THE ART OF LOGICAL THINKING/PART 17
CHAPTER XVII.
VARIETIES OF SYLLOGISMS
The
authorities in Logic hold that with the four kinds of propositions grouped in
every possible order of arrangement, it is possible to form nineteen different
kinds of valid arguments, which are called the nineteen moods of the
syllogism. These are classified by division into what are called the
four figures, each of which figures may be known by the position of the
middle term in the premises. Logicians have arranged elaborate and curious
tables constructed to show what kinds of propositions when joined in a
particular order of arrangement will make sound and valid syllogisms. We shall
not set forth these tables here, as they are too technical for a popular
presentation of the subject before us, and because they are not necessary to
the student who will thoroughly familiarize himself with the above stated Laws
of the Syllogism and who will therefore be able to determine in every case
whether any given argument is a correct syllogism, or otherwise.
In
many instances of ordinary thought and expression the complete syllogistic
form is omitted, or not stated at full length. It is common usage to omit one
premise of a syllogism, in ordinary expression, the missing premise being
inferred by the speaker and hearer. A syllogism with one premise unexpressed is
sometimes called an Enthymene, the term meaning "in the
mind." For instance, the following: "We are a free people, therefore
we are happy," the major premise "All free people are happy"
being omitted or unexpressed. Also in "Poets are imaginative, therefore
Byron was imaginative," the minor premise "Byron was a poet" is
omitted or unexpressed. Jevons says regarding this phase of the subject:
"Thus in the Sermon on the Mount, the verses known as the Beatitudes
consist each of one premise and a conclusion, and the conclusion is put first.
'Blessed are the merciful: for they shall obtain mercy.' The subject and the
predicate of the conclusion are here inverted, so that the proposition is
really 'The merciful are blessed.' It is evidently understood that
'All who shall obtain mercy are blessed,' so that the syllogism, when stated at
full length, becomes: 'All who shall obtain mercy are blessed; All who are
merciful shall obtain mercy; Therefore, all who are merciful are blessed.' This
is a perfectly good syllogism."
Whenever
we find any of the words: "because, for, therefore, since,"
or similar terms, we may know that there is an argument, and usually a
syllogism.
We
have seen that there are three special kinds of Propositions, namely, (1)
Categorical Propositions, or propositions in which the affirmation or denial is
made without reservation or qualification; (2) Hypothetical Propositions, in
which the affirmation or denial is made to depend upon certain conditions,
circumstances, or suppositions; and (3) Disjunctive Propositions, in which is
implied or asserted an alternative.
The
forms of reasoning based upon these three several classes of propositions bear
the same names as the latter. And, accordingly the respective syllogisms
expressing these forms of reasoning also bear the class name or term.
Thus, a Categorical Syllogism is one containing only categorical propositions;
a Hypothetical Syllogism is one containing one or more hypothetical
propositions; a Disjunctive Syllogism is one containing a disjunctive
proposition in the major premise.
Categorical
Syllogisms, which are far more common than the other two kinds, have been
considered in the previous chapter, and the majority of the examples of
syllogisms given in this book are of this kind. In a Categorical Syllogism the
statement or denial is made positively, and without reservation or
qualification, and the reasoning thereupon partakes of the same positive
character. In propositions or syllogisms of this kind it is asserted or assumed
that the premise is true and correct, and, if the reasoning be logically
correct it must follow that the conclusion is correct, and the new proposition
springing therefrom must likewise be Categorical in its nature.
Hypothetical
Syllogisms, on the contrary, have as one or more of their premises a
hypothetical proposition which affirms or asserts something provided, or
"if," something else be true. Hyslop says of this: "Often we
wish first to bring out, if only conditionally, the truth upon which a
proposition rests, so as to see if the connection between this conclusion and
the major premise be admitted. The whole question will then depend upon the
matter of treating the minor premise. This has the advantage of getting the
major premise admitted without the formal procedure of proof, and the minor
premise is usually more easily proved than the major. Consequently, one is made
to see more clearly the force of the argument or reasoning by removing the
question of the material truth of the major premise and concentrating attention
upon the relation between the conclusion and its conditions, so that we know
clearly what we have first to deny if we do not wish to accept it."
By
joining a hypothetical proposition with an ordinary proposition we create a
Hypothetical Proposition. For instance: "If York contains a
cathedral it is a city; York does contain a cathedral;
therefore, York is a city." Or: "If dogs have four
feet, they are quadrupeds; dogs do have four feet; therefore
dogs are quadrupeds." The Hypothetical Syllogism may be
either affirmative or negative; that is, its hypothetical proposition may
either hypothetically affirm or hypothetically deny.
The part of the premise of a Hypothetical Syllogism which conditions or
questions (and which usually contains the little word "if") is called
the Antecedent. The major premise is the one usually thus conditioned. The
other part of the conditioned proposition, and which part states what will
happen or is true under the conditional circumstances, is called the
Consequent. Thus, in one of the above examples: "If dogs have four
feet" is the Antecedent; and the remainder of the proposition: "they
are quadrupeds" is the Consequent. The Antecedent is indicated by the
presence of some conditional term as: if, supposing, granted
that, provided that, although, had, were,
etc., the general sense and meaning of such terms being that of the little word
"if." The Consequent has no special indicating term.
Jevons
gives the following clear and simple Rules regarding the Hypothetical
Syllogism:
I.
"If the Antecedent be affirmed, the consequent may be affirmed. If the
Consequent be denied, the Antecedent may be denied."
II.
"Avoid the fallacy of affirming the consequent, or denying the antecedent.
This is a fallacy because of the fact that the conditional statement made in
the major premise may not be the only one determining the
consequent." The following is an example of "Affirming the
Consequent:" "If it is raining, the sky is overclouded;
the sky is overclouded; therefore, it is raining."
In truth, the sky may be overclouded, and still it may not be
raining. The fallacy is still more apparent when expressed in symbols, as
follows: "If A is B, C is D; C is D;
therefore, A is B." The fallacy of denying the Antecedent is shown by the
following example: "If Radium were cheap it would be useful;
Radium is not cheap; therefore Radium is not useful."
Or, expressed in symbols: "If A is B, C is D; A is not B;
therefore C is not D." In truth Radium may be useful
although not cheap. Jevons gives the following examples of these fallacies:
"If a man is a good teacher, he thoroughly understands his subject; but
John Jones thoroughly understands his subject; therefore, he is a good
teacher." Also, "If snow is mixed with salt it melts; the snow
on the ground is not mixed with salt; therefore it
does not melt."
Jevons
says: "To affirm the consequent and then to infer that we can affirm the
antecedent, is as bad as breaking the third rule of the syllogism, and allowing
an undistributed middle term.... To deny the antecedent is really to break the
fourth rule of the syllogism, and to take a term as distributed in the
conclusion which was not so in the premises."
Hypothetical
Syllogisms may usually be easily reduced to or converted into Categorical
Syllogisms. As Jevons says: "In reality, hypothetical propositions and
syllogisms are not different from those which we have more fully
considered. It is all a matter of the convenience of stating the
propositions." For instance, instead of saying: "If Radium were
cheap, it would be useful," we may say "Cheap Radium would be
useful;" or instead of saying: "If glass is thin, it breaks
easily," we may say "Thin glass breaks easily." Hyslop gives the
following Rule for Conversion in such cases: "Regard the
antecedent of the hypothetical proposition as the subject of
the categorical, and the consequent of the hypothetical proposition as the
predicate of the categorical. In some cases this change is a very simple one;
in others it can be effected only by a circumlocution."
The
third class of syllogisms, known as The Disjunctive Syllogism, is
the exception to the law which holds that all good syllogisms must fit in and
come under the Rules of the Syllogism, as stated in the preceding chapter. Not
only does it refuse to obey these Rules, but it fails to resemble the ordinary
syllogism in many ways. As Jevons says: "It would be a great mistake to
suppose that all good logical arguments must obey the rules of the syllogism,
which we have been considering. Only those arguments which connect two terms
together by means of a middle term, and are therefore syllogisms, need obey
these rules. A great many of the arguments which we daily use are of this
nature; but there are a great many other kinds of arguments, some of which have
never been understood by logicians until recent years. One important kind of
argument is known as the Disjunctive Syllogism, though it does not obey the
rules of the syllogism, or in any way resemble syllogisms."
The
Disjunctive Syllogism is one having a disjunctive proposition in its major
premise. The disjunctive proposition also appears in the conclusion when the
disjunction in the major premise happens to contain more than two terms. A
disjunctive proposition, we have seen, is one which possesses alternative
predicates for the subject in which the conjunction "or"
(sometimes accompanied by "either") appears. As for instance:
"Lightning is sheet or forked;" or, "Arches
are either round or pointed;" or, "Angles are either
obtuse, or right angled, or acute." The different things joined together
by "or" are called Alternatives, the term indicating that we may
choose between the things, and that if one will not answer our purpose we may
take the other, or one of the others if there be more than one other.
The Rule
regarding the Use of Disjunctive Syllogisms is that: "If one or
more alternatives be denied, the rest may still be affirmed." Thus if we
say that "A is B or C," or that "A is either B or C," we
may deny the B but still affirm the C. Some authorities
also hold that "If we affirm one alternative, we must deny the
remainder," but this view is vigorously disputed by other authorities. It
would seem to be a valid rule in cases where the term "either"
appears as: "A is either B or C,"
because there seems to be an implication that one or the other alone can be
true. But in cases like: "A is B or C," there may be
a possibility of both being true. Jevons takes this latter view,
giving as an example the proposition: "A Magistrate is a
Justice-of-the-Peace, a Mayor, or a Stipendiary Magistrate," but it does
not follow that one who is a Justice-of-the-Peace may not be at the same time a
Mayor. He states: "After affirming one alternative we can only deny the
others if there be such a difference between them that they could not
be true at the same time." It would seem that both contentions are at
the same time true, the example given by Jevons illustrating his contention,
and the proposition "The prisoner is either guilty or innocent"
illustrating the contentions of the other side.
A Dilemma is
a conditional syllogism whose Major Premise presents some sort of
alternative. Whately defines it as: "A conditional syllogism with two or
more antecedents in the major, and a disjunctive minor." There being two
mutually exclusive propositions in the Major Premise, the reasoner is compelled
to admit one or the other, and is then caught between "the two horns of
the dilemma."
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