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: