_CONCERNING THE PROBABILITIES OF TESTIMONIES._
The majority of our opinions being founded on the probability of proofs
it is indeed important to submit it to calculus. Things it is true often
become impossible by the difficulty of appreciating the veracity of
witnesses and by the great number of circumstances which accompany the
deeds they attest; but one is able in several cases to resolve the
problems which have much analogy with the questions which are proposed
and whose solutions may be regarded as suitable approximations to guide
and to defend us against the errors and the dangers of false reasoning
to which we are exposed. An approximation of this kind, when it is well
made, is always preferable to the most specious reasonings. Let us try
then to give some general rules for obtaining it.
A single number has been drawn from an urn which contains a thousand of
them. A witness to this drawing announces that number 79 is drawn; one
asks the probability of drawing this number. Let us suppose that
experience has made known that this witness deceives one time in ten, so
that the probability of his testimony is 1/10. Here the event observed
is the witness attesting that number 79 is drawn. This event may result
from the two following hypotheses, namely: that the witness utters the
truth or that he deceives. Following the principle that has been
expounded on the probability of causes drawn from events observed it is
necessary first to determine _à priori_ the probability of the event in
each hypothesis. In the first, the probability that the witness will
announce number 79 is the probability itself of the drawing of this
number, that is to say, 1/1000. It is necessary to multiply it by the
probability 6/10 of the veracity of the witness; one will have then
9/10000 for the probability of the event observed in this hypothesis. If
the witness deceives, number 79 is not drawn, and the probability of
this case is 999/1000. But to announce the drawing of this number the
witness has to choose it among the 999 numbers not drawn; and as he is
supposed to have no motive of preference for the ones rather than the
others, the probability that he will choose number 79 is 1/999;
multiplying, then, this probability by the preceding one, we shall have
1/1000 for the probability that the witness will announce number 79 in
the second hypothesis. It is necessary again to multiply this
probability by 1/10 of the hypothesis itself, which gives 1/10000 for
the probability of the event relative to this hypothesis. Now if we form
a fraction whose numerator is the probability relative to the first
hypothesis, and whose denominator is the sum of the probabilities
relative to the two hypotheses, we shall have, by the sixth principle,
the probability of the first hypothesis, and this probability will be
9/10; that is to say, the veracity itself of the witness. This is
likewise the probability of the drawing of number 79. The probability of
the falsehood of the witness and of the failure of drawing this number
is 1/10.
If the witness, wishing to deceive, has some interest in choosing number
79 among the numbers not drawn,—if he judges, for example, that having
placed upon this number a considerable stake, the announcement of its
drawing will increase his credit, the probability that he will choose
this number will no longer be as at first, 1/999, it will then be ½, ⅓,
etc., according to the interest that he will have in announcing its
drawing. Supposing it to be 1/9, it will be necessary to multiply by
this fraction the probability 999/1000 in order to get in the hypothesis
of the falsehood the probability of the event observed, which it is
necessary still to multiply by 1/10, which gives 111/10000 for the
probability of the event in the second hypothesis. Then the probability
of the first hypothesis, or of the drawing of number 79, is reduced by
the preceding rule to 9/120. It is then very much decreased by the
consideration of the interest which the witness may have in announcing
the drawing of number 79. In truth this same interest increases the
probability 9/10 that the witness will speak the truth if number 79 is
drawn. But this probability cannot exceed unity or 10/10; thus the
probability of the drawing of number 79 will not surpass 10/121. Common
sense tells us that this interest ought to inspire distrust, but
calculus appreciates the influence of it.
The probability _à priori_ of the number announced by the witness is
unity divided by the number of the numbers in the urn; it is changed by
virtue of the proof into the veracity itself of the witness; it may then
be decreased by the proof. If, for example, the urn contains only two
numbers, which gives ½ for the probability _à priori_ of the drawing of
number 1, and if the veracity of a witness who announces it is 4/10,
this drawing becomes less probable. Indeed it is apparent, since the
witness has then more inclination towards a falsehood than towards the
truth, that his testimony ought to decrease the probability of the fact
attested every time that this probability equals or surpasses ½. But if
there are three numbers in the urn the probability _à priori_ of the
drawing of number 1 is increased by the affirmation of a witness whose
veracity surpasses ⅓.
Suppose now that the urn contains 999 black balls and one white ball,
and that one ball having been drawn a witness of the drawing announces
that this ball is white. The probability of the event observed,
determined _à priori_ in the first hypothesis, will be here, as in the
preceding question, equal to 9/10000. But in the hypothesis where the
witness deceives, the white ball is not drawn and the probability of
this case is 999/1000. It is necessary to multiply it by the probability
1/10 of the falsehood, which gives 999/10000 for the probability of the
event observed relative to the second hypothesis. This probability was
only 1/10000 in the preceding question; this great difference results
from this—that a black ball having been drawn the witness who wishes to
deceive has no choice at all to make among the 999 balls not drawn in
order to announce the drawing of a white ball. Now if one forms two
fractions whose numerators are the probabilities relative to each
hypothesis, and whose common denominator is the sum of these
probabilities, one will have 9/1008 for the probability of the first
hypothesis and of the drawing of a white ball, and 999/1008 for the
probability of the second hypothesis and of the drawing of a black ball.
This last probability strongly approaches certainty; it would approach
it much nearer and would become 999999/1000008 if the urn contained a
million balls of which one was white, the drawing of a white ball
becoming then much more extraordinary. We see thus how the probability
of the falsehood increases in the measure that the deed becomes more
extraordinary.
We have supposed up to this time that the witness was not mistaken at
all; but if one admits, however, the chance of his error the
extraordinary incident becomes more improbable. Then in place of the two
hypotheses one will have the four following ones, namely: that of the
witness not deceiving and not being mistaken at all; that of the witness
not deceiving at all and being mistaken; the hypothesis of the witness
deceiving and not being mistaken at all; finally, that of the witness
deceiving and being mistaken. Determining _à priori_ in each of these
hypotheses the probability of the event observed, we find by the sixth
principle the probability that the fact attested is false equal to a
fraction whose numerator is the number of black balls in the urn
multiplied by the sum of the probabilities that the witness does not
deceive at all and is mistaken, or that he deceives and is not mistaken,
and whose denominator is this numerator augmented by the sum of the
probabilities that the witness does not deceive at all and is not
mistaken at all, or that he deceives and is mistaken at the same time.
We see by this that if the number of black balls in the urn is very
great, which renders the drawing of the white ball extraordinary, the
probability that the fact attested is not true approaches most nearly to
certainty.
Applying this conclusion to all extraordinary deeds it results from it
that the probability of the error or of the falsehood of the witness
becomes as much greater as the fact attested is more extraordinary. Some
authors have advanced the contrary on this basis that the view of an
extraordinary fact being perfectly similar to that of an ordinary fact
the same motives ought to lead us to give the witness the same credence
when he affirms the one or the other of these facts. Simple common sense
rejects such a strange assertion; but the calculus of probabilities,
while confirming the findings of common sense, appreciates the greatest
improbability of testimonies in regard to extraordinary facts.
These authors insist and suppose two witnesses equally worthy of belief,
of whom the first attests that he saw an individual dead fifteen days
ago whom the second witness affirms to have seen yesterday full of life.
The one or the other of these facts offers no improbability. The
reservation of the individual is a result of their combination; but the
testimonies do not bring us at all directly to this result, although the
credence which is due these testimonies ought not to be decreased by the
fact that the result of their combination is extraordinary.
But if the conclusion which results from the combination of the
testimonies was impossible one of them would be necessarily false; but
an impossible conclusion is the limit of extraordinary conclusions, as
error is the limit of improbable conclusions; the value of the
testimonies which becomes zero in the case of an impossible conclusion
ought then to be very much decreased in that of an extraordinary
conclusion. This is indeed confirmed by the calculus of probabilities.
In order to make it plain let us consider two urns, A and B, of which
the first contains a million white balls and the second a million black
balls. One draws from one of these urns a ball, which he puts back into
the other urn, from which one then draws a ball. Two witnesses, the one
of the first drawing, the other of the second, attest that the ball
which they have seen drawn is white without indicating the urn from
which it has been drawn. Each testimony taken alone is not improbable;
and it is easy to see that the probability of the fact attested is the
veracity itself of the witness. But it follows from the combination of
the testimonies that a white ball has been extracted from the urn A at
the first draw, and that then placed in the urn B it has reappeared at
the second draw, which is very extraordinary; for this second urn,
containing then one white ball among a million black balls, the
probability of drawing the white ball is 1/1000001. In order to
determine the diminution which results in the probability of the thing
announced by the two witnesses we shall notice that the event observed
is here the affirmation by each of them that the ball which he has seen
extracted is white. Let us represent by 9/10 the probability that he
announces the truth, which can occur in the present case when the
witness does not deceive and is not mistaken at all, and when he
deceives and is mistaken at the same time. One may form the four
following hypotheses:
1st. The first and second witness speak the truth. Then a white ball has
at first been drawn from the urn A, and the probability of this event is
½, since the ball drawn at the first draw may have been drawn either
from the one or the other urn. Consequently the ball drawn, placed in
the urn B, has reappeared at the second draw; the probability of this
event is 1/1000001, the probability of the fact announced is then
1/2000002. Multiplying it by the product of the probabilities 9/10 and
9/10 that the witnesses speak the truth one will have 81/200000200 for
the probability of the event observed in this first hypothesis.
2d. The first witness speaks the truth and the second does not, whether
he deceives and is not mistaken or he does not deceive and is mistaken.
Then a white ball has been drawn from the urn A at the first draw, and
the probability of this event is ½. Then this ball having been placed in
the urn B a black ball has been drawn from it: the probability of such
drawing is 1000000/1000001; one has then 1000000/2000002 for the
probability of the compound event. Multiplying it by the product of the
two probabilities 9/10 and 1/10 that the first witness speaks the truth
and that the second does not, one will have 9000000/200000200 for the
probability for the event observed in the second hypothesis.
3d. The first witness does not speak the truth and the second announces
it. Then a black ball has been drawn from the urn B at the first
drawing, and after having been placed in the urn A a white ball has been
drawn from this urn. The probability of the first of these events is ½
and that of the second is 1000000/1000001; the probability of the
compound event is then 1000000/2000002. Multiplying it by the product of
the probabilities 1/10 and 9/10 that the first witness does not speak
the truth and that the second announces it, one will have
9000000/200000200 for the probability of the event observed relative to
this hypothesis.
4th. Finally, neither of the witnesses speaks the truth. Then a black
ball has been drawn from the urn B at the first draw; then having been
placed in the urn A it has reappeared at the second drawing: the
probability of this compound event is 1/2000002. Multiplying it by the
product of the probabilities 1/10 and 1/10 that each witness does not
speak the truth one will have 1/200000200 for the probability of the
event observed in this hypothesis.
Now in order to obtain the probability of the thing announced by the two
witnesses, namely, that a white ball has been drawn at each draw, it is
necessary to divide the probability corresponding to the first
hypothesis by the sum of the probabilities relative to the four
hypotheses; and then one has for this probability 81/18000082, an
extremely small fraction.
If the two witnesses affirm the first, that a white ball has been drawn
from one of the two urns A and B; the second that a white ball has been
likewise drawn from one of the two urns A´ and B´, quite similar to the
first ones, the probability of the thing announced by the two witnesses
will be the product of the probabilities of their testimonies, or
81/100; it will then be at least a hundred and eighty thousand times
greater than the preceding one. One sees by this how much, in the first
case, the reappearance at the second draw of the white ball drawn at the
first draw, the extraordinary conclusion of the two testimonies
decreases the value of it.
We would give no credence to the testimony of a man who should attest to
us that in throwing a hundred dice into the air they had all fallen on
the same face. If we had ourselves been spectators of this event we
should believe our own eyes only after having carefully examined all the
circumstances, and after having brought in the testimonies of other eyes
in order to be quite sure that there had been neither hallucination nor
deception. But after this examination we should not hesitate to admit it
in spite of its extreme improbability; and no one would be tempted, in
order to explain it, to recur to a denial of the laws of vision. We
ought to conclude from it that the probability of the constancy of the
laws of nature is for us greater than this, that the event in question
has not taken place at all—a probability greater than that of the
majority of historical facts which we regard as incontestable. One may
judge by this the immense weight of testimonies necessary to admit a
suspension of natural laws, and how improper it would be to apply to
this case the ordinary rules of criticism. All those who without
offering this immensity of testimonies support this when making recitals
of events contrary to those laws, decrease rather than augment the
belief which they wish to inspire; for then those recitals render very
probable the error or the falsehood of their authors. But that which
diminishes the belief of educated men increases often that of the
uneducated, always greedy for the wonderful.
There are things so extraordinary that nothing can balance their
improbability. But this, by the effect of a dominant opinion, can be
weakened to the point of appearing inferior to the probability of the
testimonies; and when this opinion changes an absurd statement admitted
unanimously in the century which has given it birth offers to the
following centuries only a new proof of the extreme influence of the
general opinion upon the more enlightened minds. Two great men of the
century of Louis XIV.—Racine and Pascal—are striking examples of this.
It is painful to see with what complaisance Racine, this admirable
painter of the human heart and the most perfect poet that has ever
lived, reports as miraculous the recovery of Mlle. Perrier, a niece of
Pascal and a day pupil at the monastery of Port-Royal; it is painful to
read the reasons by which Pascal seeks to prove that this miracle should
be necessary to religion in order to justify the doctrine of the monks
of this abbey, at that time persecuted by the Jesuits. The young Perrier
had been afflicted for three years and a half by a lachrymal fistula;
she touched her afflicted eye with a relic which was pretended to be one
of the thorns of the crown of the Saviour and she had faith in instant
recovery. Some days afterward the physicians and the surgeons attest the
recovery, and they declare that nature and the remedies have had no part
in it. This event, which took place in 1656, made a great sensation, and
"all Paris rushed," says Racine, "to Port-Royal. The crowd increased
from day to day, and God himself seemed to take pleasure in authorizing
the devotion of the people by the number of miracles which were
performed in this church." At this time miracles and sorcery did not yet
appear improbable, and one did not hesitate at all to attribute to them
the singularities of nature which could not be explained otherwise.
This manner of viewing extraordinary results is found in the most
remarkable works of the century of Louis XIV.; even in the Essay on the
Human Understanding by the philosopher Locke, who says, in speaking of
the degree of assent: "Though the common experience and the ordinary
course of things have justly a mighty influence on the minds of men, to
make them give or refuse credit to anything proposed to their belief;
yet there is one case, wherein the strangeness of the fact lessens not
the assent to a fair testimony of it. For where such supernatural events
are suitable to ends aimed at by him who has the power to change the
course of nature, there, under such circumstances, they may be the
fitter to procure belief, by how much the more they are beyond or
contrary to ordinary observation." The true principles of the
probability of testimonies having been thus misunderstood by
philosophers to whom reason is principally indebted for its progress, I
have thought it necessary to present at length the results of calculus
upon this important subject.
There comes up naturally at this point the discussion of a famous
argument of Pascal, that Craig, an English mathematician, has produced
under a geometric form. Witnesses declare that they have it from
Divinity that in conforming to a certain thing one will enjoy not one or
two but an infinity of happy lives. However feeble the probability of
the proofs may be, provided that it be not infinitely small, it is clear
that the advantage of those who conform to the prescribed thing is
infinite since it is the product of this probability and an infinite
good; one ought not to hesitate then to procure for oneself this
advantage.
This argument is based upon the infinite number of happy lives promised
in the name of the Divinity by the witnesses; it is necessary then to
prescribe them, precisely because they exaggerate their promises beyond
all limits, a consequence which is repugnant to good sense. Also
calculus teaches us that this exaggeration itself enfeebles the
probability of their testimony to the point of rendering it infinitely
small or zero. Indeed this case is similar to that of a witness who
should announce the drawing of the highest number from an urn filled
with a great number of numbers, one of which has been drawn and who
would have a great interest in announcing the drawing of this number.
One has already seen how much this interest enfeebles his testimony. In
evaluating only at ½ the probability that if the witness deceives he
will choose the largest number, calculus gives the probability of his
announcement as smaller than a fraction whose numerator is unity and
whose denominator is unity plus the half of the product of the number of
the numbers by the probability of falsehood considered _à priori_ or
independently of the announcement. In order to compare this case to that
of the argument of Pascal it is sufficient to represent by the numbers
in the urn all the possible numbers of happy lives which the number of
these numbers renders infinite; and to observe that if the witnesses
deceive they have the greatest interest, in order to accredit their
falsehood, in promising an eternity of happiness. The expression of the
probability of their testimony becomes then infinitely small.
Multiplying it by the infinite number of happy lives promised, infinity
would disappear from the product which expresses the advantage resultant
from this promise which destroys the argument of Pascal.
Let us consider now the probability of the totality of several
testimonies upon an established fact. In order to fix our ideas let us
suppose that the fact be the drawing of a number from an urn which
contains a hundred of them, and of which one single number has been
drawn. Two witnesses of this drawing announce that number 2 has been
drawn, and one asks for the resultant probability of the totality of
these testimonies. One may form these two hypotheses: the witnesses
speak the truth; the witnesses deceive. In the first hypothesis the
number 2 is drawn and the probability of this event is 1/100. It is
necessary to multiply it by the product of the veracities of the
witnesses, veracities which we will suppose to be 9/10 and 7/10: one
will have then 63/10000 for the probability of the event observed in
this hypothesis. In the second, the number 2 is not drawn and the
probability of this event is 99/100. But the agreement of the witnesses
requires then that in seeking to deceive they both choose the number 2
from the 99 numbers not drawn: the probability of this choice if the
witnesses do not have a secret agreement is the product of the fraction
1/99 by itself; it becomes necessary then to multiply these two
probabilities together, and by the product of the probabilities 1/10 and
3/10 that the witnesses deceive; one will have thus 1/330000 for the
probability of the event observed in the second hypothesis. Now one will
have the probability of the fact attested or of the drawing of number 2
in dividing the probability relative to the first hypothesis by the sum
of the probabilities relative to the two hypotheses; this probability
will be then 2079/2080, and the probability of the failure to draw this
number and of the falsehood of the witnesses will be 1/2080.
If the urn should contain only the numbers 1 and 2 one would find in the
same manner 21/22 for the probability of the drawing of number 2, and
consequently 1/22 for the probability of the falsehood of the witnesses,
a probability at least ninety-four times larger than the preceding one.
One sees by this how much the probability of the falsehood of the
witnesses diminishes when the fact which they attest is less probable in
itself. Indeed one conceives that then the accord of the witnesses, when
they deceive, becomes more difficult, at least when they do not have a
secret agreement, which we do not suppose here at all.
In the preceding case where the urn contained only two numbers the _à
priori_ probability of the fact attested is ½, the resultant probability
of the testimonies is the product of the veracities of the witnesses
divided by this product added to that of the respective probabilities of
their falsehood.
It now remains for us to consider the influence of time upon the
probability of facts transmitted by a traditional chain of witnesses. It
is clear that this probability ought to diminish in proportion as the
chain is prolonged. If the fact has no probability itself, such as the
drawing of a number from an urn which contains an infinity of them, that
which it acquires by the testimonies decreases according to the
continued product of the veracity of the witnesses. If the fact has a
probability in itself; if, for example, this fact is the drawing of the
number 2 from an urn which contains an infinity of them, and of which it
is certain that one has drawn a single number; that which the
traditional chain adds to this probability decreases, following a
continued product of which the first factor is the ratio of the number
of numbers in the urn less one to the same number, and of which each
other factor is the veracity of each witness diminished by the ratio of
the probability of his falsehood to the number of the numbers in the urn
less one; so that the limit of the probability of the fact is that of
this fact considered _à priori_, or independently of the testimonies, a
probability equal to unity divided by the number of the numbers in the
urn.
The action of time enfeebles then, without ceasing, the probability of
historical facts just as it changes the most durable monuments. One can
indeed diminish it by multiplying and conserving the testimonies and the
monuments which support them. Printing offers for this purpose a great
means, unfortunately unknown to the ancients. In spite of the infinite
advantages which it procures the physical and moral revolutions by which
the surface of this globe will always be agitated will end, in
conjunction with the inevitable effect of time, by rendering doubtful
after thousands of years the historical facts regarded to-day as the
most certain.
Craig has tried to submit to calculus the gradual enfeebling of the
proofs of the Christian religion; supposing that the world ought to end
at the epoch when it will cease to be probable, he finds that this ought
to take place 1454 years after the time when he writes. But his analysis
is as faulty as his hypothesis upon the duration of the moon is bizarre.