Some AB is C or D,
Some AC or AD is B,
Some B is AC or AD,
Some C or D is AB,
Some BC or BD is A,
Something is ABC or ABD.
There is no inference by conversion from a universal affirmative or from a particular negative.
456. The Contraposition of Complex Propositions.—According to our original definition of contraposition, we contraposit a proposition when we infer from it a new proposition having the contradictory of the old predicate for its subject. Adopting this definition, the contrapositive of All A is B or C is All bc is a.
The process can be applied to universal affirmatives and to particular negatives. By obversion, conversion, and then again obversion, it is clear that in each of these cases we may obtain a legitimate contrapositive by taking as a new subject the contradictory of the old predicate, and as a new predicate the contradictory of the old subject, the proposition retaining its original quality. For example: All A is BC, therefore, Whatever is b or c is a ; Some A is not either B or C, therefore, Some bc is not a.
The above may be called the full contrapositive of a complex proposition. It should be observed that any proposition and its full contrapositive are equivalent to each other; we can pass back from the full contrapositive to the original proposition.
In dealing with complex propositions, however, it is convenient to give to the term contraposition an extended meaning. We may say that we have a process of contraposition when from a given proposition we infer a new one in which the contradictory of any term that appeared in the predicate of the original proposition now appears 491 in the subject, or the contradictory of any term that appeared in the subject of the original proposition now appears in the predicate.
Three operations may be distinguished all of which are included under the above definition, and all of which leave us with a full equivalent of the original proposition, so that there is no loss of logical power.
(1) The operation of obtaining the full contrapositive of a given proposition, as above described and defined.[496]
[496] In some cases we may desire to drop part of the information given by the complete contrapositive. Thus, from All A is BC or E may infer Whatever is be or ce is a ; but in a given application it may be sufficient for us to know that All be is a.
(2) An operation which may be described as the generalisation of the subject of a proposition by the addition of one or more alternants in the predicate. Thus, from All AB is C we may infer All A is b or C ; from Some AB is not either C or D we may infer Some A is not either b or C or D.