Complete Works of Lewis Carroll (139 page)

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’.”]

Henceforwards 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 sub-divide 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 sub-divide 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 sub-divided 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 roll up together the Substantive “Thing” and the Adjectives “material, living, belonging to the Animal Kingdom,” so as to make the new Substantive “Animal,” we get the phrase “Animal having two hands and two feet,” which is a Name (representing the same Thing as before) expressed in Form (
b
).

And, if we choose to roll up the whole phrase into one word, so as to make the new Substantive “Man,” we get a Name (still representing the very same Thing) expressed in Form (
c
).]

A Name, whose Substantive is in the
plural
number, may be used to represent either

(1) Members of a Class,
regarded as separate Things
;

or (2) a whole Class,
regarded as one single Thing
.

[Thus, when I say “Some soldiers of the Tenth Regiment are tall,” or “The soldiers of the Tenth Regiment are brave,” I am using the Name “soldiers of the Tenth Regiment” in the
first
sense; and it is just the same as if I were to point to each of them
separately
, and to say “
This
soldier of the Tenth Regiment is tall,” “
That
soldier of the Tenth Regiment is tall,” and so on.

But, when I say “The soldiers of the Tenth Regiment are formed in square,” I am using the phrase in the
second
sense; and it is just the same as if I were to say “The
Tenth Regiment
is formed in square.”]

CHAPTER V.

DEFINITIONS.

It is evident that every Member of a
Species
is
also
a Member of the
Genus
out of which that Species has been picked, and that it possesses the
Differentia
of that Species.
Hence it may be represented by a Name consisting of two parts, one being a Name representing any Member of the
Genus
, and the other being the
Differentia
of that Species.
Such a Name is called a ‘
Definition
’ of any Member of that Species, and to give it such a Name is to ‘
define
’ it.

[Thus, we may define a “Treasure” as a “valuable Thing.”
In this case we regard “Things” as the
Genus
, and “valuable” as the
Differentia
.]

The following Examples, of this Process, may be taken as models for working others.

[Note that, in each Definition, the Substantive, representing a Member (or Members) of the
Genus
, is printed in Capitals.]

1.
Define “a Treasure.”

Ans.
“a valuable Thing.”

2.
Define “Treasures.”

Ans.
“valuable Things.”

3.
Define “a Town.”

Ans.
“a material artificial Thing, consisting of houses and streets.”

4.
Define “Men.”

Ans.
“material, living Things, belonging to the Animal Kingdom, having two hands and two feet”;

or else

“Animals having two hands and two feet.”

5.
Define “London.”

Ans.
“the material artificial Thing, which consists of houses and streets, and has four million inhabitants”;

or else

“the Town which has four million inhabitants.”

[Note that we here use the article “the” instead of “a”, because we happen to know that there is only
one
such Thing.

The Reader can set himself any number of Examples of this Process, by simply choosing the Name of any common Thing (such as “house,” “tree,” “knife”), making a Definition for it, and then testing his answer by referring to any English Dictionary.]

BOOK II.

PROPOSITIONS.

CHAPTER I.

PROPOSITIONS GENERALLY.

§ 1.

Introductory.

Note that the word “some” is to be regarded, henceforward, as meaning “one or more.”

The word ‘Proposition,’ as used in ordinary conversation, may be applied to
any
word, or phrase, which conveys any information whatever.

[Thus the words “yes” and “no” are Propositions in the ordinary sense of the word; and so are the phrases “you owe me five farthings” and “I don’t!”

Such words as “oh!”
or “never!”, and such phrases as “fetch me that book!”
“which book do you mean?”
do not seem, at first sight, to convey any
information
; but they can easily be turned into equivalent forms which do so, viz.
“I am surprised,” “I will never consent to it,” “I order you to fetch me that book,” “I want to know which book you mean.”]

But a ‘
Proposition
,’ as used in this First Part of “Symbolic Logic,” has a peculiar form, which may be called its ‘
Normal form
’; and if any Proposition, which we wish to use in an argument, is not in normal form, we must reduce it to such a form, before we can use it.

A ‘
Proposition
,’ when in normal form, asserts, as to certain two Classes, which are called its ‘
Subject
’ and ‘
Predicate
,’ either

    (1) that
some
Members of its Subject are Members of its Predicate;

or (2) that
no
Members of its Subject are Members of its Predicate;

or (3) that
all
Members of its Subject are Members of its Predicate.

The Subject and the Predicate of a Proposition are called its ‘
Terms
.’

Two Propositions, which convey the
same
information, are said to be ‘
equivalent
’.

[Thus, the two Propositions, “I see John” and “John is seen by me,” are equivalent.]

§ 2.

Normal form of a Proposition.

A Proposition, in normal form, consists of four parts, viz.—

(1) The word “some,” or “no,” or “all.”
(This word, which tells us
how many
Members of the Subject are also Members of the Predicate, is called the ‘
Sign of Quantity
.’)

(2) Name of Subject.

(3) The verb “are” (or “is”).
(This is called the ‘
Copula
.’)

(4) Name of Predicate.

§ 3.

Various kinds of Propositions.

A Proposition, that begins with “Some”, is said to be ‘
Particular
.’
It is also called ‘a Proposition
in I
.’

[Note, that it is called ‘Particular,’ because it refers to a
part
only of the Subject.]

A Proposition, that begins with “No”, is said to be ‘
Universal Negative
.’
It is also called ‘a Proposition
in E
.’

A Proposition, that begins with “All”, is said to be ‘
Universal Affirmative
.’
It is also called ‘a Proposition
in A
.’

[Note, that they are called ‘Universal’, because they refer to the
whole
of the Subject.]

A Proposition, whose Subject is an
Individual
, is to be regarded as
Universal
.

[Let us take, as an example, the Proposition “John is not well”.
This of course implies that there is an
Individual
, to whom the speaker refers when he mentions “John”, and whom the listener
knows
to be referred to.
Hence the Class “men referred to by the speaker when he mentions ‘John’” is a one-Member Class, and the Proposition is equivalent to “
All
the men, who are referred to by the speaker when he mentions ‘John’, are not well.”]

Propositions are of two kinds, ‘Propositions of Existence’ and ‘Propositions of Relation.’

These shall be discussed separately.