These rules, and also the rules given in the preceding section, may be established by the aid of the following axioms: What is true of all (distributively) is true of every part ; What is true of part of a part is true of a part of the larger whole.
486 When we add a determinant to a term, or remove an alternant, we usually diminish, and at any rate do not increase, the extension of the term; when, on the other hand, we add an alternant, or remove a determinant, we usually increase, and at any rate do not diminish, its extension. Hence it follows that if a term is distributed we may add a determinant or remove an alternant, whilst if a term is undistributed we may add an alternant or remove a determinant. Thus,
All A is CD, therefore, All AB is C ;
No A is C, therefore, No AB is CD ;
Some AB is C, therefore, Some A is C or D ;
Some AB is not either C or D, therefore, Some A is not C.
From the above rules taken in connexion with the rules given in section [445] we may obtain the following corollaries:
(3) In universal affirmatives, any determinant may be introduced into the predicate, if it is also introduced into the subject; and any alternant may be introduced into the subject if it is also introduced into the predicate.
Given All A is C, then All AB is C by rule (1) above; and from this we obtain All AB is BC by rule (2) of section [445].
Again, given All A is C, then All A is B or C ; and therefore, by rule (3) of section [445], Whatever is A or B is B or C.
(4) In universal negatives any alternant may be introduced into subject or predicate, if its contradictory is introduced into the other term as a determinant.
Given No A is C, then No AB is C ; and, therefore, by rule (5) of section [445], No AB is b or C.
Again, given No A is C, then No A is BC ; and, therefore, by rule (6) of section [445], No A or b is BC.
In none of the inferences considered in this section is it possible to pass back from the conclusion to the original proposition.
450. Interpretation of Anomalous Forms.—It will be found that propositions which apparently involve a contradiction in terms and are thus in direct contravention of the fundamental laws of thought—for example, No AB is B, All Ab is B—sometimes result from the manipulation of complex propositions. In interpreting such propositions as these, a distinction must be drawn between universals and particulars, at any rate if particulars are interpreted as implying, while universals are not interpreted as implying, the existence of their subjects.
487 It can be shewn that a universal proposition of the form No AB is B or All Ab is B must be interpreted as implying the non-existence in the universe of discourse of the subject of the proposition. For a universal negative denies the existence of anything that comes under both its subject and its predicate; thus, No AB is B denies the existence of ABB, that is, it denies the existence of AB. Again, a universal affirmative denies the existence of anything that comes under its subject without also coming under its predicate; thus, All Ab is B denies the existence of anything that is Ab and at the same time not-B, that is, b ; but Ab is Ab and also b, and hence the existence of Ab is denied.
Since the existence of its subject is held to be part of the implication of a particular proposition, the above interpretation is obviously inapplicable in the case of particulars. Hence if a proposition of the form Some Ab is B is obtained, we are thrown back on the alternative that there is some inconsistency in the premisses; either some one individual premiss is self-contradictory, or the premisses are inconsistent with one another.
EXERCISES.
451. Shew that if No A is bc or Cd, then No A is bd. [K.]
452. Give the contradictory of each of the following propositions:—(1) Flowering plants are either endogens or exogens, but not both; (2) Flowering plants are vascular, and either endogens or exogens, but not both. [M.]