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.
3. From premises both negative no conclusion can be drawn. For it is
obvious, that the mere assertion of disagreement between each of two
things and a third, can be no reason for inferring either agreement or
disagreement between these two things. It will not be difficult to
reduce any case which falls under this rule to a breach of the first
rule: thus, No A is X, No B is X, gives
Every A is (something which is not X)
Every B is (something which is not X)
in which the middle term is not spoken of universally in either. Again,
‘No X is A, Some X is not B,’ may be converted into
Every A is (a thing which is not X)
Some (thing which is not B) is X
in which there is no middle term.
4. From premises both particular no conclusion can be drawn. This is
sufficiently obvious when the first or second rule is broken, as in
‘Some A is X, Some B is X.’ But it is not immediately obvious when the
middle term enters one of the premises universally. The following
reasoning will serve for exercise in the preceding results. Since both
premises are particular in form, the middle term can only enter one of
them universally by being the predicate of a negative proposition;
consequently (Rule 3) the other premiss must be affirmative, and, being
particular, neither of its terms is universal. Consequently both the
terms as to which the conclusion is to be drawn enter partially, and the
conclusion (Rule 2) can only be a particular _affirmative_ proposition.
But if one of the premises be negative, the conclusion must be
_negative_ (as we shall immediately see). This contradiction shews that
the supposition of particular premises producing a legitimate result is
inadmissible.
5. If one premiss be negative, the conclusion, if any, must be negative.
If one term agree with a second and disagree with a third, no agreement
can be inferred between the second and third.
6. If one premiss be particular, the conclusion must be particular. This
is not very obvious, since the middle term may be universally spoken of
in a particular proposition, as in Some B is not X. But this requires
one negative proposition, whence (Rule 3) the other must be affirmative.
Again, since the conclusion must be negative (Rule 5) its predicate is
spoken of universally, and, therefore, must enter universally; the other
term A must enter, then, in a universal affirmative proposition, which
is against the supposition.
In the preceding set of syllogisms we observe one form only which
produces A, or E, or I, but three which produce O.
Let an assertion be said to be weakened when it is reduced from
universal to particular, and strengthened in the contrary case. Thus,
‘Every A is B’ is called stronger than ‘Some A is B.’
Every form of syllogism which can give a legitimate result is either one
of the preceding six, or another formed from one of the six, either by
changing one of the assertions into its converse, if that be allowable,
or by strengthening one of the premises without altering the conclusion,
or both. Thus,
Some A is X may be written Some X is A
Every X is B „ Every X is B
What follows will still follow from _Every_ X is A
„ Every X is B
for all which is true when ‘Some X is A,’ is not less true when ‘Every X
is A.’
It would be possible also to form a legitimate syllogism by weakening
the conclusion, when it is universal, since that which is true of all is
true of some. Thus, ‘Every A is X, Every X is B,’ which yields ‘Every A
is B,’ also yields ‘Some A is B.’ But writers on logic have always
considered these syllogisms as useless, conceiving it better to draw
from any premises their strongest conclusion. In this they were
undoubtedly right; and the only question is, whether it would not have
been advisable to make the premises as weak as possible, and not to
admit any syllogisms in which more appeared than was absolutely
necessary to the conclusion. If such had been the practice, then