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The more common Humean reduction, then, adds a projectivist twist by somehow reducing causation to constant conjunction plus the internal impression of necessity. In addition to its accounting for the necessity of causation mentioned above, recall that Hume makes frequent reference to both definitions as accurate or just, and at one point even refers to D2 as constituting the essence of causation. Even considering Hume’s alternate account of definitions, where a definition is an enumeration of the constituent ideas of the definiendum, this does not change the two definitions’ reductive nature. The family of reductionist theories, often read out of Hume’s account of necessity outlined above, maintain that causation, power, necessity, and so forth, as something that exists between external objects rather than in the observer, is constituted entirely by regular succession. The unifying thread of the reductionist interpretations is that causation, as it exists in the object, is constituted by regularity. In the external world, causation simply is the regularity of constant conjunction. Given that Hume’s discussions of causation culminate in these two definitions, combined with the fact that the conception of causation they provide is used in Hume’s later philosophical arguments of the Treatise, the definitions play a crucial role in understanding his account of causation.

First, it relies on assigning the "traditional interpretation" to the Problem of induction though, as discussed above, this is not the only account. The bottom line for Hume’s Problem of induction seems to be that there is no clear way to rationally justify any causal reasoning (and therefore no inductive inference) whatsoever. However, there are philosophers (Max Black, R. B. Braithwaite, Charles Peirce, and Brian Skyrms, for instance) that, while agreeing that Hume targets the justification of inductive inference, insist that this particular justificatory circle is not vicious or that it is unproblematic for various reasons. This means that the PUN is an instance of (B), but we were invoking the PUN as the grounds for moving from beliefs of type (A) to beliefs of type (B), thus creating a vicious circle when attempting to justify type (B) matters of fact. Therefore, knowledge of the PUN must be a matter of fact. We have no ground that allows us to move from (A) to (B), to move beyond sensation and memory, so any matter of fact knowledge beyond these becomes suspect.

Hence, if we limit causation to the content provided by the two definitions, we cannot use this weak necessity to justify the PUN and therefore cannot ground predictions. If Hume’s account is intended to be epistemic, then the Problem of induction can be seen as taking Hume’s insights about our impressions of necessity to an extreme but reasonable conclusion. But given the Humean account of causation outlined above, it is not difficult to see how Hume’s writings give rise to such reductionist positions. To be two yards beyond is far better than to be two yards short, for in the first place you give the ball a chance of holing, what is billiards and in the second place it relieves the eye somewhat to turn round and putt along a different line. This will give us an opportunity of trying to find an appointment for Willie before we go. We’ll keep it up, once the precedent is established, until finally it will become a class club entirely-a Plumbers’ Club, for instance-and how absurd that would be in Hades!

They are always littering the club up with themselves. D. C. Stove maintains that, while Hume argues that inductive inference never adds probability to its conclusion, Hume’s premises actually only support "inductive fallibilism", a much weaker position that induction can never attain certainty (that is, that the inferences are never valid). Tom Beauchamp and Alexander Rosenberg agree that Hume’s argument implies inductive fallibilism, but hold that this position is adopted intentionally as a critique of the deductivist rationalism of Hume’s time. Beauchamp and Rosenberg 1981: 44) Annette Baier defends a similar account, focusing on Hume’s use of "reason" in the argument, which she insists should be used only in the narrow sense of Hume’s "demonstrative sciences". See, for instance, Beauchamp and Rosenberg 1981: 11, Goodman 1983: 60, Mounce 1999: 42, Noonan 1999: 140-145, Ott 2009: 224 or Wilson 1997: 16) Of course while this second type of reductionist agrees that the projectivist component should be included, there is less agreement as to how, precisely, it is supposed to fit into Hume’s overall causal picture. Alternatively, there are those that think that Hume claims too much in insisting that inductive arguments fail to lend probability to their conclusions. This, though the thought of the moment, did not end with the moment; for the inclination to act was awakened, and in no one more strongly than in him who was now master of the house; and who, having so much leisure as to make almost any novelty a certain good, had likewise such a degree of lively talents and comic taste, as were exactly adapted to the novelty of acting.