Ethical naturalism is the thesis that (A) moral truths are equivalent to—or reducible to—(some) truths about the natural world and (B) there is at least one objective moral truth. "Ethical naturalism" is generally characterized such that you must simultaneously believe in (A) and (B) to be an ethical naturalist.

Hume's law (otherwise known as the Is-Ought gap) is the thesis that, from a set of non-moral premises, one cannot infer a moral statement. For example, let's say we have the set of sentences:

• "Stepping on Death's foot is painful."
• "Death's socks are really expensive."

We, at least prima facie, unfortunately cannot infer "Stepping on Death's foot is immoral".

Hume didn't actually explicitly formulate some rule called "Hume's law"; he simply said the following (in his book, A Treatise of Human Nature), which inspired people to concoct "Hume's law":

Hume

In every system of morality, 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 God, or makes observations concerning human affairs; when of a sudden I am surprised 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, 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. 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 would 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 perceived by reason.

Since:

1. "Truths about the natural world" are descriptive ("rocks are solid", "cars are heavy" etc).
2. Hume's law seems reasonable/valid.

Some people have concluded that Hume's law debunks—or is incompatible with—ethical naturalism—that, in order to be an ethical naturalist, you need to reject Hume's law; however, this is false.

You do not need to reject Hume's law to be an ethical naturalist, and I will explain why.

Before we resume talking about the Is-Ought gap, I would like to introduce another type of gap: the Has-Is gap. Consider the following argument:

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[P1]: X has quills.
[P2]: X has four legs.
[P3]: X has 2 eyes.

[P4...P104]: (100+ other "X has Y" statements, where X is the specific object in question and Y is a property that hedgehogs generally have.)

[C]: X is a hedgehog.

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It is prima facie deductively invalid. Just because X has one or all of the properties commonly ascribed to hedgehogs, it does not deductively follow that Y is a hedgehog.

Nevertheless, whenstever you assess whether or not something is a hedgehog, the only thing you do is check whether or not it possesses a high number of the properties generally ascribed to hedgehogs—that, incontrovertibly, is your only criteria for verifying its status as a hedgehog.

Now, here's the important thing: hedgehogs are not considered to be strange, non-natural entities that are utterly unlike anything in the natural world—the deductive invalidity is merely a consequence of how deductive reasoning works.

Mere explanation of the Has-Is gap will suffice to convince many; however, some may regard the preceding section to be irrelevant because, for them, the root of the issue—the belief that Hume's law debunks or is incompatible with ethical naturalism—is actually the consequence of an extreme interpretation/formulation of Hume's law.

Consider the following:

The Restrictive Interpretation (HL¹):
"For any set of premises S, if S is devoid of moral predications (that is, there is not even one moral predication in S), you cannot infer a moral predication from S".

The Lenient Interpretation (HL²):
"For any set of premises S, if S makes no mention of morality, one cannot infer a moral predication from S".

Now, consider the following argument:

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[P1]: Stepping on Death's foot is pain-inducing.
[P2]: "Immoral" means "pain-inducing".
[C]: Stepping on Death's foot is immoral.

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