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 figure |
Second figure |
Third figure |
Fourth 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:
First figure: All M
men
are G
mortal,
S
Socrates
is a M
man,
∴ S
Socrates
is G
mortal.
M — G
S — M
S — G
Second figure: All
G
good citizens
love their M
country,
No S
criminal
loves his M
country,
∴ No S
criminal
is a G
good citizen.
G — M
S — M
S — G
Third figure: All
M
good citizens
are G
law abiding,
All M
good citizens S
vote,
∴ Some who S
vote
are G
law abiding.
M — G
M — S
S — G
Fourth figure: Some
G
teachers
are M
fair minded,
All who are M
fair minded
are S
just,
∴ Some S
just persons
are G
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.
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
A
E
(2) A
I
I
(3) 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
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 continued. The mood A
A
A
has stood the test of the rules for negative and 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
A
is invalid in the second, because of “undistributed middle,” and 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 should be underscored. It is now evident that A
I
I
is not valid in figures 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 be universal. A
E
O
illustrates this as the conclusion could be E. This 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) | ||||
| First figure | A A A |
E A E |
A I I |
E I O |
— — — |
— — — |
4 | ||
| Second figure | E A E |
A E E |
A O O |
E I O |
— — — |
— — — |
8 | ||
| Third figure | A A I |
I A I |
A I I |
E A O |
O A O |
E I O |
14 | ||
| Fourth figure | A A I |
A E E |
I A I |
E A O |
E I O |
— — — |
19 |
Of these nineteen moods it is not much of a tax to remember that
A
A
A
is valid only in the first figure; whereas
E
A
E
is valid in the first and second figures;
A
I
I
in the first and third; while
E
I
O
is valid in all. This 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.
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.
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 us to believe that
A
E
E
is valid in the first figure, we may recall that 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.
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, Ferioque 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 in Fesapo means that the major premise E of the mood
E
A
O
of the fourth 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.
(1) Given: A syllogism in Darapti
A
A
I
A All
M
true teachers
are G
just,
A All
M
true teachers
are S
sympathetic,
I ∴ Some
S
sympathetic persons
are G
just.
The symbols indicate that the mood is
A
A
I
or is in Darapti and that this mood is used in the third figure.
Problem: To reduce
A
A
I
of the third figure to some mood of the first 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 whose initial letter is D. This mood is in Darii, or
is A
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:
A All
M
true teachers
are G
just,
I Some
S
sympathetic persons
are M
true teachers,
I ∴ Some
S
sympathetic persons
are G
just.
The symbolization indicates that the mood is
A
I
I
of the first figure, or is in Darii.
(2) Given: A syllogism in Camestres A
E
E
A All
G
true teachers
are M
just,
E No
S
one
who shows partiality is M
just,
E ∴ No
S
one
who shows partiality is a G
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 must be reduced to the mood in Celarent
E
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:
E No
M
just person
shows G
partiality,
A All
S
true teachers
are M
just persons,
E ∴ No
S
true teacher
shows G
partiality.
Here, then, is the
E
A
E
of the first figure or the mood in Celarent.
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.