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.