_CONSEQUENCES OF THE FOREGOING DISTINCTIONS._
1. We are now in a position to explain and justify some
important conclusions which, if not direct consequences of
the distinctions laid down in the last chapter, will at any
rate be more readily appreciated and accepted after that
exposition.
In the first place, it will be seen that in Probability _time_
has nothing to do with the question; in other words, it does
not matter whether the event, whose probability we are discussing,
be past, present, or future. The problem before us,
in its simplest form, is this:--Statistics (extended by Induction,
and practically often gained by Deduction) inform us
that a certain event has happened, does happen, or will
happen, in a certain way in a certain proportion of cases.
We form a conception of that event, and regard it as possible;
but we want to do more; we want to know _how much_ we
ought to expect it (under the explanations given in a former
chapter about quantity of belief). There is therefore a
sort of relative futurity about the event, inasmuch as our
knowledge of the fact, and therefore our justification or
otherwise of the correctness of our surmise, almost necessarily
comes after the surmise was formed; but the futurity
is only relative. The evidence by which the question is to
be settled may not be forthcoming yet, or we may have it by
us but only consult it afterwards. It is from the fact of the
futurity being, as above described, only relative, that I have
preferred to speak of the conception of the event rather than
of the anticipation of it. The latter term, which in some respects
would have seemed more intelligible and appropriate,
is open to the objection, that it does rather, in popular estimation,
convey the notion of an absolute as opposed to a
relative futurity.
2. For example; a die is thrown. Once in six times
it gives ace; if therefore we assume, without examination,
that the throw is ace, we shall be right once in six times.
In so doing we may, according to the usual plan, go _forwards_
in time; that is, form our opinion about the throw beforehand,
when no one can tell what it will be. Or we might go
_backwards_; that is, form an opinion about dice that had
been cast on some occasion in time past, and then correct
our opinion by the testimony of some one who had been a
witness of the throws. In either case the mental operation
is precisely the same; an opinion formed merely on statistical
grounds is afterwards corrected by specific evidence. The
opinion may have been formed upon a past, present, or future
event; the evidence which corrects it afterwards may be our
own eyesight, or the testimony of others, or any kind of inference;
by the evidence is merely meant such subsequent
examination of the case as is assumed to set the matter at
rest. It is quite possible, of course, that this specific evidence
should never be forthcoming; the conception in that
case remains as a conception, and never obtains that degree
of conviction which qualifies it to be regarded as a 'fact.'
This is clearly the case with all past throws of dice the results
of which do not happen to have been recorded.
In discussing games of chance there are obvious advantages
in confining ourselves to what is really, as well as
relatively, future, for in that case direct information concerning
the contemplated result being impossible, all persons are
on precisely the same footing of comparative ignorance, and
must form their opinion entirely from the known or inferred
frequency of occurrence of the event in question. On the
other hand, if the event be passed, there is almost always evidence
of some kind and of some value, however slight, to
inform us what the event really was; if this evidence is not
actually at hand, we can generally, by waiting a little, obtain
something that shall be at least of some use to us in forming
our opinion. Practically therefore we generally confine ourselves,
in anticipations of this kind, to what is really future,
and so in popular estimation futurity becomes indissolubly
associated with probability.
3. There is however an error closely connected with
the above view of the subject, or at least an inaccuracy of
expression which is constantly liable to lead to error, which
has found wide acceptance, and has been sanctioned by
writers of the greatest authority. For instance, both Butler,
in his _Analogy_, and Mill, have drawn attention, under one
form of expression or another, to the distinction between improbability
before the event and improbability after the
event, which they consider to be perfectly different things.
That this phraseology indicates a distinction of importance
cannot be denied, but it seems to me that the language in
which it is often expressed requires to be amended.
Butler's remarks on this subject occur in his _Analogy_, in
the chapter on miracles. Admitting that there is a strong
presumption against miracles (his equivalent for the ordinary
expression, an 'improbability before the event') he
strives to obtain assent for them by showing that other
events, which also have a strong presumption against them,
are received on what is in reality very slight evidence. He
says, "There is a very strong presumption against common
speculative truths, and against the most ordinary facts, before
the proof of them; which yet is overcome by almost any
proof. There is a presumption of millions to one against the
story of Cæsar, or of any other man. For, suppose a number
of common facts so and so circumstanced, of which one had
no kind of proof, should happen to come into one's thoughts,
every one would without any possible doubt conclude them
to be false. And the like may be said of a single common
fact."
4. These remarks have been a good deal criticized,
and they certainly seem to me misleading and obscure in
their reference. If one may judge by the context, and by
another passage in which the same argument is afterwards
referred to,[1] it would certainly appear that Butler drew no
distinction between miraculous accounts, and other accounts
which, to use any of the various expressions in common use,
are unlikely or improbable or have a presumption against
them; and concluded that since some of the latter were instantly
accepted upon somewhat mediocre testimony, it was
altogether irrational to reject the former when similarly or
better supported.[2] This subject will come again under our
notice, and demand fuller discussion, in the chapter on the
Credibility of extraordinary stories. It will suffice here to
remark that, however satisfactory such a view of the matter
might be to some theologians, no antagonist of miracles
would for a moment accept it. He would naturally object
that, instead of the miraculous element being (as Butler
considers) "a small additional presumption" against the
narrative, it involved the events in a totally distinct class of
incredibility; that it multiplied, rather than merely added
to, the difficulties and objections in the way of accepting
the account.
Mill's remarks (_Logic_, Bk. III. ch. XXV. §4) are of a different
character. Discussing the grounds of disbelief he
speaks of people making the mistake of "overlooking the
distinction between (what may be called) improbability
before the fact, and improbability after it, two different
properties, the latter of which is always a ground of disbelief,
the former not always." He instances the throwing
of a die. It is improbable beforehand that it should turn
up ace, and yet afterwards, "there is no reason for disbelieving
it if any credible witness asserts it." So again, "the
chances are greatly against A. B.'s dying, yet if any one tells
us that he died yesterday we believe it."
5. That there is some difficulty about such problems
as these must be admitted. The fact that so many people
find them a source of perplexity, and that such various
explanations are offered to solve the perplexity, are a sufficient
proof of this.[3] The considerations of the last chapter,
however, over-technical and even scholastic as some of the
language in which it was expressed may have seemed to the
reader, will I hope guide us to a more satisfactory way of
regarding the matter.
When we speak of an improbable event, it must be
remembered that, objectively considered, an event can only
be more or less _rare_; the extreme degree of rarity being of
course that in which the event does not occur at all. Now,
as was shown in the last chapter, our position, when forming
judgments of the time in question, is that of entertaining
a conception or conjecture (call it what we will), and assigning
a certain weight of trustworthiness to it. The real
distinction, therefore, between the two classes of examples
respectively, which are adduced both by Butler and by Mill,
consists in the way in which those conceptions are obtained;
they being obtained in one case by the process of guessing,
and in the other by that of giving heed to the reports of
witnesses.
6. Take Butler's instance first. In the 'presumption
before the proof' we have represented to us a man thinking
of the story of Cæsar, that is, making a guess about certain
historical events without any definite grounds for it, and
then speculating as to what value is to be attached to the
probability of its truth. Such a guess is of course, as he
says, concluded to be false. But what does he understand
by the 'presumption after the proof'? That a story not
adopted at random, but actually suggested and supported by
witnesses, should be true. The latter might be accepted,
whilst the former would undoubtedly be rejected; but all
that this proves, or rather illustrates, is that the testimony
of almost any witness is in most cases vastly better than
a mere guess.[4] We may in both cases alike speak of 'the
event' if we will; in fact, as was admitted in the last
chapter, common language will not readily lend itself to any
other way of speaking. But it should be clearly understood
that, phrase it how we will, what is really present to
the man's mind, and what is to have its probable value
assigned to it, is the conception of an event, in the sense in
which that expression has already been explained. And
surely no two conceptions can have a much more important
distinction put between them than that which is involved in
supposing one to rest on a mere guess, and the other on the
report of a witness. Precisely the same remarks apply to
the example given by Mill. Before A. B.'s death our
opinion upon the subject was nothing but a guess of our
own founded upon life statistics; after his death it was
founded upon the evidence of some one who presumably
had tolerable opportunities of knowing what the facts really
were.
7. That the distinction before us has no essential connection
whatever with time is indeed obvious on a moment's
consideration. Conceive for a moment that some one had
opportunities of knowing whether A. B. would die or not.
If he told us that A. B. would die to-morrow, we should in
that case be just as ready to believe him as when he tells us
that A. B. _has_ died. If we continued to feel any doubt
about the statement (supposing always that we had full
confidence about his veracity in matters into which he had
duly enquired), it would be because we thought that in his
case, as in ours, it was equivalent to a guess, and nothing
more. So with the event when past, the fact of its being
past makes no difference whatever; until the credible witness
informs us of what he knows to have occurred, we
should doubt it if it happened to come into our minds, just
as much as if it were future.
The distinction, therefore, between probability before the
event and probability after the event seems to resolve itself
simply into this;--before the event we often have no better
means of information than to appeal to statistics in some
form or other, and so to guess amongst the various possible
alternatives; after the event the guess may most commonly
be improved or superseded by appeal to specific evidence,
in the shape of testimony or observation. Hence, naturally,
our estimate in the latter case is commonly of much more
value. But if these characteristics were anyhow inverted;
if, that is, we were to confine ourselves to guessing about the
past, and if we could find any additional evidence about the
future, the respective values of the different estimates would
also be inverted. The difference between these values has
no necessary connection with time, but depends entirely
upon the different grounds upon which our conception or
conjecture about the event in question rests.
8. The following imaginary example will serve to
bring out the point indicated above. Conceive a people with
very short memories, and who preserved no kind of record to
perpetuate their hold upon the events which happened
amongst them.[5] The whole region of the past would then be
to them what much of the future is to us; viz. a region of
guesses and conjectures, one in reference to which they
could only judge upon general considerations of probability,
rather than by direct and specific evidence. But conceive
also that they had amongst them a race of prophets who
could succeed in foretelling the future with as near an
approach to accuracy and trustworthiness as our various
histories, and biographies, and recollections, can attain in
respect to the past. The present and usual functions of
direct evidence or testimony, and of probability, would then
be simply inverted; and so in consequence would the present
accidental characteristics of improbability before and
after the event. It would then be the latter which would
by comparison be regarded as 'not always a ground of disbelief,'
whereas in the case of the former we should then
have it maintained that it always was so.
9. The origin of the mistake just discussed is worth
enquiring into. I take it to be as follows. It is often the
case, as above remarked, when we are speculating about
a future event, and almost always the case when that future
event is taken from a game of chance, that all persons are in
precisely the same condition of ignorance in respect to it.
The limit of available information is confined to statistics,
and amounts to the knowledge that the unknown event
must assume some one of various alternative forms. The
conjecture, therefore, of any one man about it is as valuable
as that of any other. But in regard to the past the case is
very different. Here we are not in the habit of relying
upon statistical information. Hence the conjectures of different
men are of extremely different values; in the case of
many they amount to what we call positive knowledge.
This puts a broad distinction, in popular estimation, between
what may be called the objective certainty of the past and of
the future, a distinction, however, which from the standing-point
of a science of inference ought to have no existence.
In consequence of this, when we apply to the past and
the future respectively the somewhat ambiguous expression
'the chance of the event,' it commonly comes to bear very
different significations. Applied to the future it bears its
proper meaning, namely, the value to be assigned to a conjecture
upon statistical grounds. It does so, because in this
case hardly any one has more to judge by than such conjectures.
But applied to the past it shifts its meaning,
owing to the fact that whereas some men have conjectures
only, others have positive knowledge. By the chance of the
event is now often meant, not the value to be assigned to a
conjecture founded on statistics, but to such a conjecture
derived from and enforced by any body else's conjecture, that
is by his knowledge and his testimony.
10. There is a class of cases in apparent opposition to
some of the statements in this chapter, but which will be
found, when examined closely, decidedly to confirm them.
I am walking, say, in a remote part of the country, and suddenly
meet with a friend. At this I am naturally surprised.
Yet if the view be correct that we cannot properly speak
about events in themselves being probable or improbable,
but only say this of our conjectures about them, how do we
explain this? We had formed no conjecture beforehand,
for we were not thinking about anything of the kind, but
yet few would fail to feel surprise at such an incident.
The reply might fairly be made that we _had_ formed
such anticipations tacitly. On any such occasion every
one unconsciously divides things into those which are known
to him and those which are not. During a considerable
previous period a countless number of persons had met us,
and all fallen into the list of the unknown to us. There
was nothing to remind us of having formed the anticipation
or distinction at all, until it was suddenly called out
into vivid consciousness by the exceptional event. The
words which we should instinctively use in our surprise seem
to show this:--'Who would have thought of seeing you
here?' viz. Who would have given any weight to the latent
thought if it had been called out into consciousness beforehand?
We put our words into the past tense, showing that
we have had the distinction lurking in our minds all the
time. We always have a multitude of such ready-made
classes of events in our minds, and when a thing happens to
fall into one of those classes which are very small we cannot
help noticing the fact.
Or suppose I am one of a regiment into which a shot
flies, and it strikes me, and me only. At this I am surprised,
and why? Our common language will guide us to the
reason. 'How strange that it should just have hit _me_ of all
men!' We are thinking of the very natural two-fold division
of mankind into, ourselves, and everybody else; our surprise
is again, as it were, retrospective, and in reference to this
division. No anticipation was distinctly formed, because
we did not think beforehand of the event, but the event,
when it has happened, is at once assigned to its appropriate
class.
11. This view is confirmed by the following considerations.
Tell the story to a friend, and he will be a little
surprised, but less so than we were, _his_ division in this
particular case being,--his friends (of whom we are but one),
and the rest of mankind. It is not a necessary division, but
it is the one which will be most likely suggested to him.
Tell it again to a perfect stranger, and his division being
different (viz. we falling into the majority) we shall fail to
make him perceive that there is anything at all remarkable
in the event.
It is not of course attempted in these remarks to justify
our surprise in every case in which it exists. Different
persons might be differently affected in the cases supposed,
and the examples are therefore given mainly for illustration.
Still on principles already discussed (Ch. VI. §32) we might
expect to find something like a general justification of the
amount of surprise.
12. The answer commonly given in these cases is
confined to attempting to show that the surprise should not
arise, rather than to explaining how it does arise. It takes
the following form,--'You have no right to be surprised, for
nothing remarkable has really occurred. If this particular
thing had not happened something equally improbable
must. If the shot had not hit you or your friend, it must
have hit some one else who was _à priori_ as unlikely to be
hit.'
For one thing this answer does not explain the fact
that almost every one _is_ surprised in such cases, and surprised
somewhat in the different proportions mentioned
above. Moreover it has the inherent unsatisfactoriness
of admitting that something improbable has really happened,
but getting over the difficulty by saying that all the
other alternatives were equally improbable. A natural inference
from this is that there is a class of things, in themselves
really improbable, which can yet be established upon
very slight evidence. Butler accepted this inference, and
worked it out to the strange conclusion given above. Mill
attempts to avoid it by the consideration of the very different
values to be assigned to improbability before and after
the event. Some further discussion of this point will be
found in the chapter on Fallacies, and in that on the Credibility
of Extraordinary Stories.
13. In connection with the subject at present under
discussion we will now take notice of a distinction which we
shall often find insisted on in works on Probability, but to
which apparently needless importance has been attached.
It is frequently said that probability is _relative_, in the sense
that it has a different value to different persons according
to their respective information upon the subject in question.
For example, two persons, A and B, are going to draw
a ball from a bag containing 4 balls: A knows that the
balls are black and white, but does not know more;
B knows that three are black and one white. It would
be said that the probability of a white ball to A is 1/2, and
to B 1/4.
When however we regard the subject from the material
standing point, there really does not seem to me much more
in this than the principle, equally true in every other science,
that our inferences will vary according to the data we assume.
We might on logical grounds with almost equal
propriety speak of the area of a field or the height of a
mountain being relative, and therefore having one value to
one person and another to another. The real meaning of the
example cited above is this: A supposes that he is choosing
white at random out of a series which in the long run would
give white and black equally often; B supposes that he
is choosing white out of a series which in the long run would
give three black to one white. By the application, therefore,
of a precisely similar rule they draw different conclusions;
but so they would under the same circumstances in
any other science. If two men are measuring the height of
a mountain, and one supposes his base to be 1000 feet,
whilst the other takes it to be 1001, they would of course
form different opinions about the height. The science of
mensuration is not supposed to have anything to do with
the truth of the data, but assumes them to have been correctly
taken; why should not this be equally the case with
Probability, making of course due allowance for the peculiar
character of the data with which it is concerned?
14. This view of the relativeness of probability is
connected, as it appears to me, with the subjective view of
the science, and is indeed characteristic of it. It seems a
fair illustration of the weak side of that view, that it should
lead us to lay any stress on such an expression. As was
fully explained in the last chapter, in proportion as we work
out the Conceptualist principle we are led away from the
fundamental question of the material logic, viz. Is our belief
actually correct, or not? and, if the former, to what extent
and degree is it correct? We are directed rather to ask,
What belief does any one as a matter of fact hold? And,
since the belief thus entertained naturally varies according
to the circumstances and other sources of information of the
person in question, its relativeness comes to be admitted as
inevitable, or at least it is not to be wondered at if such
should be the case.
On our view of Probability, therefore, its 'relativeness' in
any given case is a misleading expression, and it will be
found much preferable to speak of the effect produced by
variations in the nature and amount of the data which we
have before us. Now it must be admitted that there are
frequently cases in our science in which such variations are
peculiarly likely to be found. For instance, I am expecting
a friend who is a passenger in an ocean steamer. There are
a hundred passengers on board, and the crew also numbers
a hundred. I read in the papers that one person was lost by
falling overboard; my anticipation that it was my friend who
was lost is but small, of course. On turning to another
paper, I see that the man who was lost was a passenger, not
one of the crew; my slight anxiety is at once doubled. But
another account adds that it was an Englishman, and on
that line at that season the English passengers are known
to be few; I at once begin to entertain decided fears. And
so on, every trifling bit of information instantly affecting my
expectations.
15. Now since it is peculiarly characteristic of Probability,
as distinguished from Induction, to be thus at the
mercy, so to say, of every little fact that may be floating
about when we are in the act of forming our opinion, what
can be the harm (it may be urged) of expressing this state
of things by terming our state of expectation _relative_?
There seem to me to be two objections. In the first place,
as just mentioned, we are induced to reject such an expression
on grounds of consistency. It is inconsistent with the
general spirit and treatment of the subject hitherto adopted,
and tends to divorce Probability from Inductive logic instead
of regarding them as cognate sciences. We are aiming at
truth, as far as that goal can be reached by our road, and
therefore we dislike to regard our conclusions as relative in
any other sense than that in which truth itself may be said
to be relative.
In the second place, this condition of unstable assent,
this constant liability to have our judgment affected, to any
degree and at any moment, by the accession of new knowledge,
though doubtless characteristic of Probability, does
not seem to me characteristic of it in its sounder and more
legitimate applications. It seems rather appropriate to a
precipitate judgment formed in accordance with the rules,
than a strict example of their natural employment. Such
precipitate judgments may occur in the case of ordinary deductive
conclusions. In the practical exigencies of life we
are constantly in the habit of forming a hasty opinion with
nearly full confidence, at any rate temporarily, upon the
strength of evidence which we must well know at the time
cannot be final. We wait a short time, and something else
turns up which induces us to alter our opinion, perhaps to
reverse it. Here our conclusions may have been perfectly
sound under the given circumstances, that is, they may be
such as every one else would have drawn who was bound to
make up his mind upon the data before us, and they are
unquestionably 'relative' judgments in the sense now under
discussion. And yet, I think, every one would shrink from
so terming them who wished systematically to carry out the
view that Logic was to be regarded as an organon of truth.
16. In the examples of Probability which we have
hitherto employed, we have for the most part assumed that
there was a certain body of statistics set before us on which
our conclusion was to rest. It was assumed, on the one
hand, that no direct specific evidence could be got, so that
the judgment was really to be one of Probability, and to rest
on these statistics; in other words, that nothing better than
them was available for us. But it was equally assumed, on
the other hand, that these statistics were open to the observation
of every one, so that we need not have to put up with
anything inferior to them in forming our opinion. In other
words, we have been assuming that here, as in the case of
most other sciences, those who have to draw a conclusion
start from the same footing of opportunity and information.
This, for instance, clearly is or ought to be the case when
we are concerned with games of chance; ignorance or misapprehension
of the common data is never contemplated
there. So with the statistics of life, or other insurance: so
long as our judgment is to be accurate (after its fashion) or
justifiable, the common tables of mortality are all that any
one has to go by.
17. It is true that in the case of a man's prospect of
death we should each qualify our judgment by what we
knew or reasonably supposed as to his health, habits, profession,
and so on, and should thus arrive at varying estimates.
But no one could _justify_ his own estimate without
appealing explicitly or implicitly to the statistical grounds on
which he had relied, and if these were not previously available
to other persons, he must now set them before their
notice. In other words, the judgments we entertain, here as
elsewhere, are only relative so long as we rest them on
grounds peculiar to ourselves. The process of justification,
which I consider to be essential to logic, has a tendency to
correct such individualities of judgment, and to set all observers
on the same basis as regards their data.
It is better therefore to regard the conclusions of Probability
as being absolute and objective, in the same sense as,
though doubtless in a far less degree than, they are in Induction.
Fully admitting that our conclusions will in many
cases vary exceedingly from time to time by fresh accessions
of knowledge, it is preferable to regard such fluctuations of
assent as partaking of the nature of precipitate judgments,
founded on special statistics, instead of depending only on
those which are common to all observers. In calling such
judgments precipitate it is not implied that there is any
blame in entertaining them, but simply that, for one reason
or another, we have been induced to form them without
waiting for the possession of the full amount of evidence,
statistical or otherwise, which might ultimately be looked
for. This explanation will suit the facts equally well, and is
more consistent with the general philosophical position maintained
in this work.
1. "Is it not self-evident that internal improbabilities of all kinds
weaken external proof? Doubtless, but to what practical purpose can
this be alleged here, when it has been proved before, that real
internal improbabilities, which rise even to moral certainty, are
overcome by the most ordinary testimony." Part II. ch. III.
2. "Miracles must not be compared to common natural events; or to
events which, though uncommon, are similar to what we daily
experience; but to the extraordinary phenomena of nature. And then
the comparison will be between the presumption against miracles, and
the presumption against such uncommon appearances, suppose as
comets,".... Part II. ch. II.
3. For instance, Sir J. F. Stephen explains it by drawing a
distinction between chances and probabilities, which he says that
Butler has confused together; "the objection that very ordinary
proof will overcome a presumption of millions to one is based upon a
confusion between probabilities and chances. The probability of an
event is its capability of being proved. Its chance is the numerical
proportion between the number of possible cases--supposed to be
equally favourable--favourable to its occurrence; and the number of
possible cases unfavourable to its occurrence" (_General view of the
Criminal Law of England_, p. 255). Donkin, again (_Phil. Magazine_,
June, 1851), employs the terms improbability and incredibility to
mark the same distinction.
4. In the extreme case of the witness himself merely guessing, or
being as untrustworthy as if he merely guessed, the two stories will
of course stand on precisely the same footing. This case will be
noticed again in Chapter XVII. It may be remarked that there are
several subtleties here which cannot be adequately noticed without
some previous investigation into the question of the credibility of
witnesses.
5. According to Dante, something resembling this prevailed amongst the
occupants of the _Inferno_. The cardinals and others whom he there
meets are able to _give_ information about many events which were
yet to happen upon earth, but they had to _ask_ it for many events
which actually had happened.