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:

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.

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