(iv) Some S is not P,
therefore (by obversion), Some S is not-P,
therefore (by conversion), Some not-P is S,
therefore (by obversion), Some not-P is not not-S.
We can now answer the question with which we commenced this enquiry. The required proposition can be obtained only if the given proposition is universal; we then have, according as it is affirmative or negative,—
All S is P, therefore, Some not-S is not P (= Some not-S is not-P); 139
No S is P, therefore, Some not-S is P (= Some not-S is not not-P).
This form of immediate inference has been more or less casually recognised by various logicians, without receiving any distinctive name. Sometimes it has been vaguely classed under contraposition (compare Jevons, Elementary Lessons in Logic, pp. 185, 6), but it is really as far removed from the process to which that designation has been given as the latter is from ordinary conversion. The term inversion was suggested in an earlier edition of this work, and has since been adopted by some other writers. Inversion may be defined as a process of immediate inference in which from a given proposition another proposition is inferred having for its subject the contradictory of the original subject. Thus, given a proposition with S as subject and P as predicate, we obtain by inversion a new proposition with not-S as subject. The original proposition may be called the invertend, and the inferred proposition the inverse.
In the above definition it is not specified whether the inverse is to have for its predicate P or not-P. Hence two forms (each being the obverse of the other) have been obtained as in the case of contraposition. So far as it is necessary to mark the distinction, we may speak of the form in which P is the predicate as the partial inverse, and of that in which not-P is the predicate as the full inverse.
104. The Validity of Inversion.—It will be remembered that we are at present working on the assumption that each class represented by a simple term exists in the universe of discourse, while at the same time it does not exhaust that universe; in other words, we assume that S, not-S, P, not-P, all represent existing classes. This assumption is perhaps specially important in the case of inversion, and it is connected with certain difficulties that may have already occurred to the reader. In passing from All S is P to its inverse Some not-S is not P there is an apparent illicit process, which it is not quite easy either to account for or explain away. For the term P, which is undistributed in the premiss, is distributed in the conclusion, and yet if the universal validity of obversion and 140 conversion is granted, it is impossible to detect any flaw in the argument by which the conclusion is reached. It is in the assumption of the existence of the contradictory of the original predicate that an explanation of the apparent anomaly may be found. That assumption may be expressed in the form Some things are not P. The conclusion Some not-S is not P may accordingly be regarded as based on this premiss combined with the explicit premiss All S is P ; and it will be observed that, in the additional premiss, P is distributed.[145]
[145] The question of the validity of inversion under other assumptions will be considered in [chapter 8].
105. Summary of Results.—The results obtained in the preceding sections are summed up in the following table:—
| A. | E. | I. | O. | ||
| i | Original proposition | SaP | SiP | SeP | SoP |
| ii | Obverse | SePʹ | SoPʹ | SaPʹ | SiPʹ |
| iii | Converse | PiS | PiS | PeS | |
| iv | Obverted Converse | PoSʹ | PoSʹ | PaSʹ | |
| v | Partial Contrapositive[146] | PʹeS | PʹiS | PʹiS | |
| vi | Full Contrapositive[146] | PʹaSʹ | PʹoSʹ | PʹoSʹ | |
| vii | Partial Inverse[146] | SʹoP | SʹiP | ||
| viii | Full Inverse[146] | SʹiPʹ | SʹoPʹ |
[146] In previous editions what are here called the partial contrapositive and the full contrapositive respectively were called the contrapositive and the obverted contrapositive; and what are here called the partial inverse and the full inverse were called the inverse and the obverted inverse.
It may be pointed out that the following rules apply to all the above immediate inferences:— 141
Rule of Quality.—The total number of negatives admitted or omitted in subject, predicate, or copula must be even.
Rules of Quantity.—If the new subject is S, the quantity may remain unchanged; if Sʹ, the quantity must be depressed;[147] if P, the quantity must be depressed in A and O; if Pʹ, the quantity must be depressed in E and I.