Hume is quite clear about what causes the problem in the inference between 'is' and 'ought':
For as this ought, or ought not, expresses some new relation or affirmation, 'tis necessary that it shou'd be observ'd and explain'd; and at the same time that a reason should be given, for what seems altogether inconceivable, how this new relation can be a deduction from others, which are entirely different from it.
But we should stop here and think a moment. In Treatise 1.1.5, Hume lists all the philosophical relations that he thinks exists:
resemblance
identity
relations in time and space
proportion in quantity or number
degrees of quality
contrariety
causation
Only four of these are entirely relations of ideas, depending on the ideas alone, so only four can be the objects of reason alone (they are resemblance, contrariety, degree of quality, and proportion of quantity or number). But Hume seems to call 'ought' a relation in the is/ought passage. Where, in all of this, is a relation that can be called an 'ought'? The point can be made more acute. In the very same section as the is/ought passage, only seven or eight paragraphs before, in fact, Hume had used as an argument against the moral rationalist that none of the philosophical relations known by reason alone are moral relations. He gets quite sarcastic:
Shou'd it be asserted, that the sense of morality consists in the discovery of some relation, distinct from these, and that our enumeration was not compleat, when we comprehended all demonstrable relations under four general heads: To this I know not what to reply, till some one be so good as to point out to me this new relation. 'Tis impossible to refute a system, which has never yet been explain'd.
And yet any relation that is expressed by an 'ought' would be precisely the sort of answer a rationalist could use; and here in the is/ought passage we find Hume saying that the reason you need to explain how you are inferring an 'ought' from an 'is' is that it seems inconceivable how one can deduce this new relation from another relation.
The point can be made even more simply. Hume is also very clear what conclusion we should draw from the is/ought problem: "the distinction of vice and virtue is not founded merely on the relations of objects, nor is perceiv'd by reason." This implies directly that 'ought' can't be a relation. And yet the argument itself treats it as if it were. Is Hume contradicting himself?
The answer, of course, is that he is not, and there is a very simple and straightforward interpretation available, and it is this: that our conclusion from our inability to draw an 'ought' relation from an 'is' relation is that 'ought' does not indicate a relation. Remember that the is/ought passage is raising a problem for rationalists like Malebranche and Clarke, who believes that morality consists in relation perceived by reason, and "vulgar systems of morality" like The Whole Duty of Man, which are at least partially (even if not rigorously) rationalist, by thinking that unaided reason can draw conclusions about what we ought to do from facts about God or human affairs. Thus it makes sense for Hume to identify a problem that arises from within rationalism itself. The argument proceeds on rationalist assumptions, and identifies a problem that he thinks it is clear the rationalist cannot resolve. On the assumption that 'ought' is a relation, we find ourselves unable to see how we could derive an 'ought' from an 'is'. But rationalists certainly do reason as if they can derive an 'ought' from an 'is', even though such an inference is completely mysterious on rationalist assumptions.
If this is so, however, than we are led to the conclusion that what is often called Hume's Law, the unqualified claim that you can't derive an 'ought' from an 'is', is not in Hume. Hume's discussion of 'is' and 'ought' is entirely geared to showing that an inability to derive an 'ought' from an 'is' is a problem that arises if you are a moral rationalist, which he obviously is not. What is more, Hume is very clear that this is a rationalist problem in particular. It is clear from the context (a section titled "Moral distinctions not deriv'd from reason"). It is clear from the conclusion that Hume explicitly says we should draw when considering the problem ("the distinction of vice and virtue is not founded merely on the relations of objects, nor is perceiv'd by reason"). It is clear from the fact that if you don't interpret it in this restricted way, the argument contradicts an argument Hume has made earlier in the section. And it is clear from the fact that if you don't take the is/ought problem to be a purely rationalist problem, the argument would in fact contradict the very conclusion Hume says you should draw from it. Who can't draw an 'ought' from an 'is'? Only a moral rationalist.
On Hume's own account, there is no problem of drawing an 'ought' relation from an 'is' relation, because morality does not consist in relations at all. Obligations, and therefore 'oughts', aren't relations but results of moral sentiments. On Hume's account of obligations, we say an action is virtuous when it pleases us in a particular way (not just any way, but one particular way), and we say that a virtuous action is obligatory when not performing the action displeases us in the corresponding way (cf. 3.2.5.4, SBN 517). The obligation, therefore, is not a relation: it is a straightforward fact about what sentiments result when we think about it in the right way (namely, when we consider it in general, without reference to our own particular interests). Thus, at its simplest level, 'ought' expresses a matter of fact about human nature. And there is no mystery about how to get an 'ought' from an 'is' if an 'ought' is a particular kind of 'is'. Thus we have what Hume scholars have pointed out for decades, that Hume himself quite often will derive an 'ought' from an 'is' -- when, for instance, he discusses the natural and moral obligations that pertain to justice, promises, allegiance, &c., &c. If you are moral sense theorist like Hume, however, there is no problem: 'ought', not being a necessary relation between ideas, is exactly the sort of thing you explain with a causal explanation (one form of 'is'). And this is what Hume does: he explains our moral obligations causally, in just the same way he would explain our feeling that the air in the room suddenly became colder, by identifying how custom links it to antecedents with which it is conjoined.
So not only does Hume not argue for 'Hume's Law' or a general gap between fact and value; it is inconsistent with Hume's entire system. The only thing in Hume's argument that could be taken as a reason for 'Hume's Law' is the claim that it is inexplicable how a relation can be deduced from an entirely different relation. But it makes no sense whatsoever for Hume, at this point in the Treatise, to be making a general claim about 'is' and 'ought' on the basis of the claim that it's mysterious how you could get an 'ought' relation from an 'is' relation. He is explicitly arguing against the view that morality consists in relations. He knows an account of morality in which 'ought' does not indicate a relation at all, and goes on to argue for it. And so there is one and only one interpretation that makes sense: on Hume's account of 'is' and 'ought', the one and only way in which you are unable to draw an 'ought' from an 'is' is if you are already conceding too much to moral rationalism.
