Symbolic Logic

[For example, “I,” “London,” “roses,” “redness,” “old English books,” “the letter which I received yesterday.”]

[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.]

[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.”]

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.

[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.”]

[For example, the Class “towns having four million inhabitants,” which Class contains only one Member, viz. “London.”]

[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.]

[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.]

[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.]

[For example, we may divide “books” into the two Classes whose Differentiæ are “old” and “not-old.”]

[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.”]

[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.]

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.”]

[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).

(a) The Substantive “Thing”, and one or more Adjectives (or phrases used as Adjectives) conveying the ideas of the Attributes;

(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.”]

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

[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.”

[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.”]

    (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.

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.]

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

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

[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.”]

These shall be discussed separately.

Its Sign of Quantity is “Some” or “No”.

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.

The Sign of Quantity is “Some” or “No” or “All”.

It is called “a Proposition of Relation” because its effect is to assert that a certain relationship exists between its Terms.

(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.

[Thus, the Proposition “All bankers are rich men” evidently contains the smaller Proposition “Some bankers are rich men”.]

The question now arises “What is the rest of the information which this Proposition gives us?”

In order to answer this question, let us begin with the smaller Proposition, “Some Members of the Subject are Members of the Predicate,” and suppose that this is all we have been told; and let us proceed to inquire what else we need to be told, in order to know that “All Members of the Subject are Members of the Predicate”.

[Thus, we may suppose that the Proposition “Some bankers are rich men” is all the information we possess; and we may proceed to inquire what other Proposition needs to be added to it, in order to make up the entire Proposition “All bankers are rich men”.]

Let us also suppose that the ‘Univ.’ (i.e. the Genus, of which both the Subject and the Predicate are Specieses) has been divided (by the Process of Dichotomy) into two smaller Classes, viz.

(1) the Predicate;

(2) the Class whose Differentia is contradictory to that of the Predicate.

[Thus, we may suppose that the Genus “men,” (of which both “bankers” and “rich men” are Specieses) has been divided into the two smaller Classes, “rich men”, “poor men”.]

(1) “Some Members of the Subject are Members of the Predicate”;

(2) “No Members of the Subject are Members of the Class whose Differentia is contradictory to that of the Predicate”.

[Thus, the Proposition “All bankers are rich men” is a Double Proposition, and is equivalent to the two Propositions

(1) “Some bankers are rich men”;

(2) “No bankers are poor men”.]

Note that the rules, here laid down, are arbitrary, and only apply to Part I of my “Symbolic Logic.”

[Thus, the Proposition “Some rich men are invalids” is to be understood as asserting that some existing Things are “rich invalids”. Hence it implies that each of the two Classes, “rich men” and “invalids”, taken by itself, is Real.]

[Thus, the Proposition “No mermaids are milliners” is to be understood as asserting that no existing Things are “mermaid-milliners”. But this implies nothing as to the Reality, or the Unreality, of either of the two Classes, “mermaids” and “milliners”, taken by itself. In this case as it happens, the Subject is Imaginary, and the Predicate Real.]

We have seen that a Proposition of Relation, beginning with “Some,” asserts that some existing Things, being Members of its Subject, are also Members of its Predicate. Hence, it asserts that some existing Things are Members of both; i.e. it asserts that some existing Things are Members of the Class of Things which have all the Attributes of the Subject and the Predicate.

Hence, to translate it into a Proposition of Existence, we take “existing Things” as the new Subject, and Things, which have all the Attributes of the Subject and the Predicate, as the new Predicate.

Similarly for a Proposition of Relation beginning with “No”.

[For example, we might say “Let Univ. be ‘books’”; and we might imagine the Diagram to be a large table, assigned to all “books.”]

[The Reader is strongly advised, in reading this Chapter, not to refer to the above Diagram, but to draw a large one for himself, without any letters, and to have it by him while he reads, and keep his finger on that particular part of it, about which he is reading.]

[For example, we might say “Let x mean ‘old,’ so that x′ will mean ‘new’,” and we might suppose that we had divided books into the two Classes whose Differentiæ are “old” and “new,” and had assigned the North Half of the table to “old books” and the South Half to “new books.”]

[For example, we might say “Let y mean ‘English,’ so that y′ will mean ‘foreign’”, and we might suppose that we had subdivided “old books” into the two Classes whose Differentiæ are “English” and “foreign”, and had assigned the North-West Cell to “old English books”, and the North-East Cell to “old foreign books.”]

[For example, we might suppose that we had subdivided “new books” into the two Classes “new English books” and “new foreign books”, and had assigned the South-West Cell to the one, and the South-East Cell to the other.]