Hume once criticized the principle of causality, claiming that the causal connection between two objects was a projection of the mind.

To what extent is Hume's criticism of causality linked to his epistemological presuppositions (i.e. such as Hume's fork, or his belief that clear and separate sensory images are the only matters of fact)?

What criticisms can and have been given that attack Hume's criticism of causality by attacking his epistemological commitments?

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    One of the most famous responses is due to Kant: while causality is neither analytic (relation of ideas) nor can be induced empirically (matter of fact) it is no mere habit. That is where Kant's signature synthetic a priori originated, "their possibility has its ground merely in the relation of the understanding to experience, however, not in such a way that they are derived from experience, but that experience is derived from them", see Kant and Hume on Causality. – Conifold Jun 7 '17 at 19:32

Hume's criticism of causality is a direct consequence of his epistemological system. You can't really understand Hume's thoughts on necessity and the causal relations between objects (or events) unless you understand how Hume thinks we experience said objects.

Hume divides all knowledge into 'relations of ideas' and 'matters of fact' (this is, as you point out, 'Hume's fork'). The former Hume believes to be entirely the construct of humans, and to a probably lesser extent, so too does Hume believe the latter to be of human construct. For Hume, matters of fact are really private sense images, or 'impressions,' only secondarily experienced via memory in what he calls 'ideas.' Relations that persist between any two ideas are based entirely in Hume's threefold list of associations: resemblance, contiguity, and cause and effect. What is interesting is that Hume always talks about these associations in the context of human action. This is in keeping with his general commitment to viewing these associations as being entirely human constructions that in no way are founded in the basic impressions themselves.

Given this outline, it should come as no surprise that Hume sees the notion of cause and effect as falling into the category of 'relations of ideas,' and as being, in consequence, entirely of human construction:

In short, every effect is a distinct event from its cause. So it can’t be discovered in the cause, and the first invention or conception of it a priori must be wholly arbitrary. Also, even after it has been suggested, the linking of it with the cause must still appear as arbitrary, because plenty of other possible effects must seem just as consistent and natural from reason’s point of view. Enquiry Concerning Human Understanding

Since the starting blocks of knowledge are distinct, separate ideas that in no way persist beyond their material existence , Hume's criticism of causality is the natural skeptical conclusion that Hume must eventually arrive at. Hume's first founding blocks of knowledge are purged of all their ability to ground the relations that persist between them. Any relations between sensations, including cause and effect, are entirely human concoctions.

Arguments Against the Humean Project

Piercean Argument

You don't need to be a realist to take trouble with Hume's epistemological system. However, an argument might be given against Hume's epistemological program that borrows from a phenomenological argument used against nominalism by the late, great Charles Pierce. He writes:

Now, upon the nominalistic theory there is not only no absolute or numerical identity, but there are not even any real agreements of likenesses between individuals; for likeness consists merely in the calling of several individuals by one name or (in some systems) in their exciting one idea. The Revival of Realism

Using this general logical argument that Charles Pierce employed against nominalists, we can twist it slightly to point out the oddity in Hume's position of trying to make out any sense of 'impression' that doesn't welcome as well the sense of relation between 'impressions'. Pierce's argument, in short, amounts to this: how can we hope to logically preserve our ability to recognize impressions when we attack our ability to recognize the relations that persist between them? Change would be viewed on this model not in the presupposed, reductionist epistemological language as one distinct impression following upon another, a language that threatens the continuity of identity shared between impressions themselves which (according to Pierce) in turn threatens any identity of any sensory image, but as one subsistent object losing or gaining something.

Sellarsian Argument

Another recent insight that we might use to reject the radically skeptical Humean conclusions is to accept the new findings by Wilfrid Sellars (among others) regarding the 'myth oft the given.'

The short of this argument is that nothing can serve as a justification in our field of knowledge besides something that is itself conceptually significant. Put in other words, knowledge isn't an entirely passive experience. The human subject participates in knowledge. This in itself tells us that action, which is in turn a subcategory of change, is a real feature. But putting that point aside, we can apply this insight to Humean epistemology and, in doing so, see the justification of the insight itself.

For Hume to even talk about this impression compared to that impression is an activity that is loaded down with conceptuality, which is composed, in turn, of relations. The Sellarsian would say to the Humean something like this: you are already caught in commitments to relations, such as cause and effect, before you try to bring us back down to the epistemically barren wasteland of 'impressions.' This is self-evidenced in that it is not your impressions that are justifying any of your positions: it is your comparative, relational logic.

This argument, while not flatly disproving Hume's specific epistemological opinions about causality, does get after something that is probably more important: motivation. This argument asks Hume why he's trying to pull off his whole program in the face of the fact that it is relations themselves that are epistemologically more basic than sensations (at least as far as explicit knowledge goes). In Sellarsian fashion we would say to Hume that it is not sensory impressions that should compel all talk of justification: it is, on the contrary, the conceptually significant relations that are actually affirmed or negated.

Psychological Argument

Another argument against the Humean criticism of causality is pretty basic. In short, it wants to get after the normativity of causality as experienced. Why is it that we connect certain successive events in a causal way while we don't connect other successive events in a causal way?

Hume never seems to get after the full explanation of the causal phenomenon itself. When he poses this question to himself, he busies himself with tackling the problem of inference, but even this consideration implicitly assumes the phenomenon of past causal connections rather than effectively reducing them away. In other words, he is always caught taking advantage of causal connections rather than truly reducing them to distinct impressions. This is why Kant, in response to Hume, held to the irreducibility of the principle of causality, if not as a principle of reality, then as a principle of mind.

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