(1) Whatever is conceivable is possible.
(2) Whatever is distinguishable is separable.
From these two, with very little work, Hume is able to derive two more substantive claims:
(3) The mind does not perceive any real connection among distinct existences.
(4) Distinct perceptions are distinct existences.
These already have substantive ramifications. You get something even more Humean if you add to them the basic principle of Hume's empiricism, the Copy Principle:
(5) All our simple ideas in their first appearance are deriv’d from simple impressions, which are correspondent to them, and which they exactly represent.
I've noted before that Hume's arguments are often more carefully reasoned than they look. So I was thinking today about how one might go about constructing Humean philosophy synthetically. You'd start with (1), (2), and (5); you'd need to postulate the existence of at least one mind; you'd need definitions for perceptions/ideas, real connections, impressions, and the like. What else might you need to add, or at least have in the system? An obvious candidate are the principles of association.
(6) Ideas are associated by resemblance, contiguity in time and place, or causation.
Which would require definitions for association, resemblance, contiguity, and causation, and would probably need to be specified into several different principles. The definition for causation would obviously be something like Hume's own. Hume scholars often talk about Hume's account of causation in terms of 'revival sets', so if you gave a definition of a 'revival set' -- roughly, the set of ideas that can come to mind when itis faced with another idea -- you could probably use it as a basis for your association principles. In any case, you'd want Hume's Rules by which to Judge of Cause and Effect somewhere:
(7) The cause and effect must be contiguous in space and time.
(8) The cause must be prior to the effect.
(9) There must be a constant union betwixt the cause and effect.
(10) The same cause always produces the same effect, and the same effect never arises but from the same cause.
(11) Where several different objects produce the same effect, it must be by means of some quality, which we discover to be common amongst them.
(12) The difference in the effects of two resembling objects must proceed from that particular, in which they differ.
(13) When any object encreases or diminishes with the encrease or diminution of its cause, it is to be regarded as a compounded effect, derived from the union of the several different effects, which arise from the several different parts of the cause.
(14) An object, which exists for any time in its full perfection without any effect, is not the sole cause of that effect, but requires to be assisted by some other principle, which may forward its influence and operation.
The relation among these is somewhat interesting. (7), (8), and (9) are obviously just Hume's basic account of what causation is (objectively speaking). Hume says (10) is derived directly from experience, so it also is basic; it, which can be called the Experimental Principle, is also extremely important, since Hume says most philosophical reasoning (which would include most scientific reasoning) depends on it. Hume explicitly appeals to (10) in his arguments for (11), (12), and (14), and while he is somewhat vague, I think (13) is based on (9). It has been noted that (10) and (11) are very similar to principles of reasoning formulated by Isaac Newton, and this is certainly deliberate.
From (10) and (13) Hume elsewhere ("Of Interest") derives another principle,
(14) An effect always holds proportion with its cause.
And so one could continue on. One of the reasons why one would do this would be to clarify the relations between Hume and other people. For instance, I get off the boat quite early, since I don't think (2) is correct (it makes modal logic impossible, at least in any robust interpretation, which I imagine would be fine with Hume, but which is not fine with me). James Clerk Maxwell, the physicist, has an interesting passage in one of his books (Matter and Motion, I think), in which he argues that the second half of (10), "The same effects have the same causes" is a sort of general philosophical maxim from which one can derive what Maxwell calls the "Statement of the General Maxim of Science" -- "The difference between one event and another does not depend on the mere difference of the times or the places at which they occur, but only on differences in the nature, configuration, or motion of the bodies considered". So Maxwell certainly accepts the second half of (10), although he thinks it needs to be clarified and specified further. However, Maxwell also explicitly denies the first half of (10), at least as Hume would understand it: Maxwell famously insists that it is only true when the system is such that small initial variations only produce small final variations (what we would call a non-chaotic system). So we see something of how a Maxwellian account of Newtonian physics would diverge from a Humean account of Newtonian physics.
The second reason, of course, why one would do this is simply to get a better sense of Hume's philosophy itself, in much the same way that, while Heyting's logic is not the same as Brouwer's intuitionism, it gives one a better sense of what the latter implies.
ADDED LATER: Another one of note (from T 1.1.5 (SBN 14-15)):
(15) No objects will admit of comparison, but what have some degree of resemblance.
And another possibility (from T 1.3.2 (SBN 75)):
(16) There is nothing existent, either externally or internally, which is not to be consider'd either as a cause or an effect.