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.

Sec. 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: 

      All domestic animals are useful: 
    .’.  Some domestic animals (e.g., pugs) are useful.

The derivation of negative syllogisms from the Law of Contradiction (he might add) may be shown in a similar manner.

But the force of this ingenious argument depends on the participial clause—­’having once identified pugs with domestic animals.’  If this is a distinct step of the reasoning, the above syllogism cannot be reduced to one step, cannot be exhibited as mere subalternation, nor be brought directly under the law of Identity.  If ‘pug,’ ‘domestic,’ and ‘useful’ are distinct terms; and if ‘pug’ and ‘useful’ are only known to be connected because of their relations to ‘domestic’:  this is something more than the Laws of Thought provide for:  it is not Immediate Inference, but Mediate; and to justify it, scientific method requires that its conditions be generalised.  The Dictum, then, as we have seen, does generalise these conditions, and declares that when such conditions are satisfied a Mediate Inference is valid.

But, after all (to go back a little), consider again that proposition All pugs are domestic animals:  is it a distinct step of the reasoning; that is to say, is it a Real Proposition?  If, indeed, ‘domestic’ is no part of the definition of ‘pug,’ the proposition is real, and is a distinct part of the argument.  But take such a case as this: 

    All dogs are useful;
    All pugs are dogs.

Here we clearly have, in the minor premise, only a verbal proposition; to be a dog is certainly part of the definition of ‘pug.’  But, if so, the inference ‘All pugs are useful’ involves no real mediation, and the argument is no more than this: