Chapter 33
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.
pg013 § 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."
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 "no _not_-brave."
(1) The Subject has for its _Attribute_ "not-brave." But no _Substantive_ is 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 the _Times_, unless he is well-educated."
(1) The Subject is evidently persons who are not well-educated ("no _one_" evidently means "no _person_").
(2) The Verb is "takes in," for which we may substitute the phrase "are persons taking in."
(3) The Predicate is "persons taking in the _Times_."
(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 the _Times_."
(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 only _one_ Member.)
(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, the _Predicate_ comes first.) At first sight, the Subject seems to be an '_Individual_'; but on further consideration, we see that the article "the" does _not_ imply that there is only _one_ such 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."]
pg017 § 3.
_A Proposition of Relation, beginning with "All", is a Double Proposition._
A Proposition of Relation, beginning with "All", asserts (as we already know) that "_All_ Members of the Subject are Members of the Predicate". This evidently contains, as a _part_ of what it tells us, the smaller Proposition "_Some_ Members of the Subject are Members of the 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".] pg018 Now we know that _every_ Member of the Subject is (as shown at p. 6) a Member of the Univ. Hence _every_ Member of the Subject is either in Class (1) or else in Class (2).
[Thus, we know that _every_ banker is a Member of the Genus "men". Hence, _every_ banker 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, _some_ Members of the Subject are in Class (1). What _else_ do we need to be told, in order to know that _all_ of them are there? Evidently we need to be told that _none_ of them are in Class (2); i.e. that _none_ of them are Members of the Class whose Differentia is _contradictory_ to that of the Predicate.
[Thus, we may suppose we have been told that _some_ bankers are in the Class "rich men". What _else_ do we need to be told, in order to know that _all_ of them are there? Evidently we need to be told that _none_ of them are in the Class "_poor_ men".]
Hence a Proposition of Relation, beginning with "All", is a _Double_ Proposition, and is '=equivalent=' to (i.e. gives the same information as) the _two_ Propositions
(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".]
pg019 § 4.
_What is implied, in a Proposition of Relation, as to the Reality of its Terms?_
Note that the rules, here laid down, are _arbitrary_, and only apply to