We are encouraged to answer our own questions and to share our research. So I would like to share my thoughts on the question after the very helpful discussion of it.
When I asked the question, I quoted from A Treatise of Human Nature. In this answer, I have restricted myself to An Enquiry Concerning Human Understanding because one can expect the Enquiry to be self-consistent and there is no need to get involved in assessing the differences between the two for the purposes of the question.
Hume's project
Hume summarizes his project in the final section of the book (Section 12 Of the Academical or Sceptical Philosophy). It is to identify the proper scope of human knowledge and get rid whatever is useless. This means we should “commit to the flames” any volume of divinity or school metaphysics that does not contain either "abstract discussion of mathematics (quantity or number)" or "of matter of fact and existence".
Which seems clear enough, if a bit radical. Divinity and metaphysics are condemned because they try to extend the scope of human reason beyond what it can do. It gives rise to excessive scepticism and vacuous ideas with no basis in experience. Some divinity gets a pass because “It has a foundation in reason, so far as it is supported by experience.” “But” he continues “its best and most solid foundation is faith and divine revelation.” (Section 12.3). However, his critique of reason does allow for a role for reason, with limitations.
He explains the project in a different way in the body of the section, where he distinguishes between Pyrrhonism or excessive scepticism and what he calls mitigated scepticism.
Excessive scepticism is the philosophical scepticism we are familiar with from Descartes and Berkeley. Hume believes that it is difficult, if not impossible, to refute. His main objection is that no lasting good can come of it, and he prescribes “action, and employment, and the occupations of common life” as a cure. This respect for common life shows at several points in his philosophy.
Mitigated scepticism is characterized in two ways. Part of proper thinking, he says, is “a degree of doubt, and caution, and modesty …, in all kinds of scrutiny and decision.” The other characterization is the result of Pyrrhonian scepticism. It is “the limitation of our enquiries to such subjects as are best adapted to the narrow capacity of human understanding.” (Section 12.3)
The problem
In Section 10.1 Of Miracles Hume offers “uniform experience” as “proof, full and entire” for his proposition that miracles are impossible. How does he get that from his argument against induction in Section 4 Sceptical Doubts Concerning the Operations of the Understanding?
Proof
In Section 4, he presses home the key point that experience in the past cannot be extended to cover experiences in the future, or indeed any experiences not yet known. In Section 5 Sceptical Solution of these Doubts he does not abandon empirical generalizations but argues that they are customs or habits, arising from the imagination, which calls up the idea of the effect when the idea of the cause is presented; he calls this the association of ideas. He argues that this is a good thing, because it is useful. He also claims (in Section 5.2) that “we find (sc. have found in the past) a pre-established harmony between the course of nature and the succession of our ideas”. All this runs up against the sceptical objection of course, but this isn’t about what is rational. It is about what we do.
In a footnote to the title of Section 6 Of Probability Hume cites Locke’s distinction between “demonstrative” and “probable” arguments; Hume objects that this means that it is only probable that everyone must die. He proposes that we classify arguments as demonstrations, proofs and probabilities. “By proofs meaning such arguments from experience as leave no room for doubt or opposition.”
So he does explain to the attentive reader how he will use the term ‘proof' in future. One might object that the aim of proof is justification of belief, while an account of the mechanism that generates belief is not, or not necessarily, a justification of belief. So it seems odd to suggest that his account of the genesis of customs or habits is a justification of them. But presenting evidence will cause belief, so the aim of proof is achieved. In practice, this will mean that “uniform experience” will amount to a proof.
The ruling out of doubt or opposition is rather surprising, but it is hard to argue with the examples he offers later in the same section – “Fire has always burned, and water suffocated every human creature: The production of motion by impulse and gravity is an universal law, which has hitherto admitted of no exception.” (Section 6)
He seems in no doubt that he has succeeded in resolving the sceptical doubts, but without directly refuting excessive scepticism; that would certainly explain the odd title of Section 5 - Sceptical Solution of these Doubts. So the difference between his discussion of induction and “custom or habit” in Section 5 and his presentation of the argument against miracles in Section 10 is really just a change in emphasis and presentation.
Miracles
Hume tells us that “A miracle may be accurately defined, a transgression of a law of nature by a particular volition of the Deity, or by the interposition of some invisible agent.” (Footnote in Section 10.1) His long argument against the testimony for miracles culminates in his argument that because a miracle is defined as a violation of the laws of nature, which are established by “firm and unalterable experience”, they are impossible.
Knock-down arguments are very satisfying, but they are rare and I am suspicious of them. In my view this argument is very narrow and over-simplified. Hume does recognize some of the complications, but not all. To understand the topic a wider view is required. But that is beyond my scope.
He even applies his argument to the miracle of the resurrection of Jesus Christ and then declares that the real foundation of belief in that miracle is not reason, or even custom and experience. It is faith. He further says that faith is itself a miracle. So there is no more to be said. It seems that the real point of the section is to show, not only another limitation of reason, but also of custom and experience.
The appeal to faith is a surprise. There is no hint of it Section 4 or 5. On the other hand, it is not exactly incompatible with those Sections 4 and 5.
Hume is following the same strategy in his discussion of induction and of miracles and he is consistent in that. Start with an argument of excessive scepticism, separate what needs to be saved and what must be let go, and find an alternative justification for what needs to be saved. So the argument against causation is allowed to run its course; then an alternative foundation – experience - is found for empirical generalizations and the pre-established harmony between the course of nature and the succession of our ideas” justifies him referring to natural laws that are objectively true. In the same way, the argument against miracles is pressed to the end, at which point, faith saves at least the two miracles that are critical for him.
