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 certainspecial 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.