_Of the Canon of Reasoning._
& 569. The first figure was regarded by logicians as the only perfect
type of syllogism, because the validity of moods in this figure may be
tested directly by their complying, or failing to comply, with a
certain axiom, the truth of which is self-evident. This axiom is known
as the Dictum de Omni et Nullo. It may be expressed as follows--
Whatever may be affirmed or denied of a whole class may be affirmed
or denied of everything contained in that class.
§ 570. This mode of stating the axiom contemplates predication as
being made in extension, whereas it is more naturally to be regarded
as being made in intension.
§ 571. The same principle may be expressed intensively as follows--
Whatever has certain attributes has also the attributes which
invariably accompany them .[Footnote: Nota notae est nota rei
ipsius. 'Whatever has any mark has that which it is a mark of.'
Mill, vol. i, p. 201,]
§ 572. By Aristotle himself the principle was expressed in a neutral
form thus--
'Whatever is stated of the predicate will be stated also of the
subject [Footnote: [Greek: osa katà toû kategorouménou légetai pánta kaì
katà toû hypokeiménou rhaetésetai]. Cat. 3, § I].'
This way of putting it, however, is too loose.
§ 573. The principle precisely stated is as follows--
Whatever may be affirmed or denied universally of the predicate of
an affirmative proposition, may be affirmed or denied also of the
subject.
§ 574. Thus, given an affirmative proposition 'Whales are mammals,' if
we can affirm anything universally of the predicate 'mammals,' as, for
instance, that 'All mammals are warm-blooded,' we shall be able to
affirm the same of the subject 'whales'; and, if we can deny anything
universally of the predicate, as that 'No mammals are oviparous,' we
shall be able to deny the same of the subject.
§ 575. In whatever way the supposed canon of reasoning may be stated,
it has the defect of applying only to a single figure, namely, the
first. The characteristic of the reasoning in that figure is that some
general rule is maintained to hold good in a particular case. The
major premiss lays down some general principle, whether affirmative or
negative; the minor premiss asserts that a particular case falls under
this principle; and the conclusion applies the general principle to
the particular case. But though all syllogistic reasoning may be
tortured into conformity with this type, some of it finds expression
more naturally in other ways.
§ 576. Modern logicians therefore prefer to abandon the Dictum de Omni
et Nullo in any shape, and to substitute for it the following three
axioms, which apply to all figures alike.
_Three Axioms of Mediale Inference._
(1) If two terms agree with the same third term, they agree with one
another.
(2) If one term agrees, and another disagrees, with the same third
term, they disagree with one another.
(3) If two terms disagree with the same third term, they may or may
not agree with one another.
§ 577. The first of these axioms is the principle of all affirmative,
the second of all negative, syllogisms; the third points out the
conditions under which no conclusion can be drawn. If there is any
agreement at all between the two terms and the third, as in the cases
contemplated in the first and second axioms, then we have a conclusion
of some kind: if it is otherwise, we have none.
§ 578. It must be understood with regard to these axioms that, when we
speak of terms agreeing or disagreeing with the same third term, we
mean that they agree or disagree with the same part of it.
§ 579. Hence in applying these axioms it is necessary to bear in mind
the rules for the distinction of terms. Thus from
All B is A,
No C is B,
the only inference which can be drawn is that Some A is not C (which
alters the figure from the first to the fourth). For it was only part
of A which was known to agree with B. On the theory of the quantified
predicate we could draw the inference No C is some A.
§ 580. It is of course possible for terms to agree with different
parts of the same third term, and yet to have no connection with one
another. Thus
All birds fly.
All bats fly.
But we do not infer therefrom that bats are birds or vice versâ.
§ 581. On the other hand, had we said,--
All birds lay eggs,
No bats lay eggs,
we might confidently have drawn the conclusion
No bats are birds
For the term 'bats,' being excluded from the whole of the term 'lay
eggs,' is thereby necessarily excluded from that part of it which
coincides with 'birds.'
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