which are called the four figures, and every kind of syllogism in each figure is called a mood. I now put down the various moods of each figure, the letters of which will be a guide to find out those of the preceding list from which they are derived. Co means that a premiss of the preceding list has been converted; + that it has been strengthened; Co +, that both changes have taken place. Thus,
| A | Every X is B | A | Every X is B | ||
| I | Some A is X | becomes | A | Every X is A: | (Co +) |
| I | Some A is B | I | Some A is B |
And Co + abbreviates the following: If some A be X, then some X is A (Co); and all that is true when Some X is A, is true when Every X is A (+); therefore the second is legitimate, if the first be so.
| First Figure. | |||||
| A | Every X is B | A | Every X is B | ||
| A | Every A is X | I | Some A is X | ||
| A | Every A is B | I | Some A is B | ||
| E | No X is B | E | No X is B | ||
| A | Every A is X | I | Some A is X | ||
| E | No A is B | O | Some A is not B | ||
| Second Figure. | |||||
| E | No B is X | (Co) | E | No B is X (Co) | |
| A | Every A is X | I | Some A is X | ||
| E | No A is B | O | Some A is not B | ||
| A | Every B is X | A | Every B is X | ||
| E | No A is X | (Co) | O | Some A is not X | |
| E | No A is B | O | Some A is not B | ||
| Third Figure. | |||||
| A | Every X is B | E | No X is B | ||
| A | Every X is A | (Co+) | A | Every X is A (Co+) | |
| I | Some A is B | O | Some A is not B | ||
| I | Some X is B | (Co) | O | Some X is not B | |
| A | Every X is A | A | Every X is A | ||
| I | Some A is B | O | Some A is not B | ||
| A | Every X is B | E | No X is B | ||
| I | Some X is A | (Co) | I | Some X is A (Co) | |
| I | Some A is B | O | Some A is not B | ||
| Fourth Figure. | |||||
| A | Every B is X | (+) | I | Some B is X | |
| A | Every X is A | A | Every X is A | ||
| I | Some A is B | I | Some B is A | ||
| A | Every B is X | E | No B is X (Co) | ||
| E | No X is A | A | Every X is A (Co+) | ||
| E | No A is B | O | Some A is not B | ||
| E | No B is X (Co) | ||||
| I | Some X is A (Co) | ||||
| O | Some A is not B | ||||
The above is the ancient method of dividing syllogisms; but, for the present purpose, it will be sufficient to consider the six from which the rest can be obtained. And since some of the six have A in the predicate of the conclusion, and not B, we shall join to them the six other syllogisms which are found by transposing B and A. The complete list, therefore, of syllogisms with the weakest premises and the strongest conclusions, in which a comparison of A and B is obtained by comparison of both with X, is as follows:
| Every A is X | Every B is X | Some A is X | Some B is X |
| Every X is B | Every X is A | No X is B | No X is A |
| Every A is B | Every B is A | Some A is not B | Some B is not A |
| Every A is X | Every B is X | Every A is X | Every B is X |
| No X is B | No X is A | Some B is not X | Some A is not X |
| No A is B | No B is A | Some B is not A | Some A is not B |
| Some A is X | Some B is X | Every X is A | Every X is B |
| Every X is B | Every X is A | Some X is not B | Some X is not A |
| Some A is B | Some B is A | Some A is not B | Some B is not A |
In the list of page [19], there was nothing but recapitulation of forms, each form admitting a variation by interchanging A and B. This interchange having been made, and the results collected as above, if we take every case in which B is the predicate, or can be made the predicate by allowable conversion, we have a collection of all possible weakest forms in which the result is one of the four ‘Every A is B,’ ‘No A is B,’ ‘Some A is B,’ ‘Some A is not B’; as follows. The premises are written in what appeared the most natural order, without distinction of major or minor; and the letters prefixed are according to the forms of the several premises, as in page [10].
| A | Every A is X | ||||
| U | Every X is B | ||||
| A | Every A is B | ||||
| I | Some A is X | I | Some B is X | ||
| U | Every X is B | U | Every X is A | ||
| I | Some A is B | I | Some A is B | ||
| A | Every A is X | A | Every B is X | ||
| E | No B is X | E | No A is X | ||
| E | No A is B | E | No A is B | ||
| I | Some A is X | A | Every B is X | U | Every X is A |
| E | No B is X | O | Some A is not X | Y | Some X is not B |
| O | Some A is not B | O | Some A is not B | O | Some A is not B |
Every assertion which can be made upon two things by comparison with any third, that is, every simple inference, can be reduced to one of the preceding forms. Generally speaking, one of the premises is omitted, as obvious from the conclusion; that is, one premiss being named and the conclusion, that premiss is implied which is necessary to make the conclusion good. Thus, if I say, “That race must have possessed some of the arts of life, for they came from Asia,” it is obviously meant to be asserted, that all races coming from Asia must have possessed some of the arts of life. The preceding is then a syllogism, as follows:
‘That race’ is ‘a race of Asiatic origin:’