Every A is X│ All the △ is in the ○ Every X is B│ All the ○ is in the □ Therefore Every A is B│Therefore All the △ is in the □ In order to find all the possible forms of syllogism, we must make a table of all the elements of which they can consist; namely— A and X B and X Every A is X A Every B is X No A is X E No B is X Some A is X I Some B is X Some A is not X O Some B is not X Every X is A U Every X is B Some X is not A Y Some X is not B Or their synonymes, △ and ○ □ and ○ All the △ is in the ○ A All the □ is in the ○ None of the △ is in the ○ E None of the □ is in the ○ Some of the △ is in the ○ I Some of the □ is in the ○ Some of the △ is not in the ○ O Some of the □ is not in the ○ All the ○ is in the △ U All the ○ is in the □ Some of the ○ is not in the △ Y Some of the ○ is not in the □ Now, taking any one of the six relations between A and X, and combining it with either of those between B and X, we have six pairs of premises, and the same number repeated for every different relation of A and X. We have then thirty-six forms to consider: but, thirty of these (namely, all but (A, A) (E, E), &c.) are half of them repetitions of the other half. Thus, ‘Every A is X, no B is X,’ and ‘Every B is X, no A is X,’ are of the same form, and only differ by changing A into B and B into A. There are then only 15 + 6, or 21 distinct forms, some of which give a necessary conclusion, while others do not. We shall select the former of these, classifying them by their conclusions; that is, according as the inference is of the form A, E, I, or O. I. In what manner can a universal affirmative conclusion be drawn; namely, that one figure is entirely contained in the other? This we can only assert when we know that one figure is entirely contained in the circle, which itself is entirely contained in the other figure. Thus, Every A is X│ All the △ is in the ○ A Every X is B│ All the ○ is in the □ A ∴ Every A is B│∴ All the △ is in the □ A is the only way in which a universal affirmative conclusion can be drawn. II. In what manner can a universal negative conclusion be drawn; namely, that one figure is entirely exterior to the other? Only when we are able to assert that one figure is entirely within, and the other entirely without, the circle. Thus, Every A is X│All the △ is in the ○ A No B is X│None of the □ is in E │the ○ ∴ No A is B│None of the △ is in E │the □ is the only way in which a universal negative conclusion can be drawn.