IMPLICATION
A good illustration of the fact that what is called "implication" in
logic is such that a false proposition implies any other proposition,
true or false, is given by Lewis Carroll's puzzle of the three
barbers.[51]
Allen, Brown, and Carr keep a barber's shop together; so that one of
them must be in during working hours. Allen has lately had an illness of
such a nature that, if Allen is out, Brown must be accompanying him.
Further, if Carr is out, then, if Allen is out, Brown must be in for
obvious business reasons. The problem is, may Carr ever go out?
Putting _p_ for "Carr is out," _q_ for "Allen is out" and _r_ for "Brown
is out," we have:
(1) _q_ implies _r_,
(2) _p_ implies that _q_ implies not-_r_.
Lewis Carroll supposed that "_q_ implies _r_" and "_q_ implies not-_r_"
are inconsistent, and hence that _p_ must be false. But these
propositions are not inconsistent, and are, in fact, both true if _q_ is
false. The contradictory of "_q_ implies _r_" is "_q_ does not imply
_r_" which is not a consequence of "_q_ implies not-_r_." It seems to be
true theoretically that, if Mr. X is a Christian, he is not an Atheist,
but we cannot conclude from this alone that his being a Christian does
not imply that he is an Atheist, unless we assume that the class of
Christians is not null. Thus, if _p_ is true, _q_ is false; or, if Carr
is out, Allen is in. The odd part of this conclusion is that it is the
one which common-sense would have drawn in that particular case.
A distinguished philosopher (M) once thought that the logical use of the
word "implication"--any false proposition being said to "imply" any
proposition true or false--is absurd, on the grounds that it is
ridiculous to suppose that the proposition "2 and 2 make 5" implies the
proposition "M is the Pope." This is a most unfortunate instance,
because it so happens that the false proposition that 2 and 2 make 5 can
rigorously be proved to imply that M, or anybody else other than the
Pope, is the Pope. For if 2 and 2 make 5, since they also make 4, we
would conclude that 5 is equal to 4. Consequently, subtracting 3 from
both sides, we conclude that 2 would be equal to 1. But if this were
true, since M and the Pope are two, they would be one, and obviously
then M would be the Pope.
The principle that the false implies the true has very important
applications in political arguments. In fact, it is hard to find a
single principle of politics of which false propositions are not the
main support.
If _p_ and _q_ are two propositions, and _p_ implies _q_; then, if, and
only if, _q_ and _p_ are both false or both true, we also have: _q_
implies _p_. The most important applications of this invertibility were
made by the late Samuel Butler[52] and Mr. G. B. Shaw. A political
application may be made as follows: In a country where only those with
middling-sized incomes are taxed, conservative and _bourgeois_
politicians would still maintain that the proposition "the rich are
taxed" implies the proposition "the poor are taxed," and this
implication, which is true because both premiss and conclusion are
false, would be quite unnecessarily supported by many false practical
arguments. It is equally true that "the poor are taxed" implies that
"the rich are taxed." And this can be proved, in certain cases, on other
grounds. For the taxation of the poor would imply, ultimately, that the
poor could not afford to pay a little more for the necessities of life
than, in strict justice, they ought; and this would mean the cessation
of one of the chief means of production of individual wealth.
We also see why a valuable means for the discovery of truth is given by
the inversion of platitudinous implications. It may happen that another
platitude is the result of inversion; but it is the fate of any true
remark, especially if it is easy to remember by reason of a paradoxical
form, to become a platitude in course of time. There are rare cases of a
platitude remaining unrepeated for so long that, by a converse process,
it has become paradoxical. Such, for example, is Plato's remark that a
lie is less important than an error in thought.
Of late years, a method of disguising platitudes as paradoxes has been
too extensively used by Mr. G. K. Chesterton. The method is as follows.
Take any proposition _p_ which holds of an entity _a_; choose _p_ so
that it seems plausible that _p_ also holds of at least two other
entities _b_ and _c_; call _a_, _b_, _c_, and any others for which _p_
holds or seems to hold, the class A, and _p_ the "A-ness" or "A-ity" of
A; let _d_ be an entity for which _p_ does not hold; and put _d_ among
the A's when you think that nobody is looking. Then state your paradox:
"Some A's do not have A-ness." By further manipulation you can get the
proposition "No A's have A-ness." But it is possible to make a very
successful _coup_ if A is the null-class, which has the advantage that
manipulation is unnecessary. Thus, Mr. Chesterton, in his _Orthodoxy_
put A for the class of doubters who doubt the possibility of logic, and
proved that such agnostics refuted themselves--a conclusion which seems
to have pleased many clergymen.
In this way, Mr. Chesterton has been enabled readily to write many books
and to maintain, on almost every page, such theses as that simplicity is
not simple, heterodoxy is not heterodox, poets are not poetical, and so
on; thereby building up the gigantic platitude that Mr. Chesterton is
Chestertonian.
In the chapter on Identity we have illustrated the use of a case of the
principle that any proposition implies any true proposition. This
important principle may be called _the principle of the irrelevant
premiss_;[53] and is of great service in oratory, because it does not
matter what the premiss is, true or false. There is a _principle of the
irrelevant conclusion_, but, except in law-courts, interruptions of
meetings, and family life, this is seldom used, partly because of the
limitation involved in the logical impossibility for the conclusion to
be false if the premiss be true, but chiefly because the conclusion is
more important than the premiss, being usually a matter of prejudice.
Certain modern logicians, such as Frege, have found it necessary so to
extend the meaning of implication of _q_ by _p_ that it holds when _p_
is not a proposition at all. Hitherto, politicians, finding that either
identical or false propositions are sufficient for their needs, have
made no use of this principle; but it is obvious that their stock of
arguments would be vastly increased thereby.
Logical implication is often an enemy of dignity and eloquence. De
Morgan[54] relates "a tradition of a Cambridge professor who was once
asked in a mathematical discussion, 'I suppose you will admit that the
whole is greater than its part?' and who answered, 'Not I, until I see
what use you are going to make of it.'" And the care displayed by
cautious mathematicians like Poincare, Schoenflies, Borel, Hobson, and
Baire in abstaining from pushing their arguments to their logical
conclusions is probably founded on the unconscious--but no less
well-grounded--fear of appearing ridiculous if they dealt with such
extreme cases as "the series of all ordinal numbers."[55] They are,
probably, as unconscious of implication as Gibbon, when he remarked that
he always had a copy of Horace in his pocket, and often in his hand, was
of the necessary implication of these propositions that his hand was
sometimes in his pocket.
FOOTNOTES:
[51] _Md._, N. S., vol. iii., 1894, pp. 436-8. Cf. the discussions by W.
E. Johnson (_ibid._, p. 583) and Russell (_P. M._, p. 18, note, and
_Md._, N. S., vol. xiv., 1905, pp. 400-1).
[52] The inhabitants of "Erewhon" punished invalids more severely than
criminals. In modern times, one frequently hears the statement that
crime is a disease; and if so, it is surely false that criminals ought
to be punished.
[53] _Irrelevant_ in a popular sense; one would not say, speaking
loosely, that the fact that Brutus killed Caesar implies that the sea is
salt; and yet this conclusion is implied both by the above premiss, and
the premiss that Caesar killed Brutus. Cf. on such questions Venn,
_S. L._, 2nd ed., pp. 240-4.
[54] _F. L._, p. 264.
[55] Cf. Chapters XXIX and XXXVII.