In his important work, Observations on the Nature and Tendency of the Doctrine of Mr Hume, one of the earliest significant works of Hume scholarship, Thomas Brown distills Hume's causal theory into five basic propositions, which he goes on to evaluate. (Brown is important because he is one of the first people to look seriously at Hume's causal theory who doesn't interpret Hume as denying that we have an idea of cause and effect. As Brown points out, Hume's whole argument only makes sense on the assumption that he thinks we do have such an idea -- necessary connection -- and that the tricky thing is to determine which impression provides it.) The five propositions are:
(1) The relation of cause and effect cannot be discovered a priori.
(2) Even after experience, the relation of cause and effect cannot be discovered by reason.
(3) The relation of cause and effect is an object of belief alone.
(4) The relation of cause and effect is believed to exist between objects, only after their customary conjunction is known to us.
(5) When two objects have been frequently observed in succession, the mind passes readily from the idea of one to the idea of the other: from this tendency to transition, and from the greater vividness of the idea thus more readily suggested, there arises a belief of the relation of cause and effect between them; the transition in the mind itself being the impression from which the idea of the necessary connection of the objects, as cause and effect, is derived.
Brown's view is that (1), (2), and (3) stand or fall together; he also thinks Hume is right on all three counts. (4) and (5) he regards as related to each other, but independent from the first three. He also regards them as clearly wrong. His basic argument against (4) is that we can believe things to be causally connected even in the absence of customary conjunction (e.g., in inferences about single experiments). Hume, of course, has ways to handle some of these, but Brown argues against them. Thus Brown substitutes Hume's basic explanatory principle, 'custom', with his own, 'instinct' or 'instinctive belief'.
The fourth chapter of Lady Mary Shepherd's An Essay upon the Relation of Cause and Effect is explicitly devoted to Brown's Observations. Shepherd accepts Brown's criticisms against Hume on (4) and (5), but, of course, sharply criticizes Brown for accepting (1), (2), and (3), and devotes some space to showing that Brown's instinct doesn't fare much better than Hume's custom as an explanation of causal reasoning. She notes that both Hume and Brown are guilty of an ambiguity with regard to the first two propositions: they don't properly distinguish questions about the general form of reasoning from questions about particular instances of it. For instance, Shepherd admits that we can't know a priori the particular qualities that will arise under particular circumstances; but she points out that this is neither surprising nor particularly interesting. What we want to know is the answer to questions like: "Must like causes in general necessarily be connected with like effects?" Given these general forms, we just need to go to experience to fill them in with details. Her argument against the third proposition is more indirect, since she argues that neither Hume's custom nor Brown's instinct can do the work they would have to do even assuming that causation is a matter of belief and not knowledge. (This is relevant because a great deal of Brown's argument against (4) is devoted to arguing that Hume's custom cannot do this work.)
Shepherd thinks that a problem with (5) is that it makes Hume seem to argue in a circle. Hume's search for foundational impressions, the whole point of the Treatise, is a causal search -- impressions are important because they are the causes of ideas. Ideas must be derived from impressions, because there is a necessary connection between them; but then he disproves the only account of necessary connection on which the original principle could be built. She attributes this argument to Brown, but, unless I am missing something, this appears to be a mistake. Most of Brown's argument is actually an argument against Hume's account of belief, combined with the argument that if necessary connection derives from the impression of the mind's transition from idea to idea, all associations would be causal. But it is clear that Hume has to distinguish causal associations from associations due to resemblance and contiguity in order to motivate his inquiry about causes. Thus Brown accuses Hume of petitio principii, but it is a different petitio than the one Shepherd accuses Hume of here.
