than half,' or more than
50 per cent.

Still, another apparent exception is entirely logical. Suppose we are
given, the premises--_All P is M_, and _All S is M_--the middle term is
undistributed. But take the obverse of the contrapositive of both
premises:

All m is p;
All m is s:
.'. Some s is p.

Here we have a conclusion legitimately obtained; but it is not in the
terms originally given.

For Mediate Inference depending on truly logical premises, then, it is
necessary that one premise should distribute the middle term; and the
reason of this may be illustrated even by the above supposed numerical
exceptions. For in them the premises are such that, though neither of
the two premises by itself distributes the Middle, yet they always
overlap upon it. If each premise dealt with exactly half the Middle,
thus barely distributing it between them, there would be no logical
proposition inferrible. We require that the middle term, as used in one
premise, should necessarily overlap the same term as used in the other,
so as to furnish common ground for comparing the other terms. Hence I
have defined the middle term as 'that term common to both premises by
means of which the other terms are compared.'

(5) One at least of the premises must be affirmative; or, from two
negative premises nothing can be inferred (in the given terms).

The fourth Canon required that the middle term should be given
distributed, or in its whole extent, at least once, in order to afford
sure ground of comparison for the others. But that such comparison may
be effected, something more is requisite; the relation of the other
terms to the Middle must be of a certain character. One at least of them
must be, as to its extent or denotation, partially or wholly identified
with the Middle; so that to that extent it may be known to bear to the
other term, whatever relation we are told that so much of the Middle
bears to that other term. Now, identity of denotation can only be
predicated in an affirmative proposition: one premise, then, must be
affirmative.

If both premises are negative, we only know that both the other terms
are partly or wholly excluded from the Middle, or are not identical with
it in denotation: where they lie, then, in relation to one another we
have no means of knowing. Similarly, in the mediate comparison of
quantities, if we are told that A and C are both of them unequal to B,
we can infer nothing as to the rela