The Mathematical Works of Lewis Carroll. Ð›ÑŒÑŽÐ¸Ñ ÐšÑрролл
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.’