Hume's ultimate target is the New Testament miracles, especially Jesus'
resurrection. However, he does not attack them directly, but uses
indirect arguments. Hume is not saying that miracles are impossible and
do not happen. What he is saying is that the evidence will always be
insufficient to warrant belief.
1 Bayes Theorem
It has been said that one of the problems with Hume is that he was born
before Bayes' Theorem. He wrote at a time when the mathematics of
probability was in its infancy. Thus his argument is qualitative rather
than quantitative. By a strange coincidence, Bayes' Theorem was
proposed by Thomas Bayes (1702-1761), who was a British mathematician
and a Presbyterian minister. Bayes' Theorem can be stated as P(A/B) =
P(B/A) * P(A) / P(B), where P(A/B) means the probability that A will
occur given that event B has already occurred. Bayes' Theorem is most
easily understood by an example. I have taken this example from
Wikipedia, but altered it slightly to reflect modern trends.
Suppose there is a school with 60 boys and 40 girls. 50% of girls wear
trousers and 50% wear skirts. 5/6 of the boys wear trousers and (you
guessed it) 1/6 boys wear skirts. If a student (of unknown
gender) is wearing trousers, what is the probability that it is a girl?
Let A be, "The student is a girl" and B be, "The
student is wearing trousers".
P(B/A) is that probability that a student is wearing
trousers given that the student is a girl, which is 0.5 or 50%.
P(A) is the probability that a student is a girl,
which is 0.4 or 40%.
P(B) is the probability that a student is wearing
trousers. This is the probability that:
A student is a boy (0.6) * the probability that a boy is wearing
trousers (5/6) +
probability that a student is a girl (0.4) * the probability that a
girl is wearing trousers (0.5).
Thus P(B) is the probability that a student is wearing trousers =
0.6 * 5/6 + 0.4 * 0.5 = 0.7.
Thus according to Bayes' Theorem P(A/B) = 0.5 * 0.4/0.7 = 2/7. i.e.,
the probability that a student is a girl given that the student is
wearing trousers is 2 in 7.
This can be verified by the following table:
There are 60 boys and 5/6 wear trousers; therefore 50 boys wear
trousers. There are 40 girls and 50% wear trousers; hence 20 girls wear
trousers. Therefore, 20 out of 70 students who wear trousers are girls.
Therefore P(A/B) = 2/7 as derived by Bayes' Theorem. So, behold it
Now I will apply Bayes' Theorem to miracles.
2 Application of Bayes' Theorem to Miracles
Let M be the event that a miracle occurred. Let R be the event that a
report has been received that a miracle occurred. Then P(M/R) is the
probability that a miracle occurred given that a report is received.
According to Bayes' Theorem P(M/R) = P(R/M) * P(M) / P(R).
People tell the truth most of the time but occasionally what they say
is wrong, either deliberately or unintentionally. Let us be
uncharitable and denote an incorrect report by the event L (for Lie),
and let T be a true report. Therefore, P(T) = 1 – P(L).
P(R/M) is the probability that a miracle is reported given that a
miracle actually occurred. This is simply P(T), ie the witness reported
P(M) is the probability that the miracle occurred in the first place.
P(R) is a bit trickier. This is the probability that a report of a
miracle was received, whether true or false. Let M' (not M) be the
event that the miracle did not occur. Then P(R) = P(M)*P(T) +
For the purpose of this example, I will adopt Hume's assumptions on
probabilities, although I will argue against them later. Hume argues
that a miracle, by nature of the case, is highly improbable just based
on the relative frequency of occurrence. Thus P(M) is a very small
positive number. Hume would argue that there is at most one person who
has risen from the dead (the rest being resuscitations). Thus, as far
as the resurrection is concerned, P(M) is approximately 10-10
(based approximately on the total number of people who have existed
during recorded history). Hume allows that people tell the truth much
more often than not. So let us be generous (like Hume) and suppose that
people lie less than 1% of the time. Then the application of Bayes'
P(M/R) = 9.9*10-9, which is a very small number.
So Hume seems to be right. If a single witness reported that a random
person was raised from the dead, then it is highly improbable that the
event actually occurred. This is Hume's core argument that the
improbability of a miracle outweighs the reliability of human testimony.
However, what if multiple (n) independent witnesses reported the same
event? The probability that they were all lying is P(L)n,
which I will denote as Pn(L), which becomes vanishingly small as n
increases. Conversely, Pn(T) asymptotes to 1 as n increases.
The following graph shows a plot of the probability that the miracle
actually occurred against the number of independent witnesses (for the
assumed values of P(M) and P(L)).
For this example, a wise man would believe that the
miracle occurred if there were more than 5 independent witnesses.
You can argue all you like about appropriate values
for P(M) and P(L). However, provided P(M) is non-zero there is always a
value for the number of witnesses above which it would be wise to
believe the report.
The above example is simplistic. Many factors
contribute to an actual historical argument and these factors may be
independent or interdependent. In most historical arguments it is
difficult to assign agreed probabilities to each factor and derive a
reliable result. Thus most historical arguments end up being
qualitative rather than quantitative. However, the example demonstrates
that multiple attestations may be sufficient to warrant belief in an
improbable event. Thus Hume's core argument fails.
What I am saying is only common sense. It is
something that is happening in our courts every day. There is a low
probability that a random individual has committed a crime, but the
weight of evidence may still be sufficient to secure a conviction
beyond reasonable doubt.
3 The Probability of Miracles
In chapter 90 Hume states, "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." From this Hume seems to be implying:
A miracle is a violation of the laws of nature.
The laws of nature are derived from our uniform
experience and are a description of what always happens.
Thus by definition miracles never happen.
However something smells about this argument. It
simply illustrates that the term "miracle" can be defined in such a
manner as to be logically incoherent, such as a "married bachelor". On
the contrary, the above argument could be modified as follows:
A miracle is a violation of the laws of nature.
The laws of nature are a description of what
Thus a miracle is an unusual event.
Hume also seems to assign probabilities just based
on relative frequencies. However, this approach is simplistic. For
example, more people die from playing lawn bowls than from
hang-gliding. Does that make lawn bowls a more dangerous sport? The
context (e.g., age of participants) will affect the probability. In the
same manner, the probability of Jesus' resurrection should not be based
just on relative frequencies. It will be affected by background
issues/beliefs, such as whether God exists and if He is interested in
us. In this case, a person may well believe that P(M) is much greater
4 Witness Reliability
Hume's second argument is that no miracle has been
attested by a sufficient number of reliable witnesses. In paragraph 92
there is not to be found, in all history, any miracle
attested by a sufficient number of men, of such unquestioned
good-sense, education, and learning, as to secure us against all
delusion in themselves; of such undoubted integrity, as to place them
beyond all suspicion of any design to deceive others; of such credit
and reputation in the eyes of mankind, as to have a great deal to lose
in case of their being detected in any falsehood; and at the same time,
attesting facts performed in such a public manner and in so celebrated
a part of the world, as to render the detection unavoidable: All which
circumstances are requisite to give us a full assurance in the
testimony of men."
A couple of comments can be made on this argument.
Firstly, he sets the bar very high. I doubt
whether Hume would qualify himself. If we required the same witness
credibility in court then the court system would get nowhere.
Secondly, he does not consider examples. What
about the apostle Paul and Luke? I haven't got space to go into detail,
but I think they come close.
5 Conflicting Miracle Claims
In his final section Hume claims that there are
competing and conflicting historical miracle claims that essentially
defeat one another. He cites a number of examples. I have not space to
consider them all, but I will only discuss the example of the Roman
emperor Vespasian. Hume states in section 96,
"One of the
miracles in all profane history, is that which Tacitus reports of
Vespasian, who cured a blind man in Alexandria, by means of his
spittle, and a lame man by the mere touch of his foot; in obedience to
a vision of the god Serapis, who had enjoined them to have recourse to
the Emperor, for these miraculous cures."
There are 2 issues with this example:
How strong is the historical attestation for
Would this miracle be in conflict with Biblical
Vespasian (9AD to 79AD) led the Roman army in
subjugating the Jewish rebellion in 66AD. He became emperor in 69AD.
While in Alexandria in Egypt in 69 AD Vespasian is purported to have
healed a blind man and a lame man. These miraculous events were
reported by the Roman historians:
Tacitus (56AD to 117AD) in Book 4 of his Histories
(written about 100AD to 110AD),
Suetonius (69AD to 130AD) in Book 8 of the Lives
of the Caesars (written about 119AD), and
Dio Cassius (about 155AD to 229AD) in book 65 of
his Roman History (written after 200AD).
You can access these accounts at:
Hence we have 3 records by credible historians.
Tacitus even records, "Persons actually present attest both facts, even
now when nothing is to be gained by falsehood." Most historians agree
that something unusual happened. However, there are causes for doubt.
Tacitus elsewhere records that he did not believe it was a miracle and
that he believed that the healings occurred by natural means. Dio
Cassius records that the Alexandrians were unimpressed.
The Roman emperors were purported to be divine
figures and any miraculous associations assisted an aspirant in
obtaining power. Modern historians surmise that it was a setup by
Vespasian's followers to enhance his aspirations for power. If we
consider P(M) and P(L), we discover things are not so good. The
Egyptian god Serapis has long since packed his bags and departed from
public interest. So a coherent basis for the miracles is lacking. Thus
P(M) is inordinately low. The historians were probably honest in their
2nd-hand reports of the stories but the promoters and witnesses at the
time had a strong motivation for a positive account. Thus P(L) is high
and the witnesses were probably not independent. So, do I believe it?
The other issue is whether this is a conflicting
miracle claim. In Matthew 24:24 Jesus is reported to say, "False
Christs and false prophets will appear and perform great signs and
miracles to deceive even the elect — if that were possible." This opens
the possibility for miracles originating from "profane" sources.
Doubtless there have been many spurious claims
regarding miracles. However, if 2 miraculous claims are inherently
incompatible then at least one of them must lack sufficient evidence,
but the fact that they conflict does not mean that both claims are
Hume states, "A wise man proportions his belief to
the evidence." He goes on, however, to conclude that the consideration
of historical evidence for a miracle is pointless, as a matter of
I have not attempted to prove that miracles do
occur. My purpose has been more modest. I have only attempted to show
that a historical argument for the occurrence of a miracle can
potentially be sufficient to warrant belief. My argument has shown that
multiple attestations can in principle provide sufficient evidence to
warrant belief in a miracle.
Miracles and Probability
(Investigator 134, 2010
Kevin Rogers (Investigator 133) uses the inferential theorem of
Reverend Thomas Bayes (1702 – 1761) to argue falsification of David
Hume's claim that 'historical evidence' supporting miracles will
be insufficient to warrant belief in the miraculous. He accurately
summarizes Hume's argument relating to 'frequency data', illustrates
with an example using specified numbers of boys and girls, clad in
trousers and/or skirts, and equates his example with a fundamentally
different hypothetical investigation relating to undefined reported
'miracles' provided by an unspecified number of 'witnesses'. Using this
assortment of 'eclectic methodology', he claims to demonstrate
"multiple attestations may be sufficient to warrant belief in an
Note the important difference between his 'example', where 'the number'
of subjects with 'two differences' (they wear trousers or skirts), are clearly
specified whereas Kevin's 'hypothetical data' will
relate to an 'open-ended' number of subjects (the world's population?)
infinite possibility of 'differences' (miraculous happenings). Before
identifying further flaws in Kevin's rationale, a few words defining
and clarifying basic statistical terminology that might be helpful for
readers unfamiliar with 'inferential statistics' and Bayes' theorem in
A researcher should test an 'hypothesis' from a 'representative' sample
from the target 'population' (being pedantic, this is never really
possible; we can never be sure our sample is truly 'representative'
— if we could, we wouldn't need a sample for testing, in the
first place!). A 'hypothesis' normally takes the form that a given
'variable' will influence/cause subjects to behave in a particular way;
an experiment will impose that variable on sample members, recording
any resultant behavioural change. The obtained data will either
support or not support the hypothesis.
Bayes is different.
In most branches of science, statistical tests are used to 'test'
beliefs and theories against reality, but this is not the case
Bayesian inference. Bayes requires not that beliefs
theories correspond to reality, rather that they are internally
consistent (or 'cohere'). In contrast to the most commonly used
procedures for 'testing probability', a Bayesian practitioner works not
only with an hypothesis, but also with a specified predicted 'prior
probability' that his test will evoke an expected probability outcome.
As prominent Bayesian theoretician Lindley put it, "the theory…is
concerned with coherence; with how different views fit together, not
with judgments of right and wrong. Perhaps the most obvious example of
coherence is the way in which views prior to the data must cohere with
those posterior to it: namely the Bayes' theorem."(1)
The whole slant of the Bayesian approach is that obtained evidence is
not required to match the 'data' of the world 'out there', but to infer
the probability that the opening (a priori) hypothesis might be true
– 'truth' measured by its consistency with the hypothesis. Bayesian
inference does not aspire to be objective, rather it relies upon
(measurable) 'degrees of belief' — subjective probabilities. The
frequent criticism of the Bayesian approach arises from the status of
the a posterior probability; after all, researchers may differ
radically in their opinions regarding a population value. It is
accepted from the outset, that an a posterior probability remains a
statement of personal probability, that being the only kind of
probability there is – the only way out of this dilemma for the
Bayesian being that, in principle, the addition of more and
more data, should cause the two positions to approach one
the above in mind, we can now better understand some of Kevin's
(subjects) must be 'independent'
This is a basic requirement of virtually every statistical test.
Witnesses cannot be viewed as 'independent' if they are subject to
influence from a fellow witness or group, especially if sharing
critical or relevant beliefs: individual perceptions readily come
together/transcend into group perceptions. Countless studies
demonstrate the presence of others influences how an individual
perceives an event; in addition, mistaken perception may be shared by
all witnesses – how many members of the audience see the conjuror
produce an egg from his ear? It is relatively easy to manipulate
naïve Christian subjects into believing they are having a
'religious experience' when they are together at prayer. (2)
When it is reported that in Galilee, after Christ's crucifixion, Jesus
was seen "by 500 brethren at one time", that may be independent
evidence provided the person reporting the incident was himself
at the event. However, the only record comes from a Pauline
letter. Paul was not there; he was reporting what he had been
evidence is 'hearsay'. A convert to the new religion (as a result of an
experience on the road to Damascus — a 'religious experience'), he had
to believe in the resurrection. (I am not suggesting Paul
dishonest or lying — I have no reason to doubt the sincerity of any
biblical witness — such evidence is commonly reported in
studies of conversion).
Assuming there were indeed 500 people present, not all could recognize
or 'knew' Jesus (remember, a few days before, the Romans needed bribe
Judas to identify Jesus!) — if it were a mass gathering of mostly 'true
believers', the mass psychology of the occasion would engender a belief
the person in question was indeed a 'resurrected being' — possibly not
considered so unusual in those times!? There is no independent
of this event.
One of the most common errors in the application of Bayes
Theorem is a
failure by practitioners to take account of the lack of independence of
the subjects/witnesses. This is especially true of groups of people
sharing a strong belief/commitment shown a relevant 'happening'.
recognizes the need for independence, but fails to consider that none
of the Biblical resurrection 'witnesses' remotely approach
'independence' – indeed, the opening four verses of the gospel of
make it crystal clear that Luke himself purports to be doing no more
than summarize (and edit) some of the contemporary hearsay
the Miraculous details
It is acceptable for Kevin Rogers to select a statistical procedure allowing
him to play with 'belief probabilities' rather than
events; thus avoiding details of the 'miraculous event' – but he
have it 'both ways' by arguing his example "demonstrates that
attestations may be sufficient to warrant belief in an improbable
event". Remember, the Bayes' theorem focuses on 'belief
probabilities', not events that actually happen. Kevin
demonstrate 'multiple attestations justify a belief' — rather that
people might justifiably change their degree of belief, that 'prior
opinions become less and less relevant to posterior opinion with
additional data' — Bayesian inference argues a 'test' gives us 'test
probabilities', not real probabilities!
From his graph, Kevin tells us any intelligent person would require
only five independent witnesses to justify believing a miracle had
occurred. But Kevin himself does not believe this nonsense! In
discussion, almost a year ago, I referred Kevin to a Bill Bryson book
where the author describes his decision to return to the States (from
the UK) on learning that "3.7 million Americans, according to a
Poll, believed that they had been abducted by aliens at one time or
another"(3). In 1994, John E Mack, professor of Psychiatry at the
Harvard Medical School published details of face-to-face interviews
with more than eighty American citizens describing their 'alien
abductions'. Thousands of hours of interviews convinced Mack these men,
women and children were truthfully reporting authentic experiences…he
describes, in great detail, their accounts of encounters with the
'beings' who transported their immobilized 'victims' to a spacecraft.
(4) When it was pointed out to Kevin Rogers that Mack's book provides
substantive evidence from a multitude of independent witnesses
(compared with the inability of his Church to provide a single
independent witness of Christ's resurrection!), it became clear
was not prepared to discuss 'alien abductions'. Yet today he
"a wise man would believe that the miracle occurred if there were
than five independent witnesses".
All Luke's gospel (including 'Acts') evidence is hearsay. Were
document, signed by one of the two apostles who encountered Jesus on
the road to Emmaus, we might accept it as 'independent' witness
statement and acceptable as evidence…but one wonders what credence we
might give these two 'witnesses'. Consider 'the facts'. Two members of
Christ's 'inner circle', walked and discussed with Jesus for several
hours (the same Jesus they had associated with up to a few days
previously), yet failed to recognize him during their very
longish walk to Emmaus. Are these the individuals we might take to a
modern-day 'identification parade', looking for the guilty person? All
Luke's 'hearsay' reportage of encounters between disciples and the
'risen Lord' echo this same problem. For several years
previously, the disciples witnessed Jesus perform numerous 'miracles';
the raising of Lazarus, blind men given sight, devils cast out — yet
when Jesus returns, in the same physical form, as he had previously
assured them he would, albeit with wounded hands and feet, they
difficulty recognizing him. Even, as he ascends into the clouds (quite
literally) two angels are waiting to re-assure 'the eleven' that it was
indeed their late 'teacher' who ascended into the sky…and that he
I read Tom Paine's The Age of Reason in my teenage years and
astounded at his identification of the conflicting accounts of the
Resurrection in the four gospels. Christians replied to me it was
common knowledge that if a dozen people witnessed a road accident and
all twelve were obliged to write independent reports of what
place, the outcome would be twelve quite different accounts. At first I
thought this to be a very reasonable reply. But the argument has its
limits. If it was my Dad driving one of the cars, I can't imagine
imagining it was the next-door neighbour. If an earthquake opened
cracks in the road and dead bodies emerged from the nearby cemetery and
inspected the debris in their shrouds, I can't imagine most present not
hearing and feeling the earth move — hardly trivial happenings, but the
events described by Matthew, find no corroboration elsewhere. With the
absence of independent witnesses, I cannot see how Kevin can even begin
with his statistics!
(1) Lindley, D. V. Bayesian Statistics in Harper & Hooker
(eds) Foundations of Probability Theory, Statistical Inference and
Statistical Theories of Science: Reidel (1976) p 435.
(2) Pahnke, W. N. The Religious Experience in Psychedelic
(3) Bryson, Bill Notes from a Small Island Black
(4) John E Mack Abduction: Human Encounters with Aliens
& Schuster (1994)
Hume, Miracles and Probability
(Investigator 136, 2010
In Investigator #133 I presented a critique of "On Miracles" by
Scottish Enlightenment philosopher, David Hume (1711-1776). Hume's
primary argument was that we should never believe a report of a
miracle, as the likelihood of a false report would always outweigh the
chance of the miracle actually occurring. I argued that multiple
attestations to the same event from a sufficient number of independent
witnesses could provide sufficient evidence to warrant belief. In Investigator
#134 Bob Potter critiqued my argument.
Baye's Theorem states, P(A/B)*P(B) = P(B/A)*P(A), where P(A/B) means
the probability that A occurred given that B has already occurred. For
example, what is the probability that it will rain, given that there
are clouds in the sky? Baye's Theorem allows us to calculate
conditional probabilities. There are 4 terms in the equation. If we
know 3 of them then we can calculate the 4th. I demonstrated Baye's
Theorem with an example concerning boys and girls wearing trousers and
skirts at school. Bob claimed that I equated my tutorial example with
the argument on miracles. This is not true. The purpose of the example
was only to illustrate the use of Baye's Theorem. I then reapplied
Bayes's Theorem to the argument regarding miracles.
Bob claims that my example contained a known number of students whereas
miracles involve an unknown number of subjects. This is true, but Bayes
Theorem is always true, whether the number of members in the set is
defined or not. With large undefined samples the 3 "known" terms in
general have to be estimated. The answer is only as accurate as the
input data. Since my argument was an in-principle argument, the initial
estimates of the input parameters are not relevant. This supposedly was
my initial flaw and Bob claims further flaws.
Bob goes on to provide a discussion of "Bayesian Inference". Bob has
confused Baye's Theorem with Bayesian Inference. Bayesian Inference is
a form of qualitative argument that is based on Baye's Theorem, whereas
Baye's Theorem is a simple quantitative mathematical formula. If you
would like to see good examples of the use of Bayesian Inference, then
read Richard Swinburne's books on "The existence of God" and "The
Resurrection of God Incarnate".
Bob then proceeds to use a whole pile of big words, convoluted
sentences, plenty of underlining and bolding. I think we are supposed
to be impressed by his great learning. At the end he concludes, "We can
now better understand some of Kevin's statistical flaws." Can we? Did
you? Was it the bolding or underlining that did it? The conclusion
appears like a rabbit out of a hat. The conclusion is unrelated to his
Bob then goes on to claim that witnesses must be independent, as
dependent witnesses can affect each other. I wholeheartedly agree, as
this was what I stated in my original article. However, Hume's primary
argument was an "in-principle" argument and I provided an in-principle
objection. Neither Hume nor I were considering specific examples.
Fortunately Bob and I are actually in agreement. Bob doesn't believe
David Hume either, for Bob then proceeds to raise historical arguments
about the resurrection. If Bob actually believed David Hume, then he
wouldn't have bothered. The essence of Hume's argument is that the
issue of miracles is not worth discussing, but Bob then goes on to
discuss the evidence. Bob, I am on your side. I agree with you that
David Hume is wrong.
Bob goes on to argue that I do not believe my own argument because I do
not believe in alien abductions despite the large number of witnesses.
My reasons for not believing in alien abductions are the initial
implausibility and that there tend not to be independent witnesses to
the same event. It seems implausible because of the huge distances
involved. It would require a nuclear powered spaceship with nearly 100%
efficiency to accelerate a spaceship near to the speed of light. Even
then it would require a large number of years to travel from the
nearest viable star. In addition I do not know of any cases where there
are multiple independent witnesses to the same event. Instead, there
are multiple witnesses to different events. I actually do believe
Hume's argument where there is a single witness to a single event.
Given the large population of earth there are bound to be odd
individuals who will lie or be deluded and provide a strange report.
However, their stories are not generally corroborated by multiple
independent witnesses. There are numerous instances of multiple
witnesses attesting to seeing a UFO. However, this does not necessarily
mean that the UFO is an alien. I am actually agnostic on the subject.
If I could be shown that there were multiple witnesses to the same
event then I would have to reconsider my position. Apart from this, I
haven't done much research on the issue.
Bob also mentions that multiple people may see a conjurer produce an
egg from his ear. This is a bad argument. The multiple witnesses do not
actually believe that the conjurer produced an egg from their ear. They
know it is a trick. The multiple witnesses will uniformly testify that
they were tricked; and they would be right.
My argument against David Hume is nothing new. Hume's argument has been
debunked many times before and it is widely accepted within
philosophical circles that Hume's argument is wrong. However, there is
a false perception amongst sceptics that Hume has issued the final
word. You don't have to understand Baye's Theorem to appreciate that
Hume is wrong. If there are multiple independent witnesses to the same
event then this provides stronger evidence that the event actually
occurred. This is just common sense and is basic to police
investigations and for court proceedings.
I agree with Bob that the apostolic witness to the resurrection is not
a clear cut case of multiple independent witnesses. The actual
situation is far more complicated and involves a mixture of independent
and dependent testimony. However, I think we can dispense with
David Hume's in-principle argument and look at the actual nature of the
evidence, but that will have to be dealt with another time.