THE ART OF LOGICAL THINKING/PART 9
CHAPTER IX.
PROPOSITIONS
We
have seen that the first step of Deductive Reasoning is that which we call
Concepts. The second step is that which we call Propositions.
In
Logic, a Proposition is: "A sentence, or part of a
sentence, affirming or denying a connection between the terms; limited to
express assertions rather than extended to questions and commands." Hyslop
defines a Proposition as: "any affirmation or denial of an agreement
between two conceptions."
Examples
of Propositions are found in the following sentences: "The rose is a
flower;" "a horse is an animal;" "Chicago is a city;"
all of which are affirmations of agreement between the two terms involved; also
in: "A horse is not a zebra;" "pinks are not roses;"
"the whale is not a fish;" etc., which are denials of agreement
between the terms.
The Parts
of a Proposition are: (1) the Subject, or that of which
something is affirmed or denied; (2) the Predicate, or the
something which is affirmed or denied regarding the Subject;
and (3) the Copula, or the verb serving as a link between the
Subject and the Predicate.
In
the Proposition: "Man is an animal," the term man is
the Subject; the term an animal is the Predicate; and the word is,
is the Copula. The Copula is always some form of the verb to be, in
the present tense indicative, in an affirmative Proposition; and the same with
the negative particle affixed, in a negative Proposition. The Copula is not
always directly expressed by the word is or is not,
etc., but is instead expressed in some phrase which implies them. For instance,
we say "he runs," which implies "he is running." In the
same way, it may appear at times as if the Predicate was missing, as in:
"God is," by which is meant "God is existing." In some
cases, the Proposition is inverted, the Predicate appearing first in order, and
the Subject last, as in: "Blessed are the peacemakers;" or
"Strong is Truth." In such cases judgment must be used in determining
the matter, in accordance with the character and meaning of the terms.
An Affirmative
Proposition is one in which the Predicate is affirmed to
agree with the Subject. A Negative Proposition is one in which
the agreement of the Predicate and Subject is denied. Examples of
both of these classes have been given in this chapter.
Another
classification of Propositions divides them in three classes, as follows (1)
Categorical; (2) Hypothetical; (3) Disjunctive.
A Categorical
Proposition is one in which the affirmation or denial is made without
reservation or qualification, as for instance: "Man is an animal;"
"the rose is a flower," etc. The fact asserted may not be true,
but the statement is made positively as a statement of reality.
A Hypothetical
Proposition is one in which the affirmation or denial is made to
depend upon certain conditions, circumstances or suppositions, as for instance:
"If the water is boiling-hot, it will scald;" or "if the powder
be damp, it will not explode," etc. Jevons says: "Hypothetical Propositions
may generally be recognized by containing the little word 'if;' but it is
doubtful whether they really differ much from the ordinary propositions.... We
may easily say that 'boiling water will scald,' and 'damp gunpowder will not
explode,' thus avoiding the use of the word 'if.'"
A Disjunctive
Proposition is one "implying or asserting an alternative,"
and usually containing the conjunction "or," sometimes together with
"either," as for instance: "Lightning is sheet or forked;"
"Arches are either round or pointed;" "Angles are either obtuse,
right angled or acute."
Another
classification of Propositions divides them in two classes as follows: (1)
Universal; (2) Particular.
A Universal
Proposition is one in which the whole quantity of the
Subject is involved in the assertion or denial of the Predicate. For instance:
"All men are liars," by which is affirmed that all of
the entire race of men are in the category of liars, not some men
but all the men that are in existence. In the same way the
Proposition: "No men are immortal" is Universal, for it is a universal
denial.
A Particular
Proposition is one in which the affirmation or denial of the Predicate
involves only a part or portion of the whole of the Subject,
as for instance: "Some men are atheists," or "Some women
are not vain," in which cases the affirmation or denial does not
involve all or the whole of the Subject.
Other examples are: "A few men," etc.; "many people,"
etc.; "certain books," etc.; "most people,"
etc.
Hyslop
says: "The signs of the Universal Proposition, when formally expressed,
are all, every, each, any, and
whole or words with equivalent import." The signs of Particular
Propositions are also certain adjectives of quantity, such as some, certain, a
few, many, most or such others as denote at
least a part of a class.
The
subject of the Distribution of Terms in Propositions is considered very
important by Logicians, and as Hyslop says: "has much importance in
determining the legitimacy, or at least the intelligibility, of our reasoning
and the assurance that it will be accepted by others." Some authorities
favor the term, "Qualification of the Terms of
Propositions," but the established usage favors the term
"Distribution."
The
definition of the Logical term, "Distribution," is: "The
distinguishing of a universal whole into its several kinds of species; the
employment of a term to its fullest extent; the application of a term to its
fullest extent, so as to include all significations or applications." A
Term of a Proposition is distributed when it is employed in
its fullest sense; that is to say, when it is employed so as to apply
to each and every object, person or thing included under it. Thus in the
proposition, "All horses are animals," the term horses is
distributed; and in the proposition, "Some horses are thoroughbreds,"
the term horses is not distributed. Both of these examples
relate to the distribution of the subject of the proposition.
But the predicate of a proposition also may or may not be distributed. For
instance, in the proposition, "All horses are animals," the
predicate, animals, is not distributed, that is, not used
in its fullest sense, for all animals are not horses—there
are some animals which are not horses and, therefore, the
predicate, animals, not being used in its fullest sense is
said to be "not distributed." The proposition really means:
"All horses are some animals."
There
is however another point to be remembered in the consideration of Distribution
of Terms of Propositions, which Brooks expresses as follows: "Distribution
generally shows itself in the form of the expression, but sometimes it may be
determined by the thought. Thus if we say, 'Men are mortal,' we mean all
men, and the term men is distributed. But if we say 'Books are necessary to
a library,' we mean, not 'all books' but 'some books.' The test of
distribution is whether the term applies to 'each and every.'
Thus when we say 'men are mortal,' it is true of each and every man that he is
mortal."
The
Rules of Distribution of the Terms of Proposition are as follows:
1.
All universals distribute the subject.
2.
All particulars do not distribute the subject.
3.
All negatives distribute the predicate.
4.
All affirmatives do not distribute the predicate.
The
above rules are based upon logical reasoning. The reason for the first two
rules is quite obvious, for when the subject is universal, it
follows that the whole subject is involved; when the subject
is particular it follows that only a part of
the subject is involved. In the case of the third rule, it will be seen that in
every negative proposition the whole of the predicate must
be denied the subject, as for instance, when we say: "Some animals are not
horses," the whole class of horses is cut off from
the subject, and is thus distributed. In the case of the fourth
rule, we may readily see that in the affirmative proposition the whole of the
predicate is not denied the subject, as for instance, when we
say that: "Horses are animals," we do not mean that horses are all
the animals, but that they are merely a part or portion of
the class animal—therefore, the predicate, animals, is not
distributed.
In
addition to the forms of Propositions given there is another class of
Propositions known as Definitive or Substitutive Propositions, in
which the Subject and the Predicate are exactly alike in extent and rank. For
instance, in the proposition, "A triangle is a polygon
of three sides" the two terms are interchangeable; that is, may be
substituted for each other. Hence the term "substitutive." The term
"definitive" arises from the fact that the respective terms of this
kind of a proposition necessarily define each other. All
logical definitions are expressed in this last mentioned form of proposition,
for in such cases the subject and the predicate are precisely equal to each
other.
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