The Mathematical Works of Lewis Carroll. Ð›ÑŒÑŽÐ¸Ñ ÐšÑрролл
any single Thing, which we can name so as to distinguish it from all other Things, may be regarded as a one-Member Class.
[Thus “London” may be regarded as the one-Member Class, picked out from the Class “towns,” which has, as its Differentia, “having four million inhabitants.”]
A Class, containing two or more Members, is sometimes regarded as one single Thing. When so regarded, it may possess an Adjunct which is not possessed by any Member of it taken separately.
[Thus, the Class “The soldiers of the Tenth Regiment,” when regarded as one single Thing, may possess the Attribute “formed in square,” which is not possessed by any Member of it taken separately.]
CHAPTER III.
DIVISION.
§ 1.
Introductory.
‘Division’ is a Mental Process, in which we think of a certain Class of Things, and imagine that we have divided it into two or more smaller Classes.
[Thus, we might think of the Class “books,” and imagine that we had divided it into the two smaller Classes “bound books” and “unbound books,” or into the three Classes, “books priced at less than a shilling,” “shilling-books,” “books priced at more than a shilling,” or into the twenty-six Classes, “books whose names begin with A,” “books whose names begin with B,” &c.]
A Class, that has been obtained by a certain Division, is said to be ‘codivisional’ with every Class obtained by that Division.
[Thus, the Class “bound books” is codivisional with each of the two Classes, “bound books” and “unbound books.”
Similarly, the Battle of Waterloo may be said to have been “contemporary” with every event that happened in 1815.]
Hence a Class, obtained by Division, is codivisional with itself.
[Thus, the Class “bound books” is codivisional with itself.
Similarly, the Battle of Waterloo may be said to have been “contemporary” with itself.]
§ 2.
Dichotomy.
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.]
CHAPTER IV.
NAMES.
The word “Thing”, which conveys the idea of a Thing, without any idea of an Adjunct, represents any single Thing. Any other word (or phrase), which conveys the idea of a Thing, with the idea of an Adjunct represents any Thing which possesses that Adjunct; i.e., it represents any Member of the Class to which that Adjunct is peculiar.
Such a word (or phrase) is called a ‘Name’; and, if there be an existing Thing which it represents, it is said to be a Name of that Thing.
[For example, the words “Thing,” “Treasure,” “Town,” and the phrases “valuable Thing,” “material artificial Thing consisting of houses and streets,” “Town lit with gas,” “Town paved with gold,” “old English Book.”]
Just as a Class is said to be Real, or Unreal, according as there is, or is not, an existing Thing in it, so also a Name is said to be Real, or Unreal, according as there is, or is not, an existing Thing represented by it.
[Thus, “Town lit with gas” is a Real Name: “Town paved with gold” is an Unreal Name.]
Every Name is either a Substantive only, or else a phrase consisting of a Substantive and one or more Adjectives (or phrases used as Adjectives).
Every Name, except “Thing”, may usually be expressed in three different forms:—
(a) The Substantive “Thing”, and one or more Adjectives (or phrases used as Adjectives) conveying the ideas of the Attributes;
(b) A Substantive, conveying the idea of a Thing with the ideas of some of the Attributes, and one or more Adjectives (or phrases used as Adjectives) conveying the ideas of the other Attributes;
(c) A Substantive conveying the idea of a Thing with the ideas of all the Attributes.
[Thus, the phrase “material living Thing, belonging to the Animal Kingdom, having two hands and two feet” is a Name expressed in Form (a).
If we choose to