Imagine Hume's remarks but with reference to the usual disjunction introduction:
In every system of conjunction, which I have hitherto met with, I have always remarked, that the author proceeds for some time in the ordinary way of reasoning, and establishes the being of a primal unity, or makes observations concerning the integrity of human affairs; when of a sudden I am surprised to find, that instead of the usual copulations of propositions, and, and not and, I meet with no proposition that is not connected with an or, or an or not. This change is imperceptible; but is, however, of the last consequence. For as this or, or or not, expresses some new relation or affirmation, it's necessary that it should be observed and explained; and at the same time that a reason should be given, for what seems altogether inconceivable, how this new relation can be a deduction from others, which are entirely different from it.
In other words, would inferring "ought" from "is" be similar to going from, "I am a narwhal," to, "I am a narwhal or Jupiter is larger than the sun," etc.? Because otherwise, I'm at a loss now as to what Hume's point is (I know what it's supposed to be, but for now, he seems to fail in making it):
- (Hume's guillotine): A ⊬ OBB unless A = OBC (trivially if OBB = OBC)
- (Hume's antiknife?): A ⊬ (A ∨ B) unless A = A ∨ B
... so that (for those of us who are not relevance logicians) going from an "is" to an "ought" can just be an axiom of some system, and a sense of these axiomatics underscored whichever targets Hume had in mind (retrospectively, as with natural-law theories (I suppose), or prospectively, as with Rawls' "acting from our true nature"):
- ∃ABC((A ∧ B) ⊢ OBC) and A, B are stipulated to not be OB-sentences
- Alternatively: ∃ABC((A ∧ B) → OBC) (this would probably be a better format for an axiom)
Note: I tried doing a search "is-ought gap and relevance logic" on Google, and nothing clearly, exactly came up that was relevant(!). To be sure, some of the results were for extended discussions (essays) in which both topics were analyzed, but from the snippets it wasn't evident how directly these matters were related, if directly enough at all.