Part 4

Every X is A│ All the ○ is in the △ A

Every X is B│ All the ○ is in the □ A

∴ Some A is B│∴ Some of the △ is in I │ the □

Every X is A│ All the ○ is in the △ A

Some X is B│ Some of the ○ is in I │ the □

Some A is B│ Some of the △ is in I │ the □

The second of these contains all that is strictly necessary to the conclusion, and the first may be omitted. That which follows when an assertion can be made as to some, must follow when the same assertion can be made of all.

IV. How can a particular negative proposition be inferred; namely, that part, or all of one figure, is not contained in the other? It would seem at first sight, whenever we are able to assert that part or all of one figure is in the circle, and that part or all of the other figure is not. The weakest syllogism from which such an inference can be drawn would then seem to be as follows.

Some A is X │ Some of the △ is in the ○ Some B is not X│ Some of the □ is not in the ○ ∴ Some B is not A│∴ Some of the △ is not in the □

But here it will appear, on a little consideration, that the conclusion is only thus far true; that those As which are Xs cannot be those Bs which are not Xs; but they may be other Bs, about which nothing is asserted when we say that some Bs are not Xs. And further consideration will make it evident, that a conclusion of this form can only be arrived at when one of the figures is entirely within the circle, and the whole or part of the other without; or else when the whole of one of the figures is without the circle, and the whole or part of the other within; or lastly, when the circle lies entirely within one of the figures, and not entirely within the other. That is, the following are the distinct forms which allow of a particular negative conclusion, in which it should be remembered that a particular proposition in the premises may always be changed into a universal one, without affecting the conclusion. For that which necessarily follows from “some,” follows from “all.”

Every A is X │ All the △ is in the ○ A Some B is not X│ Some of the □ is not in the ○ O ∴ Some B is not A│ Some of the □ is not in the △ O

No A Is X │ None of the △ is in the ○ E Some B is X │ Some of the □ is in the ○ I ∴ Some B is not A│ Some of the □ is not in the △ O

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.

1. The middle term must enter universally into one or the other premiss. If it were not so, the one premiss might speak of one part of the middle term, and the other of the other; so that there would, in fact, be no middle term. Thus, ‘Every A is X, Every B is X,’ gives no conclusion: it may be thus stated;

All the As make up a part of the Xs All the Bs make up a part of the Xs

And, before we can know that there is any common term of comparison at all, we must have some means of shewing that the two parts are the same; or the preceding premises by themselves are inconclusive.

2. No term must enter the conclusion more generally than it is found in the premises; thus, if A be spoken of partially in the premises, it must enter partially into the conclusion. This is obvious, since the conclusion must assert no more than the premises imply.