The Universe contains ‘Things.’
[For example, “I,” “London,” “roses,” “redness,” “old English books,” “the letter which I received yesterday.”]
[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.”]
[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.]
[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.”]
[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.”]
‘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 Adjunctnotpossessed 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’.
pg002As this Process is entirelyMental, we can perform it whether thereis, oris not, anexistingThing which possesses that Adjunct.If thereis, 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 aGenus, “Towns” as aSpeciesof Things, and “material, artificial, consisting of houses and streets” as itsDifferentia.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 theImaginaryClass “Things that weigh a ton and are easily lifted by a baby.”]
[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 aGenus, “Towns” as aSpeciesof Things, and “material, artificial, consisting of houses and streets” as itsDifferentia.
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 theImaginaryClass “Things that weigh a ton and are easily lifted by a baby.”]
(3) We may think of a certain Class,notthe Class “Things,” and may imagine that we have picked out from it all the Members of it which possess a certain Adjunctnotpossessed 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 aGenus, “Towns lit with gas” as aSpeciesof Towns, and “lit with gas” as itsDifferentia.If, in the above example, we were to alter “lit with gas” into “paved with gold,” we should get theImaginaryClass “towns paved with gold.”]
[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 aGenus, “Towns lit with gas” as aSpeciesof Towns, and “lit with gas” as itsDifferentia.
If, in the above example, we were to alter “lit with gas” into “paved with gold,” we should get theImaginaryClass “towns paved with gold.”]
A Class, containing onlyoneMember is called an ‘Individual.’
[For example, the Class “towns having four million inhabitants,” which Class contains onlyoneMember, viz. “London.”]
[For example, the Class “towns having four million inhabitants,” which Class contains onlyoneMember, viz. “London.”]
pg002½Hence, 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.”]
[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 asone single Thing. When so regarded, it may possess an Adjunct which isnotpossessed by any Member of it taken separately.
[Thus, the Class “The soldiers of the Tenth Regiment,” when regarded asone single Thing, may possess the Attribute “formed in square,” which isnotpossessed by any Member of it taken separately.]
[Thus, the Class “The soldiers of the Tenth Regiment,” when regarded asone single Thing, may possess the Attribute “formed in square,” which isnotpossessed by any Member of it taken separately.]
‘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 withA,” “books whose names begin withB,” &c.]
[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 withA,” “books whose names begin withB,” &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.]
[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.]
[Thus, the Class “bound books” is codivisional with itself.
Similarly, the Battle of Waterloo may be said to have been “contemporary” with itself.]
If we think of a certain Class, and imagine that we have picked out from it a certain smaller Class, it is evident that theRemainderof the large Class doesnotpossess the Differentia of that smaller Class. Hence it may be regarded asanothersmaller 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 havedividedthe Class first thought of intotwosmaller Classes, whose Differentiæ arecontradictory. This kind of Division is called ‘Dichotomy’.
[For example, we may divide “books” into the two Classes whose Differentiæ are “old” and “not-old.”]
[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 decidewhichof the Things belong to the one Class andwhichto the other. In such a case, it would be necessary to lay down some arbitraryrule, as towherethe one Class should end and the other begin.
[Thus, in dividing “books” into “old” and “not-old,” we may say “Let all books printed beforea.d.1801, be regarded as ‘old,’ and all others as ‘not-old’.”]
[Thus, in dividing “books” into “old” and “not-old,” we may say “Let all books printed beforea.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.”]
[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.”]
pg004After dividing a Class, by the Process ofDichotomy, 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 gettingfourClasses, 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.]
[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 gettingfourClasses, 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.]
The word “Thing”, which conveys the idea of a Thing,withoutany idea of an Adjunct, representsanysingle Thing. Any other word (or phrase), which conveys the idea of a Thing,withthe idea of an Adjunct representsanyThing which possesses that Adjunct; i.e., it represents any Member of the Class to which that Adjunct ispeculiar.
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.”]
[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 beReal, orUnreal, according as thereis, oris not, an existing Thing in it, so also a Name is said to beReal, orUnreal, according as thereis, oris not, an existing Thing represented by it.
[Thus, “Town lit with gas” is aRealName: “Town paved with gold” is anUnrealName.]
[Thus, “Town lit with gas” is aRealName: “Town paved with gold” is anUnrealName.]
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;
pg005(b)A Substantive, conveying the idea of a Thing with the ideas ofsomeof the Attributes, and one or more Adjectives (or phrases used as Adjectives) conveying the ideas of theotherAttributes;
(c)A Substantive conveying the idea of a Thing with the ideas ofallthe 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).]
[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 thepluralnumber, 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 thefirstsense; and it is just the same as if I were to point to each of themseparately, and to say “Thissoldier of the Tenth Regiment is tall,” “Thatsoldier 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 thesecondsense; and it is just the same as if I were to say “TheTenth Regimentis formed in square.”]
[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 thefirstsense; and it is just the same as if I were to point to each of themseparately, and to say “Thissoldier of the Tenth Regiment is tall,” “Thatsoldier 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 thesecondsense; and it is just the same as if I were to say “TheTenth Regimentis formed in square.”]
It is evident that every Member of aSpeciesisalsoa Member of theGenusout of which that Species has been picked, and that it possesses theDifferentiaof that Species. Hence it may be represented by a Name consisting of two parts, one being a Name representing any Member of theGenus, and the other being theDifferentiaof 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 theGenus, and “valuable” as theDifferentia.]
[Thus, we may define a “Treasure” as a “valuable Thing.” In this case we regard “Things” as theGenus, and “valuable” as theDifferentia.]
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 theGenus, is printed in Capitals.]
[Note that, in each Definition, the Substantive, representing a Member (or Members) of theGenus, is printed in Capitals.]
1. Define “a Treasure.”
Ans.“a valuableThing.”
2. Define “Treasures.”
Ans.“valuableThings.”
3. Define “a Town.”
Ans.“a material artificialThing, consisting of houses and streets.”
pg0074. Define “Men.”
Ans.“material, livingThings, belonging to the Animal Kingdom, having two hands and two feet”;
or else
“Animalshaving two hands and two feet.”
5. Define “London.”
Ans.“the material artificialThing, which consists of houses and streets, and has four million inhabitants”;
or else
“theTownwhich has four million inhabitants.”
[Note that we here use the article “the” instead of “a”, because we happen to know that there is onlyonesuch 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.]
[Note that we here use the article “the” instead of “a”, because we happen to know that there is onlyonesuch 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.]
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 toanyword, 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 anyinformation; 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.”]
[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 anyinformation; 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 ‘Normalpg009form’; 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)thatsomeMembers of its Subject are Members of its Predicate;
or (2)thatnoMembers of its Subject are Members of its Predicate;
or (3)thatallMembers of its Subject are Members of its Predicate.
The Subject and the Predicate of a Proposition are called its ‘Terms.’
Two Propositions, which convey thesameinformation, are said to be ‘equivalent’.
[Thus, the two Propositions, “I see John” and “John is seen by me,” are equivalent.]
[Thus, the two Propositions, “I see John” and “John is seen by me,” are equivalent.]
A Proposition, in normal form, consists of four parts, viz.—
(1)The word “some,” or “no,” or “all.” (This word, which tells ushow manyMembers 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.
A Proposition, that begins with “Some”, is said to be ‘Particular.’ It is also called ‘a Propositionin I.’
[Note, that it is called ‘Particular,’ because it refers to apartonly of the Subject.]
[Note, that it is called ‘Particular,’ because it refers to apartonly of the Subject.]
A Proposition, that begins with “No”, is said to be ‘Universal Negative.’ It is also called ‘a Propositionin E.’
A Proposition, that begins with “All”, is said to be ‘Universal Affirmative.’ It is also called ‘a Propositionin A.’
[Note, that they are called ‘Universal’, because they refer to thewholeof the Subject.]
[Note, that they are called ‘Universal’, because they refer to thewholeof the Subject.]
A Proposition, whose Subject is anIndividual, is to be regarded asUniversal.
[Let us take, as an example, the Proposition “John is not well”. This of course implies that there is anIndividual, to whom the speaker refers when he mentions “John”, and whom the listenerknowsto 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 “Allthe men, who are referred to by the speaker when he mentions ‘John’, are not well.”]
[Let us take, as an example, the Proposition “John is not well”. This of course implies that there is anIndividual, to whom the speaker refers when he mentions “John”, and whom the listenerknowsto 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 “Allthe 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.
A ‘Proposition of Existence’, when in normal form, has, for itsSubject, the Class “existing Things”.
Its Sign of Quantity is “Some” or “No”.
[Note that, though its Sign of Quantity tells ushow manyexisting Things are Members of its Predicate, it doesnottell us theexactnumber: in fact, it only deals withtwonumbers, which are, in ascending order, “0” and “1 or more.”]
[Note that, though its Sign of Quantity tells ushow manyexisting Things are Members of its Predicate, it doesnottell us theexactnumber: in fact, it only deals withtwonumbers, which are, in ascending order, “0” and “1 or more.”]
It is called “a Proposition of Existence” because its effect is to assert theReality(i.e. the realexistence), or else theImaginariness, of its Predicate.
[Thus, the Proposition “Some existing Things are honest men” asserts that the Class “honest men” isReal.This is thenormalform; 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” isImaginary.This is thenormalform; 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.”]
[Thus, the Proposition “Some existing Things are honest men” asserts that the Class “honest men” isReal.
This is thenormalform; 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” isImaginary.
This is thenormalform; 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.”]
AProposition 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 Attributenotconveyed 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 isnotconveyed by the other Name.But the Proposition “Some dogs are setters” isnotof the right kind, since, although it is true that “dogs” and “setters” are Specieses of the same Genus “animals”, it isnottrue that the Name “dogs” conveys the idea of any Attribute not conveyed by the Name “setters”. Such Propositions will be discussed in Part II.]
[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 isnotconveyed by the other Name.
But the Proposition “Some dogs are setters” isnotof the right kind, since, although it is true that “dogs” and “setters” are Specieses of the same Genus “animals”, it isnottrue 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 ushow manyMembers of its Subject arealsoMembers of its Predicate, it does not tell us theexactnumber: in fact, it only deals withthreenumbers, which are, in ascending order, “0”, “1 or more”, “the total number of Members of the Subject”.]
[Note that, though its Sign of Quantity tells ushow manyMembers of its Subject arealsoMembers of its Predicate, it does not tell us theexactnumber: in fact, it only deals withthreenumbers, 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 certainrelationshipexists between its Terms.
The Rules, for doing this, are as follows:—
(1) Ascertain what is theSubject(i.e., ascertain what Class we aretalking about);
(2) If the verb, governed by the Subject, isnotthe verb “are” (or “is”), substitute for it a phrase beginning with “are” (or “is”);
(3) Ascertain what is thePredicate(i.e., ascertain what Class it is, which is asserted to containsome, ornone, orall, of the Members of the Subject);
(4) If the Name of each Term iscompletely expressed(i.e. if it contains a Substantive), there is no need to determine the ‘Univ.’; but, if either Name isincompletely expressed, and containsAttributesonly, it is then necessary to determine a ‘Univ.’, in order to insert its Name as the Substantive.
(5) Ascertain theSign 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.”pg014(2)“None of my speculations have brought me as much as 5 per cent.”(1) The Subject is “my speculations.”(2) The Verb is “have brought,” for which we substitute the phrase “are * * * that have brought”.(3) The Predicate is “* * * that have brought &c.”(4) Let Univ. be “transactions.”(5) The Sign of Quantity is “none of.”(6) The Proposition now becomes“None of | my speculations | are | transactions that have brought me as much as 5 per cent.”(3)“None but the brave deserve the fair.”To begin with, we note that the phrase “none but the brave” is equivalent to “nonot-brave.”(1) The Subject has for itsAttribute“not-brave.” But noSubstantiveis supplied. So we express the Subject as “not-brave * * *.”(2) The Verb is “deserve,” for which we substitute the phrase “are deserving of”.(3) The Predicate is “* * * deserving of the fair.”(4) Let Univ. be “persons.”(5) The Sign of Quantity is “no.”(6) The Proposition now becomes“No | not-brave persons | are | persons deserving of the fair.”(4)“A lame puppy would not say “thank you” if you offered to lend it a skipping-rope.”(1) The Subject is evidently “lame puppies,” and all the rest of the sentence must somehow be packed into the Predicate.(2) The Verb is “would not say,” &c., for which we may substitute the phrase “are not grateful for.”(3) The Predicate may be expressed as “* * * not grateful for the loan of a skipping-rope.”(4) Let Univ. be “puppies.”(5) The Sign of Quantity is “all.”(6) The Proposition now becomes“All | lame puppies | are | puppies not grateful for the loan of a skipping-rope.”pg015(5)“No one takes in theTimes, unless he is well-educated.”(1) The Subject is evidently persons who are not well-educated (“noone” evidently means “noperson”).(2) The Verb is “takes in,” for which we may substitute the phrase “are persons taking in.”(3) The Predicate is “persons taking in theTimes.”(4) Let Univ. be “persons.”(5) The Sign of Quantity is “no.”(6) The Proposition now becomes“No | persons who are not well-educated | are | persons taking in theTimes.”(6)“My carriage will meet you at the station.”(1) The Subject is “my carriage.” This, being an ‘Individual,’ is equivalent to the Class “my carriages.” (Note that this Class contains onlyoneMember.)(2) The Verb is “will meet”, for which we may substitute the phrase “are * * * that will meet.”(3) The Predicate is “* * * that will meet you at the station.”(4) Let Univ. be “things.”(5) The Sign of Quantity is “all.”(6) The Proposition now becomes“All | my carriages | are | things that will meet you at the station.”(7)“Happy is the man who does not know what ‘toothache’ means!”(1) The Subject is evidently “the man &c.” (Note that in this sentence, thePredicatecomes first.) At first sight, the Subject seems to be an ‘Individual’; but on further consideration, we see that the article “the” doesnotimply that there is onlyonesuch man. Hence the phrase “the man who” is equivalent to “all men who”.(2) The Verb is “are.”(3) The Predicate is “happy * * *.”(4) Let Univ. be “men.”(5) The Sign of Quantity is “all.”(6) The Proposition now becomes“All | men who do not know what ‘toothache’ means | are | happy men.”pg016(8)“Some farmers always grumble at the weather, whatever it may be.”(1) The Subject is “farmers.”(2) The Verb is “grumble,” for which we substitute the phrase “are * * * who grumble.”(3) The Predicate is “* * * who always grumble &c.”(4) Let Univ. be “persons.”(5) The Sign of Quantity is “some.”(6) The Proposition now becomes“Some | farmers | are | persons who always grumble at the weather, whatever it may be.”(9)“No lambs are accustomed to smoke cigars.”(1) The Subject is “lambs.”(2) The Verb is “are.”(3) The Predicate is “* * * accustomed &c.”(4) Let Univ. be “animals.”(5) The Sign of Quantity is “no.”(6) The Proposition now becomes“No | lambs | are | animals accustomed to smoke cigars.”(10)“I ca’n’t understand examples that are not arranged in regular order, like those I am used to.”(1) The Subject is “examples that,” &c.(2) The Verb is “I ca’n’t understand,” which we must alter, so as to have “examples,” instead of “I,” as the nominative case. It may be expressed as “are not understood by me.”(3) The Predicate is “* * * not understood by me.”(4) Let Univ. be “examples.”(5) The Sign of Quantity is “all.”(6) The Proposition now becomes“All | examples that are not arranged in regular order like those I am used to | are | examples not understood by me.”]
[Let us work a few Examples, to illustrate these Rules.
“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.”
“None of my speculations have brought me as much as 5 per cent.”
(1) The Subject is “my speculations.”
(2) The Verb is “have brought,” for which we substitute the phrase “are * * * that have brought”.
(3) The Predicate is “* * * that have brought &c.”
(4) Let Univ. be “transactions.”
(5) The Sign of Quantity is “none of.”
(6) The Proposition now becomes
“None of | my speculations | are | transactions that have brought me as much as 5 per cent.”
“None but the brave deserve the fair.”
To begin with, we note that the phrase “none but the brave” is equivalent to “nonot-brave.”
(1) The Subject has for itsAttribute“not-brave.” But noSubstantiveis supplied. So we express the Subject as “not-brave * * *.”
(2) The Verb is “deserve,” for which we substitute the phrase “are deserving of”.
(3) The Predicate is “* * * deserving of the fair.”
(4) Let Univ. be “persons.”
(5) The Sign of Quantity is “no.”
(6) The Proposition now becomes
“No | not-brave persons | are | persons deserving of the fair.”
“A lame puppy would not say “thank you” if you offered to lend it a skipping-rope.”
(1) The Subject is evidently “lame puppies,” and all the rest of the sentence must somehow be packed into the Predicate.
(2) The Verb is “would not say,” &c., for which we may substitute the phrase “are not grateful for.”
(3) The Predicate may be expressed as “* * * not grateful for the loan of a skipping-rope.”
(4) Let Univ. be “puppies.”
(5) The Sign of Quantity is “all.”
(6) The Proposition now becomes
“All | lame puppies | are | puppies not grateful for the loan of a skipping-rope.”
“No one takes in theTimes, unless he is well-educated.”
(1) The Subject is evidently persons who are not well-educated (“noone” evidently means “noperson”).
(2) The Verb is “takes in,” for which we may substitute the phrase “are persons taking in.”
(3) The Predicate is “persons taking in theTimes.”
(4) Let Univ. be “persons.”
(5) The Sign of Quantity is “no.”
(6) The Proposition now becomes
“No | persons who are not well-educated | are | persons taking in theTimes.”
“My carriage will meet you at the station.”
(1) The Subject is “my carriage.” This, being an ‘Individual,’ is equivalent to the Class “my carriages.” (Note that this Class contains onlyoneMember.)
(2) The Verb is “will meet”, for which we may substitute the phrase “are * * * that will meet.”
(3) The Predicate is “* * * that will meet you at the station.”
(4) Let Univ. be “things.”
(5) The Sign of Quantity is “all.”
(6) The Proposition now becomes
“All | my carriages | are | things that will meet you at the station.”
“Happy is the man who does not know what ‘toothache’ means!”
(1) The Subject is evidently “the man &c.” (Note that in this sentence, thePredicatecomes first.) At first sight, the Subject seems to be an ‘Individual’; but on further consideration, we see that the article “the” doesnotimply that there is onlyonesuch man. Hence the phrase “the man who” is equivalent to “all men who”.
(2) The Verb is “are.”
(3) The Predicate is “happy * * *.”
(4) Let Univ. be “men.”
(5) The Sign of Quantity is “all.”
(6) The Proposition now becomes
“All | men who do not know what ‘toothache’ means | are | happy men.”
“Some farmers always grumble at the weather, whatever it may be.”
(1) The Subject is “farmers.”
(2) The Verb is “grumble,” for which we substitute the phrase “are * * * who grumble.”
(3) The Predicate is “* * * who always grumble &c.”
(4) Let Univ. be “persons.”
(5) The Sign of Quantity is “some.”
(6) The Proposition now becomes
“Some | farmers | are | persons who always grumble at the weather, whatever it may be.”
“No lambs are accustomed to smoke cigars.”
(1) The Subject is “lambs.”
(2) The Verb is “are.”
(3) The Predicate is “* * * accustomed &c.”
(4) Let Univ. be “animals.”
(5) The Sign of Quantity is “no.”
(6) The Proposition now becomes
“No | lambs | are | animals accustomed to smoke cigars.”
“I ca’n’t understand examples that are not arranged in regular order, like those I am used to.”
(1) The Subject is “examples that,” &c.
(2) The Verb is “I ca’n’t understand,” which we must alter, so as to have “examples,” instead of “I,” as the nominative case. It may be expressed as “are not understood by me.”
(3) The Predicate is “* * * not understood by me.”
(4) Let Univ. be “examples.”
(5) The Sign of Quantity is “all.”
(6) The Proposition now becomes
“All | examples that are not arranged in regular order like those I am used to | are | examples not understood by me.”]
A Proposition of Relation, beginning with “All”, asserts (as we already know) that “AllMembers of the Subject are Members of the Predicate”. This evidently contains, as apartof what it tells us, the smaller Proposition “SomeMembers of the Subject are Members of the Predicate”.
[Thus, the Proposition “Allbankers are rich men” evidently contains the smaller Proposition “Somebankers are rich men”.]
[Thus, the Proposition “Allbankers are rich men” evidently contains the smaller Proposition “Somebankers are rich men”.]
The question now arises “What is therestof the information which this Proposition gives us?”
In order to answer this question, let us begin with the smaller Proposition, “SomeMembers of the Subject are Members of the Predicate,” and suppose that this isallwe have been told; and let us proceed to inquire whatelsewe need to be told, in order to know that “AllMembers of the Subject are Members of the Predicate”.
[Thus, we may suppose that the Proposition “Somebankers are rich men” is all the information we possess; and we may proceed to inquire whatotherProposition needs to be added to it, in order to make up the entire Proposition “Allbankers are rich men”.]
[Thus, we may suppose that the Proposition “Somebankers are rich men” is all the information we possess; and we may proceed to inquire whatotherProposition needs to be added to it, in order to make up the entire Proposition “Allbankers 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 ofDichotomy) into two smaller Classes, viz.
(1)the Predicate;
(2)the Class whose Differentia iscontradictoryto 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”.]
[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”.]
pg018Now we know thateveryMember of the Subject is (as shown atp. 6) a Member of the Univ. HenceeveryMember of the Subject is either in Class (1) or else in Class (2).
[Thus, we know thateverybanker is a Member of the Genus “men”. Hence,everybanker is either in the Class “rich men”, or else in the Class “poor men”.]
[Thus, we know thateverybanker is a Member of the Genus “men”. Hence,everybanker is either in the Class “rich men”, or else in the Class “poor men”.]
Also we have been told that, in the case we are discussing,someMembers of the Subject are in Class (1). Whatelsedo we need to be told, in order to know thatallof them are there? Evidently we need to be told thatnoneof them are in Class (2); i.e. thatnoneof them are Members of the Class whose Differentia iscontradictoryto that of the Predicate.
[Thus, we may suppose we have been told thatsomebankers are in the Class “rich men”. Whatelsedo we need to be told, in order to know thatallof them are there? Evidently we need to be told thatnoneof them are in the Class “poormen”.]
[Thus, we may suppose we have been told thatsomebankers are in the Class “rich men”. Whatelsedo we need to be told, in order to know thatallof them are there? Evidently we need to be told thatnoneof them are in the Class “poormen”.]
Hence a Proposition of Relation, beginning with “All”, is aDoubleProposition, and is ‘equivalent’ to (i.e. gives the same information as) thetwoPropositions
(1)“SomeMembers of the Subject are Members of the Predicate”;
(2)“NoMembers of the Subject are Members of the Class whose Differentia iscontradictoryto that of the Predicate”.
[Thus, the Proposition “Allbankers are rich men” is aDoubleProposition, and is equivalent to thetwoPropositions(1)“Somebankers are rich men”;(2)“Nobankers arepoormen”.]
[Thus, the Proposition “Allbankers are rich men” is aDoubleProposition, and is equivalent to thetwoPropositions
(1)“Somebankers are rich men”;
(2)“Nobankers arepoormen”.]
Note that the rules, here laid down, arearbitrary, and only apply to Part I of my “Symbolic Logic.”
A Proposition of Relation, beginning with “Some”, is henceforward to be understood as asserting that there aresome existing Things, which, being Members of the Subject, are also Members of the Predicate; i.e. thatsome existing Thingsare Members ofbothTerms at once. Hence it is to be understood as implying thateachTerm, taken by itself, isReal.
[Thus, the Proposition “Some rich men are invalids” is to be understood as asserting thatsome existing Thingsare “rich invalids”. Hence it implies thateachof the two Classes, “rich men” and “invalids”, taken by itself, isReal.]
[Thus, the Proposition “Some rich men are invalids” is to be understood as asserting thatsome existing Thingsare “rich invalids”. Hence it implies thateachof the two Classes, “rich men” and “invalids”, taken by itself, isReal.]
A Proposition of Relation, beginning with “No”, is henceforward to be understood as asserting that there areno existing Thingswhich, being Members of the Subject, are also Members of the Predicate; i.e. thatno existing Thingsare Members ofbothTerms at once. But this implies nothing as to theRealityof either Term taken by itself.
[Thus, the Proposition “No mermaids are milliners” is to be understood as asserting thatno existing Thingsare “mermaid-milliners”. But this implies nothing as to theReality, or theUnreality, of either of the two Classes, “mermaids” and “milliners”, taken by itself. In this case as it happens, the Subject isImaginary, and the PredicateReal.]
[Thus, the Proposition “No mermaids are milliners” is to be understood as asserting thatno existing Thingsare “mermaid-milliners”. But this implies nothing as to theReality, or theUnreality, of either of the two Classes, “mermaids” and “milliners”, taken by itself. In this case as it happens, the Subject isImaginary, and the PredicateReal.]
A Proposition of Relation, beginning with “All”, contains (see§ 3) a similar Proposition beginning with “Some”. Hence it is to be understood as implying thateachTerm, taken by itself, isReal.
[Thus, the Proposition “All hyænas are savage animals” contains the Proposition “Some hyænas are savage animals”. Hence it implies thateachof the two Classes, “hyænas” and “savage animals”, taken by itself, isReal.]
[Thus, the Proposition “All hyænas are savage animals” contains the Proposition “Some hyænas are savage animals”. Hence it implies thateachof the two Classes, “hyænas” and “savage animals”, taken by itself, isReal.]
We have seen that a Proposition of Relation, beginning with “Some,” asserts thatsome existing Things, being Members of its Subject, arealsoMembers of its Predicate. Hence, it asserts that some existing Things are Members ofboth; i.e. it asserts that some existing Things are Members of the Class of Things which haveallthe Attributes of the Subject and the Predicate.
Hence, to translate it into a Proposition of Existence, we take “existing Things” as the newSubject, and Things, which haveallthe Attributes of the Subject and the Predicate, as the new Predicate.
Similarly for a Proposition of Relation beginning with “No”.
A Proposition of Relation, beginning with “All”, is (as shown in§ 3) equivalent totwoPropositions, one beginning with “Some” and the other with “No”, each of which we now know how to translate.
[Let us work a few Examples, to illustrate these Rules.(1)“Some apples are not ripe.”Here we arrange thus:—“Some”Sign of Quantity.“existing Things”Subject.“are”Copula.“not-ripe apples”Predicate.or thus:—“Some | existing Things | are | not-ripe apples.”pg021(2)“Some farmers always grumble at the weather, whatever it may be.”Here we arrange thus:—“Some | existing Things | are | farmers who always grumble at the weather, whatever it may be.”(3)“No lambs are accustomed to smoke cigars.”Here we arrange thus:—“No | existing Things |are | lambs accustomed to smoke cigars.”(4)“None of my speculations have brought me as much as 5 per cent.”Here we arrange thus:—“No | existing Things | are | speculations of mine, which have brought me as much as 5 per cent.”(5)“None but the brave deserve the fair.”Here we note, to begin with, that the phrase “none but the brave” is equivalent to “no not-brave men.” We then arrange thus:—“No | existing Things | are | not-brave men deserving of the fair.”(6)“All bankers are rich men.”This is equivalent to the two Propositions “Some bankers are rich men” and “No bankers are poor men.”Here we arrange thus:—“Some | existing Things | are | rich bankers”; and “No | existing Things | are | poor bankers.”]
[Let us work a few Examples, to illustrate these Rules.
“Some apples are not ripe.”
Here we arrange thus:—
“Some”Sign of Quantity.“existing Things”Subject.“are”Copula.“not-ripe apples”Predicate.
or thus:—
“Some | existing Things | are | not-ripe apples.”
“Some farmers always grumble at the weather, whatever it may be.”
Here we arrange thus:—
“Some | existing Things | are | farmers who always grumble at the weather, whatever it may be.”
“No lambs are accustomed to smoke cigars.”
Here we arrange thus:—
“No | existing Things |are | lambs accustomed to smoke cigars.”
“None of my speculations have brought me as much as 5 per cent.”
Here we arrange thus:—
“No | existing Things | are | speculations of mine, which have brought me as much as 5 per cent.”
“None but the brave deserve the fair.”
Here we note, to begin with, that the phrase “none but the brave” is equivalent to “no not-brave men.” We then arrange thus:—
“No | existing Things | are | not-brave men deserving of the fair.”
“All bankers are rich men.”
This is equivalent to the two Propositions “Some bankers are rich men” and “No bankers are poor men.”
Here we arrange thus:—
“Some | existing Things | are | rich bankers”; and “No | existing Things | are | poor bankers.”]
[Work Examples §1, 1–4 (p. 97).]
[Work Examples §1, 1–4 (p. 97).]
An annotated biliteral diagram
First, let us suppose that the above Diagram is an enclosure assigned to a certain Class of Things, which we have selected as our ‘Universe of Discourse.’ or, more briefly, as our ‘Univ’.
[For example, we might say “Let Univ. be ‘books’”; and we might imagine the Diagram to be a large table, assigned to all “books.”]
[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,notto 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 particularpartof it, about which he is reading.]
[The Reader is strongly advised, in reading this Chapter,notto 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 particularpartof it, about which he is reading.]
pg023Secondly, let us suppose that we have selected a certain Adjunct, which we may call “x,” and have divided the large Class, to which we have assigned the whole Diagram, into the two smaller Classes whose Differentiæ are “x” and “not-x” (which we may call “x′”),and that we have assigned theNorthHalf of the Diagram to the one (which we may call “the Class ofx-Things,” or “thex-Class”), and theSouthHalf to the other (which we may call “the Class ofx′-Things,” or “thex′-Class”).
[For example, we might say “Letxmean ‘old,’ so thatx′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 theNorthHalf of the table to “oldbooks” and theSouthHalf to “newbooks.”]
[For example, we might say “Letxmean ‘old,’ so thatx′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 theNorthHalf of the table to “oldbooks” and theSouthHalf to “newbooks.”]
Thirdly, let us suppose that we have selected another Adjunct, which we may call “y”, and have subdivided thex-Class into the two Classes whose Differentiæ are “y” and “y′”,and that we have assigned the North-WestCell to the one (which we may call “thexy-Class”), and the North-EastCell to the other (which we may call “thexy′-Class”).
[For example, we might say “Letymean ‘English,’ so thaty′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-WestCell to “oldEnglishbooks”, and the North-EastCell to “oldforeignbooks.”]
[For example, we might say “Letymean ‘English,’ so thaty′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-WestCell to “oldEnglishbooks”, and the North-EastCell to “oldforeignbooks.”]
Fourthly, let us suppose that we have subdivided thex′-Class in the same manner,and have assigned the South-WestCell to thex′y-Class, and the South-EastCell to thex′y′-Class.
[For example, we might suppose that we had subdivided “new books” into the two Classes “newEnglishbooks” and “newforeignbooks”, and had assigned the South-WestCell to the one, and the South-EastCell to the other.]
[For example, we might suppose that we had subdivided “new books” into the two Classes “newEnglishbooks” and “newforeignbooks”, and had assigned the South-WestCell to the one, and the South-EastCell to the other.]
It is evident that, if we had begun by dividing foryandy′, and had then subdivided forxandx′, we should have got thepg024samefour Classes. Hence we see that we have assigned theWestHalf to they-Class, and theEastHalf to they′-Class.
Diagram representing books[Thus, in the above Example, we should find that we had assigned theWestHalf of the table to “Englishbooks” and theEastHalf to “foreignbooks.”We have, in fact, assigned the four Quarters of the table to four different Classes of books, as here shown.]
Diagram representing books
[Thus, in the above Example, we should find that we had assigned theWestHalf of the table to “Englishbooks” and theEastHalf to “foreignbooks.”
We have, in fact, assigned the four Quarters of the table to four different Classes of books, as here shown.]
The Reader should carefully remember that, in such a phrase as “the x-Things,” the word “Things” means that particularkindof Things, to which the whole Diagram has been assigned.
[Thus, if we say “Let Univ. be ‘books’,” we mean that we have assigned the whole Diagram to “books.” In that case, if we took “x” to mean “old”, the phrase “thex-Things” would mean “the old books.”]
[Thus, if we say “Let Univ. be ‘books’,” we mean that we have assigned the whole Diagram to “books.” In that case, if we took “x” to mean “old”, the phrase “thex-Things” would mean “the old books.”]
The Reader should not go on to the next Chapter until he isquite familiarwith theblankDiagram I have advised him to draw.
He ought to be able to name,instantly, theAdjunctassigned to any Compartment named in the right-hand column of the following Table.
Also he ought to be able to name,instantly, theCompartmentassigned to any Adjunct named in the left-hand column.
To make sure of this, he had better put the book into the hands of some genial friend, while he himself has nothing but the blank Diagram, and get that genial friend to question him on this Table,dodgingabout as much as possible. The Questions and Answers should be something like this:—