David Hume believed that “laws of nature” were generalizations about nature that are based upon our experiences of what repeatedly happens in similar circumstances. When a sufficiently large number of experiences have been collected which yield the same result, a law-like summary of those experiences can be expressed in the form of All A’s are B’s. The process of forming generalizations based on specific results is called induction.
Hume defined a miracle as a “violation” of the laws of nature. A miracle is a violation because its occurrence conflicts with the massive body of evidence we have that nature operates in a regular and seamless way, without exceptions. An event such as a resurrection from the dead would count as a miracle because it conflicts which all the evidence we have that corpses do not come back to life.
Hume’s definition of miracles is distinct from his definition of so-called marvels. Marvels are bizarre and unusual events which are initially improbable but can still be rationally believed on the basis of new evidence because they do not conflict with nature’s laws. An example of a marvel might be one’s very first observation of an airplane in flight, without any prior knowledge of aerodynamics.
According to Hume, there are no circumstances under which it would be reasonable to believe in a miracle because any evidence that it occurred would be outweighed by the evidence that a natural law has not been violated. His reasoning is based on his so-called straight rule of induction which stipulates that if a sufficient number of A’s are found to be B’s, then the probability that all A’s are B’s is 1 and the probability that an A is not-B is 0.
In his book Hume’s Abject Failure (2000), John Earman criticizes the straight rule by considering the parable of the Indian Prince, a story already known in Hume’s day. This Prince lived in a part of India where the climate did not allow water to freeze. As a result, he had never seen water take solid form in freezing temperatures. Now, suppose that the Prince heard reports from other parts of the world that water could be transformed into a solid. These reports (which he would otherwise consider reliable) stated that water could bear the entire weight of an elephant by being frozen into large blocks of ice! How should the Prince respond to this new information? Well, if Hume’s straight rule is correct, the Prince should always reject those reports because the generalization that “water does not freeze” yields a probability of 1 – which implies that reports to the contrary have a probability of zero. But clearly this reasoning is mistaken. The Prince can (and should) reject his belief once he is provided with adequate reports to the contrary. So something must be wrong with Hume’s original straight rule.
In all fairness, perhaps Hume’s straight rule can be adapted to avoid the Prince’s mistake. Perhaps it can be restated to include variables such as reports from other people and evidence of low temperature climates. Unfortunately, not matter how many variables the rule tries to include, it will never yield probabilities equal to 1. Why? Because no rule of induction can specify in advance all the variables that are relevant to whether a generalization holds. It is always possible that we will encounter new variables which impact the probability of the generalization by yielding a conflicting result. Therefore, at most, Hume’s rule can only assign a probability of less than 1 to a generalization.
According to Earman, this conclusion undermines Hume’s original distinction between miracles and marvels. There are two reasons for this: first, if an inductive generalization G about the regular operations of nature cannot have a probability equal to 1, then a miraculous exception M does not have a probability equal to zero. G renders M improbable, not impossible. Second, the occurrence of a marvel S is similarly improbable. Although the occurrence of S does not conflict with G, its occurrence is still discordant with other past experiences; otherwise the marvel would not appear bizarre. In other words, both M and S are improbable with respect to past experience – it’s just that M poses an exception to G whereas S does not. M might be (antecedently) much more improbable than S, but for Earman, such a distinction is “a matter of degree rather than kind” (p.32).
By way of reply, Hume could object that the prior probability of a miracle is so much lower than for a marvel that belief in the former can never be overcome by favourable evidence. This reply would reinforce Hume’s original distinction between miracles and marvels, but it still remains to be seen just how low the prior probability of an event has to be to guarantee that it can never be rationally believed.
 In fact, according to the Bayes-Laplace rule of probability, for any inductive generalization of the form All A’s are B’s, if all future instances of A are specified, their probability of being B is near zero no matter how many past instances have yielded B’s. See John Earman, Hume’s Abject Failure: the Argument Against Miracles (2000) p.30.