Hume famously divided knowledge into “relations of ideas” and “matters of fact.” In more modern terminology, he divided knowledge into the analytic a priori and synthetic a posteriori. This idea is captured by Hume’s fork in An Enquiry Concerning Human Understanding:
If we take in our hand any volume; of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames: for it can contain nothing but sophistry and illusion.
This fork was later adopted by the logical positivists and it was noticed that it was self-refuting because this fork is neither a relation of ideas nor a matter of fact; so, by its own criteria, it is nothing but sophistry and illusion—or meaningless.
I think I spotted another lapse in Hume. In A Treatise of Human Nature, Hume says:
To form a clear idea of anything, is an undeniable argument for its possibility, and is alone a refutation of any pretended demonstration against it.
I’m not sure, but this may be the earliest form of the popular principle that conceivability is a guide to metaphysical possibility. (Some think there are counterexamples to this, so David Chalmers makes a distinction between primary and secondary conceivability to mitigate that.)
The lapse, as I see it, is that this conceivability-to-possibility principle is neither a relation of ideas nor a matter of fact, so it must be meaningless by his own lights. For the sake of argument, let’s say that it is a relation of ideas. The problem doesn’t go away because analytic a priori statements are just relations between ideas and say nothing informative about the world. Saying that something is metaphysically possible is an informative statement about the world. The better option is to say that it is a matter of fact; we’d at least be saying something substantive then. But how could it be? After all, conceiving is done from the armchair. We don’t see metaphysical possibilities anymore than we see causation.
The strange thing is that in Dialogues concerning Natural Religion (9.6), Hume says:
The words, therefore, necessary existence, have no meaning …
If necessary existence has no meaning then how can contingent existence have meaning? I suspect that Bertrand Russell may have already noticed the point that I’m making, for in his famous debate with Fredrick Copleston he says:
I don’t admit the idea of a Necessary Being and I don’t admit that there is any particular meaning in calling other beings “contingent.” These phrases don’t for me have a significance except within a logic that I reject.
I think the consistent thing for a full-blown Humean to say is that contingent existence doesn’t have any meaning either.