The Mathematical Works of Lewis Carroll. Ð›ÑŒÑŽÐ¸Ñ ÐšÑрролл
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.
CHAPTER II.
PROPOSITIONS OF EXISTENCE.
A ‘Proposition of Existence’, when in normal form, has, for its Subject, the Class “existing Things”.
Its Sign of Quantity is “Some” or “No”.
[Note that, though its Sign of Quantity tells us how many existing Things are Members of its Predicate, it does not tell us the exact number: in fact, it only deals with two numbers, which are, in ascending order, “0” and “1 or more.”]
It is called “a Proposition of Existence” because its effect is to assert the Reality (i.e. the real existence), or else the Imaginariness, of its Predicate.
[Thus, the Proposition “Some existing Things are honest men” asserts that the Class “honest men” is Real.
This is the normal form; but it may also be expressed in any one of the following forms:—
(1) “Honest men exist”;
(2) “Some honest men exist”;
(3) “The Class ‘honest men’ exists”;
(4) “There are honest men”;
(5) “There are some honest men”.
Similarly, the Proposition “No existing Things are men fifty feet high” asserts that the Class “men 50 feet high” is Imaginary.
This is the normal form; but it may also be expressed in any one of the following forms:—
(1) “Men 50 feet high do not exist”;
(2) “No men 50 feet high exist”;
(3) “The Class ‘men 50 feet high’ does not exist”;
(4) “There are not any men 50 feet high”;
(5) “There are no men 50 feet high.”]
CHAPTER III.
PROPOSITIONS OF RELATION.
§ 1.
Introductory.
A Proposition of Relation, of the kind to be here discussed, has, for its Terms, two Specieses of the same Genus, such that each of the two Names conveys the idea of some Attribute not conveyed by the other.
[Thus, the Proposition “Some merchants are misers” is of the right kind, since “merchants” and “misers” are Specieses of the same Genus “men”; and since the Name “merchants” conveys the idea of the Attribute “mercantile”, and the name “misers” the idea of the Attribute “miserly”, each of which ideas is not conveyed by the other Name.
But the Proposition “Some dogs are setters” is not of the right kind, since, although it is true that “dogs” and “setters” are Specieses of the same Genus “animals”, it is not true that the Name “dogs” conveys the idea of any Attribute not conveyed by the Name “setters”. Such Propositions will be discussed in Part II.]
The Genus, of which the two Terms are Specieses, is called the ‘Universe of Discourse,’ or (more briefly) the ‘Univ.’
The Sign of Quantity is “Some” or “No” or “All”.
[Note that, though its Sign of Quantity tells us how many Members of its Subject are also Members of its Predicate, it does not tell us the exact number: in fact, it only deals with three numbers, which are, in ascending order, “0”, “1 or more”, “the total number of Members of the Subject”.]
It is called “a Proposition of Relation” because its effect is to assert that a certain relationship exists between its Terms.
§ 2.
Reduction of a Proposition of Relation to Normal form.
The Rules, for doing this, are as follows:—
(1) Ascertain what is the Subject (i.e., ascertain what Class we are talking about);
(2) If the verb, governed by the Subject, is not the verb “are” (or “is”), substitute for it a phrase beginning with “are” (or “is”);
(3) Ascertain what is the Predicate (i.e., ascertain what Class it is, which is asserted to contain some, or none, or all, of the Members of the Subject);
(4) If the Name of each Term is completely expressed (i.e. if it contains a Substantive), there is no need to determine the ‘Univ.’; but, if either Name is incompletely expressed, and contains Attributes only, it is then necessary to determine a ‘Univ.’, in order to insert its Name as the Substantive.
(5) Ascertain the Sign of Quantity;
(6) Arrange in the following order:—
Sign of Quantity,
Subject,
Copula,
Predicate.
[Let us work a few Examples, to illustrate these Rules.
(1)
“Some apples are not ripe.”
(1) The Subject is “apples.”
(2) The Verb is “are.”
(3) The Predicate is “not-ripe *.” (As no Substantive is expressed, and we have not yet settled what the Univ. is to be, we are forced to leave a blank.)
(4) Let Univ. be “fruit.”
(5) The Sign of Quantity is “some.”
(6) The Proposition now becomes
“Some | apples | are | not-ripe fruit.”
(2)
“None of my speculations have brought me as much