Complete Works of Lewis Carroll
Page 99
Mental recreation is a thing that we all of us need for our mental health; and you may get much healthy enjoyment, no doubt, from Games, such as Back-gammon, Chess, and the new Game “Halma”. But, after all, when you have made yourself a first-rate player at any one of these Games, you have nothing real to show for it, as a result! You enjoyed the Game, and the victory, no doubt, at the time: but you have no result that you can treasure up and get real good out of. And, all the while, you have been leaving unexplored a perfect mine of wealth. Once master the machinery of Symbolic Logic, and you have a mental occupation always at hand, of absorbing interest, and one that will be of real use to you in any subject you may take up. It will give you clearness of thought——the ability to see your way through a puzzle——the habit of arranging your ideas in an orderly and get-at-able form——and, more valuable than all, the power to detect fallacies, and to tear to pieces the flimsy illogical arguments, which you will so continually encounter in books, in newspapers, in speeches, and even in sermons, and which so easily delude those who have never taken the trouble to master this fascinating Art. Try it. That is all I ask of you!
L. C.
29, Bedford Street, Strand.
February 21, 1896.
BOOK I.
THINGS AND THEIR ATTRIBUTES.
CHAPTER I.
INTRODUCTORY.
The Universe contains ‘Things.’
[For example, “I,” “London,” “roses,” “redness,” “old English books,” “the letter which I received yesterday.”]
Things have ‘Attributes.’
[For example, “large,” “red,” “old,” “which I received yesterday.”]
One Thing may have many Attributes; and one Attribute may belong to many Things.
[Thus, the Thing “a rose” may have the Attributes “red,” “scented,” “full-blown,” &c.; and the Attribute “red” may belong to the Things “a rose,” “a brick,” “a ribbon,” &c.]
Any Attribute, or any Set of Attributes, may be called an ‘Adjunct.’
[This word is introduced in order to avoid the constant repetition of the phrase “Attribute or Set of Attributes.”
Thus, we may say that a rose has the Attribute “red” (or the Adjunct “red,” whichever we prefer); or we may say that it has the Adjunct “red, scented and full-blown.”]
CHAPTER II.
CLASSIFICATION.
‘Classification,’ or the formation of Classes, is a Mental Process, in which we imagine that we have put together, in a group, certain Things. Such a group is called a ‘Class.’
This Process may be performed in three different ways, as follows:—
(1) We may imagine that we have put together all Things. The Class so formed (i.e. the Class “Things”) contains the whole Universe.
(2) We may think of the Class “Things,” and may imagine that we have picked out from it all the Things which possess a certain Adjunct not possessed by the whole Class. This Adjunct is said to be ‘peculiar’ to the Class so formed. In this case, the Class “Things” is called a ‘Genus’ with regard to the Class so formed: the Class, so formed, is called a ‘Species’ of the Class “Things”: and its peculiar Adjunct is called its ‘Differentia’.
As this Process is entirely Mental, we can perform it whether there is, or is not, an existing Thing which possesses that Adjunct. If there is, the Class is said to be ‘Real’; if not, it is said to be ‘Unreal’, or ‘Imaginary.’
[For example, we may imagine that we have picked out, from the Class “Things,” all the Things which possess the Adjunct “material, artificial, consisting of houses and streets”; and we may thus form the Real Class “towns.” Here we may regard “Things” as a Genus, “Towns” as a Species of Things, and “material, artificial, consisting of houses and streets” as its Differentia.
Again, we may imagine that we have picked out all the Things which possess the Adjunct “weighing a ton, easily lifted by a baby”; and we may thus form the Imaginary Class “Things that weigh a ton and are easily lifted by a baby.”]
(3) We may think of a certain Class, not the Class “Things,” and may imagine that we have picked out from it all the Members of it which possess a certain Adjunct not possessed by the whole Class. This Adjunct is said to be ‘peculiar’ to the smaller Class so formed. In this case, the Class thought of is called a ‘Genus’ with regard to the smaller Class picked out from it: the smaller Class is called a ‘Species’ of the larger: and its peculiar Adjunct is called its ‘Differentia’.
[For example, we may think of the Class “towns,” and imagine that we have picked out from it all the towns which possess the Attribute “lit with gas”; and we may thus form the Real Class “towns lit with gas.” Here we may regard “Towns” as a Genus, “Towns lit with gas” as a Species of Towns, and “lit with gas” as its Differentia.
If, in the above example, we were to alter “lit with gas” into “paved with gold,” we should get the Imaginary Class “towns paved with gold.”]
A Class, containing only one Member is called an ‘Individual.’
[For example, the Class “towns having four million inhabitants,” which Class contains only one Member, viz. “London.”]
½Hence, 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’.”]
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.”]