Suppose, next, that one of the premises is negative and one affirmative, say, M is not P, S is M (or M is P, S is not M). So long as we are dealing with singular terms and with relations of identity, such premises will imply the negative conclusion S is not P. For in such cases the two premises between them say that the two terms, S and P, are related in opposed ways to the same middle term, M, one of them (no matter which) affirming a relationship of identity, the other denying it (and so asserting a relationship of difference). Consequently, the two terms, S and P, must be different from each other, in other words, S is not P. In this case, as in the previous one, the rejection of this conclu sion would violate the law of contradiction. Lastly, suppose both premises are negative, M is not P and S is not M. In this case no inference can be drawn, for we are only told that S and P are both different from M, and their common difference from M is compatible with either identity or difference between S and P.
The foregoing results hold good also when the relationship between the singular terms of a mediate argument is not one of identity, but some other transitive relation. A transitive rela tionship is such that if it holds good between one term and another, and also between this other and a third, then it holds good between the first and third terms. Identity is one such relationship, equality is another, "less than," "greater than," "parallel with," "brother of," "sister of" are other transitive rela tionships. All such cases can be expressed in the formula M r P, S r M, therefore S r P, when r stands for the affirmative of a transitive relation. In all such cases, too, when both premises are negative, no inference is justified. But the cases are rather vari able when one premise is affirmative and the other is negative. Some cases, such as those dealing with relations of equality or parallelism, are like those concerned with identity, and warrant a negative conclusion. It is different with other cases, and, un fortunately, no general rule can be laid down—one must use his common sense and his knowledge of the systems of relationship concerned in the cases which arise.
Besides transitive relations there are certain other relations which commonly occur in mediate inference, and which may be called dovetail relations. In these cases one premise asserts one kind of relationship, the other asserts a different relationship, and the conclusion asserts yet a third kind of relationship by dove tailing or integrating the other two; e.g., "M is the wife of P,"
"S is the brother of M," therefore "S is the brother-in-law of P"; or again, "M is due south of P," "S is due east of M," there fore "S is south-east of P." Such inferences likewise can only be made with safety if one is conversant with the system of relations involved. But the mediate character of the inferences is essentially the same in all the cases considered so far; and the system of relations most frequently concerned are such as people of ordinary intelligence are familiar with. A suitable for mula for the last type of mediate inference would be M ri P, S M, therefore S P.
Mediate Inference with General Middle Terms.—When the middle term is general instead of singular, certain complica tions arise which require special care. The middle term must be the same in the two premises. But the mere fact that the same general name occurs in two propositions does not necessarily mean that the same things are referred to. The things meant in the one proposition may not be the same as those referred to in the other, and in that case, although the two propositions have a common name occurring in them, they really have no middle term or mediating link. For example, the general term "British sub jects" occurs in both propositions "All Englishmen are British subjects" and "All Australians are British subjects," but the "British subjects" are different in the two cases, and so the com mon term does not furnish a basis for mediate inference. Only when the common term is distributed in at least one of the two propositions have they really got a middle term in the strict sense. For in that case they are sure to have something in common. When the middle term is singular, it is distributed in any case ; not so when it is general. Again, when the middle term is singular the conclusion can only be singular, but when it is general the conclusion may be general or only particular, according to cir cumstances. Here the general rule already mentioned must be applied—no term may be distributed in the conclusion unless it is distributed in its premise. And so the conclusion must not be general (only particular) unless the minor term is distributed in the minor premise. Moreover, in a negative conclusion the major term is distributed, and this, for the same reason, is only permissible when the major term is distributed in the major premise. So long as the terms are all singular they are all dis tributed in any case ; but when they are general it is different, and these precautions must be borne in mind.