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."