_ON THE CREDIBILITY OF EXTRAORDINARY STORIES._
1. It is now time to recur for fuller investigation to an
enquiry which has been already briefly touched upon more
than once; that is, the validity of testimony to establish,
as it is frequently expressed, an otherwise improbable story.
It will be remembered that in a previous chapter (the
twelfth) we devoted some examination to an assertion by
Butler, which seemed to be to some extent countenanced
by Mill, that a great improbability before the proof might
become but a very small improbability after the proof. In
opposition to this it was pointed out that the different
estimates which we undoubtedly formed of the credibility
of the examples adduced, had nothing to do with the
fact of the event being past or future, but arose from a
very different cause; that the conception of the event
which we entertain at the moment (which is all that is then
and there actually present to us, and as to the correctness
of which as a representation of facts we have to make up our
minds) comes before us in two very different ways. In one
instance it was a mere guess of our own which we knew
from statistics would be right in a certain proportion of
cases; in the other instance it was the assertion of a witness,
and therefore the appeal was not now primarily to statistics of
the event, but to the trustworthiness of the witness. The conception,
or 'event' if we will so term it, had in fact passed
out of the category of guesses (on statistical grounds), into
that of assertions (most likely resting on some specific evidence),
and would therefore be naturally regarded in a very
different light.
2. But it may seem as if this principle would lead us
to somewhat startling conclusions. For, by transferring the
appeal from the frequency with which the event occurs to
the trustworthiness of the witness who makes the assertion,
is it not implied that the probability or improbability of an
assertion depends solely upon the veracity of the witness?
If so, ought not any story whatever to be believed when
it is asserted by a truthful person?
In order to settle this question we must look a little
more closely into the circumstances under which such
testimony is commonly presented to us. As it is of course
necessary, for clearness of exposition, to take a numerical
example, let us suppose that a given statement is made by
a witness who, on the whole and in the long run, is right
in what he says nine times out of ten.[1] Here then is an
average given to us, an average veracity that is, which
includes all the particular statements which the witness has
made or will make.
3. Now it has been abundantly shown in a former
chapter (Ch. IX. §§14-32) that the mere fact of a particular
average having been assigned, is no reason for our
being forced invariably to adhere to it, even in those cases
in which our most natural and appropriate ground of judgment
is found in an appeal to statistics and averages.
The general average may constantly have to be corrected
in order to meet more accurately the circumstances of particular
cases. In statistics of mortality, for instance; instead
of resorting to the wider tables furnished by people in
general of a given age, we often prefer the narrower tables
furnished by men of a particular profession, abode, or mode
of life. The reader may however be conveniently reminded
here that in so doing we must not suppose that we are able,
by any such device, in any special or peculiar way to secure
truth. The general average, if persistently adhered to
throughout a sufficiently wide and varied experience, would
in the long run tend to give us the truth; all the advantage
which the more special averages can secure for us is to give
us the same tendency to the truth with fewer and slighter
aberrations.
4. Returning then to our witness, we know that if
we have a very great many statements from him upon all
possible subjects, we may feel convinced that in nine out of
ten of these he will tell us the truth, and that in the tenth
case he will go wrong. This is nothing more than a matter
of definition or consistency. But cannot we do better than
thus rely upon his general average? Cannot we, in almost
any given case, specialize it by attending to various characteristic
circumstances in the nature of the statement
which he makes; just as we specialize his prospects of
mortality by attending to circumstances in his constitution
or mode of life?
Undoubtedly we may do this; and in any of the practical
contingencies of life, supposing that we were at all guided
by considerations of this nature, we should act very foolishly
if we did not adopt some such plan. Two methods of thus
correcting the average may be suggested: one of them being
that which practical sagacity would be most likely to employ,
the other that which is almost universally adopted by writers
on Probability. The former attempts to make the correction
by the following considerations: instead of relying upon the
witness' general average, we assign to it a sort of conjectural
correction to meet the case before us, founded on our experience
or observation; that is, we appeal to experience to
establish that stories of such and such a kind are more or
less likely to be true, as the case may be, than stories in
general. The other proceeds upon a different and somewhat
more methodical plan. It is here endeavoured to
show, by an analysis of the nature and number of the
sources of error in the cases in question, that such and such
kinds of stories must be more or less likely to be correctly
reported, and this in certain numerical proportions.
5. Before proceeding to a discussion of these methods
a distinction must be pointed out to which writers upon the
subject have not always attended, or at any rate to which
they have not generally sufficiently directed their readers'
attention.[2] There are, broadly speaking, two different ways
in which we may suppose testimony to be given. It may, in
the first place, take the form of a reply to an alternative
question, a question, that is, framed to be answered by _yes_
or _no_. Here, of course, the possible answers are mutually
contradictory, so that if one of them is not correct the other
must be so:--Has A happened, yes or no? The common
mode of illustrating this kind of testimony numerically is by
supposing a lottery with a prize and blanks, or a bag of balls
of two colours only, the witness knowing that there are only
two, or at any rate being confined to naming one or other of
them. If they are black and white, and he errs when black
is drawn, he must say 'white,' The reason for the prominence
assigned to examples of this class is, probably, that
they correspond to the very important case of verdicts of
juries; juries being supposed to have nothing else to do than
to say 'guilty' or 'not guilty.'
On the other hand, the testimony may take the form of a
more original statement or piece of information. Instead of
saying, Did A happen? we may ask, What happened? Here
if the witness speaks truth he must be supposed, as before,
to have but one way of doing so; for the occurrence of some
specific event was of course contemplated. But if he errs he
has many ways of going wrong, possibly an infinite number.
Ordinarily however his possible false statements are assumed
to be limited in number, as must generally be more or less
the result in practice. This case is represented numerically
by supposing the balls in the bag not to be of two colours
only, but to be all distinct from each other; say by their
being all numbered successively. It may of course be objected
that a large number of the statements that are made
in the world are not in any way answers to questions, either
of the alternative or of the open kind. For instance, a man
simply asserts that he has drawn the seven of spades from a
pack of cards; and we do not know perhaps whether he had
been asked 'Has that card been drawn?' or 'What card has
been drawn?' or indeed whether he had been asked anything
at all. Still more might this be so in the case of any ordinary
historical statement.
This objection is quite to the point, and must be recognized
as constituting an additional difficulty. All that we
can do is to endeavour, as best we may, to ascertain, from
the circumstances of the case, what number of alternatives
the witness may be supposed to have had before him. When
he simply testifies to some matter well known to be in dispute,
and does not go much into detail, we may fairly consider
that there were practically only the two alternatives
before him of saying 'yes' or 'no.' When, on the other hand,
he tells a story of a more original kind, or (what comes to
much the same thing) goes into details, we must regard him
as having a wide comparative range of alternatives before
him.
These two classes of examples, viz. that of the black and
white balls, in which only one form of error is possible, and
the numbered balls, in which there may be many forms of
error, are the only two which we need notice. In practice it
would seem that they may gradually merge into each other,
according to the varying ways in which we choose to frame
our question. Besides asking, Did you see A strike B? and,
What did you see? we may introduce any number of intermediate
leading questions, as, What did A do? What did
he do to B? and so on. In this way we may gradually narrow
the possible openings to wrong statement, and so approach
to the direct alternative question. But it is clear that all
these cases may be represented numerically by a supposed
diminution in the number of the balls which are thus distinguished
from each other.
6. Of the two plans mentioned in §4 we will begin
with the latter, as it is the only methodical and scientific one
which has been proposed. Suppose that there is a bag with
1000 balls, only one of which is white, the rest being all
black. A ball is drawn at random, and our witness whose
veracity is 9/10 reports that the white ball was drawn. Take
a great many of his statements upon this particular subject,
say 10,000; that is, suppose that 10,000 balls having been
successively drawn out of this bag, or bags of exactly the
same kind, he makes his report in each case. His 10,000
statements being taken as a fair sample of his general average,
we shall find, by supposition, that 9 out of every 10 of
them are true and the remaining one false. What will be
the nature of these false statements? Under the circumstances
in question, he having only one way of going wrong,
the answer is easy. In the 10,000 drawings the white ball
would come out 10 times, and therefore be rightly asserted
9 times, whilst on the one of these occasions on which he
goes wrong he has nothing to say but 'black.' So with the
9990 occasions on which black is drawn; he is right and
says black on 8991 of them, and is wrong and therefore says
white on 999 of them. On the whole, therefore, we conclude
that out of every 1008 times on which he says that white is
drawn he is wrong 999 times and right only 9 times. That
is, his special veracity, as we may term it, for cases of this
description, has been reduced from 9/10 to 9/1008. As it would
commonly be expressed, the latter fraction represents the
chance that this particular statement of his is true.[3]
7. We will now take the case in which the witness
has many ways of going wrong, instead of merely one. Suppose
that the balls were all numbered, from 1 to 1,000, and
the witness knows this fact. A ball is drawn, and he tells
me that it was numbered 25, what are the odds that he is
right? Proceeding as before, in 10,000 drawings this ball
would be obtained 10 times, and correctly named 9 times.
But on the 9990 occasions on which it was not drawn there
would be a difference, for the witness has now many openings
for error before him. It is, however, generally considered
reasonable to assume that his errors will all take the form of
announcing wrong numbers; and that, there being no apparent
reason why he should choose one number rather than another,
he will be likely to announce all the wrong ones equally
often. Hence his 999 errors, instead of all leading him now
back again to one spot, will be uniformly spread over as
many distinct ways of going wrong. On one only of these
occasions, therefore, will he mention 25 as having been
drawn. It follows therefore that out of every 10 times that
he names 25 he is right 9 times; so that in this case his
average or general truthfulness applies equally well to the
special case in point.
8. With regard to the truth of these conclusions, it
must of course be admitted that if we grant the validity of
the assumptions about the limits within which the blundering
or mendacity of the witness are confined, and the complete
impartiality with which his answers are disposed within
those limits, the reasoning is perfectly sound. But are not
these assumptions extremely arbitrary, that is, are not our
lotteries and bags of balls rendered perfectly precise in many
respects in which, in ordinary life, the conditions supposed to
correspond to them are so vague and uncertain that no such
method of reasoning becomes practically available? Suppose
that a person whom I have long known, and of whose measure
of veracity and judgment I may be supposed therefore
to have acquired some knowledge, informs me that there is
something to my advantage if I choose to go to certain
trouble or expense in order to secure it. As regards the
general veracity of the witness, then, there is no difficulty;
we suppose that this is determined for us. But as regards
his story, difficulty and vagueness emerge at every point.
What is the number of balls in the bag here? What in fact
are the nature and contents of the bag out of which we suppose
the drawing to have been made? It does not seem
that the materials for any rational judgment exist here.
But if we are to get at any such amended figure of veracity
as those attained in the above example, these questions must
necessarily be answered with some degree of accuracy; for
the main point of the method consists in determining how
often the event must be considered _not_ to happen, and thence
inferring how often the witness will be led wrongly to assert
that it has happened.
It is not of course denied that considerations of the kind
in question have some influence upon our decision, but only
that this influence could under any ordinary circumstances
be submitted to numerical determination. We are doubtless
liable to have information given to us that we have
come in for some kind of fortune, for instance, when no
such good luck has really befallen us; and this not once
only but repeatedly. But who can give the faintest intimation
of the nature and number of the occasions on which,
a blank being thus really drawn, a prize will nevertheless
be falsely announced? It appears to me therefore that
numerical results of any practical value can seldom, if ever,
be looked for from this method of procedure.
9. Our conclusion in the case of the lottery, or, what
comes to the same thing, in the case of the bag with black
and white balls, has been questioned or objected to[4] on the
ground that it is contrary to all experience to suppose that
the testimony of a moderately good witness could be so
enormously depreciated under such circumstances. I should
prefer to base the objection on the ground that experience
scarcely ever presents such circumstances as those supposed;
but if we postulate their existence the given conclusion seems
correct enough. Assume that a man is merely required to
say _yes_ or _no_; assume also a group or succession of cases in
which _no_ should rightly be said very much oftener than
_yes_. Then, assuming almost any general truthfulness of the
witness, we may easily suppose the rightful occasions for
denial to be so much the more frequent that a majority of
his affirmative answers will actually occur as false 'noes'
rather than as correct 'ayes.' This of course lowers the
average value of his 'ayes,' and renders them comparatively
untrustworthy.
Consider the following example. I have a gardener whom
I trust as to all ordinary matters of fact. If he were to
tell me some morning that my dog had run away I should
fully believe him. He tells me however that the dog has
gone mad. Surely I should accept the statement with much
hesitation, and on the grounds indicated above. It is not
that he is more likely to be wrong when the dog _is_ mad;
but that experience shows that there are other complaints
(e.g. fits) which are far more common than madness, and
that most of the assertions of madness are erroneous assertions
referring to these. This seems a somewhat parallel
case to that in which we find that most of the assertions
that a white ball had been drawn are really false assertions
referring to the drawing of a black ball. Practically I do
not think that any one would feel a difficulty in thus exorbitantly
discounting some particular assertion of a witness
whom in most other respects he fully trusted.
10. There is one particular case which has been regarded
as a difficulty in the way of this treatment of the
problem, but which seems to me to be a decided confirmation
of it; always, be it understood, within the very narrow
and artificial limits to which we must suppose ourselves to
be confined. This is the case of a witness whose veracity is
just one-half; that is, one who, when a mere _yes_ or _no_ is
demanded of him, is as often wrong as right. In the case of
any other assigned degree of veracity it is extremely difficult
to get anything approaching to a confirmation from practical
judgment and experience. We are not accustomed to
estimate the merits of witnesses in this way, and hardly appreciate
what is meant by his numerical degree of truthfulness.
But as regards the man whose veracity is one-half, we
are (as Mr C. J. Monro has very ingeniously suggested) only
too well acquainted with such witnesses, though under a
somewhat different name; for this is really nothing else than
the case of a person confidently answering a question about
a subject-matter of which he knows nothing, and can therefore
only give a mere guess.
Now in the case of the lottery with one prize, when the
witness whose veracity is one-half tells us that we have
gained the prize, we find on calculation that his testimony
goes for absolutely nothing; the chances that we have got
the prize are just the same as they would be if he had never
opened his lips, viz. 1/1000. But clearly this is what ought
to be the result, for the witness who knows nothing about
the matter leaves it exactly as he found it. He is indeed,
in strictness, scarcely a witness at all; for the natural function
of a witness is to examine the matter, and so to add
confirmation, more or less, according to his judgment and
probity, but at any rate to offer an improvement upon the
mere guesser. If, however, we will give heed to his mere
guess we are doing just the same thing as if we were to guess
ourselves, in which case of course the odds that we are right
are simply measured by the frequency of occurrence of the
events.
We cannot quite so readily apply the same rule to the
other case, namely to that of the numbered balls, for there
the witness who is right every other time may really be a
very fair, or even excellent, witness. If he has many ways
of going wrong, and yet is right in half his statements, it is
clear that he must have taken some degree of care, and cannot
have merely guessed. In a case of _yes_ or _no_, any one can
be right every other time, but it is different where truth
is single and error is manifold. To represent the case of a
simply worthless witness when there were 1000 balls and
the drawing of one assigned ball was in question, we should
have to put his figure of veracity at 1/1000. If this were
done we should of course get a similar result.
11. It deserves notice therefore that the figure of
veracity, or fraction representing the general truthfulness
of a witness, is in a way relative, not absolute; that is, it
depends upon, and varies with, the general character of the
answer which he is supposed to give. Two witnesses of
equal intrinsic veracity and worth, one of whom confined
himself to saying _yes_ and _no_, whilst the other ventured to
make more original assertions, would be represented by
different fractions; the former having set himself a much
easier task than the latter. The real caution and truthfulness
of the witness are only one factor, therefore, in his
actual figure of veracity; the other factor consists of the
nature of his assertions, as just pointed out. The ordinary
plan therefore, in such problems, of assigning an average
truthfulness to the witness, and accepting this alike in the
case of each of the two kinds of answers, though convenient,
seems scarcely sound. This consideration would however
be of much more importance were not the discussions upon
the subject mainly concerned with only one description of
answer, namely that of the 'yes or no' kind.
12. So much for the methodical way of treating such
a problem. The way in which it would be taken in hand by
those who had made no study of Probability is very different.
It would, I apprehend, strike them as follows. They would
say to themselves, Here is a story related by a witness who
tells the truth, say, nine times out of ten. But it is a story
of a kind which experience shows to be very generally made
untruly, say 99 times out of 100. Having then these opposite
inducements to belief, they would attempt in some way
to strike a balance between them. Nothing in the nature of
a strict rule could be given to enable them to decide how
they might escape out of the difficulty. Probably, in so far
as they did not judge at haphazard, they would be guided
by still further resort to experience, or unconscious recollections
of its previous teachings, in order to settle which
of the two opposing inductions was better entitled to carry
the day in the particular case before them. The reader
will readily see that any general solution of the problem,
when thus presented, is impossible. It is simply the now
familiar case (Chap. IX. §§14-32) of an individual which
belongs equally to two distinct, or even, in respect of their
characteristics, opposing classes. We cannot decide off-hand
to which of the two its characteristics most naturally and
rightly refer it. A fresh induction is needed in order to
settle this point.
13. Rules have indeed been suggested by various
writers in order to extricate us from the difficulty. The controversy
about miracles has probably been the most fertile
occasion for suggestions of this kind on one side or the
other. It is to this controversy, presumably, that the phrase
is due, so often employed in discussions upon similar subjects,
'a contest of opposite improbabilities.' What is meant
by such an expression is clearly this: that in forming a
judgment upon the truth of certain assertions we may find
that they are comprised in two very distinct classes, so that,
according as we regarded them as belonging to one or the
other of these distinct classes, our opinion as to their truth
would be very different. Such an assertion belongs to one
class, of course, by its being a statement of a particular
witness, or kind of witness; it belongs to the other by its
being a particular kind of story, one of what is called an
improbable nature. Its belonging to the former class is so
far favourable to its truth, its belonging to the latter is so far
hostile to its truth. It seems to be assumed, in speaking of
a contest of opposite improbabilities, that when these different
sources of conviction co-exist together, they would each in
some way retain their probative force so as to produce a
contest, ending generally in a victory to one or other of
them. Hume, for instance, speaks of our _deducting_ one
probability from the other, and apportioning our belief to
the remainder.[5] Thomson, in his _Laws of Thought_, speaks
of one probability as entirely superseding the other.
14. It does not appear to me that the slightest philosophical
value can be attached to any such rules as these.
They doubtless may, and indeed will, hold in individual
cases, but they cannot lay claim to any generality. Even
the notion of a contest, as any necessary ingredient in the
case, must be laid aside. For let us refer again to the way
in which the perplexity arises, and we shall readily see, as
has just been remarked, that it is nothing more than a particular
exemplification of a difficulty which has already been
recognized as incapable of solution by any general _à priori_
method of treatment. All that we are supposed to have before
us is a statement. On this occasion it is made by a witness
who lies, say, once in ten times in the long run; that is, who
mostly tells the truth. But on the other hand, it is a statement
which experience, derived from a variety of witnesses on
various occasions, assures us is mostly false; stated numerically
it is found, let us suppose, to be false 99 times in a hundred.
Now, as was shown in the chapter on Induction, we are
thus brought to a complete dead lock. Our science offers no
principles by which we can form an opinion, or attempt to
decide the matter one way or the other; for, as we found,
there are an indefinite number of conclusions which are all
equally possible. For instance, all the witness' extraordinary
assertions may be true, or they may all be false, or they may
be divided into the true and the false in any proportion
whatever. Having gone so far in our appeal to statistics as
to recognize that the witness is generally right, but that his
story is generally false, we cannot stop there. We ought to
make still further appeal to experience, and ascertain how it
stands with regard to his stories when they are of that
particular nature: or rather, for this would be to make a
needlessly narrow reference, how it stands with regard to
stories of that kind when advanced by witnesses of his
general character, position, sympathies, and so on.[6]
15. That extraordinary stories are in many cases, probably
in a great majority of cases, less trustworthy than
others must be fully admitted. That is, if we were to make
two distinct classes of such stories respectively, we should
find that the same witness, or similar witnesses, were proportionally
more often wrong when asserting the former than
when asserting the latter. But it does not by any means
appear to me that this must always be the case. We may
well conceive, for instance, that with some people the mere
fact of the story being of a very unusual character may make
them more careful in what they state, so as actually to add
to their veracity. If this were so we might be ready to
accept their extraordinary stories with even more readiness
than their ordinary ones.
Such a supposition as that just made does not seem to me
by any means forced. Put such a case as this: let us suppose
that two persons, one of them a man of merely ordinary
probity and intelligence, the other a scientific naturalist,
make a statement about some common event. We believe
them both. Let them now each report some extraordinary
_lusus naturæ_ or monstrosity which they profess to have seen.
Most persons, we may presume, would receive the statement
of the naturalist in this latter case almost as readily as in
the former: whereas when the same story came from the
unscientific observer it would be received with considerable
hesitation. Whence arises the difference? From the conviction
that the naturalist will be far more careful, and
therefore to the full as accurate, in matters of this kind as
in those of the most ordinary description, whereas with the
other man we feel by no means the same confidence. Even
if any one is not prepared to go this length, he will probably
admit that the difference of credit which he would attach to
the two kinds of story, respectively, when they came from
the naturalist, would be much less than what it would be
when they came from the other man.
16. Whilst we are on this part of the subject, it must
be pointed out that there is considerable ambiguity and
consequent confusion about the use of the term 'an extraordinary
story.' Within the province of pure Probability it
ought to mean simply a story which asserts an _unusual_ event.
At least this is the view which has been adopted and maintained,
it is hoped consistently, throughout this work. So
long as we adhere to this sense we know precisely what we
mean by the term. It has a purely objective reference; it
simply connotes a very low degree of relative statistical
frequency, actual or prospective. Out of a great number of
events we suppose a selection of some particular kind to be
contemplated, which occurs relatively very seldom, and this
is termed an unusual or extraordinary event. It follows, as
was abundantly shown in a former chapter, that owing to
the rarity of the event we are very little disposed to expect
its occurrence in any given case. Our guess about it, in case
we thus anticipated it, would very seldom be justified, and
we are therefore apt to be much surprised when it does
occur. This, I take it, is the only legitimate sense of 'extraordinary'
so far as Probability is concerned.
But there is another and very different use of the word,
which belongs to Induction, or rather to the science of
evidence in general, more than to that limited portion of it
termed Probability. In this sense the 'extraordinary,' and
still more the 'improbable,' event is not merely one of
extreme statistical rarity, which we could not expect to
guess aright, but which on moderate evidence we may pretty
readily accept; it is rather one which possesses, so to say, an
actual evidence-resisting power. It may be something which
affects the credibility of the witness at the fountain-head,
which makes, that is, his statements upon such a subject
essentially inferior to those on other subjects. This is the
case, for instance, with anything which excites his prejudices
or passions or superstitions. In these cases it would seem
unreasonable to attempt to estimate the credibility of the
witness by calculating (as in §6) how often his errors would
mislead us through his having been wrongly brought to an
affirmation instead of adhering correctly to a negation. We
should rather be disposed to put our correction on the witness'
average veracity at once.
17. In true Probability, as has just been remarked,
every event has its own definitely recognizable degree of
frequency of occurrence. It may be excessively rare, rare to
any extreme we like to postulate, but still every one who
understands and admits the data upon which its occurrence
depends will be able to appreciate within what range of
experience it may be expected to present itself. We do not
expect it in any individual case, nor within any brief range,
but we do confidently expect it within an adequately extensive
range. How therefore can miraculous stories be similarly
taken account of, when the disputants, on one side at
least, are not prepared to admit their actual occurrence anywhere
or at any time? How can any arrangement of bags
and balls, or other mechanical or numerical illustrations of
unlikely events, be admitted as fairly illustrative of miraculous
occurrences, or indeed of many of those which come
under the designation of 'very extraordinary' or 'highly
improbable'? Those who contest the occurrence of a particular
miracle, as reported by this or that narrator, do not
admit that miracles are to be confidently expected sooner or
later. It is not a question as to whether what must happen
sometimes has happened some particular time, and therefore
no illustration of the kind can be regarded as apposite.
How unsuitable these merely rare events, however excessive
their rarity may be, are as examples of miraculous
events, will be evident from a single consideration. No one,
I presume, who admitted the occasional occurrence of an exceedingly
unusual combination, would be in much doubt if
he considered that he had actually seen it himself.[7] On the
other hand, few men of any really scientific turn would
readily accept a miracle even if it appeared to happen under
their very eyes. They might be staggered at the time, but
they would probably soon come to discredit it afterwards,
or so explain it as to evacuate it of all that is meant by
miraculous.
18. It appears to me therefore, on the whole, that
very little can be made of these problems of testimony in
the way in which it is generally intended that they should
be treated; that is, in obtaining specific rules for the estimation
of the testimony under any given circumstances.
Assuming that the veracity of the witness can be measured,
we encounter the real difficulty in the utter impossibility of
determining the limits within which the failures of the event
in question are to be considered to lie, and the degree of
explicitness with which the witness is supposed to answer
the enquiry addressed to him; both of these being characteristics
of which it is necessary to have a numerical estimate
before we can consider ourselves in possession of the
requisite data.
Since therefore the practical resource of most persons,
viz. that of putting a direct and immediate correction, of
course of a somewhat conjectural nature, upon the general
trustworthiness of the witness, by a consideration of the
nature of the circumstances under which his statement is
made, is essentially unscientific and irreducible to rule; it
really seems to me that there is something to be said in
favour of the simple plan of trusting in all cases alike to
the witness' general veracity.[8] That is, whether his story
is ordinary or extraordinary, we may resolve to put it on
the same footing of credibility, provided of course that the
event is fully recognized as one which does or may occasionally
happen. It is true that we shall thus go constantly
astray, and may do so to a great extent, so that if there
were any rational and precise method of specializing his
trustworthiness, according to the nature of his story, we
should be on much firmer ground. But at least we may
thus know what to expect on the average. Provided we
have a sufficient number and variety of statements from
him, and always take them at the same constant rate or
degree of trustworthiness, we may succeed in balancing and
correcting our conduct in the long run so as to avoid any
ruinous error.
19. A few words may now be added about the combination
of testimony. No new principles are introduced here,
though the consequent complication is naturally greater. Let
us suppose two witnesses, the veracity of each being 9/10.
Now suppose 100 statements made by the pair; according to
the plan of proceeding adopted before, we should have them
both right 81 times and both wrong once, in the remaining
18 cases one being right and the other wrong. But since
they are both supposed to give the same account, what we
have to compare together are the number of occasions on
which they agree and are right, and the total number on which
they agree whether right or wrong. The ratio of the former
to the latter is the fraction which expresses the trustworthiness
of their combination of testimony in the case in question.
In attempting to decide this point the only difficulty is
in determining how often they will be found to agree when
they are both wrong, for clearly they must agree when they
are both right. This enquiry turns of course upon the number
of ways in which they can succeed in going wrong.
Suppose first the case of a simple _yes_ or _no_ (as in §6), and
take the same example, of a bag with 1000 balls, in which
one only is white. Proceeding as before, we should find that
out of 100,000 drawings (the number required in order to
obtain a complete cycle of all possible occurrences, as well as
of all possible reports about them) the two witnesses agree
in a correct report of the appearance of white in 81, and
agree in a wrong report of it in 999. The Probability therefore
of the story when so attested is 81/1080; the fact therefore
of two such witnesses of equal veracity having concurred
makes the report nearly 9 times as likely as when it rested
upon the authority of only one of them.[9]
20. When however the witnesses have many ways of
going wrong, the fact of their agreeing makes the report far
more likely to be true. For instance, in the case of the 1000
numbered balls, it is very unlikely that when they both mistake
the number they should (without collusion) happen to
make the same misstatement. Whereas, in the last case,
every combined misstatement necessarily led them both to
the assertion that the event in question had happened, we
should now find that only once in 999 × 999 times would
they both be led to assert that _some given number_ (say, as
before, 25) had been drawn. The odds in favour of the
event in fact now become 80919/80920, which are enormously
greater than when there was only one witness.
It appears therefore that when two, and of course still
more when many, witnesses agree in a statement in a matter
about which they might make many and various errors, the
combination of their favourable testimony adds enormously
to the likelihood of the event; provided always that there
is no chance of collusion. And in the extreme case of the
opportunities for error being, as they well may be, practically
infinite in number, such combination would produce almost
perfect certainty. But then this condition, viz. absence of
collusion, very seldom can be secured. Practically our main
source of error and suspicion is in the possible existence of
some kind of collusion. Since we can seldom entirely get
rid of this danger, and when it exists it can never be submitted
to numerical calculation, it appears to me that combination
of testimony, in regard to detailed accounts, is yet
more unfitted for consideration in Probability than even that
of single testimony.
21. The impossibility of any adequate or even appropriate
consideration of the credibility of miraculous stories
by the rules of Probability has been already noticed in §17.
But, since the grounds of this impossibility are often very
insufficiently appreciated, a few pages may conveniently be
added here with a view to enforcing this point. If it be
regarded as a digression, the importance of the subject and
the persistency with which various writers have at one time
or another attempted to treat it by the rules of our science
must be the excuse for entering upon it.
A necessary preliminary will be to decide upon some definition
of a miracle. It will, we may suppose, be admitted by
most persons that in calling a miracle 'a suspension of a law
of causation,' we are giving what, though it may not amount
to an adequate definition, is at least true as a description.
It is true, though it may not be the whole truth. Whatever
else the miracle may be, this is its physical aspect: this is the
point at which it comes into contact with the subject-matter
of science. If it were not considered that any suspension of
causation were involved, the event would be regarded merely
as an ordinary one to which some special significance was
attached, that is, as a type or symbol rather than a miracle.
It is this aspect moreover of the miracle which is now exposed
to the main brunt of the attack, and in support of
which therefore the defence has generally been carried on.
Now it is obvious that this, like most other definitions or
descriptions, makes some assumption as to matters of fact,
and involves something of a theory. The assumption clearly
is, that laws of causation prevail universally, or almost universally,
throughout nature, so that infractions of them are
marked and exceptional. This assumption is made, but it
does not appear that anything more than this is necessarily
required; that is, there is nothing which need necessarily
make us side with either of the two principal schools which
are divided as to the nature of these laws of causation. The
definition will serve equally well whether we understand by
_law_ nothing more than uniformity of antecedent and consequent,
or whether we assert that there is some deeper and
more mysterious tie between the events than mere sequence.
The use of the term 'causation' in this minimum of signification
is common to both schools, though the one might
consider it inadequate; we may speak, therefore, of 'suspensions
of causation' without committing ourselves to either.
22. It should be observed that the aspect of the question
suggested by this definition is one from which we can
hardly escape. Attempts indeed have been sometimes made
to avoid the necessity of any assumption as to the universal
prevalence of law and order in nature, by defining a miracle
from a different point of view. A miracle may be called, for
instance, 'an immediate exertion of creative power,' 'a sign
of a revelation,' or, still more vaguely, an 'extraordinary
event.' But nothing would be gained by adopting any such
definitions as these. However they might satisfy the theologian,
the student of physical science would not rest content
with them for a moment. He would at once assert his own
belief, and that of other scientific men, in the existence of
universal law, and enquire what was the connection of the
definition with this doctrine. An answer would imperatively
be demanded to the question, Does the miracle, as you have
described it, imply an infraction of one of these laws, or does
it not? And an answer must be given, unless indeed we
reject his assumption by denying our belief in the existence
of this universal law, in which case of course we put ourselves
out of the pale of argument with him. The necessity
of having to recognize this fact is growing upon men day by
day, with the increased study of physical science. And since
this aspect of the question has to be met some time or other,
it is as well to place it in the front. The difficulty, in its
scientific form, is of course a modern one, for the doctrine out
of which it arises is modern. But it is only one instance, out
of many that might be mentioned, in which the growth of
some philosophical conception has gradually affected the
nature of the dispute, and at last shifted the position of the
battle-ground, in some discussion with which it might not at
first have appeared to have any connection whatever.
23. So far our path is plain. Up to this point disciples
of very different schools may advance together; for in laying
down the above doctrine we have carefully abstained from
implying or admitting that it contains the whole truth. But
from this point two paths branch out before us, paths as
different from each other in their character, origin, and
direction, as can well be conceived. As this enquiry is only
a digression, we may confine ourselves to stating briefly what
seem to be the characteristics of each, without attempting to
give the arguments which might be used in their support.
(I.) On the one hand, we may assume that this principle
of causation is the ultimate one. By so terming it, we do
not mean that it is one from which we consciously start in
our investigations, as we do from the axioms of geometry,
but rather that it is the final result towards which we find
ourselves drawn by a study of nature. Finding that,
throughout the scope of our enquiries, event follows event in
never-failing uniformity, and finding moreover (some might
add) that this experience is supported or even demanded by
a tendency or law of our nature (it does not matter here how
we describe it), we may come to regard this as the one
fundamental principle on which all our enquiries should rest.
(II.) Or, on the other hand, we may admit a class of
principles of a very different kind. Allowing that there is
this uniformity so far as our experience extends, we may yet
admit what can hardly be otherwise described than by
calling it a Superintending Providence, that is, a Scheme or
Order, in reference to which Design may be predicated
without using merely metaphorical language. To adopt an
aptly chosen distinction, it is not to be understood as _over-ruling_
events, but rather as _underlying_ them.
24. Now it is quite clear that according as we come to
the discussion of any particular miracle or extraordinary
story under one or other of these prepossessions, the question
of its credibility will assume a very different aspect. It
is sometimes overlooked that although a difference about
_facts_ is one of the conditions of a _bonâ fide_ argument, a difference
which reaches to ultimate principles is fatal to all
argument. The possibility of present conflict is banished in
such a case as absolutely as that of future concord. A large
amount of popular literature on the subject of miracles
seems to labour under this defect. Arguments are stated
and examined for and against the credibility of miraculous
stories without the disputants appearing to have any
adequate conception of the chasm which separates one side
from the other.
25. The following illustration may serve in some
degree to show the sort of inconsistency of which we are
speaking. A sailor reports that in some remote coral island
of the Pacific, on which he had landed by himself, he had
found a number of stones on the beach disposed in the exact
form of a cross. Now if we conceive a debate to arise about
the truth of his story, in which it is attempted to decide the
matter simply by considerations about the validity of testimony,
without introducing the question of the existence of
inhabitants, and the nature of their customs, we shall have
some notion of the unsatisfactory nature of many of the
current arguments about miracles. All illustrations of this
subject are imperfect, but a case like this, in which a supposed
trace of human agency is detected interfering with the
orderly sequence of other and non-intelligent natural causes,
is as much to the point as any illustration can be. The
thing omitted here from the discussion is clearly the one important
thing. If we suppose that there is no inhabitant, we
shall probably disbelieve the story, or consider it to be
grossly exaggerated. If we suppose that there are inhabitants,
the question is at once resolved into a different and
somewhat more intricate one. The credibility of the witness
is not the only element, but we should necessarily have to
take into consideration the character of the supposed inhabitants,
and the object of such an action on their part.
26. Considerations of this character are doubtless
often introduced into the discussion, but it appears to me
that they are introduced to a very inadequate extent. It is
often urged, after Paley, 'Once believe in a God, and miracles
are not incredible.' Such an admission surely demands
some modification and extension. It should rather be stated
thus, Believe in a God whose working may be traced throughout
the whole moral and physical world. It amounts, in
fact, to this;--Admit that there may be a _design_ which we
can trace somehow or other in the course of things; admit
that we are not wholly confined to tracing the connection of
events, or following out their effects, but that we can form
some idea, feeble and imperfect though it be, of a _scheme_.[10]
Paley's advice sounds too much like saying, Admit that there
are fairies, and we can account for our cups being cracked.
The admission is not to be made in so off-hand a manner.
To any one labouring under the difficulty we are speaking
of, this belief in a God almost out of any constant relation to
nature, whom we then imagine to occasionally manifest himself
in a perhaps irregular manner, is altogether impossible.
The only form under which belief in the Deity can gain entrance
into his mind is as the controlling Spirit of an infinite
and orderly system. In fact, it appears to me, paradoxical
as the suggestion may appear, that it might even be more
easy for a person thoroughly imbued with the spirit of Inductive
science, though an atheist, to believe in a miracle
which formed a part of a vast system, than for such a person,
as a theist, to accept an isolated miracle.
27. It is therefore with great prudence that Hume,
and others after him, have practically insisted on commencing
with a discussion of the credibility of the single miracle,
treating the question as though the Christian Revelation
could be adequately regarded as a succession of such events.
As well might one consider the living body to be represented
by the aggregate of the limbs which compose it. What is to
be complained of in so many popular discussions on the subject
is the entire absence of any recognition of the different
ground on which the attackers and defenders of miracles are
so often really standing. Proofs and illustrations are produced
in endless number, which involving, as they almost all
do in the mind of the disputants on one side at least, that
very principle of causation, the absence of which in the case
in question they are intended to establish, they fail in the
single essential point. To attempt to induce any one to disbelieve
in the existence of physical causation, in a given
instance, by means of illustrations which to him seem only
additional examples of the principle in question, is like trying
to make a dam, in order to stop the flow of a river, by
shovelling in snow. Such illustrations are plentiful in times
of controversy, but being in reality only modified forms of
that which they are applied to counteract, they change their
shape at their first contact with the disbeliever's mind, and
only help to swell the flood which they were intended to
check.
1. Reasons were given in the last chapter against the propriety of
applying the rules of Probability with any strictness to such
examples as these. But although all approach to numerical accuracy
is unattainable, we do undoubtedly recognize in ordinary life a
distinction between the credibility of one witness and another; such
a rough practical distinction will be quite sufficient for the
purposes of this chapter. For convenience, and to illustrate the
theory, the examples are best stated in a numerical form, but it is
not intended thereby to imply that any such accuracy is really
attainable in practice.
2. I must plead guilty to this charge myself, in the first edition of
this work. The result was to make the treatment of this part of the
subject obscure and imperfect, and in some respects erroneous.
3. The generalized algebraical form of this result is as follows. Let
p be the _à priori_ probability of an event, and x be the
credibility of the witness. Then, if he asserts that the event
happened, the probability that it really did happen is
px/(px + (1 - p)(1 - x)); whilst if he asserts that it did _not_
happen the probability that it did happen is
p(1 - x)/(p(1 - x) + (1 - p)x).
In illustration of some remarks to be presently made, the reader
will notice that on making either of these expressions = p, we
obtain in each case x = 1/2. That is, a witness whose veracity = 1/2
leaves the _à priori_ probability of an event (of this kind)
unaffected.
If, on the other hand, we make these expressions equal to x and 1 - x
respectively, we obtain in each case p = 1/2. That is, when an event
(of this kind) is as likely to happen as not, the ordinary veracity
of the witness in respect of it remains unaffected.
4. Todhunter's _History_, p. 400. _Philosophical Magazine_, July, 1864.
5. "When therefore these two kinds of experience are contrary, we have
nothing to do but subtract the one from the other, and embrace an
opinion, either on one side or the other, with that assurance which
arises from the remainder." (Essay on Miracles.)
6. Considerations of this kind have indeed been introduced into the
mathematical treatment of the subject. The common algebraical
solution of the problem in §5 (to begin with the simplest case) is
of course as follows. Let p be the antecedent probability of the
event, and t the measure of the truthfulness of the witness; then
the chance of his statement being true is pt/(pt + (1 - p)(1 - t)).
This supposes him to lie as much when the event does not happen as
when it does. But we may meet the cases supposed in the text by
assuming that t' is the measure of his veracity when the event does
not happen, so that the above formula becomes pt/(pt + (1 - p)(1 - t')).
Here t' and t measure respectively his trustworthiness in usual and
unusual events. As a formal solution this certainly meets the
objections stated above in §§14 and 15. The determination however
of t' would demand, as I have remarked, continually renewed appeal
to experience. In any case the practical methods which would be
adopted, if any plans of the kind indicated above were resorted to,
seem to me to differ very much from that adopted by the
mathematicians, in their spirit and plan.
7. Laplace, for instance (_Essai_, ed. 1825, p. 149), says that if we
saw 100 dies (known of course to be fair ones) all give the same
face, we should be bewildered at the time, and need confirmation
from others, but that, after due examination, no one would feel
obliged to postulate hallucination in the matter. But the chance of
this occurrence is represented by a fraction whose numerator is 1,
and denominator contains 77 figures, and is therefore utterly
inappreciable by the imagination. It must be admitted, though, that
there is something hypothetical about such an example, for we could
not really know that the dies were fair with a confidence even
distantly approaching such prodigious odds. In other words, it is
difficult here to keep apart those different aspects of the question
discussed in Chap. XIV. §§28-33.
8. In the first edition this was stated, as it now seems to me, in
decidedly too unqualified a manner. It must be remembered, however,
that (as was shown in §7) this plan is really the best theoretical
one which can be adopted in certain cases.
9. It is on this principle that the remarkable conclusion mentioned on
p. 405 is based. Suppose an event whose probability is p; and that,
of a number of witnesses of the same veracity (y), m assert that it
happened, and n deny this. Generalizing the arithmetical reasoning
given above we see that the chance of the event being asserted
varies as
py^{m} (1 - y)^{n} + (1 - p) y^{n} (1 - y)^{m};
(viz. as the chance that the event happens, and that m are right and
n are wrong; _plus_ the chance that it does not happen, and that n
are right and m are wrong). And the chance of its being rightly
asserted as py^{m} (1 - y)^{n}. Therefore the chance that when we
have an assertion before us it is a true one is
py^{m} (1 - y)^{n}
----------------------------------------------
py^{m} (1 - y)^{n} + (1 - p) y^{n} (1 - y)^{m},
which is equal to
py^{m-n}
--------------------------------
py^{m-n} + (1 - p) (1 - y)^{m-n}.
But this last expression represents the probability of an assertion
which is unanimously supported by m - n such witnesses.
10. The stress which Butler lays upon this notion of a scheme is, I
think, one great merit of his _Analogy_.