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Whether the Problem of induction is in fact separable from Hume’s account of necessary connection, he himself connects the two by arguing that "…the knowledge of this relation is not, in any instance, attained by reasonings a priori; but arises entirely from experience, when we find that any particular objects are constantly conjoined with each other." (EHU 4.6; SBN 27) Here, Hume invokes the account of causation explicated above to show that the necessity supporting (B) is grounded in our observation of constant conjunction. Since the Problem of Induction demands that causal connections cannot be known a priori, and that our access is only to constant conjunction, what is billiards the Problem seems to require the most crucial components of his account of necessity. Gimbals are also useful for components like solar panels. But if this is right, then Hume should be able to endorse both D1 and D2 as vital components of causation without implying that he endorses either (or both) as necessary and sufficient for causation. In the Treatise, however, a version of the Problem appears after Hume’s insights about experience limiting causation to constant conjunction but before the explication of the projectivist necessity and his presenting of the two definitions.

The second of Hume’s influential causal arguments is known as the problem of induction, a skeptical argument that utilizes Hume’s insights about experience limiting our causal knowledge to constant conjunction. Ergo, the idea of necessity that supplements constant conjunction is a psychological projection. As we experience enough cases of a particular constant conjunction, our minds begin to pass a natural determination from cause to effect, adding a little more "oomph" to the prediction of the effect every time, a growing certitude that the effect will follow again. D1 reduces causation to proximity, continuity, and constant conjunction, and D2 similarly reduces causation to proximity, continuity, and the internal mental determination that moves the first object or idea to the second. For Hume, the necessary connection invoked by causation is nothing more than this certainty. In fact, later in the Treatise, Hume states that necessity is defined by both, either as the constant conjunction or as the mental inference, that they are two different senses of necessity, and Hume, at various points, identifies both as the essence of connection or power. But again, (A) by itself gives us no predictive power. A web version of Rod Cross's paper together with a PDF download gives experimental measurements of mallet-ball-ball collisions.

The only apparent answer is the assumption of some version of the Principle of the Uniformity of Nature (PUN), the doctrine that nature is always uniform, so unobserved instances of phenomena will resemble the observed. Because of this, our notion of causal law seems to be a mere presentiment that the constant conjunction will continue to be constant, some certainty that this mysterious union will persist. For instance, D1 can be seen as tracing the external impressions (that is, the constant conjunction) requisite for our idea of causation while D2 traces the internal impressions, both of which are important to Hume in providing a complete account. Two objects can be constantly conjoined without our mind determining that one causes the other, and it seems possible that we can be determined that one object causes another without their being constantly conjoined. I got so much help that it turned out to be one grand mess, talking about being confused. I wish you wouldn't take it so hard, I can't help it.

An expert pool player, on the other hand -- well, he can just look at the billiards table and imagine lines drawn out, marking the trajectory the cue ball must take to make the shot. That's very fine, I dare say, but it's sad enough to make one cry. Ott 2009: 239) This way of dismissing the nonequivalence of the two definitions becomes more problematic, however, when we realize that Hume does not make the distinction between natural and philosophical relations in the Enquiry, yet provides approximately the same two definitions. It seems to be the laws governing cause and effect that provide support for predictions, as human reason tries to reduce particular natural phenomena "… Some scholars have argued for ways of squaring the two definitions (Don Garrett, for instance, argues that the two are equivalent if they are both read objectively or both read subjectively), while others have given reason to think that seeking to fit or eliminate definitions may be a misguided project. However, Hume has just given us reason to think that we have no such satisfactory constituent ideas, hence the "inconvenience" requiring us to appeal to the "extraneous." This is not to say that the definitions are incorrect.