Every X is A │ All the ○ is in the △ A Some X is not B│ Some of the ○ is not in the □ O ∴ Some A is not B│ Some of the △ is not in the □ O It appears, then, that there are but six distinct syllogisms. All others are made from them by strengthening one of the premises, or converting one or both of the premises, where such conversion is allowable; or else by first making the conversion, and then strengthening one of the premises. And the following arrangement will shew that two of them are universal, three of the others being derived from them by weakening one of the premises in a manner which does not destroy, but only weakens, the conclusion. 1. Every A is X 3. Every A is X Every X is B No B is X ......... ———————————— ———————————— Every A is B No A is B │ │ │ ┌─────────┴─────────┐ │ │ │ 2. Some A is X 4. Some A is X 5. Every A is X 6. Every X is A Every X is B No B is X Some B is not X Some X is not B ———————————— ——————————————— ———————————————— ——————————————— Some A is B Some A is not B Some B is not A Some A is not B We may see how it arises that one of the partial syllogisms is not immediately derived, like the others, from a universal one. In the preceding, AEE may be considered as derived from AAA, by changing the term in which X enters universally into its contrary. If this be done with the other term instead, we have No A is X│from which universal premises we cannot deduce a universal │ conclusion, but only Some B is not A. Every X is B│ „ If we weaken one and the other of these premises, as they stand, we obtain Some A is not X No A is X Every X is B and Some X is B ———————————————— ——————————————— No conclusion Some B is not A equivalent to the fourth of the preceding: but if we convert the first premiss, and proceed in the same manner, No X is A Some X is not A From Every X is B we obtain Every X is B ———————————————— ———————————————— Some B is not A Some B is not A which is legitimate, and is the same as the last of the preceding list, with A and B interchanged. Before proceeding to shew that all the usual forms are contained in the preceding, let the reader remark the following rules, which may be proved either by collecting them from the preceding cases, or by independent reasoning.