Few arguments have excited greater attention, and produced more attempts at refutation, than the celebrated one of David Hume, respecting miracles; and it might be added, that more sophistry has been advanced against it, than its author employed in the whole of his writings.

It must be admitted that in the argument, as originally developed by its author, there exists some confusion between personal experience and that which is derived from testimony; and that there are several other points open to criticism and objection; but the main [120/121] argument, divested of its less important adjuncts, never has, and never will be refuted. Dr. Johnson seems to have been of this opinion, as the following extract from his life by Boswell proves:—

"Talking of Dr. Johnson's unwillingness to believe extraordinary things, I ventured to say— "'Sir, you come near to Hume's argument against miracles — That it is more probable witnesses should lie, or be mistaken, than that they should happen.'

"Johnson.— Why, Sir, Hume, taking the proposition simply, is right.
But the Christian revelation is not proved by miracles alone, but as connected
with prophecies, and' with the doctrines in confirmation of which miracles
were' wrought.'"^{1}

Hume contends that a miracle is a violation of the laws of nature; and as a firm and unalterable experience has established these laws', the proof against a miracle from the very nature of the fact, is as entire as any argument from experience can possibly be imagined. [121/122]

"The plain consequence is (and it is a general maxim worthy of our attention),
that no testimony is sufficient to establish a miracle, unless the testimony
be of such a kind, that its falsehood would be more miraculous than the
fact which it endeavours to establish: and even in that case there is a mutual
destruction of arguments, and the superior " only gives us an assurance suitable
to that degree of force " which remains after deducting the inferior."^{2}

The word *miraculous* employed in this passage is evidently equivalent
to *improbable,* although the improbability is of a very high degree.

The
condition, therefore, which, it is asserted by the argument of Hume, must be
fulfilled with regard to the testimony, is that the *improbability* of
its falsehood must be greater than the *improbability* of the occurrence
of the fact.

This is a condition which, when the terms in which it is expressed
are understood, immediately commands our assent. It is in the
[122/123] subsequent stage of the reasoning that the fallacy is introduced. Hume asserts,
that this condition cannot be fulfilled by the evidence of *any number* of
witnesses, because our experience of the truth of human testimony is not uniform
and without any exceptions; whereas, our experience of the course of nature,
or our experience against miracles, is uniform and uninterrupted.

The only sound
way of trying the validity of this assertion is to *measure* the numerical
value of the two improbabilities, one of which it is admitted must be greater
than the other; and to ascertain whether, by making any hypothesis respecting
the veracity of each witness, it is possible to fulfil that condition by any
finite number of such witnesses.

Hume appears to have been but very slightly
acquainted with the doctrine of probabilities, and, indeed, at the period when
he wrote, the details by which the conclusions [123/124] he had arrived at could be proved or refuted were yet to be examined and arranged.
It is, however, remarkable that the opinion he maintained respecting our knowledge
of causation is one which eminently brings the whole question within the province
of the calculus of probabilities. In fact, its solution can only be *completely* understood by those who are acquainted with that most difficult branch of
science. By those who are not so prepared, certain calculations, which will
be found more fully developed in the Note (E), must be taken for granted; and
all that can be attempted will be, to convey to them a general outline of the
nature of the principles on which these enquiries depend.

A miracle is, according to Hume, an event which has never happened within the experience of the whole human race. Now, the improbability of the future happening of such an occurrence may be calculated according to two different views. [124/125]

We may conceive an urn, containing *only* black and white balls, from
which *m* black balls have been successively drawn and replaced, one by
one; and we may calculate the probability of the appearance of a white ball
at the next drawing. This would be analogous to the case of one human being
raised from the dead after *m* instances to the contrary.

Looking, in another point of view, at a miracle, we may imagine an urn to contain a very large number of tickets, on each of which is written one of the series of natural numbers. These being thoroughly mixed together, a single ticket is drawn: the prediction of the particular number inscribed on the ticket about to be drawn may be assimilated to the occurrence of a miracle.

According to either of these views, the probability
of the occurrence of such an event by mere accident may be calculated. Now,
the reply to Hume's argument is this: [125/126] Admitting at once the essential point, viz. that the improbability of the concurrence
of the witnesses in falsehood must be *greater* than the improbability
of the miracle, it may be denied that this does not take place. Hume has asserted
that, in order to prove a miracle, a certain improbability must be *greater
* than another; and he has also asserted that this *never* can take
place.

Now, as each improbability can be truly measured by number, the *only
* way to refute Hume's argument is by examining the *magnitude* of these
numbers. This examination depends on known and admitted principles, for which
the reader, who is prepared by previous study, may refer to the work of Laplace,
*Théorie Analytique des Probabilités;* Poisson, *Recherches
sur la Probabilité des Jugements,* 1837; or he may consult the article
*Probabilities,* by Mr. De Morgan, in the Encyclopaedia Metropolitana,
in which he will find this subject examined. [126/127]

One of the most important principles on which the question rests, is the concurrence
of the testimony of independent witnesses. This principle has been stated by
Campbell, and has been employed by the Archbishop of Dublin,^{3} and also by Dr.
Chalmers.^{4} It requires however to be combined with another principle,
in order to obtain the numerical values of the quantities spoken of in the argument.
The following example may be sufficient for a popular illustration.

Let us suppose that there are witnesses who will speak the truth, and who are not themselves deceived in ninety-nine cases out of a hundred. Now, let us examine what is the probability of the falsehood of a statement about to be made by two such persons absolutely unknown to and unconnected with each other.

Since the order in which independent [127/128] witnesses give their testimony does not affect their credit, we may suppose that, in a given number of statements, both witnesses tell the truth in the ninety-nine first cases, and the falsehood in the hundredth.

Then the first time the second witness B testifies, he will agree with the testimony of the first witness A, in the ninety-nine first cases, and differ from him in the hundredth. Similarly, in the second testimony of B, he will again agree with A in ninety-nine cases, and differ in the hundredth, and so on for ninety-nine times; so that, after A has testified a hundred, and B ninety-nine times, we shall have

99 X 99 cases in which both agree,

99 cases in which they differ, A being
wrong.

Now, in the hundredth case in which B testifies, he is wrong; and, if we combine this with the testimony of A, we have ninety-nine cases in which A will be right and B wrong; and one case only in which both A and B will . [129] agree in error. The whole number of cases, which amounts to ten thousand, may be thus divided: — .

99 x 99 =9801 cases in which A and B agree in truth,

1 x 99 = 99 cases in which B is true and A is false,

99 x 1 = 99 cases in which A is true and B false,

1 x 1 = 1 cases in which bth A and B agree in a falsehood.

-------

10,000 cases

As there is only one case in ten thousand in which two such independent witnesses can agree in error, the probability of their future testimony being false is

1/10,000 or 1/(100)^{2}

The reader will already perceive how great a reliance is due to the *future
*concurring testimony of two independent witnesses of tolerably good character
and understanding. It appears that, previously to the testimony, the chance
of one such witness being in error is that of two concurring in the same error
(1/100)^{1} is (1/100)^{2} and
if the same reasoning be applied to three independent witnesses, it will be
[129/130] found that the probability of their agreeing in error is (1/100)^{3}; or that the odds are 999,999 to 1 against
the agreement.

Pursuing the same reasoning, the probability of the falsehood
of a fact which six such independent witnesses attest is, previously to the
testimony, (1/100)^{6 }or it is, in round numbers,
.

The improbability of the miracle of a dead man being restored, is, on the
principles stated by Hume, 1/20 (100)^{5}or it is —

200,000,000.000 to 1 against its occurrence.

It follows, then, that the chances of accidental or other independent concurrence
of only *six* such independent witnesses, is already *five times* as
great as the improbability against [130/131] the miracle of a dead man's being restored to life, deduced from Hume's method
of estimating its probability solely from experience.

This illustration shows the great accumulation of probability arising from the concurrence of independent witnesses: we must however combine this principle with another, before we can arrive at the real numerical value of the improbabilities referred to in the argument.

The calculation of the numerical values of these improbabilities I
have given in Note (E.) From this it results that, provided we assume that independent
witnesses can be found of whose testimony it can be stated that it is more probable
that it is true than that it is false, *we can always assign a number of witnesses
which will, according to Hume's argument, prove the truth of a miracle. *

12 December 2008