# Towards plausible moral naturalism

In ["Generalizing zombie arguments"](https://unstableontology.com/2025/07/15/generalizing-zombie-arguments/), I hinted at the idea of applying a Chalmers-like framework to morality. Here I develop this idea further.

Suppose we are working in an axiomatic system rich enough to express physics and physical facts. Can this system include moral facts as well? Perhaps moral statements such as "homicide is never morally permissible" can be translated into the axiomatic system or an extension of it.

It would be difficult to argue that a realistic axiomatic system *must* be able to express moral statements. So I'll bracket the alternative possibility: Perhaps moral claims do not translate into well-formed statements in the theory at all. This would be a type of moral non-realism, of considering moral claims to be meaningless.

There's another possibility to bracket: moral trivialism, according to which moral statements do have truth values, but only trivial ones. For example, perhaps all actions are morally permissible. Or perhaps no actions are. This is a roughly moral nonrealist possibility, especially compatible with error theory.

The point of this post is not to argue against moral meaninglessness or moral trivialism, but rather to explore alternatives to see which are most realistic. Even if moral meaninglessness or trivialism is likely, the combination of uncertainty and disagreement regarding their truth could motivate examining alternative theories.

Let's start with a thought experiment. Suppose there are two possible universes that are physically identical. They both contain versions of a certain physical person, who takes the same actions in each possible universe. However, in one possible universe, this person's actions are morally wrong, while in another, they are morally right.

We could elaborate on this by imagining that in the actual universe, this person's actions are morally wrong, although they are materially helpful and match normal concepts of moral behavior. This person would be a P-evildoer, analogous to a P-zombie; they would be physically identical to a morally acting person, but still an evildoer nonetheless.

Conversely, we could imagine that in the actual universe, this person's actions are morally good, although they are materially malicious and match normal concepts of immoral behavior. They would be a P-saint. (Scott Alexander has written about a similar scenario in ["The Consequentialism FAQ"](https://web.archive.org/web/20160306082537/https://raikoth.net/consequentialism.html).)

I, for one, balk at imagining such a scenario. Something about it seems inconceivable. It seems easier to imagine that I am so wrong about morality that intuitive judgments of moral goodness are anti-correlated with real moral goodness, than that physically identical persons in different possible worlds have different moral properties. Somehow, P-evildoer scenarios seem even worse than P-zombie scenarios. To be clear, it's not too hard to imagine that the P-evildoer's actions could have negative supernatural consequences (while their physically identical moral twin's actions have positive supernatural consequences), but moral judgments seem like they should evaluate agents relative to their possible epistemic state(s), rather than depending on un-knowable supernatural consequences.

As motivation for considering such deeply unfair scenarios, consider the common idea that atheists and materialists cannot ground their morality, as morality must be grounded in a supernatural entity such as a god. Then, depending on the whims of this god, a given physical person may act morally well or morally badly, despite taking the same actions in either case. And, unless the whims of the god are so finite and predictable that humans can comprehend them in general, different possible gods (and accordingly moral judgments) are logically possible and conceivable from a human perspective.

Broadly, this idea could be called "moral supernaturalism", which says that moral properties are not determined by the combination of physical properties, mathematical properties, and metaphysical constraints on conceivability; moral properties are instead "further facts". I'm not sure how to argue the inconceivability of P-evildoers to someone who doesn't already agree, but if someone accepts a principle of no P-evildoers, then this is a strong argument against moral supernaturalism.

The remaining alternative to moral meaninglessness, moral trivialism, and moral supernaturalism, would seem to be reasonably called "moral naturalism". I'll take some preliminary steps towards formalizing it.

Let us work in a typed variant of first-order logic, with three types: *W* (a type of possible worlds), *P* (a type of physical world trajectory), and *M* (a type of moral world trajectory). Each possible world has a physical and a moral trajectory, denoted $p(w)$ and $m(w)$ respectively. To state a no P-evildoers principle:

$$\neg \exists (w_1, w_2 : W), p(w_1) = p(w_2) \wedge m(w_1) \neq m(w_2)$$

In other words, any two possible worlds with identical physical properties must have the same moral properties. (As an aside, modal logic may provide alternative formulations of possibility without reifying possible worlds as "existent" first-order logical entities, though I don't present a modal formulation here.) I would like to demonstrate a more helpful statement from the no P-evildoers principle:

$$\forall (p^* : P) \exists (m^* : M) \forall (w : W), p(w) = p^* \rightarrow m(w) = w^*$$

This says that any physical trajectory has a corresponding moral trajectory that holds across possible worlds. To prove this, start by letting $p^* : P$ be arbitrary. Either there is some possible world with this physical trajectory, or not. If not, we can let $m^*$ be arbitrary, and $\forall (w : W), p(w) = p^* \rightarrow m(w) = m^*$ will hold vacuously.

If so, then let $w^*$ be some such world and set $m^* = m(w^*)$. Now consider some arbitrary possible world *w* for which $p(w) = p^*$. Either it is true that $m(w) = w^*$ or not. If it is, we are done; we have that $p(w) = p^* \rightarrow m(w) = w^*$. If not, then observe that *w* and $w^*$ have identical physical trajectories, but different moral trajectories. This contradicts the no P-evildoers principle.

So, the no P-evildoers principle implies the alternative statement, which expresses something of a functional relationship between physical trajectories and moral ones; for any possible physical trajectory, there is only one possible corresponding moral trajectory. With a bit of set theory, and perhaps axiom of choice shenanigans, we may be able to show the existence of a function *f* mapping physical trajectories to corresponding moral trajectories.

Now *f* expresses a *necessary* (working across all possible worlds) mapping from physical to moral trajectories, a sort of multiversal moral judgment function. Its necessity across possible worlds demonstrates its a priori truth, showing Kant was right about morality being derivable from the a priori. We should return to Kant to truly comprehend morality itself.

...This is perhaps jumping to conclusions. Even though we have an *f* expressing a necessary functional relationship between physical and moral trajectories, this does not show the epistemic derivability of *f* from any realistic perspective. Perhaps different alien species across different planets develop different beliefs about *f*, having no way to effectively resolve their disagreements.

We don't have the framework to rule this out, so far. Yet something seems suspiciously morally supernaturalist about this scenario. In the P-evildoer scenario, we could imagine that God decides all physical facts, and then adds additional moral facts, but potentially attributes different moral facts to physically identical universes. With the no P-evildoers principle, we have constrained God a little, but it still seems God might need to add facts about *f* even after determining all physical facts, to yield moral judgments.

So perhaps we can consider a stronger rejection of moral supernaturalism, to rule out a supernatural God of the moral gaps. We would need to re-formulate a negation of moral supernaturalism, distinct from our original no P-evildoers axiom.

One possibility is logical supervenience of moral facts on physical ones; the axiomatic system would be sufficient to prove all moral facts from all physical facts. This could be called moral reductionism: moral statements are logically equivalent to perhaps complex statements about particle trajectories and so on. This would imply that there are conceptually straightforward physical definitions of morality, even if we haven't found them yet. However, assuming moral reductionism might be overly dogmatic, even if moral supernaturalism is rejected. Perhaps there are more constraints to "possibility" or "conceivability" than just logical coherence; perhaps there are additional metaphysical properties that must be satisfied, and morality cannot be in general derived without such constraints.

As an example, Kant suggests that geometric facts are not analytic. While there are axiomatizations of geometry such as Euclidean geometry, any one may or may not correspond to the a priori intuition of space. Perhaps systems weaker than Euclidean geometry allow many logically consistent possibilities, but some of these do not match a priori spatial intuition, so some of these logically consistent geometric trajectories are inconceivable.

Let us start with an axiomatic system *T* capable of expressing physical and moral statements. Rather than the previous possible-world theory, we will instead consider it to have a set of physical statements *P* and a set of moral statements *M*, which are all about the actual universe, rather than any other possible worlds.

Let us, for formality, suppose that these additional metaphysical conceivability constraints can be added to our axiomatic system *T*; call the augmented system *T'*. Now we can apply model theory to *T'*. Are there models of *T'* that have the same physical facts, yet different moral facts?

We must handle some theoretical thorniness: [Gödel's first incompleteness theorem](https://scottaaronson.blog/?p=710) shows that there are multiple models of Peano arithmetic which yield different assignments of truth values to statements about the natural numbers, assuming PA is consistent. There is some intuition that these statements (which exist in the [arithmetical hierarchy](https://en.wikipedia.org/wiki/Arithmetical_hierarchy)) nonetheless have real truth values, although it's hard to be sure. Even if they don't, it seems hard to formalize a system similar to Peano arithmetic that is consistent and complete; the doubter of the meaningfulness of statements in higher levels of the arithmetical hierarchy is going to have to work in an impoverished axiomatic system until such a system is formalized.

So we augment *T'* with additional mathematical axioms, expressing, for example, all true PA statements according to the standard natural model; call this further-augmented system $T^*$. The axioms of $T^*$ are, of course, uncomputable, but this is not an obstacle to model theory. These mathematical axioms disambiguate cases where the truth values of these arithmetic hierarchy statements are morally relevant somehow.

In a meta-theory capable of model-theoretic analysis of $T^*$ (such as ZFC), we can express a stronger form of moral naturalism:

"Any two models of $T^*$ having the same assignment of truth values to statements in *P* have the same assignment of truth values to statements in *M*."

Recall that *P* is the set of physical statements, while *M* is the set of moral statements. What this expresses is that there are no "further facts" to morality in addition to physics, metaphysical conceivability considerations, and mathematical oracle facts, ruling out a broader class of moral supernaturalism than our previous formulation.

Using [Gödel's completeness theorem](https://unstableontology.com/2024/05/27/understanding-godels-completeness-theorem/), we can show from the moral naturalism statement:

"$T^*$ augmented with an infinite axiom schema assigning any logically consistent truth values to all statements in *P* will eventually prove binary truth values for all statements in *M*".

The infinite axiom schema here is somewhat unsatisfying; it's not really necessary if *P* is finite, but perhaps we want to consider countably infinite *P* as well. Luckily, since all proofs of individual *M*-statements (or their negations) are finite, they only use a finite subset of the axiom schema. Hence we can show:

"For any finite subset of *M*, $T^*$ augmented with some finite subset of an infinite axiom schema assigning any logically consistent truth values to all statements in *P* will eventually prove binary truth values for all statements in this subset of *M*".

This is more satisfying: a finite subset of moral statements only requires a finite subset of physical statements to prove their truth values. Likewise, the proofs need only use a finite subset of the axioms of $T^*$, e.g. they only require a finite number of mathematical oracle axioms.

To summarize, constraining models of an augmented axiomatic system, with metaphysical conceivability and mathematical axioms, to never imply different moral facts given the same physical facts, yields a stronger form of moral non-supernaturalism, preventing there from being "further facts" to morality beyond physics, metaphysical conceivability constraints, and mathematics. Using model theory, this would imply that the truth values of any finite subset of moral statements are logically derivable from the system and from some finite subset of physical statements. This implies not just a theoretical existence of a functional relationship between physics and morality (*f* from before), but hypothetical *a priori* epistemic efficacy to such a derivation, albeit making use of an uncomputable mathematical oracle.

The practical upshot is not entirely clear. Aliens might have different beliefs about uncomputable mathematical statements, or different metaphysical conceivability axioms, yielding different moral beliefs. But disagreements over metaphysical conceivability and mathematics seem more epistemically weighted than fully general moral disagreements; they would relate to epistemic cognitive architecture and so on, rather than being orthogonal to epistemology.

Physically instantiated agents also might not generally be *motivated* to act morally, even if they have specific beliefs about morality. This doesn't contradict moral realism per se, but it indicates practical obstacles to the efficacy of moral theory. As an example, the aliens may be moral realists, but additionally have beliefs about "schmorality", a related but distinct concept, which they are more motivated to put into effect.

While it would take more work to determine the likelihood of Kantian morality conditioned on moral naturalism, in broad strokes they seem quite compatible. In the *Critique of Practical Reason*, Kant argues:

> If a rational being can think of its maxims as practical  universal laws, he can do so only by considering them as principles which contain the determining grounds of the will because of their form and not because of their matter.
>
> The material of a practical principle is the object of the will. This object either is the determining ground of the will or it is not. If it is, the rule of the will is subject to an empirical condition (to the relation of the determining idea to feelings of pleasure or displeasure), and therefore it is not a practical law. If all material of a law, i.e., every object of the will considered as a ground of its determination, is abstracted from it, nothing remains except the mere form of giving universal law. Therefore, a rational being either cannot think of his subjectively practical principles (maxims) as universal laws, or he must suppose that their mere form, through which they are fitted for being universal laws, is alone that which makes them a practical law.

Mainly, he is emphasizing that universal moral laws must be determined *a priori*, rather than subject to empirical determination; hence, different rational agents would derive the same universal moral laws given enough reflection (though, of course, this assumes some metaphysical agreement among the rational agents, such as regarding the a priori synthetic). While my formulation of moral naturalism allows moral judgments to depend on physical facts, the general functional mapping from physical to moral judgments is itself *a priori*, not depending on the specific physical facts, but instead derivable from an axiomatic system capable of expressing physical facts, plus metaphysical conceivability constraints and mathematical axioms.

This is meant to be more of a starting point for moral naturalism than a definitive treatment. It is a sort of *what if* exercise: what if moral meaninglessness, moral trivialism, and moral supernaturalism are all false? It is difficult to decisively argue *for* moral naturalism, so I am more focused on exploring the implications *of* moral naturalism; this will make it easier to have an idea of the scope of plausible moral theories, and how they compare with each other.