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# An outline of logic online

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Usually, however, a set of rules are offered to which
the syllogism must conform. These rules merely for-
mulate what the diagrams or circles may be made to
demonstrate. They are as follows:

(1) In every syllogism there should be three and
not more than three terms, and these terms must be
used throughout in the same sense.

THE CATEGORICAL SYLLOGISM 71

(2) The middle term must be distributed at least
once in the premises.

(3) No term must be distributed in the conclusion
that was not distributed in one of the premises.

(4) No conclusion may be drawn from two negative
premises.

(5) If one premise is negative, the conclusion must
be negative.

(6) No conclusion may be drawn from two par-
ticular premises.

(7) If one of the premises is particular, the con-
clusion must be particular.

The reason for the first of these rules has already
appeared. Unless there are three and only three terms,
we have no standard for comparison, i.e., no common
point of reference, by which we can determine the
relation of the two terms in the conclusion to each
other. Inference is consequently impossible. Certain
qualifications of this rule will be discussed a little later
in this chapter under the heading, ' Sorites.'

The second rule, that the middle term must be dis-
tributed at least once, is based upon the fact that
the middle term is the medium of comparison for
the other two terms. If neither premise makes an
assertion about the whole class designated by the
middle term, it may happen that one premise applies
to one part of that class, while the other premise
applies to some other part. This is illustrated in
the following syllogism :

All good citizens are interested in politics;
These men are interested in politics;
Therefore these men are good citizens.

72 AN OUTLINE OF LOGIC

It may be, of course, that these men are interested in
politics for purely selfish reasons. That the conclu-
sion does not necessarily follow appears from the
accompanying diagram. ' These men ' may fall either

inside or outside the class, ' good citizens.' If we
were to perform a false conversion of the major
premise and say, ' All who are interested in politics
are good citizens,' the conclusion would be valid, but
the truth of this new major premise would be more
than questionable. A violation of this second rule is
known as an Undistributed Middle Term, or more
briefly, as an Undistributed Middle.

The third rule, which forbids us to distribute in
the conclusion a term that was not distributed in the
premises, implies the obvious truth that if the asser-
tion made in the premises is meant to apply to some
part of a class only, it must not be construed in the
conclusion as though it applied to the class as a
whole. It means that a term must not be used more
widely in the conclusion than it was used in the
premises. This rule may be violated in connection
with either the major or the minor term, and the fal-
lacies which result are known in logic parlance as the
Illicit Process of the major and minor terms, respect-
ively, or, more briefly, as illicit major or illicit minor,

-

THE CATEGORICAL SYLLOGISM 73

An example of an illicit major is given in the
syllogism below:

All the democrats voted against this measure;

This man is not a Democrat;

Therefore this man did not vote against this measure.

The conclusion makes an assertion regarding the whole
class of ' Those who voted against this measure,' viz.,
that the class excludes ' this man.' But, as is shown
in the diagram, the fact that ' this man ' is outside the

Those voting
against

[Democrats] *Tnis

class ' Democrats ' does not determine whether he
is inside or outside the class, ' Those who voted against
this measure.' The inference gains whatever plausi-
bility it may possess from the tacit conversion of
the major premise to, ' All who voted against this
measure are Democrats.' A diagram will show that
if the major is thus converted, the conclusion is
valid.

The next argument illustrates the fallacy of illicit
minor :

All good citizens pay their taxes;
All good citizens vote at elections;
Therefore all who vote at elections pay their taxes.

74

AN OUTLINE OF LOGIC

This inference, as the accompanying diagram shows, is
unwarranted. If the premises are true, the two classes

must indeed overlap, and so it must be true that some
who vote at elections are good citizens. But since
the premises say nothing about the whole class of
' those who vote at elections,' we are not permitted to
do so in the conclusion. The conclusion is valid only
if we convert the minor premise to ' All who vote at
elections are good citizens.'

That no conclusion can be drawn from negative
premises, as is stated in the fourth rule, is evident
if we note that the premises inform us merely that
S and P both fall outside of M and thus give us no
sort of clue as to the relation which they may sustain
to each other. This appears from the following argu-
ment:

No pauper has a vote;
John is no pauper;
Therefore John has a vote.

If we represent this syllogism by circles, we see that
while ' John ' is outside the class of paupers, he is not
necessarily included in the class of ' those who have a
vote. ' The appearance of validity seems to be due to

THE CATEGORICAL SYLLOGISM 75

the fact that ' no pauper has a vote ' bears a certain
resemblance to the proposition, ' Those who are not

paupers have a vote,' i.e., there is a certain tempta-
tion to perform a false obversion.

The rule that two negative premises cannot yield
a valid conclusion is apt to mislead occasionally, unless
we distinguish between propositions which are nega-
tive in form only and those which are negative in
meaning. The following is an example of a syllogism
that appears to have two negative premises but never-
theless permits us to draw a valid conclusion:

Nobody who is not thirsty is suffering from fever,

This person is not thirsty,

Therefore this person is not suffering from fever (Minto).

This is a valid conclusion, for the reason that the
inference is based not upon any assigned quality, but
upon the absence of a quality, viz., ' thirsty.' The
major is equivalent to ' Wherever thirst is absent
fever is absent.' Such a syllogism can be made cor-
rect in form by making the minor premise read, ' This
person is one who is not thirsty.' The ' not ' is thus
made to belong to the predicate and not to the copula.

76 AN OUTLINE OF LOGIC

The remaining rules may be dismissed more briefly.
Rule five, which states that if one premise is negative
the conclusion must be negative, is necessarily true,
because under such conditions one of the other two
terms agrees, while the other does not agree, with the
middle term. Hence the two do not agree with each
other, i.e., the conclusion must be negative. The sixth
and seventh rules are based upon the fact that [as
stated in rules (2) and (3)], the middle term must
be distributed at least once, and that no term must
be distributed in the conclusion if it was not dis-
tributed in the premises. It will be found that if
both premises are particular one or the other of these
two rules is violated, and that the same is true if
we attempt to draw a universal conclusion when one
of the premises is particular. (3 / //?

The Figures of the Syllogism. — By ' figure ' is
meant the form or arrangement of the syllogism as
determined by the position of the middle term. Since
the middle term may function in the premises as
either the subject or the predicate, four arrangements
or forms are possible. These arrangements are ex-
emplified in the following syllogisms :

FIGURE I

All men are mortal; M — P

Socrates is a man; S — M

Therefore Socrates is mortal. . " . S — P

FIGURE II

All men are mortal; P — M

Inanimate beings are not mortal; S — M

Therefore inanimate beings are not men. . ' . S — P

THE CATEGORICAL SYLLOGISM 77

FIGURE III

All men are mortal; M — P

All men are fallible; M — S

Therefore some fallible beings are mortal. . * . S — P

FIGURE IV

All men are mortal; P — M

All mortals are fallible; M — S

Therefore some fallible beings are men. . ' . S — P

In the first figure the middle term is the subject
of the major premise and the predicate of the minor
premise. In the second figure it is the predicate of
both premises. In the third figure it is the subject
of both premises. In the fourth figure it is the predi-
cate of the major premise and the subject of the minor
premise. It may be remarked that while the fourth
figure is theoretically possible, it is of no particular
practical importance and is not recognized in Aris-
totle's doctrine of the syllogism.

Reduction. — The first figure was regarded by Aris-
totle as the most direct and convincing, and was
called by him the perfect figure. In the first figure
the major premise is a universal proposition, and the
minor points out something which this universal in-
cludes. The other figures were called the imperfect
figures, and the process of changing these to the first
figure is called reduction. Elaborate rules governing
the process of reduction were formulated during the
Middle Ages. Reduction is accomplished through cer-
tain processes of obversion and conversion, and
through the transposition of the premises whenever
necessary. The syllogism, for example, in the third

78 AN OUTLINE OF LOGIC

figure above, may be reduced to the first figure by
simply converting the minor premise so as to makj
it read, ' Some fallible beings are men '; and the
syllogism in the fourth figure may be reduced by trans-
posing the premises and converting the conclusion.
Reduction is of interest chiefly because it shows that
whenever we put an argument in the form of a syl-
logism, the particular form of syllogism that we adopt
is more or less a matter of accident.

Sorites, or Chain of Reasoning. — A Sorites is a
chain of reasoning in which the two terms of the
conclusion are united through the mediation of more
than one intervening or connecting term. It may
assume either of the two following forms :

A is B;

All negroes are men;

B is C;

All men are vertebrates;

C is D;

All vertebrates are animals;

D is E;

All animals are mortal;

A is E. ,

. ' . All negroes are mortal.

II

D is E;

All animals are mortal;

C is D;

All vertebrates are animals;

B is C;

All men are vertebrates;

A is B;

All negroes are men;

A is E.

. ' . All negroes are mortal.

It is possible to treat a Sorites as an abbreviated
form of syllogistic inference, because the chain of
reasoning may be resolved into a series of syllogisms,
each of which, except the last, yields a conclusion

THE CATEGORICAL SYLLOGISM 79

that serves as a premise in the succeeding syllogism.
From this point of view, the first of the above in-
ferences is equivalent to three complete syllogisms,
as follows:

I II III

B is C;

C is D;

D is E;

A is B;

A is C;

A is D;

A is C.

. ' . A is D.

. \ A is E.

Inferences in Quantitative Relations. — Certain in-
ferences that deal with quantitative relations give
valid conclusions, in spite of the fact that they seem
to violate the rules of the syllogism. The following
are examples:

A is greater than B;

B is greater than Cj

.*, A is greater than C.

II

A is north of B;

B is north of C;

. " . A is north of C.

In form these arguments are exactly the same as:

A is the landlord of B;

B is the landlord of C;

. * . A is the landlord of C.

Yet this latter conclusion does not follow from the
premises. All of these syllogisms, it will be noticed,

80 AN OUTLINE OF LOGIC

have four terms. What requires explanation, there-
fore, is the fact that, in spite of this apparent ir-
regularity, it is possible to draw valid conclusions
when the subject-matter concerns relations of quantity.
The explanation of this fact is, in brief, that the
valid conclusions are possible because they rest upon
a true major premise which does not appear in the
argument. If A is north of B, and B is north of C,
we can infer the relation of A and C, because we
are familiar with the nature of space relations. To
state the law or the generalization which underlies
the inference is a matter of some difficulty. Accord-
ing to some writers the inference, in correct syllogistic

Whatever is north of that which is north of another

is north of that other;
A is something that is north of that which is north of C;
. * . A is north of C.

It is true that we never formulate the major premise
of this inference, and that we usually do not even
suspect its presence. But, as we shall see a little
later (see Chapter VIII), the suppression of one of
our premises is a frequent occurrence in everyday
reasoning. This major premise is not formulated, just
because the relationship which it expresses is so simple
and obvious. This relationship is peculiar to the
realm of quantity, and so the recognition of this
relationship enables us to make inferences in this
realm which have no precise parallel in other fields.

CHAPTER VII

HYPOTHETICAL AND DISJUNCTIVE
SYLLOGISMS

In a previous chapter propositions were distin-
guished as categorical and conditional. The latter
kind again presents two forms, the hypothetical and
the disjunctive. Corresponding to the two kinds of
conditional propositions, we have two kinds of con-
ditional syllogisms, the hypothetical and the dis-
junctive syllogism, just as the syllogism discussed in
the preceding chapter corresponds to the categorical
proposition.

The Hypothetical Syllogism. — It was pointed out
that the hypothetical proposition expresses a condition
and a result, e.g., ' If it storms, the boat will capsize.'
The part that expresses the condition, ' if it storms,' is
called the antecedent ; the part that expresses the result,
' the boat will capsize, ' is called the consequent. The
reason for this distinction will appear in a moment.

In hypothetical syllogisms both hypothetical and
categorical propositions are employed. The hypo-
thetical syllogism consists of a hypothetical major
premise and a categorical minor premise. The follow-
ing is an example:

If the strike has been called off, the men are back at work;
The strike has been called off;
Therefore the men are back at work.

81

82 AN OUTLINE OF LOGIC

In this case the minor affirms the antecedent. A
valid conclusion may also be drawn if the minor denies
the consequent, as in:

If the strike has been called off, the men are back at work;

The men are not back at work;

Therefore the strike has not been called off.

These two illustrations represent the gist of the
hypothetical sjdlogism. The rule that governs this
syllogism is : The minor premise must either affirm the
antecedent or deny the consequent. If the antecedent
exists, the consequent must also exist. Conversely, if
the consequent does not exist, it follows that the ante-
cedent, which it invariably accompanies, does not exist
either.

Error arises if we attempt to draw conclusions from
a syllogism in which the minor premise denies the
antecedent or affirms the consequent. Thus with the
same major premise as above, and with the minor,
1 The strike has not been called off,' we cannot infer
that the men are not back at work, for they may have
returned to work without the formality of calling off
the strike. If we ignore this possibility, we commit
the fallacy of denying the antecedent. Nor can we
draw a valid inference with, ' The men are back at
work,' as a minor. If we conclude from this that
' the strike has been called off,' we again over-
look the possibility which has just been mentioned.
This fallacy is the fallacy of affirming the con-
sequent.

It should be noticed that the antecedent may be
affirmed by a minor premise which is negative in form

SYLLOGISMS 83

and that the consequent may be denied by a minor
which is affirmative in form. For example:

If war is not declared commerce will increase;
But war will not be declared;
Therefore commerce will increase.

This minor is affirmative in force, because the absence
of a condition, ' war,' asserted by it is a basis for an
inference. Or we may say that the absence of war
is a state that conditions the increase of commerce.
Similarly, in the argument:

If he is the right kind of man he will not use money to
secure his election;

But he will use money to secure his election;
Therefore he is not the right kind of man.

the affirmation that he used money is equivalent to
a denial that he did not use money, and hence it denies
the consequent.

It is evident, if we reflect a moment, that the fallacy
of denying the antecedent is in principle the same
as that of affirming the consequent. In both cases
we disregard the fact that the antecedent is not nec-
essarily the only antecedent upon which the conse-
quent depends. While it may be true, therefore, that
if A is B then C is D, it need not be true that if
A is not B then C is not D ; nor need it be true that
if C is D then A is B. If every consequent had but
one antecedent, it would not be fallacious to draw a
conclusion by denying the antecedent or affirming the
consequent.

If we analyze the matter a little further, we find

84 AN OUTLINE OF LOGIC

that these fallacies are both due to false obversion.
This can be best shown by an illustration. Suppose
we take for major premise the proposition, ' If he is
ambitious, he will work.' With either ' He is not
ambitious ' or ' He will work ' as a minor, we can-
not draw a conclusion. He may decide to work,
because this is a less unpleasant alternative than
starvation. Nor can we draw a conclusion by sub-
stituting for the original major premise its true ob-
verse, which is, ' If he is ambitious, he is not a person
who will not work.' But in case we take the obverse
to be, ' If he is not ambitious, he will not work,' we
can draw a valid conclusion with either of the above
premises as a minor. The conclusion is then valid
because, through a false obversion, the original major
premise is interpreted as equivalent to something that
is entirely different. In this false obversion, ambi-
tion is assumed to be the only condition upon
which the consequent, ' willingness to work,' is
dependent.

The Disjunctive Syllogism. — A disjunctive syl-
logism is characterized by the fact that its major
premise is a disjunctive proposition, while its minor
is categorical. The function of the major premise is
to state the different alternatives of which one or the
other must be true. The minor then either specifies
which of the alternatives is true, or it specifies which
of the alternatives are not true. In symbolic form
the disjunctive syllogism may be expressed as follows:

A is either B or C; Either A is B or C is D;
A is B; A is B;

. * . A is not C. . ' . C is not D.

SYLLOGISMS 85

He acquired his money either honestly or by fraud ;

He acquired it honestly;

Therefore he did not acquire it by fraud.

It is assumed that the alternatives mentioned in the
major premise are exclusive of each other. Unless
this is the case, we cannot infer that because A is
B it is not C, although we can be sure that if A is
not B it is necessarily C. ' He is either a knave
or a fool,' is an example. If he is not a knave, he
must be a fool, but if we should find that he is a
knave, we should have to allow for the possibility
that he is a fool as well. This possibility invalidates
the following inference :

He is either elected or the ballotbox was stuffed;

He is elected;

Therefore the ballotbox was not stuffed.

It is evident that the alternatives here mentioned do
not exclude each other. The candidate might be fairly
elected, in the sense that he received a majority of the
votes which were cast, but there might be fraud at
the same time. If, however, the minor premise denied
one of the alternatives, the conclusion would be valid.
The Dilemma. — In practical life a dilemma means
any situation that offers two or more alternatives of
action, all of which, however, are disagreeable. In
logic a dilemma is an argument whose premises are
made up of hypothetical and disjunctive propositions.
Its major consists of two or more hypothetical proposi-
tions, while the minor is a disjunctive proposition.
The following is an example:

86 AN OUTLINE OF LOGIC

If A is B, C is D; and if E is F, C is D;
But either A is B, or E is F;
Therefore C is D.

The dilemmatie argument may at times be somewhat
more complex, but the principle remains the same.
Thus:

If A is B, C is D; and if E is F, G is H;
But either C is not D or G is not H;
Therefore either A is not B, or E is not F.

" If this man were wise he would not speak ir-
reverently of Scripture in jest; and if he were good,
he would not do so in earnest. But he does it either
in jest or in earnest; therefore he is either not wise
or not good." (Whately.)

In the course of the Lincoln-Douglas debate a
question was put by Lincoln to Douglas, as follows:
" Can the people of a United States territory in any
lawful way, against the wish of any citizens of the
United States, exclude slavery from its limits, prior
to the formation of a state constitution? " The ques-
tion may be viewed as the source of a dilemma, both
in the practical and in the syllogistic sense of the
term. In fact it involved a situation which, syl-
logistically, comprised more than one dilemma. They
may be stated as follows :

I. If Douglas answers yes, he offends the South, and if
he answers no, he offends the Noi'th;
But he must answer either yes or no;
Therefore he will offend either the South or the North.

SYLLOGISMS 87

II. If Douglas offends the South, he loses the nomination
for the Presidency in the next convention ; and if he offends
the North, he loses the election to the United States Senator-
ship (and his chances for the Presidency).

But he must offend either the South or the North;

Therefore he loses either the Presidency or the Senator-
ship.

Or, III. If Douglas offends the South, he cannot become
President; and if he offends the North, he cannot become
President ;

But he must offend either the South or the North;

Therefore he cannot become President.

When rightly used the dilemma is an extremely
effective form of argument. Its effectiveness, how-
ever, like that of the disjunctive syllogism, depends
upon the exhaustiveness with which the alternatives
are stated. In many, perhaps most, dilemmas some
of the alternatives are overlooked; so that the argu-
ment can be blocked by simply pointing out this fact.
" Thus if we were, to argue that ' if a pupil is fond
of learning he needs no stimulus, and that if he dis-
likes learning no stimulus will be of any avail, but
that, as he is either fond of learning or dislikes it,
a stimulus is either useless or of no avail,' we evi-
dently assume improperly the disjunctive minor
premise. Fondness and dislike are not the only two
possible alternatives, for there may be some who are
neither fond of learning nor dislike it, and to these
a stimulus in the shape of rewards may be desir-
able."*

* Jevons, Lessons in Logic, p. 168.

88 AN OUTLINE OF LOGIC

The Relation of Categorical and Conditional
Syllogisms. — A comparison of the categorical syl-
logism with conditional syllogisms shows that the two
differ widely in their emphasis or mode of procedure.
The categorical syllogism, as traditionally interpreted,
concerns itself altogether with the relations of in-
clusion and exclusion of classes, i.e., with the aspect
of denotation or extension. This works very well as a
rule, although the interpretation in terms of classes
becomes rather artificial in some instances. For ex-
ample, it is undeniably stilted to interpret, ' Pride
goeth before a fall, ' as meaning, ' The class of things
characterized by pride falls within the class of things
that go before a fall.' It is stilted for the reason
that when the statement is made, we are not thinking
of the relations of classes at all. Our attention is
occupied with the connotative or intensive side of the
proposition; or, to put it differently, with the rela-
tions of the abstract attributes or characters ' pride '
and ' fall. ' Extension and intension are both genuine
aspects of propositions, but they do not necessarily
receive the same emphasis in our thinking. Condi-
tional syllogisms differ from the categorical in that
they emphasize this intensive character. Conditional

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