(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.