Читать книгу Selected Mathematical Works: Symbolic Logic + The Game of Logic + Feeding the Mind: by Charles Lutwidge Dodgson, alias Lewis Carroll - Lewis Carroll - Страница 10

If we think of a certain Class, and imagine that we have picked out from it a certain smaller Class, it is evident that the Remainder of the large Class does not possess the Differentia of that smaller Class. Hence it may be regarded as another smaller Class, whose Differentia may be formed, from that of the Class first picked out, by prefixing the word “not”; and we may imagine that we have divided the Class first thought of into two smaller Classes, whose Differentiæ are contradictory. This kind of Division is called ‘Dichotomy’.

[For example, we may divide “books” into the two Classes whose Differentiæ are “old” and “not-old.”]

In performing this Process, we may sometimes find that the Attributes we have chosen are used so loosely, in ordinary conversation, that it is not easy to decide which of the Things belong to the one Class and which to the other. In such a case, it would be necessary to lay down some arbitrary rule, as to where the one Class should end and the other begin.

[Thus, in dividing “books” into “old” and “not-old,” we may say “Let all books printed before a.d. 1801, be regarded as ‘old,’ and all others as ‘not-old’.”]

Hence forwards let it be understood that, if a Class of Things be divided into two Classes, whose Differentiæ have contrary meanings, each Differentia is to be regarded as equivalent to the other with the word “not” prefixed.

[Thus, if “books” be divided into “old” and “new” the Attribute “old” is to be regarded as equivalent to “not-new,” and the Attribute “new” as equivalent to “not-old.”]

After dividing a Class, by the Process of Dichotomy, into two smaller Classes, we may subdivide each of these into two still smaller Classes; and this Process may be repeated over and over again, the number of Classes being doubled at each repetition.

[For example, we may divide “books” into “old” and “new” (i.e. “not-old”): we may then subdivide each of these into “English” and “foreign” (i.e. “not-English”), thus getting four Classes, viz.

(1) old English;

(2) old foreign;

(3) new English;

(4) new foreign.

If we had begun by dividing into “English” and “foreign,” and had then subdivided into “old” and “new,” the four Classes would have been

(1) English old;

(2) English new;

(3) foreign old;

(4) foreign new.

The Reader will easily see that these are the very same four Classes which we had before.]