It is true that in what has been called the “numerically definite syllogism,” an inference may be drawn, though our canon seems to be violated.  Thus: 

        60 sheep in 100 are horned;
        60 sheep in 100 are blackfaced: 
    .’. at least 20 blackfaced sheep in 100 are horned.

But such an argument, though it may be correct Arithmetic, is not Logic at all; and when such numerical evidence is obtainable the comparatively indefinite arguments of Logic are needless.  Another apparent exception is the following: 

      Most men are 5 feet high;
      Most men are semi-rational: 
    .’.  Some semi-rational things are 5 feet high.

Here the Middle Term (men) is distributed in neither premise, yet the indisputable conclusion is a logical proposition.  The premises, however, are really arithmetical; for ‘most’ means ‘more 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.