[136] The process of immediate inference discussed in this section has been called by a good many different names. The term obversion, which is used by Professor Bain, is the most convenient. Other names which have been used are permutation (Fowler), aequipollence (Ueberweg), infinitation (Bowen), immediate inference by private conception (Jevons), contraversion (De Morgan), contraposition (Spalding). Professor Bain distinguishes between formal obversion and material obversion. By formal obversion is meant the kind of obversion discussed in the above section, and this is the only kind of obversion that can properly be recognised by the formal logician. Material obversion is described as the process of making “obverse inferences which are justified only on an examination of the matter of the proposition” (Logic, vol. i., p. 111); and the following are given as examples—“Warmth is agreeable; therefore, cold is disagreeable. War is productive of evil; therefore, peace is productive of good. Knowledge is good; therefore, ignorance is bad.” It is very doubtful if these are legitimate inferences, formal or otherwise. The conclusions appear to require quite independent investigations to establish them. Apart from this, however, it is a mistake to regard the process as analogous to formal obversion. In the latter, the inferred proposition has the same subject as the original proposition, whilst its quality is different; but neither of these conditions is fulfilled in the above examples. The process is really more akin to the immediate inference presently to be discussed under the name of inversion.

We have the following table:—

Obvertend.Obverse.
All S is P.  A.No S is not-P.  E
Some S is P.  I.Some S is not not-P.  O.
No S is P.  E.All S is not-P.  A.
Some S is not P.  O.Some S is not-P.  I.

134 It will be observed that the obversion of All S is P depends upon the principle of contradiction, which tells us that if anything is P then it is not not-P; but that we pass back from No S is not-P to All S is P by the principle of excluded middle, which tells us that if anything is not not-P then it is P. The remaining inferences by obversion also depend upon one or other of these two principles.

102. The Contraposition of Categorical Propositions.[137]—Contraposition 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 predicate. Thus, given a proposition having S for its subject and P for its predicate, we seek to obtain by immediate inference a new proposition having not-P for its subject.

[137] This form of immediate inference is called by some logicians conversion by negation ; Miss Jones suggests the name contraversion. More strictly we might speak of conversion by contraposition. The word contrapositive was used by Boethius for the opposite of a term (e.g., not-A), the word contradictory being confined to propositional forms; and the passage from All S is P to All not-P is not-S was called Conversio per contrapositionem terminorum. In this usage Boethius was followed by the medieval logicians. Compare Minto, Logic, pp. 151, 153.

It will be observed that in the above definition it is left an open question whether the contrapositive of a proposition has the original subject or the contradictory of the original subject for its predicate; and every proposition which admits of contraposition will accordingly have two contrapositives, each of which is the obverse of the other. For example, in the case of All S is P there are the two forms No not-P is S and All not-P is not-S. For many purposes the distinction may be practically neglected without risk of confusion. It will be observed, however, that when not-S is taken as the predicate of the contrapositive, the quality of the original proposition is preserved and there is greater symmetry.[138] On the other hand, 135 if we regard contraposition as compounded out of obversion and conversion in the manner indicated in the following paragraph, the form with S as predicate is the more readily obtained. Perhaps the best solution (in cases in which it is necessary to mark the distinction) is to speak of the form with not-S as predicate as the full contrapositive, and the form with S as predicate as the partial contrapositive.[139]

[138] The following is from Mansel’s Aldrich, p. 61,—“Conversion by contraposition, which is not employed by Aristotle, is given by Boethius in his first book, De Syllogismo Categorico. He is followed by Petrus Hispanus. It should be observed, that the old logicians, following Boethius, maintain that in conversion by contraposition, as well as in the others, the quality should remain unchanged. Consequently the converse of ‘All A is B’ is ‘All not-B is not-A,’ and of ‘Some A is not B,’ ‘Some not-B is not not-A.’ It is simpler, however, to convert A into E, and O into I, (‘No not-B is A,’ ‘Some not-B is A’), as is done by Wallis and Archbishop Whately; and before Boethius by Apuleius and Capella, who notice the conversion, but do not give it a name. The principle of this conversion may be found in Aristotle, Top. II. 8. 1, though he does not employ it for logical purposes.”

[139] In previous editions the form with S as predicate was called the contrapositive, and the form with not-S as predicate was called the obverted contrapositive.

The following rule may be adopted for obtaining the full contrapositive of a given proposition:—Obvert the original proposition, then convert the proposition thus obtained, and then once more obvert. For given a proposition with S as subject and P as predicate, obversion will yield an equivalent proposition with S as subject and not-P as predicate; the conversion of this will make not-P the subject and S the predicate; and a repetition of the process of obversion will yield a proposition with not-P as subject and not-S as predicate.