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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. It is the internal impression of this "oomph" that gives rise to our idea of necessity, the mere feeling of certainty that the conjunction will stay constant. Since the Problem of Induction demands that causal connections cannot be known a priori, and that our access is only to constant conjunction, the Problem seems to require the most crucial components of his account of necessity. It is therefore not entirely clear how Hume views the relationship between his account of necessity and the Problem. Stathis Psillos, for instance, views Hume’s inductive skepticism as a corollary to his account of necessary connection. Stove presents a math-heavy critique of Hume’s inductive skepticism by insisting that Hume claims too much. The principle here is much the same, with the railguns behaving somewhat like discretized versions of thrusters, providing instantaneous changes in velocity as opposed to sustained steady change. 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.

Once more, all we can come up with is an experienced constant conjunction. How can you know, sir? Although Hume does the best that can be expected on the subject, he is dissatisfied, but this dissatisfaction is inevitable. We always do our best to make sure our customers receive the best prices available. Perhaps for this reason, Jonathan Bennett suggests that it is best to forget Hume’s comment of this correspondence. For these reasons, Hume’s discussion leading up to the two definitions should be taken as primary in his account of causation rather than the definitions themselves. 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. 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.

J.A. Robinson is perhaps the staunchest proponent of the position that the two are nonequivalent, arguing that there is a nonequivalence in meaning and that they fail to capture the same extension. There are several interpretations that allow us to meaningfully maintain the distinction (and therefore the nonequivalence) between the two definitions unproblematically. As Hume says, the definitions are "presenting a different view of the same object." (T 1.3.14.31; SBN 170) Supporting this, Harold Noonan holds that D1 is "what is going on in the world" and that D2 is "what goes on in the mind of the observer" and therefore, "the problem of nonequivalent definitions poses no real problem for understanding Hume." (Noonan 1999: 150-151) Simon Blackburn provides a similar interpretation that the definitions are doing two different things, externally and internally. One alternative to fitting the definitions lies in the possibility that they are doing two separate things, and it might therefore be inappropriate to reduce one to the other or claim that one is more significant than the other.

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), what is billiards while others have given reason to think that seeking to fit or eliminate definitions may be a misguided project. Having approached Hume’s account of causality by this route, we are now in a position to see where Hume’s two definitions of causation given in the Treatise come from. 1. If one of the two given numbers is a multiple of the other, what is the shape of the arithmetic billiard path? If the definitions were meant to separately track the philosophical and natural relations, we might expect Hume to have explained that distinction in the Enquiry rather than dropping it while still maintaining two definitions. Another method is to cash out the two definitions in terms of the types of relation. One way of defining an ellipse is in terms of two points, each of which is called a focus point. But Hume is at pains to point out that the definitions are inadequate. Basically, the point here is that modelling impacts like these is a tricky business. The game was played on a huge strip of land, in this case about 1000 yards long and so was more like golf than Croquet - players took great swings at the balls in an effort to hoof them as far along the pitch as possible.