Posterior Analytics (Bouchier)/Book I/Chapter XV

Chapter XV: On immediate negative propositions

 * Yet demonstration is possible in the other figures, and if of a negative character is as valid in the second figure as in the first.

Just as the quality A may inhere in B without the intervention of a middle term, so it may not inhere without such intervention. By these expressions I mean that there is no middle term connecting A and B. In that case inherence and non-inherence will no longer depend on the presence of a third term. When then either A or B, or both, are true of the whole of a third term, it is impossible that A should not be true of B immediately. We may suppose all C to be A. Then if all C is not B (for it is possible that all of a subject should be A, but none of it B) the conclusion will follow that B is not A. For if all A is C, and no B is C, then no B is A.

The same proof will be adopted if both terms are distributively predicable of a third. That B need not be predicable of a subject of which A is distributively predicable, and conversely that A need not be predicable of a third term of which B is distributively predicable may be seen clearly from a consideration of those series of terms wherein no term of the one series can be interchanged with one in the other series. Thus if none of the terms in the series A, C, D are predicable of any in the series B, E, F; if further A is distributively predicable of G, a term belonging to the same series, then it is clear that no G will be B, for otherwise these distinct series would have interchangeable terms. So too if B is distributively predicable of some other subject. If, however, neither A nor B is distributively predicable of any third term, and if A is not predicable of B, A must be not predicable of B immediately. This is so because if any middle term were present, one of the two terms named would have to be distributively predicable of a third term, since the syllogism must be either in the first or the second figure. Now if it be in the first, B will be distributively predicable of a third term, for in this case the premise must be affirmative; if it be in the second A or B may be distributively predicable of a third term, for when either premise is of a negative character a conclusion may be attained, though this is impossible when both premises are negative.

It is plain therefore that one term may be proved to be deniable of another immediately, and we have now shewn when and how this may happen.