For inferences of this type the following general rule may be given: Any determinant may be dropped from the subject of a universal affirmative or a particular negative proposition, if its contradictory is at the same time added as an alternant in the predicate.
This rule may be established as follows: Given All AB is C (or Some AB is not C)—and these may be taken, so far as the rule in question is concerned, as types of universal affirmatives and particular negatives respectively—we have by obversion No AB is c (or Some AB is c), and thence, by the rule for conversion given in section [455], No A is Bc (or Some A is Bc); then again obverting we have All A is either b or C (or Some A is not either b or C), the required result.
It will be observed that, as stated at the outset, these operations leave us with a proposition that is equivalent to our original proposition. There is, therefore, no loss of logical power.
By the application of the above rule with regard to all the explicit determinants of the subject any universal affirmative proposition may be brought to the form Everything is X1 or X2 … or Xn ; and it will be found that by means of this transformation, complex inferences are in many cases materially simplified.
(3) An operation which may be described as the particularisation of the subject of a proposition by the omission of one or more alternants in the predicate. Thus, from All A is B or C we may infer All Ab is C ; from Some A is not either B or C we may infer Some Ab is not C.
492 For inferences of this type the following general rule may be given: Any alternant may be dropped from the predicate of a universal affirmative or a particular negative proposition, if its contradictory is at the same time introduced as a determinant of the subject.[497]
[497] The application of this rule again leaves us with a proposition equivalent to our original proposition. The following rule, which may be regarded as a corollary from the above rule, or which may be arrived at independently, does not necessarily leave us with an equivalent: If a new determinant is introduced into the subject of a universal affirmative proposition (see section [449]) every alternant in the predicate which contains the contradictory of the determinant may be omitted. Thus, from Whatever is A or B is C or DX or Ex, we may infer Whatever is AX or BX is C or D.
The application of this rule may sometimes result in the disappearance of all the alternants from the predicate; and the meaning of such a result is that we now have a non-existent subject.
Thus, given All P is ABCD or Abcd or aBCd, if we particularise the subject by making it PbC, we find that all the alternants in the predicate disappear. The interpretation is that the class PbC is non-existent, that is, No P is bC ; a conclusion which might of course have been obtained directly from the given proposition.
This rule is the converse of that given under the preceding head; and it follows from the fact that the application of that rule leaves us with an equivalent proposition.