In a footnote to chapter 6 (“Of Probability”) of the first Enquiry, Hume writes,
Mr. Locke divides all arguments into demonstrative and probable. In this view, we must say, that it is only probable all men must die, or that the sun will rise to-morrow. But to conform our language more to common use, we ought to divide arguments into demonstrations, proofs, and probabilities. By proofs meaning such arguments from experience as leave no room for doubt or opposition.
This echoes a similar passage in the Treatise (T 22.214.171.124), where Hume divides “the several degrees of evidence” into knowledge, proof, and probability.
These passages, as well as letters such as the anonymously penned Letter from a Gentleman seem to suggest that Hume divides probable arguments into those that do not produce certainty (probabilities) and those that can or do (proofs). Presumably, proofs produce “moral certainty” in the early modern sense. (Hume appeals to “moral certainty” in the Letter to counter accusations that he undermines demonstrations of the existence of God and is therefore a skeptic and atheist).
I find Hume’s defense in the Letter less than convincing for a number of reasons, but let me offer two here. Hume’s opponents were correct to recognize that Hume doesn’t really have a place for “proofs” in this technical sense. First, Hume is clear in T 1.3.1 that knowledge comes from intuition and demonstration. Other than the passages mentioned, Hume only talks about certainty in connection to knowledge. Furthermore, the reason given in T 1.3.1 for why we can be certain is that intuition and demonstration rely on unalterable relations. (See also T 126.96.36.199-3.) The argument isn’t terribly clear, but it seems that if the relations between our ideas are alterable (“so long as the ideas remain the same”) then we couldn’t be certain on the matter. Because probable arguments (presumably; I don’t see him argue that they must) rely on alterable relations, they could not produce certainty, even moral certainty.
Second, Hume is careful in the Treatise and again in the Enquiry to note that philosophers should only distinguish demonstrations and probable arguments. It is only “to conform our language more to common use” (Enquiry) or when “in common discourse” (Treatise) to separate out a third category of evidence. (Note: he makes no effort to fix the potential misreading in the Enquiry.) Philosophers do and should recognize two categories of arguments. This might disturb the common folk, so we should speak of very high probabilities as “proofs,” but this is only for common use, not when doing philosophy.
I am convinced by these two reasons (and some other, less important ones) that Hume does not have a distinct category of proofs (probable arguments which produce moral certainty) when speaking philosophically; such use is only an accommodation to those common folk worried that it is not certain that the sun will rise tomorrow.
However, this seems to cut against a currently popular reading of Hume, which claims that Hume considers certain many things (such as the causal maxim) that are not established via intuition or demonstration. Often these readers appeal to Hume’s unpublished letters, which I find problematic. On my opponents’ behalf, I wonder if something could be made using different kinds of certainty (epistemic vs. psychological, say), which has some traction in the texts. Hume’s use of the terms is frustratingly inconsistent, so it is difficult to know how to proceed on this point. (One starting point could be Of the Passions, where he separates out two kinds of probability.) Until we have a fuller working out of the notions of certainty and probability, I think we should follow Hume’s suggestion and take talk of “proofs” as an accommodation to common use.