FIGURES AND MOODS OF THE SYLLOGISM.
=1. THE FOUR FIGURES OF THE SYLLOGISM.=
By a figure of a syllogism is meant some particular arrangement of the
three terms in the two premises. The conclusion is eliminated from this
discussion, because in it the arrangement of the terms is constant, the
major term always being used as the predicate of the conclusion and the
minor as the subject. Using the symbols M, G and S, we find that there
are four possible arrangements and, therefore, but _four figures_.
These may be represented as follows:
First Second Third Fourth
figure figure figure figure
M ― G G ― M M ― G G ― M
S ― M S ― M M ― S M ― S
――――― ――――― ――――― ―――――
S ― G S ― G S ― G S ― G
No matter what the syllogism, if it is to be proved “_logical_,” it
should be made to fit one of the four figure-types. To be sure, it may
fit the figure without being logical, but it cannot be strictly logical
without fitting the figure. The following valid syllogisms conform to
the four figures as will be seen by the symbolized terms:
M G
First figure: All men are mortal,
S M
Socrates is a man,
S G
∴ Socrates is mortal.
M ― G
S ― M
―――――
S ― G
G M
Second figure: All good citizens love their country,
S M
No criminal loves his country,
S G
∴ No criminal is a good citizen.
G ― M
S ― M
―――――
S ― G
M G
Third figure: All good citizens are law abiding,
M S
All good citizens vote,
S G
∴ Some who vote are law abiding.
M ― G
M ― S
―――――
S ― G
G M
Fourth figure: Some teachers are fair minded,
M S
All who are fair minded are just,
S G
∴ Some just persons are teachers.
G ― M
M ― S
―――――
S ― G
Here, then, are the types that represent all the syllogisms which
mediate inference may use. _Logic recognizes no other._ Since every
successful student of logic must be familiar with the four figures, the
following may be used as a suggestive aid to reproducing the figures at
will:
First. It is easy for any one to remember this syllogism:
All men are mortal,
Socrates is a man,
∴ Socrates is mortal.
In fact, it comes down to us from the time of Aristotle, and is
therefore a patriot of many generations to whom the faithful should
touch their hats. Let us, then, be ready to reproduce this syllogism
with automatic precision, since it will enable us to know at once the
_position of the terms_ in the first figure. Second. Converting the
terms of the _major premise_ of the first figure gives the second
figure, as, e. g.:
First figure. Second figure.
M ― G (Convert) G ― M
S ― M S ― M
――――― ―――――
S ― G S ― G
Third. Converting the terms of the _minor premise_ of the first figure
gives the third figure, as, e. g.:
First figure. Third figure.
M ― G M ― G
S ― M (Convert) M ― S
――――― ―――――
S ― G S ― M
Fourth. Converting the terms of _both the major and minor premises_ of
the first figure gives the fourth, as, e. g.:
First figure. Fourth figure.
M ― G (Convert) G ― M
S ― M (Convert) M ― S
――――― ―――――
S ― G S ― G
To summarize: _The second, third and fourth figures may be derived from
the first. Converting the major premise of the first figure gives the
second figure; converting the minor premise gives the third figure; and
converting both premises gives the fourth figure._
=2. THE MOODS OF THE SYLLOGISM.=
By the mood of a syllogism is meant some particular arrangement of
the propositions which compose the syllogisms. “_Mood_” stands for
an arrangement of the _propositions_, while “_figure_” represents an
arrangement of the _terms_ in any syllogism.
Combining any three of the four logical propositions gives a mood, as,
e. g.,
(1) E (2) A (3) E
A I I
E I O
are moods. The first one has an E proposition for the major premise, an
A for the minor and an E for the conclusion. This syllogism represents
the first mood given above:
E No men are trees,
A All Americans are men,
E ∴ No Americans are trees.
It would not be difficult to determine by actual experiment, just how
many moods could be formed, and of these, how many would admit of valid
conclusions. It may be seen that there are sixty-four permutations of
the four logical propositions, taken three at a time. These are in part:
(1) (2) (3) (4) (5) (6) (7) (8)
A A A A A A A A
A A A A E E E E
A E I O A E I O
(9) (10) (11) (12) (13) (14) (15) (16)
A A A A A A A A
I I I I O O O O
A E I O A E I O
And so the permutations could be continued. Substituting E for the
major premise of the above group would give another group of sixteen,
while a like substitution of I and O would result in two more groups,
sixteen in each. This gives sixty-four in all.[10]
=3. TESTING THE VALIDITY OF THE MOODS.=
In order to put the moods to good use, it is necessary to ascertain
which ones yield a valid conclusion in any figure. If each were valid
in all of the four figures, there would be 256. But it is obvious that
such is not the case.
Referring to the sixteen permutations given above, we find that the
_“negative-conclusion” rule_ makes invalid 2, 4, 5, 7, 10, 12, 13 and
15; whereas the rule for particulars throws out 9 and 14. This leaves
the following as the probable valid moods in one or more of the figures:
1, 3, 6, 8, 11, 16. But to be certain of this the investigation must be
A
continued. The mood A has stood the test of the rules for negative and
A
particular conclusions; now let us test this mood from the standpoint
of the distribution of terms, using it in all four figures:
_First_ _Second_ _Third_ _Fourth_
A M ― G G ― M M ― G G ― M
― ― ― ―
A S ― M S ― M M ― S M ― S
― ― ― ―
――――――― ――――――― ――――――― ―――――――
A S ― G S ― G S ― G S ― G
― ― ― ―
As an A proposition distributes its subject only, we underscore the
subject of each proposition in all the figures. (This underscoring is
a simple way to indicate distribution.)
We now find that the mood is valid in the first figure, because the
middle term is distributed at least once; namely, in the major premise,
and there is no term distributed in the conclusion which is not already
distributed in the premise where it occurs. On the other hand, the mood
A
A is invalid in the second, because of “_undistributed middle_,” and
A
invalid in the third and fourth, because S is distributed in the
conclusion but not distributed in the premise where it occurs (illicit
minor).
Let us try AII in the four figures:
A M ― G G ― M M ― G G ― M
― ― ― ―
I S ― M S ― M M ― S M ― S
――――― ――――― ――――― ―――――
I S ― G S ― G S ― G S ― G
We underscore the subject of the A proposition in each of the four
figures. As I distributes neither subject nor predicate, no other term
A
should be underscored. It is now evident that I is not valid in figures
I
two and four, because in both figures the middle term is undistributed
(undistributed middle).
In a like manner all the other moods might be tested. Logicians, who
have done this, have found 24 to be valid. Five of these have weakened
conclusions; i. e., a particular conclusion when it could just as well
A
be universal. E illustrates this as the conclusion _could be E_. This
O
syllogism exemplifies the weakened conclusion:
A All trees grow,
E No sticks are trees,
O ∴ Some sticks do not grow.
This conclusion is true, since “some” means “some at least.” Yet the
conclusion is weak, because there is nothing to interfere with the
broader and stronger conclusion that, “No sticks grow.” There are,
therefore, only 19 valid and serviceable moods. These are as follows:
(1) (2) (3) (4) (5) (6)
{ A E A E ― ― }
First figure { A A I I ― ― } 4
{ A E I O ― ― }
{ E A A E ― ― }
Second figure { A E O I ― ― } 8
{ E E O O ― ― }
{ A I A E O E }
Third figure { A A I A A I } 14
{ I I I O O O }
{ A A I E E ― }
Fourth figure { A E A A I ― } 19
{ I E I O O ― }
A
Of these nineteen moods it is not much of a tax to remember that A is
A
E
valid only in the first figure; whereas A is valid in the first and
E
A E
second figures; I in the first and third; while I is valid in all. This
I O
knowledge, however, should be used only as one would employ the answers
in arithmetic. Testing the validity of a mood in the four figures is an
exceedingly valuable thought-exercise, which a knowledge of the final
result might easily vitiate. It is, no doubt, best to test the value
of any mood without such knowledge, and then compare the result by
referring to the foregoing list of valid moods. It is not always wise
to work with the answer in mind, yet it is most satisfying to know of
a _certainty_ that one’s reasoning has led to a truth which others have
verified.
=4. SPECIAL CANONS OF THE FOUR FIGURES.=
As a deductive exercise in clear, logical thought, the indirect proof
involved in establishing certain principles underlying the four figures,
is of immense value. On no account should this section be omitted. The
mere fact that it appears to be a difficult section is proof positive
that the student is in need of just such exercises.
_Canons of the first figure._
(1) The minor premise must be affirmative.
(2) The major premise must be universal.
_Problem: The minor premise must be affirmative._
_Data_: Given the form of the first figure, which is,
M ― G
S ― M
―――――
S ― G
_Proof_: (1) If the minor premise is not affirmative then it must
be negative; because affirmative and negative propositions, being
contradictory in nature, admit of no middle ground.
(2) If the minor premise is negative, the conclusion must be negative;
for the reason that a negative premise necessitates a negative
conclusion.
(3) If the conclusion is negative then its predicate, G, must be
distributed; since all negatives distribute their predicates.
(4) If the predicate of the conclusion, which is the major term, is
distributed, then it must be distributed in the premise where it occurs,
which is the major premise; for any term which is distributed in the
conclusion must be distributed in the premise where it occurs.
(5) If the major term, which is the predicate of the major premise,
is distributed, then the major premise must be negative; because only
negatives distribute their predicates.
(6) The result of this argument, then, gives _two_ negative premises,
and we know from rule 3 that a conclusion from two negatives is
untenable.
(7) Since the minor premise cannot be negative, it must be affirmative.
_Problem: To prove that the major premise must be universal._
_Data_: Given the form of the first figure:
M ― G
S ― M
―――――
S ― G
_Proof_: (1) The predicate of the minor premise, M, which is the middle
term, is undistributed; because no affirmative proposition distributes
its predicate.
(2) The middle term must be distributed in the major premise; since in
any syllogism the middle term must be distributed at least once.
(3) As the middle term, M, used as the subject of the major premise,
must be distributed, then the major premise must be universal; because
only universals distribute their subjects.
_Epitome._
_In the first figure, the minor premise must be affirmative, since
making it negative necessitates making the major premise negative also;
the major premise must be universal in order to distribute the middle
term at least once._
_Special canons of the second figure._
(1) One premise must be negative.
(2) The major premise must be universal.
_Problem: To prove that one premise must be negative._
_Data_: Given the form of the second figure:
G ― M
S ― M
―――――
S ― G
_Proof_: (1) The middle term, M, is the predicate of both premises.
(2) The middle term must be distributed at least once, according to
rule 3.
(3) Hence one premise must be negative; since only negatives distribute
their predicates.
_Problem: To prove that the major premise must be universal._
_Data_: Given the form of the second figure:
G ― M
S ― M
―――――
S ― G
_Proof_: (1) As one premise must be negative, it follows that the
conclusion must be negative according to rule 6.
(2) If the conclusion is negative, then its predicate, G, the major
term, must be distributed; since all negatives distribute their
predicates.
(3) When distributed in the conclusion, the major term, G, must also be
distributed in the major premise, where it is used as the subject. See
rule 4.
(4) Hence the major premise must be universal; for only universals
distribute their subjects.
_Epitome._
_In the second figure one premise must be negative in order to
distribute the middle term at least once; and the major premise must be
universal that the major term, which is distributed in the conclusion,
may be distributed in the premise where it occurs._
_Canons of the third figure._
(1) The minor premise must be affirmative.
(2) The conclusion must be particular.
_Problem: To prove that the minor premise must be affirmative._
_Data_: Given the form of the third figure, which is,
M ― G
M ― S
―――――
S ― G
_Proof_: (1) Suppose the minor premise were negative, then the
conclusion would have to be negative, and this would distribute the
predicate G.
(2) A distributed predicate would necessitate its being distributed in
the major premise.
(3) But G, being the conclusion of the major premise, could be
distributed only by a negative proposition.
(4) This would result in two negatives; therefore no conclusion could
be drawn, if the minor premise were negative.
_Problem: To prove that the conclusion must be particular._
_Data_: Given the form of the third figure:
M ― G
M ― S
―――――
S ― G
_Proof_: (1) The minor term, which is the predicate of the affirmative
minor premise, is undistributed; because no affirmative distributes its
predicate.
(2) If undistributed in the premise, then the minor term must remain
undistributed in the conclusion, where it is used as the subject.
(3) The conclusion must, then, be particular; since all universals
distribute their subjects.
_Epitome._
_In the third figure, unless the minor premise be affirmative, there
can be no conclusion; since a negative minor would necessitate a
negative major. An affirmative minor compels a particular conclusion,
in order that the minor term, in the conclusion, may remain
undistributed._
_Canons of the fourth figure._
(1) If the major premise is affirmative, the minor premise must be
universal.
(2) If the minor premise is affirmative, the conclusion must be
particular.
(3) If either premise is negative, the major must be universal.
_Problem: To prove that if the major is affirmative, the minor must be
universal._
_Data_: Given the form of the fourth figure:
G ― M
M ― S
―――――
S ― G
_Proof_: (1) If the major premise is affirmative, then its predicate
which is the middle term, M, is undistributed; for no affirmative
distributes its predicate.
(2) The middle term must then be distributed in the “minor” according
to rule 3.
(3) Then the “minor” must be universal; since only universals
distribute their subjects.
_Problem: To prove that if the minor is affirmative, the conclusion
must be particular._
_Data_: Given the form of the fourth figure:
G ― M
M ― S
―――――
S ― G
_Proof_: (1) If the minor premise be affirmative, then S, its
predicate, must be undistributed; because no affirmative distributes
its predicate.
(2) Since S is undistributed in the minor premise, it must remain
undistributed in the conclusion where it is used as the subject.
_Problem: To prove that if either premise is negative, the major must
be universal._
_Data_: Given the form of the fourth figure:
G ― M
M ― S
―――――
S ― G
_Proof_: (1) If one of the premises is negative, then the conclusion
must be negative according to rule 6.
(2) If the conclusion is negative, then the predicate, G, must be
distributed.
(3) If G is distributed in the conclusion, it must be distributed in
the major premise.
(4) The major premise must be universal; as G is used as its subject,
and only universals distribute their subjects.
_Epitome._
_In the fourth figure, if the “major” is affirmative, the “minor” must
be universal in order to distribute the middle term. If the minor is
affirmative, the conclusion must be particular; otherwise the fallacy
of illicit minor would result. If either premise is negative, the major
must be universal to avoid the fallacy of illicit major._
=5. SPECIAL CANONS RELATED.=
After a particular mood has been tested in the regular way, it has been
intimated that the student may refer to the tabulated list of valid
moods to ascertain, with a certainty, the validity of his reasoning.
This is equivalent to referring to the answers in arithmetic; for
if the student is unable to find the mood in the figure in which he
has proved it valid, then he knows that he has made some mistake in
his reasoning. A second check, though not absolute, is to recall the
special canons of section four. If, for example, our reasoning has led
A
us to believe that E is valid in the first figure, we may recall that
E
the minor premise of the first figure must be affirmative and therefore
AEE cannot be valid.
A few suggestions relative to memorizing the special canons may not
be out of place. The two canons of the first figure must be committed,
and then it may be remembered that the _second figure is the negative
figure of logic_. Other figures _may_ yield a negative conclusion,
but the second _must_ yield a negative conclusion. Since a negative
conclusion necessitates a negative premise, it follows that the second
figure must always appear with one premise negative. The other canon
which pertains to the major premise is the same as the “major premise”
canon of the first figure.
_The third figure is the particular figure of logic._ Other figures may
yield particular conclusions, but the third _must_ do so. This helps us
to remember the canon that the conclusion of the third figure must be
particular. The other canon which relates to the minor premise is the
same as the “minor premise” canon of the first figure. The canons of
the fourth figure are in reality a summary of the canons of the other
three figures.
=6. MNEMONIC LINES.=
As a device for remembering the 19 valid moods, the logicians of an
earlier day originated a combination of coined words which, though
rather unscientific, may be easily committed to memory. Since, however,
it is of much more value to test the moods by means of the general
rules of the syllogism than it is to try to remember these moods, the
mnemonic lines are of slight value. They are treated here merely as an
item of historical interest.
(1) _Barbara_, _Celarent_, _Darii_, _Ferio_que prioris;
(2) _Cesare_, _Camestres_, _Festino_, _Baroko_, secundæ;
(3) Tertia, _Darapti_, _Disamis_, _Datisi_, _Felapton_, _Bokardo_,
_Ferison_, habet; Quarta insuper addit
(4) _Bramantip_, _Camenes_, _Dimaris_, _Fesapo_, _Fresison_.
The only letters in these lines which mean nothing are l, n, r, t and
small b and d; all the others have a signification. For example, the
_vowels_ of the italicized words signify the various valid moods, as
e. g., the first line indicates the moods AAA, EAE, AII, EIO. The Latin
words, printed in ordinary type, are intended to make evident that the
moods indicated by the artificial italicized words of the first line,
belong to the _first_ figure; that the moods of the next four words,
belong to the _second_ figure; while the _third_ figure includes the
next six, and the _fourth_ figure the last five. It is now seen that
Festino, for example, stands for that mood of the second figure which
has an E for its major premise, an I for its minor premise, and an O
for its conclusion.
The first figure was called by Aristotle the _perfect figure_, whereas
the second and third were the _imperfect figures_. The fourth figure
was given no place in the works of Aristotle; its discovery is credited
to Galen, a celebrated teacher of medicine of the second century.
According to Aristotle, the first figure is the most serviceable and
the most convincing and, therefore, as a final test of their validity,
the moods of the other figures should be changed to the _first_.
This process in logic is termed _Reduction_. In this reduction of
the imperfect figures to the perfect, the _capital letters_ of the
artificial words, together with s, p, m, and k, have a definite meaning.
The capital letters indicate that certain moods of the imperfect
figures can be reduced to the corresponding moods of the first figure;
e. g., Festino (eio) of the second figure, Felapton (eao) of the third
figure, and Fesapo (eao) of the fourth figure may all be reduced to
Ferio (eio) of the first figure. This is known because _F_ is the
initial letter of each word. _s_ signifies that the proposition denoted
by the preceding vowel is to be converted simply. To illustrate: _s_
E
in Fesapo means that the major premise E of the mood A of the fourth
O
figure must be converted _simply_ in order to change the mood to Ferio
of the first figure. _p_ indicates that the proposition represented
by the vowel which precedes p must be converted by _limitation_ (per
accidens). _m_ (mutare) makes evident that the premises are to be
interchanged, the major of the old becoming the minor of the new,
and the minor of the old becoming the major of the new. _k_ denotes
that the mood, such as Baroko, must be reduced by a special process
known as _indirect reduction_. These directions may now be followed
as illustrative of the process of reduction.
A
(1) _Given: A syllogism in Darapti_ A
I
M G
A All true teachers are just,
M S
A All true teachers are sympathetic,
S G
I ∴ Some sympathetic persons are just.
A
The symbols indicate that the mood is A or is in Darapti and that this
I
mood is used in the third figure.
A
_Problem_: To reduce A of the third figure to some mood of the first
I
figure.
_Process_: D, being the initial letter of Darapti, suggests that its
mood must be reduced to one indicated by a word of the first figure
A
whose initial letter is D. This mood is in Darii, or is I.
I
The _p_ in Darapti indicates that the proposition represented by the
preceding vowel must be converted by limitation. This proposition is
the minor premise; converting it by limitation gives: “Some sympathetic
persons are true teachers.” As there are no other significant letters
the reduction is complete and we have this:
M G
A All true teachers are just,
S M
I Some sympathetic persons are true teachers,
S G
I ∴ Some sympathetic persons are just.
A
The symbolization indicates that the mood is I of the first figure, or
I
is in Darii.
A
(2) _Given: A syllogism in Camestres_ E
E
G M
A All true teachers are just,
S M
E No one who shows partiality is just,
S G
E ∴ No one who shows partiality is a true teacher.
The symbols show that the mood is AEE of the second figure or in
Camestres. Judging from the initial letter C, the mood in Camestres
E
must be reduced to the mood in Celarent A.
E
The letter m between a and e indicates that the major and minor
premises of the given syllogism must be interchanged. The letters
following both e’s suggest that the minor premise and the conclusion
of the syllogism must be converted simply.
This is the resulting syllogism:
M G
E No just person shows partiality,
S M
A All true teachers are just persons,
S G
E ∴ No true teacher shows partiality.
E
Here, then, is the A of the first figure or the mood in Celarent.
E
According to the ancient theory, reduction is necessary as a matter of
_final_ and _absolute proof_ that the conclusion follows from the given
premises. But, as this claim has been satisfactorily refuted by modern
logicians, we need not give more space to the process. The meaning of
k, as related to “_indirect reduction_,” is explained in most of the
earlier works on logic. See Hyslop, page 193.
=7. RELATIVE VALUE OF THE FOUR FIGURES.=
_The first figure._
The first figure is known as the _perfect_ figure; because it is the
only one which proves _all_ of the four logical propositions. Recalling
the moods of the first figure makes this evident:
A E A E
A A I I
_A_ _E_ _I_ _O_
It is likewise the more natural figure; because it is the only one
which uses both the subject and predicate of the conclusion in the same
relative places as they appear in the premises. Symbolizing the figure
makes this apparent:
M ― _G_
_S_ ― M
_S_ ― _G_
The first figure, being the only figure which proves a “universal
affirmation” (A), is used most by the _scientist_; as the object of
science is to establish _universal affirmative_ truths.
_The second figure._
As the second figure conditions negative conclusions only, it is called
the figure of _disproof_, or the exclusive figure. It is easy to see
how negative conclusions may be used to narrow the inquiry down to one
definite theory. For example, suppose it is desired to ascertain which
boy of the five broke the window; by a series of deductions the teacher
may be able to prove that the culprit is not A, not B, not C and not D;
hence the guilty one must be E. This figure is virtually the one used
in diagnosing most diseases.
_The third figure._
The third figure admits of particular conclusions only, and in
consequence is of little value to the scientist. Since, however, the
easiest way to contradict a universal affirmative (A) or a universal
negative (E), is to prove the truth, respectively, of a particular
negative (O) and a particular affirmative (I), it follows that the
third figure serves a purpose.
_The fourth figure._
This figure is so nearly like the first that it is of little value; in
fact, it may be changed to the first by simply interchanging the major
and minor premises. Some authorities refuse to recognize the fourth
figure.
=8. OUTLINE.=
FIGURES AND MOODS OF THE SYLLOGISM.
(1) The four figures of the syllogism.
Definition――symbolization.
Illustrations――device for remembering.
(2) The moods of the syllogism.
_Twenty-four_ valid.
(3) Testing the validity of the moods.
Application of the general rules of the syllogism.
Weakened conclusion――five.
_Nineteen_ useful moods.
A thought exercise.
(4) Special canons of the four figures.
Proof of the two canons of the first figure.
Proof of the two canons of the second figure.
Proof of the two canons of the third figure.
Proof of the three canons of the fourth figure.
(5) Special canons related.
Used as checks.
(6) Mnemonic lines.
Their use explained.
Reduction.
(7) Relative value of the four figures.
=9. SUMMARY.=
(1) By a syllogistic figure is meant some particular arrangement of the
three terms in the two premises.
This arrangement yields four figures which are designated by the
position of the middle term.
To be logical, any syllogism must conform to _one_ of the four figures.
The first figure is suggested by the position of the terms of the
“Socrates is mortal” syllogism. The second is derived by converting
the _major premise_ of the first; while the third figure results
from converting the _minor premise_ of the first, and the fourth by
converting _both_ major and minor of the first.
(2) By a mood of a syllogism is meant some particular arrangement of
the propositions which compose it.
There are 64 moods but only 24 are valid.
(3) The validity of the various moods may be tested by applying to them
the rules of the syllogism. No mood is valid if it violates any one of
the eight rules.
A “weakened conclusion” is a particular conclusion which could just as
well be universal.
Of the 24 valid moods five have weakened conclusions. This leaves but
19 useful moods.
Testing the validity of the various moods in the four figures is a most
valuable thought exercise.
(4) The deductive exercise involved in establishing certain special
canons of the four figures is of immense value and should not be
omitted.
In the first figure it may be proved (1) that the minor premise must
be affirmative; since making it negative necessitates making the major
premise negative, and no conclusion can be drawn from two negatives;
(2) that the major premise must be universal in order to distribute the
middle term at least once.
In the second figure it may be proved (1) that one premise must be
negative in order to distribute the middle term; (2) that the major
premise must be universal in order to distribute its subject, which
is distributed in the negative conclusion where it appears as the
predicate.
In the third figure it may be proved (1) that the minor premise must
be affirmative in order to prevent the “two negative” fallacy; (2) that
an affirmative minor necessitates a particular conclusion, because the
minor term in the conclusion must remain undistributed.
In the fourth figure it may be proved (1) that if the major is
affirmative, the minor must be universal in order to distribute the
middle term; (2) that if the minor is affirmative, the conclusion must
be particular in order to avoid committing the fallacy of illicit minor;
(3) that if either premise is negative, the major must be universal to
avoid the fallacy of illicit major.
(5) A knowledge of the special canons is helpful in that it may be used
to check fallacious reasoning.
(6) Certain mnemonic lines were used by the Schoolmen as an aid in
recalling the nineteen valid moods, and also as a suggestive device to
aid in the process known as Reduction.
The process of reduction is merely a matter of changing to the first
figure the moods of the other figures. This process is no longer
thought to be necessary.
(7) The first figure, called the perfect figure, is the one used
most by scientists, as it is the only figure which proves a universal
affirmative truth. The second figure is the negative, or figure of
disproof, and is used mainly for the purpose of eliminating all the
conditions of the inquiry save _one_. The third figure serves a purpose
in affording an easy way to contradict a universal assertion; this is
the figure of particulars. The fourth figure, because it so closely
resembles the first, is of little value.
=10. ILLUSTRATIVE EXERCISES.=
Question 1a. By making use of the rules for negatives and particulars,
O A A
test the validity of the following moods: I I A
A A I
Answer: The first mood has the negative O as its major premise, and
the affirmative A as its conclusion; the mood is thus invalid; because
a negative premise necessitates a negative conclusion according to
rule 6.
The second mood contains the particular proposition I as its minor
premise, and thus should have a particular conclusion according to
rule 8. But the conclusion A is universal and, therefore, the mood is
invalid.
The premises of the third mood are universal and the conclusion
particular. The mood, however, is valid, because rule 8 does not work
both ways, as does rule 6. When a universal can just as well be drawn,
then the particular becomes a weakened conclusion.
(1b) Using the rules for negatives and particulars, test the validity
A E E
of the following: A O A
E O O
(2a) Paying no regard to “figure,” derive as many conclusions as
possible from the following sets of premises: E A
I E
Answer: _E_. The major premise of this mood, being negative,
_I_
necessitates a negative conclusion, according to rule 6, and the minor
premise, being particular, compels a particular conclusion, according
to rule 8. Since the conclusion must be negative and particular, then
E
O is the only one which can be drawn. The completed mood is I.
O
_A_. This mood must have a negative conclusion, because the minor
_E_
premise is negative; this would necessitate either E or O; but O
as a conclusion would be, in this case, a _weakened_ one; since E
distributing both terms would necessarily distribute the minor; which
fact would permit the minor to be distributed in the conclusion. Thus
the conclusion could just as well be universal as particular. The
A
completed mood is E.
E
(2b) From the following sets of premises derive as many conclusions as
possible paying no attention to figure: E A O
A A A
(3a) Making use of all the general rules of the syllogism, test the
A
validity of the following mood in all the figures: A.
I
Answer: 1 2 3 4
A M ― G G ― M M ― G G ― M
― ― ― ―
A S ― M S ― M M ― S M ― S
― ― ― ―
――――― ――――― ――――― ―――――
I S ― G S ― G S ― G S ― G
An underscored symbol indicates a distributed term. Since A distributes
its subject, the subjects of both premises are underscored in all the
figures. No term is underscored in the conclusions; since I distributes
neither term. In the first figure the middle term is distributed in the
major premise, and no term is distributed in the conclusion. Since both
premises are affirmative, the rules for negatives are not applicable;
and as a particular may be drawn from two universals, if there is no
violation of the rules for distribution, this mood seems to be valid in
the first figure. It is, however, a _weakened_ conclusion; since an A
could just as well be drawn. The mood is invalid in the second figure
because of undistributed middle, but valid in both the third and fourth;
since in both cases the middle term is distributed at least once.
(3b) Determine the validity of the attending moods in all the figures
I A E
giving reasons: A O A.
I O O
=11. REVIEW QUESTIONS.=
(1) Define a logical figure and illustrate by means of some ordinary
syllogistic argument.
(2) Symbolize the four figures and give suggestions for remembering
them.
(3) Write syllogisms which illustrate each of the four figures.
(4) Define mood as it is used in logic. Illustrate.
(5) How many moods are valid?
(6) Explain by illustration a “weakened conclusion.”
A E
(7) Test the validity of E in the third figure; of I in the third.
E O
(8) Independent of all helps, prove the truth of the canons of the
first figure.
(9) In a similar way prove the canons of the second, third and fourth
figures.
(10) So far as testing arguments is concerned, what use may be made
of the special canons of the syllogism?
(11) Offer a few suggestions for remembering the special canons.
(12) Why did Aristotle attach so much importance to reduction in
logic?
(13) Justify calling the first figure the “perfect figure,” and the
others the “imperfect figures.”
(14) Treat of the relative value of the four figures.
(15) Show by illustration that the second figure is the exclusive
figure.
E O I
(16) Test the following moods in all the figures: I A A
A O I
A E E A A E A A A
E I A E I E O A I.
O O O O E I I I I
=12. QUESTIONS FOR ORIGINAL THOUGHT AND INVESTIGATION.=
(1) Give an illustration of a syllogism in the fourth figure which
might just as well be written in the first figure.
(2) May a syllogism, which is invalid in the fourth figure, be made
valid by writing it in the form of the first figure? Prove it.
(3) Show why it is impossible to apply all the rules of the ***
(4) Show the difference between a direct and an indirect proof.
A
(5) Show that A is valid in the first figure when the major premise
O
(A) is co-extensive.
(6) The third figure is known as the figure of particular
conclusions. Why should not the second canon of that figure be,
“_One premise_ must be particular” rather than “The conclusion
must be particular?”
(7) Show that there is some ground for thinking that, as a final
test, moods in the other figures should be reduced to the first.
(8) Illustrate the fact that the second figure is the figure of
disproof; whereas the third is the figure of contradictions.
(9) “To be logical a syllogism must conform to one of the four
figures, but this does not mean, necessarily, that _all
arguments_ must conform to some figure.” Explain this.