of the following Table.
Also he ought to be able to name, instantly, the Compartment assigned to any Adjunct named in the left-hand column.
To make sure of this, he had better put the book into the hands of some genial friend, while he himself has nothing but the blank Diagram, and get that genial friend to question him on this Table, dodging about as much as possible. The Questions and Answers should be something like this:—
TABLE I.
| Adjuncts of Classes. | Compartments, or Cells, assigned to them. |
| x | North Half. |
| x′ | South 〃 |
| y | North Half. |
| y′ | East 〃 |
| xy | North - West Cell. |
| xy′ | 〃 East 〃 |
| x′y | South - West 〃 |
| x′y′ | 〃 East 〃 |
| Q. “Adjunct for West Half?” A. “y.” Q. “Compartment for xy′?” A. “North-East Cell.” Q. “Adjunct for South-West Cell?” A. “x′y.” &c., &c. | |
After a little practice, he will find himself able to do without the blank Diagram, and will be able to see it mentally (“in my mind’s eye, Horatio!”) while answering the questions of his genial friend. When this result has been reached, he may safely go on to the next Chapter.
CHAPTER II.