A Problem in Hume
Early in the Treatise Hume sets out to establish what he calls a “general proposition”, namely: “That all our simple ideas in their first appearance are deriv’d from simple impressions, which are correspondent to them, and which they exactly represent” (Book I, Section I, p.52).  What kind of proposition is this? It is evidently a causal proposition, to the effect that ideas are caused by impressions, and not vice versa: the word “deriv’d” indicates causality. So Hume’s general proposition concerns a type of mental causation linking impressions and ideas; accordingly, it states a psychological causal law. It is not like a mathematical generalization that expresses mere “relations of ideas”, so it is not known a priori. As if to confirm this interpretation of his meaning, Hume goes on to say: “The constant conjunction of our resembling perceptions [impressions and ideas], is a convincing proof, that the one are the causes of the other; and this priority of the impressions is an equal proof, that our impressions are the causes of our ideas, not our ideas of our impressions” (p. 53). Thus we observe the constant conjunction of impressions and ideas, as well as the temporal priority of impressions over ideas, and we infer that the two are causally connected, with impressions doing the causing. In Hume’s terminology, we believe his general proposition on the basis of “experience”—our experience of constant conjunction.
But this means that Hume’s own critique of causal belief applies to his guiding principle. In brief: our causal beliefs are not based on insight into the real powers of cause and effect but on mere constant conjunctions that could easily have been otherwise, and which interact with our instincts to produce non-rational beliefs of an inductive nature. It is like our knowledge of the actions of colliding billiard balls: the real powers are hidden and our experience of objects is consistent with anything following anything; we are merely brought by custom and instinct to expect a particular type of effect when we experience a constant conjunction (and not otherwise). Thus induction is not an affair of reason but of our animal nature (animals too form expectations based on nothing more than constant conjunction). Skepticism regarding our inductive inferences is therefore indicated: induction has no rational foundation. For example, prior to our experience of constant conjunction ideas might be the cause of impressions, or ideas might have no cause, or the impression of red might cause the idea of blue, or impressions might cause heart palpitations. We observe no “necessary connexion” between cause and effect and associate the two only by experience of regularity—which might break down at any moment. Impressions have caused ideas so far but we have no reason to suppose that they will continue to do so—any more than we have reason to expect billiard balls to impart motion as they have hitherto. Hume’s general proposition is an inductive generalization and hence falls under his strictures regarding our causal knowledge (so called); in particular, it is believed on instinct not reason.
Why is this a problem for Hume? Because his own philosophy is based on a principle that he himself is committed to regarding as irrational—mere custom, animal instinct, blind acceptance. He accepts a principle—a crucial principle–that he has no reason to accept. It might be that the idea of necessary connexion, say, is an exception to the generalization Hume has arrived at on the basis of his experience of constant conjunction between impressions and ideas—the equivalent of a black swan. Nothing in our experience can logically rule out such an exception, so we cannot exclude the idea based on anything we have observed. The missing shade of blue might also simply be an instance in which the generalization breaks down. There is no necessity in the general proposition Hume seeks to establish, by his own lights–at any rate, no necessity we can know about. Hume’s philosophy is therefore self-refuting. His fundamental empiricist principle—all ideas are derived from impressions—is unjustifiable given his skepticism about induction. Maybe we can’t help accepting his principle, but that is just a matter of our animal tendencies not a reflection of any foundation in reason. It is just that when we encounter an idea our mind suggests the existence of a corresponding impression because that is what we have experienced so far—we expect to find an impression. But that is not a rational expectation, merely the operation of brute instinct. Hume’s entire philosophy thus rests on a principle that he himself regards as embodying an invalid inference.
It is remarkable that Hume uses the word “proof” as he does in the passage quoted above: he says there that the constant conjunction of impressions and ideas gives us “convincing proof” that there is a causal relation that can be relied on in new cases. Where else would Hume say that constant conjunction gives us “convincing proof” of a causal generalization? His entire position is that constant conjunction gives us no such “proof” but only inclines us by instinct to have certain psychological expectations. And it is noteworthy that in the Enquiry, the more mature work, he drops all such talk of constant conjunction, causality, and proof in relation to his basic empiricist principle, speaking merely of ideas as “derived” from impressions. But we are still entitled to ask what manner of relation this derivation is, and it is hard to see how it could be anything but causality given Hume’s general outlook. Did he come to see the basic incoherence of his philosophy and seek to paper over the problem? He certainly never directly confronts the question of whether his principle is an inductive causal generalization, and hence is subject to Humean scruples about such generalizations.
It is clear from the way he writes that Hume does not regard his principle as a fallible inference from constant conjunctions with no force beyond what experience has so far provided. He seems to suppose that it is something like a conceptual or necessary truth: there could not be a simple idea that arose spontaneously without the help of an antecedent sensory impression—as (to use his own example) a blind man necessarily cannot have ideas of color. The trouble is that nothing in his official philosophy allows him to assert such a thing: there are only “relations of ideas” and “matters of fact”, with causal knowledge based on nothing but “experience”. His principle has to be a causal generalization, according to his own standards, and yet to admit that is to undermine its power to do the work Hume requires of it. Why shouldn’t the ideas of space, time, number, body, self, and necessity all be exceptions to a generalization based on a past constant conjunction of impressions and ideas? Sometimes ideas are copies of impressions but sometimes they may not be—there is no a priori necessity about the link. That is precisely what a rationalist like Descartes or Leibniz will insist: there are many simple ideas that don’t stem from impressions; it is simply a bad induction to suppose otherwise.
According to Hume’s general theory of causation, we import the idea of necessary connexion from somewhere “extraneous and foreign”  to the causal relation itself, i.e. from the mind’s instinctual tendency to project constant conjunctions. This point should apply as much to his general proposition about ideas and impressions as to any other causal statement: but then his philosophy rests upon the same fallacy–he has attributed to his principle a necessity that arises from within his own mind. He should regard the principle as recording nothing more than a constant conjunction that he has so far observed, so that his philosophy might collapse at any time. Maybe tomorrow ideas will not be caused by impressions but arise in the mind ab initio. Nowhere does Hume ever confront such a possibility, but it is what his general position commits him to.