Lady Mary Shepherd on causal reasoning [An Essay upon the Relation of Cause and Effect, pp. 107-108]:
"If the human body is in the same state on any occasion, as on that when bread nourished it; there is as great a necessity it should again nourish as that it should be white.
"Thus all experimental reasoning consists in an observation, and a demonstration, as has before been shown;--an observation, whether teh circumstances from which an object is produced, and in which it is placed, are the same upon one occasion as upon another;--and a demonstration, that if it is so, all its exhibitions will be the same."
Shepherd's account of causal reasoning is opposed to Hume's. Hume thinks of induction as the most general sort of reasoning about matters of fact possible. Shepherd has denied this, and with considerable plausibility. One way to see Shepherd's insight is to phrase it in terms of suppositional reasoning, where we identify what follows, what cannot follow, and what is consistent with, a given set of suppositions. This sort of reasoning is (1) deductive; and (2) a more general form of reasoning about matters of fact, since it bridges the gap, which Hume considered to be insurmountable, between reasoning about relations of ideas and reasoning about matters of fact. We clear argue suppositionally in mathematics, for instance; and we also clearly do it in physics. And Shepherd considers all such reasoning to be causal. In mathematics, we reason about how one set of supposed things (e.g., a particular acute angle in a right triangle) causes another set of things to be what it is (e.g., the other acute angle in the triangle). On Shepherd's view causal reasoning in physics is exactly the same as this. The difference between physics and mathematics is not the form of reasoning, which is logically the same; the difference is that physics requires an additional step before we get to the causal reasoning. In mathematics you can suppose whatever you will. In physics, we want our reasoning to tell us something about a given state of affairs. So physics, in addition to its causal reasoning, requires a step in which we match our suppositions to the facts.
Given this move, Shepherd is able to disentangle two issues: the problem of induction and the problem of causal reasoning. Causal reasoning is deductive; it is, however, ex suppositione. When we expect that a new case will be like an old case, our reasoning moves from experience to expectation in the following way:
(1) Experience: We have analyzed previous instances of such-and-such conditions.
(2) The Observation Step, which is probabilistic: that the current conditions are the same as we have found in previous cases.
(3) The Demonstration Step, which is certain: that if they are the same, the conditions must exhibit themselves in the same way.
(4) Expectation: We therefore expect the new to be like the old. If it is not, the problem is not with our reasoning (3), assuming it has been done carefully, but with our matching of supposition to reality (2).
Thus there is no Humean mystery about why we believe that the future will resemble the past; we believe it will because the current conditions have implications, which we have discovered by reasoning about the same conditions that have come before. There is therefore no vicious circle in the justification of induction. Since induction is not the most general form of reasoning about matters of fact, we can justify induction in terms of a more general form of reasoning. There are still issues involved in induction -- for example, the Observation Step will necessarily only be probabilistic -- but these are not justificatory problems for induction itself. And Shepherd recognizes quite correctly the right way to deal with these problems: try out our suppositions. It is noteworthy that Shepherd, unlike Hume, talks a great deal about testing, since it plays an important role in her account. Having developed an expectation on the basis of certain suppositions, we test it experimentally against facts, and if it fails, we go back and reconsider our suppositions. As she notes, it is her account of reasoning, not Hume's, that more clearly takes scientific practice into account.