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 .  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.
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, .1,000,000,000,000 to 1 against the falsehood of their testimony.
The improbability of the miracle of a dead man being restored, is, on the principles stated by Hume, 1/20 (100)5or 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