The is-ought gap is commonly construed as the inability to validly deduce an ought from a set of purely is premises. That we cannot get an ought from an is at a glance is an intuitive truth to some, me included. However, the gap isn’t so easily established. On the account given here there are counterexamples rendering it false. I'll proceed to give the counter-examples, and then give an explication that Gillian Russell constructs of the is-ought gap that renders these counter-examples inert.

Supposed Counter-Examples.

Ought implies Can

Moral philosophers commonly endorse a principle known as Ought implies Can. That is to say if you ought do P then it is semantically entailed you can do P. The principle seems intuitive, to say you should do something seems to prima facie indicate that it’s possible to do.

and since contra-position preserves validity, we have the following counter-example to the gap.

Our contra-posed version of the Ought implies Can principle seems to violate the is-ought gap. Given a single descriptive premise, a normative premise is semantically entailed.

Prior’s Dilemma

The dilemma give by A.N. Prior is as follows:

1. Tea Drinking is common in England.
2. Therefore, Tea Drinking is common in England or all New-Zealanders ought be shot.

This inference is valid in classical logic using the inferential rule Disjunctive Addition. I think we would agree that (1) is descriptive. Now consider the following dilemma. Either  (2) is normative or descriptive. If it's normative this inference is a violation of the gap. If it's descriptive consider the following deduction:

1. Tea Drinking is common in England or all New-Zealanders ought be shot.
2. Tea Drinking is not common in England
3. Therefore, all New-Zealanders ought be shot.

This inference is valid in classical logic using the inferential rule Disjunctive Syllogism. Both (1) and (2) are descriptive, and (3) is normative. This would be a violation of the gap. The general form of Prior's dilemma is as follows: Let D refer to a descriptive proposition and N a normative proposition.

1. D
2. Therefore, D or N

(D or N) is either descriptive or normative, if normative it's a violation of the gap, if descriptive the following inference is a violation:

1. D or N
2. Not D
3. therefore N 

Explosion

I'll be the first to admit this counter-example is kind of cheeky, but it would constitute a violation of the Gap. Let D be a descriptive premise (and by extension Not D is descriptive). Consider the following inference:

1.  D and Not D
2. therefore N

Where N is normative. Now I'll be the first to admit, this inference isn't sound. But Explosion is a valid inference, and serves as a counter-example to the is-ought gap as we've construed it.

Logical Truths

Consider a deontic logic with a standard Kripke Semantic. Recall the Obligatory operator is closed under Logical consequence. This is to say that if OA is true and that A entails B, that OB is true. Given that logical truths are logically entailed by everything, it follow that all logical truths are obligatory. Consider the following:

This would seem to be a violation of the gap.

Non-Formal Counter-Examples

There are a list of purported counter-examples that are given in Natural language by opponents of the Gap. To be frank I won’t be considering them, it seems to me they can be viewed as enthymemes and don’t actually constitute a violation of the gap.

Formal construction of a Deontic Modal Logic

These counterexamples seem devastating for Hume’s Gap. In what sense could we defend it? We should explicate Hume’s gap. To do so we should first ask, what does it mean for a sentence to be normative? T. Karmo^[1] gives us an answer, what it means for a sentence to be normative is for its truth value to be dependent upon the normative standards. That is to say, we would expect normative sentences to be the sentences that are breakable (truth value changes) with changes in the normative standards. That’s still unclear but a step in the right to direction. To clarify this, Gillian Russell constructs^[2] a Deontic Modal Logic called DML. The primitive expressions of DML consist of the modal operators of necessity and possibility, the deontic operators of obligation and permission, and sentence letters. The DML model is a tuple:

where:

1. W is a non-empty set of worlds
2. S is a non-empty subset of W (the superb worlds)
3. @ is an element of W (The “actual” world)
4. I is an interpretation function mapping pairs of sentences letters and members of W onto truth values.

Modal operators range over all worlds, the deontic operators O and P are interpreted as quantifying over the set of superb worlds:

Where V is the truth at some model M. Truth in a model is:

Logical consequence is:

Now to clarify our notion of normative sentences breaking when normative standards change: We will define S-shifting, where a DML-model N is an S-shift of another DML-model M iff it is the same except possibly for the identity of the set of superb worlds S which may be a different non-empty subset of W. 

Formally, Where M and N are DML models, N is an S-shift of M iff: 

1. The non-empty set of worlds at model M are equal to the non-empty set of worlds at model N
2. The actual world at model M is equal to the actual world at model N
3. For all sentence letters, alpha, and worlds w

And so we have a formal definition for normative, that is:

Before we continue, let’s provide a more general result.

Limited General Barrier

Suppose we have a logic, L. This is to say that we have:

1. A formal language.
2. A set of models, U, with respect to which sentences in the formal language are true or false.
3. A definition of consequence.

Furthermore, suppose R is a binary relation on U. We’ll say a sentence Φ is R-breakable if there is a pair of models:

That is to say, a sentence is R-breakable if it’s true in a model M, false in a model N, and M is in relation with N. If a sentence is not R-breakable, we’ll call it R-preserved. Now we can give the following theorem:

Russell provides the proof.^[3]

Refuting the Counter-Examples

A Corollary of the limited general barrier theorem is the limited normative barrier:

We've succeeded in explicating Hume's Gap! Now we'll analyze our supposed counter-examples and see if they succeed against our limited normative barrier.

Ought implies Can

On our definitions, ¬◊Φ is non-normative as S-shifting does not change the modal accessibility of any worlds and as such if it is true in a model it is true in all S-shifts of the model. 

The conclusion is also normative under our definition. However, this argument meets our unless-clause as all DML models of ¬◊Φ are such that W contains no p worlds. As such no amount of S-shifting can result in a model where S contains a p-world and so as a model that makes ¬OΦ false. The argument fails as a counterexample.

Prior’s Dilemma

One our definitions. Φ∨Oψ is normative as it’s S-shift breakable and Φ is descriptive, however the inference meets our unless clause as any model of the premise Φ has no S-shift model that makes Φ false and so no S-shifts that make Φ∨Oψ false. As such this does not constitute a counterexample.

Let's consider the second inference:

It is valid and the conclusion is normative, the premise is also normative as we’ve concluded above. As such this does not constitute a counterexample.

Explosion

Let’s take the standard explosion inference:

This is valid as the premises are unsatisfiable. Unsatisfiable premises are also descriptive on our definition as they are not S-shift breakable. It meets our unless clause trivially however, any model of the premises is such that all its S-shifts are models of the conclusion. So this doesn’t constitute a counterexample.

Logical Truths

No logical truth is S-shift breakable, since it's true in all models. That would mean the sentences on the right are not normative, and therefore this isn't a counter-example.

Conclusion

Now that we’ve established that all of the purported counterexamples don’t succeed on the stricter construction of Hume’s Gap as the limited normative barrier, and that we can formally prove the limited normative barrier as true, why should we accept this route over some other approach?

For example, a possible route to saving Hume’s Gap from these counter-examples is brute-force adding exception clauses for all of these counter-examples.

The answer is it’s independently motivated and that it’s entailed by a more general theorem that applies for other purported barriers like the barriers between the present and past and future, indexicals and non-indexicals, and particulars and universals.^[4]

Otherwise, I think the implications of this for realist debates in metaethics is clear. Disputes over the is-ought gap is an approach certain realists and anti-realists go to. Establishing a limited normative barrier seems to limit the realist to abduction over philosophical datum as their methodological approach in metaethics.

To finish this, I want to point to an insight that Russell has^[5]. Barriers like Hume’s Law as commonly constructed may serve as intuitive and accessible rules of reasoning classified as “Folk Logic”. Similar to claims of the inability to prove negative claims, they are intuitive claims that over-state the truth. Formal logic shows such claims to be incorrect and hyperbolic. We can prove negatives, for example a proof that there are no square circles is trivial. The insight however is that there is more difficulty in proving negative claims, which is probably true despite being over-stated. The same can be said for Hume’s law.

1. ^{^}

   https://philpapers.org/rec/KARSVB

2. ^{^}

   https://philpapers.org/archive/RUSHTP-2 p. 22

3. ^{^}

   ibid, p. 12

4. ^{^}

   You can find the variety of these barrier proofs in her paper. 

5. ^{^}

   ibid, p. 28