§ 3. First Principle or Axiom of the Syllogism.—Hitherto in this chapter we have been analysing the conditions of valid mediate inference. We have seen that a single step of such inference, a Syllogism, contains, when fully expressed in language, three propositions and three terms, and that these terms must stand to one another in the relations required by the fourth, fifth, and sixth Canons. We now come to a principle which conveniently sums up these conditions; it is called the Dictum de omni et nullo, and may be stated thus:
Whatever is predicated (affirmatively or negatively) of a term distributed,
With which term another term can be (partly or wholly) identified,
May be predicated in like manner (affirmatively or negatively) of the latter term (or part of it).
Thus stated (nearly as by Whately in the introduction to his Logic) the Dictum follows line by line the course of a Syllogism in the First Figure (see [chap. x. § 2]). To return to our former example: All authors are vain is the same as—Vanity is predicated of all authors; Cicero is an author is the same as—Cicero is identified as an author; therefore Cicero is vain, or—Vanity may be predicated of Cicero. The Dictum then requires: (1) three propositions; (2) three terms; (3) that the middle term be distributed; (4) that one premise be affirmative, since only by an affirmative proposition can one term be identified with another; (5) that if one premise be negative the conclusion shall be so too, since whatever is predicated of the middle term is predicated in like manner of the minor.
Thus far, then, the Dictum is wholly analytic or verbal, expressing no more than is implied in the definitions of 'Syllogism' and 'Middle Term'; since (as we have seen) all the General Canons (except the third, which is a still more general condition of formal proof) are derivable from those definitions. However, the Dictum makes a further statement of a synthetic or real character, namely, that when these conditions are fulfilled an inference is justified; that then the major and minor terms are brought into comparison through the middle, and that the major term may be predicated affirmatively or negatively of all or part of the minor. It is this real assertion that justifies us in calling the Dictum an Axiom.
§ 4. Whether the Laws of Thought may not fully explain the Syllogism without the need of any synthetic principle has, however, been made a question. Take such a syllogism as the following:
All domestic animals are useful;
All pugs are domestic animals:
∴ All pugs are useful.
Here (an ingenious man might urge), having once identified pugs with domestic animals, that they are useful follows from the Law of Identity. If we attend to the meaning, and remember that what is true in one form of words is true in any other form, then, all domestic animals being useful, of course pugs are. It is merely a case of subalternation: we may put it in this way: