103. The Inversion of Categorical Propositions.—In discussing conversion and contraposition we have enquired in what cases it is possible, having given a proposition with S as subject and P as predicate, to infer (a) a proposition with P as subject, (b) a proposition with not-P as subject. We may now enquire further in what cases it is possible to infer (c) a proposition with not-S as subject.
If such a proposition can be inferred at all, it will be obtainable by a certain combination of the more elementary processes of ordinary conversion and obversion.[144] We will, therefore, take each of the fundamental forms of proposition and see what can be inferred (1) by first converting it, and then performing alternately the operations of obversion and conversion; (2) by first obverting it, and then performing alternately the operations of conversion and obversion. It will be found that in each case the process can be continued until a particular negative proposition is reached whose turn it is to be converted.
[144] It might also be obtained directly; by the aid, for example, of Euler’s circles. See the following [chapter].
(1) The results of performing alternately the processes of conversion and obversion, commencing with the former, are as follows:—
(i) All S is P,
therefore (by conversion), Some P is S,
therefore (by obversion), Some P is not not-S.
Here comes the turn for conversion; but as we have to deal with an O proposition, we can proceed no further.
(ii) Some S is P,
therefore (by conversion), Some P is S,
therefore (by obversion), Some P is not not-S ;
and again we can go no further. 138
(iii) No S is P,
therefore (by conversion), No P is S,
therefore (by obversion), All P is not-S,
therefore (by conversion), Some not-S is P,
therefore (by obversion), Some not-S is not not-P.
In this case either of the propositions in italics is the immediate inference that was sought.
(iv) Some S is not P.
In this case we are not able even to commence our series of operations.
(2) The results of performing alternately the processes of conversion and obversion, commencing with the latter, are as follows:—
(i) All S is P,
therefore (by obversion), No S is not-P,
therefore (by conversion), No not-P is S,
therefore (by obversion), All not-P is not-S,
therefore (by conversion), Some not-S is not-P,
therefore (by obversion), Some not-S is not P.
Here again we have obtained the desired form.
(ii) Some S is P,
therefore (by obversion), Some S is not not-P.
(iii) No S is P,
therefore (by obversion), All S is not-P,
therefore (by conversion), Some not-P is S,
therefore (by obversion), Some not-P is not not-S.