Selected Mathematical Works: Symbolic Logic + The Game of Logic + Feeding the Mind: by Charles Lutwidge Dodgson, alias Lewis Carroll. Lewis Carroll
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Next, let us suppose that we find two Red Counters placed in the North Half, one in each Cell.
We know that this represents the Double Proposition “Some x are y and some are y′”.
Similarly we may interpret two Red Counters, when placed in the South, or West, or East Half.
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Next, let us suppose that we find two Grey Counters placed in the North Half, one in each Cell.
We know that this represents the Proposition “No x exist”.
Similarly we may interpret two Grey Counters, when placed in the South, or West, or East Half.
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Lastly, let us suppose that we find a Red and a Grey Counter placed in the North Half, the Red in the North-West Cell, and the Grey in the North-East Cell.
We know that this represents the Proposition, “All x are y”.
[Note that the Half, occupied by the two Counters, settles what is to be the Subject of the Proposition, and that the Cell, occupied by the Red Counter, settles what is to be its Predicate.]
Similarly we may interpret a Red and a Grey counter, when placed in any one of the seven similar positions
Red in North-East, Grey in North-West;
Red in South-West, Grey in South-East;
Red in South-East, Grey in South-West;
Red in North-West, Grey in South-West;
Red in South-West, Grey in North-West;
Red in North-East, Grey in South-East;
Red in South-East, Grey in North-East.
Once more the genial friend must be appealed to, and requested to examine the Reader on Tables II and III, and to make him not only represent Propositions, but also interpret Diagrams when marked with Counters.
The Questions and Answers should be like this:—
Q. Represent “No x′ are y′.” A. Grey Counter in S.E. Cell. Q. Interpret Red Counter on E. partition. A. “Some y′ exist.” Q. Represent “All y′ are x.” A. Red in N.E. Cell; Grey in S.E. Q. Interpret Grey Counter in S.W. Cell. A. “No x′y exist” = “No x′ are y” = “No y are x′”. &c., &c.
At first the Examinee will need to have the Board and Counters before him; but he will soon learn to dispense with these, and to answer with his eyes shut or gazing into vacancy.
[Work Examples § 1, 5–8 (p. 97).]
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.]
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. (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 СКАЧАТЬ