MODERN LOGIC AND SOME PHILOSOPHICAL ARGUMENTS
The most noteworthy reformation of recent years in logic is the
discovery and development by Mr. Bertrand Russell of the fact that the
paradoxes--of Burali-Forti, Russell, Koenig, Richard, and others--which
have appeared of late years in the mathematical theory of aggregates and
have just been referred to, are of an entirely _logical_ nature, and
that their avoidance requires us to take account of a principle which
has been hitherto unrecognized, and which renders invalid several
well-known arguments in refutation of scepticism, agnosticism, and the
statement of a man that he asserts nothing.
Dr. Whitehead and Mr. Russell say:[113] "The principle which enables us
to avoid illegitimate totalities may be stated as follows: 'Whatever
involves _all_ of a collection must not be one of the collection,' or
conversely: 'If, provided a certain collection had a total, it would
have members only definable in terms of that total, then the said
collection has no total.' We shall call this the 'vicious-circle
principle,' because it enables us to avoid the vicious circles involved
in the assumption of illegitimate totalities. Arguments which are
condemned by the vicious-circle principle will be called 'vicious-circle
fallacies.' Such arguments, in certain circumstances, may lead to
contradictions, but it often happens that the conclusions to which they
lead are in fact true, though the arguments are fallacious. Take, for
example, the law of excluded middle in the form 'all propositions are
true or false.' If from this law we argue that, because the law of
excluded middle is a proposition, therefore the law of excluded middle
is true or false, we incur a vicious-circle fallacy. 'All propositions'
must be in some way limited before it becomes a legitimate totality, and
any limitation which makes it legitimate must make any statement about
the totality fall outside the totality. Similarly the imaginary sceptic
who asserts that he knows nothing and is refuted by being asked if he
knows that he knows nothing, has asserted nonsense, and has been
fallaciously refuted by an argument which involves a vicious-circle
fallacy. In order that the sceptic's assertion may become significant it
is necessary to place some limitation upon the things of which he is
asserting his ignorance; the proposition that he is ignorant of every
member of this collection must not itself be one of the collection.
Hence any significant scepticism is not open to the above form of
refutation."
In fact, the world of things falls into various sets of things of the
same "type." For every propositional function [Greek: ph](_x_) there is
a range of values of _x_ for which [Greek: ph](_x_) has a signification
as a true or a false proposition. Until this theory was brought forward,
there were occasionally discussions as to whether an object which did
not belong to the range of a certain propositional function possessed
the corresponding property or not. Thus, Jevons, in early days,[114] was
of opinion that virtue is neither black nor not-black because it is not
coloured, but rather later[115] he admitted that virtue is not
triangular.[116]
FOOTNOTES:
[113] _Pa. Ma._, p. 40.
[114] _S. o. S._ pp. 36-7.
[115] _E. L. L._, pp. 120-1.
[116] [It may perhaps be added that, some years after Mr. R*ss*ll's
death, Dr. Whitehead stated, in an address delivered in 1916 and
reprinted in his book on _The Organisation of Thought_ (London, 1917, p.
120), that "the specific heat of virtue is 0.003 is, I should imagine,
not a proposition at all, so that it is neither true nor
false...."--ED.]