Hume’s Is–Ought Gap and Ought-Implies-Can

David Hume is credited with formulating the is–ought gap (also known as Hume’s Law) in Treatise 3.1.1.27:

I cannot forbear adding to these reasonings an observation, which may, perhaps, be found of some importance. In every system of morality, which I have hitherto met with, I have always remark'd, that the author proceeds for some time in the ordinary way of reasoning, and establishes the being of a God, or makes observations concerning human affairs; when of a sudden I am surpriz'd to find, that instead of the usual copulations of propositions, is, and is not, I meet with no proposition that is not connected with an ought, or an ought not. This change is imperceptible; but is, however, of the last consequence. For as this ought, or ought not, expresses some new relation or affirmation, 'tis necessary that it shou'd be observ'd and explain'd; 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. But as authors do not commonly use this precaution, I shall presume to recommend it to the readers; and am persuaded, that this small attention wou'd subvert all the vulgar systems of morality, and let us see, that the distinction of vice and virtue is not founded merely on the relations of objects, nor is perceiv'd by reason.

This passage is often interpreted as Hume positing a complete bifurcation between the worlds of “is” and “ought”, fact and value, description and prescription. Critics have responded with examples that seem to bridge both worlds, such as promissory obligation (the fact that one has made a promise implies that one is obligated to fulfil that promise) and how modality circumscribes morality (the Ought Implies Can principle and its equivalent, Impossible Implies Omissible).

Ought Implies Can:

\begin{equation*} O \phi \vDash \Diamond \phi \end{equation*}

Impossible Implies Omissible:

\begin{equation*} \neg \Diamond \phi \vDash \neg O \phi \end{equation*}

Against these challenges, Russell (2021) has recently worked to prove Hume’s Law. But, as a mere incoming Year 2 undergraduate, her logical work is beyond me. Nonetheless, it appears to me that a more modest version of the is–ought gap can meet these challenges. After all, the is–ought gap is often invoked, especially by social progressives, to support the view that just because something is the case doesn’t mean it should be the case. This invocation is not an attack on all connections (if any) between fact and value, but on logical inferences of a specific kind, which I formulate as Modest Is–Ought Gap.

Modest Is–Ought Gap:

\begin{equation*} \phi \not \vDash O \phi \end{equation*}

There is no contradiction between Modest Is–Ought Gap and Ought Implies Can. Looking back at Treatise 3.1.1.27, Hume is evidently sceptical about logical leaps from “is” to “ought”, but he says nothing about modality: “can” or “must”. Indeed, facts and descriptions include not only “is” claims, but also “can” and “must” claims. Therefore, on this modest view, it is an oversimplification to say that Hume’s is–ought gap separates all fact from value, or all description from prescription. Hume’s is–ought gap, at its core, separates a particular type of fact from value, and a particular type of description from prescription.

That, I can raise a glass of Scotch to. Sláinte! 🥃 ∎