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, test the validity of the following moods: O
I
A A
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 of the following: A
A
E E
O
O E
O
O.
(2a) Paying no regard to “figure,” derive as many conclusions as possible from the following sets of premises: E
I A
E.
Answer:
E
I.
The major premise of this mood, being negative, 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 O is the only one which can be drawn. The completed mood
is E
I
O.
A
E.
This mood must have a negative conclusion, because the minor 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 completed mood
is A
E
E.
(2b) From the following sets of premises derive as many conclusions as possible paying no attention to figure: E
A A
A O
A.
(3a) Making use of all the general rules of the syllogism, test the validity of the following mood in all the
figures: A
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 giving reasons: I
A
I A
O
O E
A
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.”
(7) Test the validity of
A
E
E
in the third figure; of E
I
O
in the third.
(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.
(16) Test the following moods in all the figures: E
I
A
O
A
O
I
A
I
A
E
O
E
I
O
E
A
O
A
E
O
A
I
E
E
E
I
A
O
I
A
A
I
A
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.
(5) Show that A
A
O
is valid in the first figure when the major premise (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.