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: All dogs are useful; ∴ Some dogs (e.g., pugs) are useful. Similarly, if the major premise be verbal, thus: All men are rational; Socrates is a man-- to conclude that 'Socrates is rational' is no Mediate Inference; for so much was implied in the minor premise, 'Socrates is a man,' and the major premise adds nothing to this. Hence we may conclude (as anticipated in chap. vii. § 3) that 'any apparent syllogism, having one premise a verbal proposition, is really an Immediate Inference'; but that, if both premises are real propositions, the Inference is Mediate, and demands for its explanation something more than the Laws of Thought. The fact is that to prove the minor to be a case of the middle term may be an exceedingly difficult operation (chap. xiii. § 7). The difficulty is disguised by ordinary examples, used for the sake of convenience. § 5. Other kinds of Mediate Inference exist, yielding valid conclusions, without being truly syllogistic. Such are mathematical inferences of Equality, as-- A = B = C ∴ A = C. Here, according to the usual logical analysis, there are strictly four terms--(1) A, (2) equal to B, (3) B, (4) equal to C. Similarly with the argument _a fortiori_, A > B > C ∴ (much more) A > C. This also is said to contain four terms: (1) A, (2) greater than B, (3) B, (4) greater than C. Such inferences are nevertheless intuitively sound, may be verified by trial (within the limits of sense-perception), and are generalised in appropriate axioms of their own, corresponding to the _Dictum_ of the syllogism; as 'Things equal to the same thing are equal to one another,' etc. Now, surely, this is an erroneous application of the usual logical analysis of propositions. Both Logic and Mathematics treat of the _relations_ of terms; but whilst Mathematics employs the sign = for only one kind of relation, and for that relation exclusive of the terms; Logic employs the same signs (_is_ or _is not_) for all relations, recognising only a difference of quality in predication, and treating every other difference of relation as belonging to one of the terms related. Thus Logicians read _A--is--equal to B_: as if _equal to B_ could possibly be a term co-relative with A. Whence it follows that the argument _A = B = C ∴ A = C_ contains four terms; though everybody sees that there are only three.