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.  Sec. 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.  Sec. 7).  The difficulty is disguised by ordinary examples, used for the sake of convenience.

Sec. 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.

In fact (as observed in chap. ii.  Sec. 2) the sign of logical relation (is or is not), whilst usually adequate for class-reasoning (coinherence) and sometimes extensible to causation (because a cause implies a class of events), should never be stretched to include other relations in such a way as to sacrifice intelligence to formalism.  And, besides mathematical or quantitative relations, there are others (usually considered qualitative because indefinite) which cannot be justly expressed by the logical copula.  We ought to read propositions expressing time-relations (and inferences drawn accordingly) thus: