THE SYNTHETIC NATURE OF DEDUCTION
Doubt has often been expressed as to whether a syllogism can add to
our knowledge in any way. John Stuart Mill and Henri Poincare, in
particular, held the opinion that the conclusion of a syllogism is an
"analytic" judgment in the sense of Kant, and therefore could be
obtained by the mere dissection of the premisses. Any one, then, who
maintains that mathematics is founded solely on logical principles would
appear to maintain that mathematics, in the last instance, reduces to a
huge tautology.
Mill, in Chapter III of Book II of his _System of Logic_, said that "it
must be granted that in every syllogism, considered as an argument to
prove the conclusion, there is a _petitio principii_. When we say
All men are mortal,
Socrates is a man,
therefore
Socrates is mortal,
it is unanswerably urged by the adversaries of the syllogistic theory,
that the proposition, Socrates is mortal, is presupposed in the more
general assumption, All men are mortal; that we cannot be assured of the
mortality of all men unless we are already certain of the mortality of
every individual man; that if it be still doubtful whether Socrates, or
any other individual we choose to name, be mortal or not, the same
degree of uncertainty must hang over the assertion, All men are mortal;
that the general principle, instead of being given as evidence of the
particular case, cannot itself be taken for true without exception until
every shadow of doubt which could affect any case comprised with it is
dispelled by evidence _aliunde_; and then what remains for the syllogism
to prove? That, in short, no reasoning from general to particular can,
as such, prove anything, since from a general principle we cannot infer
any particulars but those which the principle itself assumes as known.
This doctrine appears to me irrefragable...."
But it is not difficult to see that in certain cases at least deduction
gives us _new_ knowledge.[56] If we already know that two and two always
make four, and that Asquith and Lloyd George are two and so are the
German Emperor and the Crown Prince, we can deduce that Asquith and
Lloyd George and the German Emperor and the Crown Prince are four. This
is new knowledge, not contained in our premisses, because the general
proposition, "two and two are four," never told us there were such
people as Asquith and Lloyd George and the German Emperor and the Crown
Prince, and the particular premisses did not tell us that there were
four of them, whereas the particular proposition deduced does tell us
both these things. But the newness of the knowledge is much less certain
if we take the stock instance of deduction that is always given in books
on logic, namely "All men are mortal; Socrates is a man, therefore
Socrates is mortal." In this case what we really know beyond reasonable
doubt is that certain men, A, B, C, were mortal, since, in fact, they
have died. If Socrates is one of these men, it is foolish to go the
roundabout way through "all men are mortal" to arrive at the conclusion
that _probably_ Socrates is mortal. If Socrates is not one of the men on
whom our induction is based, we shall still do better to argue straight
from our A, B, C, to Socrates, than to go round by the general
proposition, "all men are mortal." For the probability that Socrates is
mortal is greater, on our data, than the probability that all men are
mortal. This is obvious, because if all men are mortal, so is Socrates;
but if Socrates is mortal, it does not follow that all men are mortal.
Hence we shall reach the conclusion that Socrates is mortal, with a
greater approach to certainty if we make our argument purely inductive
than if we go by way of "all men are mortal" and then use deduction.
Many years ago there appeared, principally owing to the initiative of
Dr. F. C. S. Schiller of Oxford, a comic number of _Mind_. The idea was
extraordinarily good, not so the execution. A German friend of Dr.
Schiller was puzzled by the appearance of the advertisements, which were
doubtfully humorous. However, by a syllogistic process, he acquired
information which was new and useful to him, and thus incidentally
refuted Mill. Presumably he started from the title of the magazine
(_Mind!_), for a mark of exclamation seems nearly always in German to be
a sign of an intended joke (including of course the mark after the
politeness expressed in the first sentence of a private letter or a
public address). There would be, then, the following syllogism:
This is a book of would-be jokes (i.e. everything in this book is a
would-be joke);
This advertisement is in this book;
Therefore, this advertisement is a would-be joke.
Thus the syllogism may be almost as powerful an agent in the detection
of humour as M. Bergson's criterion, to be described in a future
chapter.[57]
FOOTNOTES:
[56] [The following passage is almost word for word the same as a
passage on pp. 123-5 of Mr. Russell's _Problems of Philosophy_, first
published in 1912, a year after Mr. R*ss*ll's death. It is easy hastily
to conclude that Mr. Russell was indebted to Mr. R*ss*ll to a greater
degree than is usually supposed. But an examination of the internal
evidence leads us to another conclusion. The two texts, it will be
found, differ only in the names of the German Emperor, the Crown Prince
and the other personages being replaced, in the book of 1912, by those
of Messrs. Brown, Jones, Smith, and Robinson. Now, Mr. Russell, in a new
edition of his _Problems_ issued near the beginning of the European war
and before the Russian revolution, substituted "the Emperor of Russia"
for "the Emperor of China" of the first edition. Hence it seems quite
likely that Mr. Russell, who has always shown a tendency to substitute
existents for nonentities, wrote Mr. R*ss*ll's notes.--<sc>Ed.</sc>]
[57] [See Chapter XLII.--ED.]