Resurrecting the Hume’s Dictum argument against metaethical non-naturalism

I argue for the viability of one neglected way of developing supervenience-based objections to metaethical non-naturalism. This way goes through a principle known as ‘Hume’s Dictum’, according to which there are no necessary connections between distinct existences. I challenge several objections to the Hume’s Dictum-based argument. In the course of doing so, I formulate and motivate modest and precise versions of Hume’s Dictum, illustrate how arguments employing these principles might proceed, and argue that the Hume’s Dictum argument enjoys some advantages relative to other supervenience-based objections to non-naturalism.

Getting more precise would require a theoretical framework for understanding metaphysical reduction and kindhood for properties. As I've said, I don't want to commit to a particular stance here. But for those who might find it helpful, here are tentative suggestions for how non-naturalism might be cashed out in different metaphysical frameworks. In the ideology of fundamentality, non-naturalism might be the view that some purely normative entities are absolutely fundamental. In the ideology of grounding, non-naturalism might be the view that some normative facts are not fully grounded in non-normative facts. In the ideology of real definition, nonnaturalism might be the view that some normative properties have no real definition in non-normative terms. In the ideology of essence, non-naturalism might be the view that some normative properties have essences that do not involve non-normative properties.
A useful analogy for grasping non-naturalism comes from philosophy of mind. Some take a position called 'non-reductive physicalism', according to which mental properties do not metaphysically reduce to physical properties, e.g. because mental properties are multiply realizable, but they are still broadly of a kind with physical properties. The non-reductive physicalist therefore makes the analogue of the first nonnaturalist commitment but not the analogue of the second. 9 By contrast, the property dualist goes all the way in claiming that mental properties are simply of a different metaphysical kind than the physical ones. On this taxonomy, the metaethical nonnaturalist is more like the property dualist than the non-reductive physicalist.
One traditional line of challenge to non-naturalism concerns the modal relationship between the normative and the non-normative. Normative properties supervene on nonnormative properties. For now it will suffice to intuitively gloss supervenience as the claim that if you fix the non-normative properties for some entity, you thereby fix the normative properties for that entity. Equivalently, supervenience says that there can be no difference in the normative properties for an entity without a difference in the nonnormative properties of that entity. That some suitably formulated supervenience thesis holds with metaphysical necessity is a rare point of consensus amongst metaethicists. 10 Two broad families of arguments have emerged as popular ways of pressing the supervenience challenge against non-naturalism. These two families leverage the supervenience premise differently. First, there is the direct argument, which uses supervenience and some auxiliary assumptions to establish the intensional equivalence (necessary coextension) of normative properties with some non-normative properties. The direct argument then invokes a coarse-grained conception of properties, according to which intensionally equivalent properties are identical, and thereby concludes that normative properties are in fact identical to non-normative ones.
The second type of argument is the explanatory argument. In contrast with the direct argument, the explanatory argument invokes the supervenience of the normative on the non-normative as a datum which requires explanation. I call this theoretical challenge of giving a metaphysical explanation of supervenience 'the explanatory challenge'. 9 Similarly, the non-reductive naturalist in metaethics makes the first commitment but not the second; they hold that normative properties are of kind with non-normative properties but are not reducible to other non-normative properties. 10 A small contingent has recently begun to challenge this premise. See, e.g., Hattiangadi (2018) and Rosen (2020). I do not respond to this objection here and simply assume supervenience as a starting point.
Proponents of the explanatory argument then claim that non-naturalism does not have the theoretical resources to meet the explanatory challenge.
The direct argument has a faithful group of adherents. 11 The explanatory argument also has its supporters, 12 which has given rise to a substantial and growing literature of increasingly sophisticated and exotic versions of non-naturalism being wheeled out to explain the supervenience of the normative on the non-normative, 13 and in turn challenges to whether these forms of non-naturalism meet the explanatory challenge. 14 By contrast, though it appears to be a distinct way of leveraging supervenience against non-naturalism, one is hard-pressed to find defenders of what I have called the HD argument.
This paper attempts to resuscitate the HD argument by examining some reasons to dismiss the HD argument as a way of pressing the supervenience challenge and showing that these challenges can be met. I suspect that one of these reasons has been primarily operative in the literature on supervenience, and so I attempt to diagnose a mistake that occurred during some influential formulations of the explanatory argument that led to the HD argument being set aside as an implausible form of the explanatory argument. Along the way, I motivate some modest versions of HD and illustrate how they can be employed in arguments against non-naturalism.

The HD argument and the explanatory argument: distinct existences
On my diagnosis, the central reason for the neglect of the HD argument is that it has been regarded as an inferior version of the explanatory argument. In this section, I trace the line of reasoning that led to this conclusion and show that it neglects a broader picture of HD as an attractive metaphysical principle even independently of explanatory motivations. I then argue that this neglect is problematic since it implies that a non-naturalist explanation of supervenience would put to rest the supervenience challenge for non-naturalism (assuming one rejects the direct argument). By contrast, I suggest that even with a prima facie adequate explanation of supervenience, nonnaturalism would still come with a metaphysical price.

The explanatory picture of HD
Mark Schroeder is among the first to revive the explanatory argument in recent discussions. 15 Schroeder is also the first to consider the HD argument as a form of the 11 Among these are Jackson (1998), Streumer (2008), and Brown (2011. These arguments are obviously descendants of the equivalent arguments for physicalism about the mental. See Kim (1993) for a collection of some of this work. 12 Such as McPherson (2012), Schroeder (2007), Schroeder (2014, and Dreier (2019). 13 For one flavor of explanation, see the versions of non-naturalism defended by Enoch (2011) and Scanlon (2014). For another, see Bader (2017). For yet more, see Leary (2017) and Moberger (2022). 14 Against the versions of non-naturalism defended by Enoch, Scanlon, and Bader, see Elliott (2014) and Leary (2017). Against the version defended by Leary, see Faraci (2017) and Toppinen (2018). 15 In Schroeder (2007Schroeder ( , 2014.
supervenience challenge to non-naturalism. But in Schroeder's classification, the HD argument falls under the general banner of the explanatory argument. This is because he views HD as being motivated by explanatory principles: Although Hume's Dictum doesn't itself mention explanation, it is natural to think of it as motivated as a special case of the idea that necessities require explanation -together with the idea that all necessities are explained by the lack of full distinctness among the entities involved. Take, for example, the necessity that something cannot be a red square without being red. As a necessity, this is a strong claim, because it rules out even the possibility of a non-red red square. But it is not puzzling why this is necessary -it is necessary because being red is just part of what it is to be a red square. This example both illustrates how a necessity can be explained, and why at least this sort of explanation seems to depend on a lack of distinctness between the entities involved, and correspondingly why it seems to be the sort of explanation that a non-reductive realist could not give of the necessities involved in supervenience. 16 On Schroeder's metaphysical picture, all necessities require explanation. Given this strong explanatory principle, there must be some explanation for the necessary connections entailed by supervenience. But now Schroeder surveys a few examples of necessities: first those involving conjunctive properties, and then the necessary connection between being water and being H 2 0. Each of these necessities seems to have a perfectly adequate reductive explanation: an explanation that involves the non-distinctness (in some sense) of the entities in question. Given these successes, Schroeder concludes that there is some support for the conjecture that all necessities are ultimately explained by the entities involved not being wholly distinct. On this picture, it is quite natural to think of the HD argument as a form of the explanatory argument. The challenge for the non-naturalist is to give an adequate non-reductive explanation of the necessary connections between the normative and the non-normative. Though these connections would violate the letter of HD, the violation would be benign because the conjecture that all necessities are explained by non-distinctness would be defeated. Therefore the explanatory motivation for HD would fail to apply, leaving us no good reason to dismiss these necessary connections from our metaphysics.
This picture of HD as being motivated only by explanatory concerns was picked up by Tristram McPherson's influential and arguably canonical formulation of the explanatory argument. 17 McPherson centers his argument around a principle which he calls a 'methodological cousin' of HD: MODEST HUMEAN: Commitment to brute necessary connections between discontinuous properties counts significantly against a view. 18 McPherson says that he prefers MODEST HUMEAN over HD, and thus the explanatory argument over the HD argument, for several reasons. Some of these do not neatly fit into the explanatory picture, so I address these later. But one concerns the restriction of MODEST HUMEAN to brute necessary connections: MODEST HUMEAN only applies to brute necessary connections. Thus, if two properties are discontinuous, but we are nonetheless able to offer a compelling explanation of the necessary connection between them, MODEST HUMEAN makes the reasonable suggestion that having made such a connection intelligible would be enough to defeat the Humean presumption. 19 The idea that it is reasonable to permit necessary connections between discontinuous properties when these necessary connections are adequately explained falls naturally out of the picture on which HD is ultimately motivated by explanatory concerns. On that picture, when one successfully explains the necessary connection in question, one defeats the Humean presumption since that Humean presumption is ultimately motivated by an aversion to inexplicable necessities.
So we have a clear conception of one reason to prefer the explanatory argument over the HD argument. This reason stems from a picture on which HD is motivated only by broad explanatory principles. On this picture, a non-naturalist explanation for supervenience defeats the explanatory motivation for HD. Therefore all the action is on whether the non-naturalist can explain supervenience, or in other words, answer the challenge put forth by the explanatory argument. The HD argument therefore looks like an obtuse way of putting forward the explanatory argument.

Two non-explanatory routes to HD
The fundamental problem with this picture is that HD can be motivated on grounds quite independent of such explanatory principles. To be clear, I needn't be opposed to explanatory motivations for HD. 20 But explanatory considerations do not exhaust the list of reasons to like HD as a metaphysical principle. On this flipping of the script, the explanatory argument looks more like a special case of the HD argument for those who like the explanatory motivation for HD. But if the motivations for HD are not exhausted by explanatory ones, then an otherwise reasonable non-naturalist explanation of supervenience may be condemned on the grounds that it violates HD. Since HD is plausible on non-explanatory grounds, the fact that the explanatory motivation for HD would be overcome (by a reasonable non-naturalist explanation of supervenience) would not mean that there is no cost to violating HD. To make this challenge more vivid, I here present two potential routes to HD that do not turn on explanatory considerations.
My purposes here are more illustrative than argumentative. That is, I want to overturn the explanatory picture by sketching out some potential non-explanatory ways of motivating HD. But it will not be possible here to give a complete defense of these motivations. Rather, by showcasing some promising directions in this area, I hope to bolster my suggestion that the HD argument is a fruitful line of argument to pursue against non-naturalism. If I can show that these motivations are at least prima facie plausible, then given the dialectical benefits of abandoning the the explanatory picture (c.f. Sect. 2.3), the HD argument should be further developed, and these motivations for HD should be considered at greater depth elsewhere.
First, one might think that versions of HD are entailed by our best accounts of the extent of metaphysical possibility. Many theorists would like an elegant, informative, simple, and systematic account of what is metaphysically possible. Some theorists would like such an account to not invoke any ineliminable modal concepts, since they aim to give a reductive analysis of modality. 21 In this context and with these desiderata in mind, recombination principles have emerged as leading candidates for giving us a satisfactory account of what is metaphysically possible. 22 Notably, in an otherwise skeptical take on the motivations for HD, Jessica Wilson takes the prima facie advantages of the recombinatorial account of modality to be a potentially promising route to HD. 23 In the broadest terms, recombination principles generate a modal space by constructing possibilities out of combinations of basic elements. It is plausible that each version of a recombination principle entails some corresponding version of HD. For example, consider the recombination principle for properties advocated by David Armstrong 24 : PATTERN: Any pattern of instantiation of any fundamental properties and relations is metaphysically possible. 25 PATTERN is meant to guarantee that all combinations of instantiations of fundamental properties and relations are possible. So, for example, if our modal space contained just two objects, a and b, and two fundamental properties P and Q, PAT-TERN would entail that there are worlds where both a and b are P but not Q, worlds where both a and b are Q but not P, worlds where a is P but not Q and b is Q but not P, etc. Now consider two arbitrary collections of distinct, fundamental properties C 1 and C 2 . By this I mean that each of the members of C 1 and C 2 is a fundamental property which is pairwise-distinct from every other member of both C 1 and C 2 . By PATTERN, any pattern of instantiation of the members of C 1 is possible without any member of C 2 being instantiated at all. The following principle thereby follows from PATTERN: HD fundamental : For any collections of distinct, fundamental properties C 1 and C 2 and any pattern of instantiation P, it is possible that the members of C 1 are instantiated in P and no member of C 2 is instantiated. 21 See Sider (2003) on motivations for eliminating primitive modality. 22 See Lewis (1986, pp. 88-92) for an example of the application of recombination principles to this project. 23 Wilson (2010, pp. 634-635). To be clear, Wilson's way of motivating HD via recombination principles is slightly different from mine. Wilson takes HD itself to serve as the motivation for recombination principles, and then takes the advantages of recombination principles to somehow render support back to HD. Whereas I argue that a version of HD is directly entailed by recombination principles and is thereby supported by the advantages of recombination principles. In other words, HD is the bedrock on Wilson's suggestion, whereas the advantages of recombination are the bedrock on my suggestion. These two strategies for motivating HD, however, clearly share much in common and needn't be in competition. 24 In Armstrong (1989Armstrong ( , 1997. I call this principle a version of HD because it rules out necessary connections between distinct collections of properties of a certain type. In particular, it denies that some pattern of instantiation for some fundamental properties necessitates the instantiation of other distinct, fundamental properties. 26 So a version of HD restricted to fundamental properties follows from this recombinatorial principle about properties. Those who like recombination principles as theoretically virtuous generators of modal space are thereby committed to accepting a version of HD. And what matters here is that this motivation for HD is independent of the premises that undergird the explanatory motivation for HD. (Namely, that all necessities require metaphysical explanation, and that these necessities are to be metaphysically explained by non-distinctness.) The second non-explanatory route to HD goes through principles that link conceivability and metaphysical possibility. This is a traditional motivation for HD, going back to Hume himself and discussed more recently by Gibbs (2019, ch. 5). Contemporary defenders of the conceivability-possibility link have distinguished various types of conceivability, including such distinctions as primary/secondary and positive/negative conceivability. 27 A natural motivation for the conceivability-possibility link is the idea that the conceivability-based approach is the best overall account of our modal knowledge. A fuller defense of this idea would need to consider alternative approaches to modal epistemology. But my purpose here is merely to showcase some potential non-explanatory routes to HD, not to defend them at length. Just as I cannot here undertake an extensive consideration of the plausibility of combinatorial accounts of modality, I also cannot here undertake an extensive consideration of conceivability-based approaches to modal epistemology.
Given the assumption that conceivability (of a certain type) entails or is a good guide to possibility, there is pressure to endorse a version of HD. The reason is that if it is conceivable that some collection of properties are instantiated in some pattern P, then it is conceivable that they are instantiated in P without any property discontinuous with those properties being instantiated. This means that insofar as some pattern of instantiation is conceivable, it is also conceivable in the absence of any other properties which radically differ in metaphysical kind. The picturesque idea is that when some properties have their own unique metaphysical glow, and we conceive of some pattern of instantiation, we can conceive that pattern in the absence of that special glow. This idea seems to be what lies behind the dualist's claim that we can conceive of certain sterile patterns of physical properties totally lacking any mental coloring at all (i.e. zombies). There seems to be a link between discontinuity and the conceivability of absence. 28 26 The route from PATTERN to HD fundamental is merely supposed to be a simple illustration of the route from recombination principles to versions of HD. For a considerably more modest property recombination principle from which a version of HD suitable for our purposes here also follows, see Saucedo (2011, Sect. 3). 27 See specifically Chalmers (2010, ch. 6) for these distinctions. See also Gregory (2004), Geirsson (2005), Ichikawa and Jarvis (2013), and Kung (2010) for examples of defenses of the conceivability-possibility link. 28 Later I will suggest that this link should be qualified if there are any necessarily instantiated properties. C.f. Section 4.3.
Given the above discontinuity-conceivability link, and the familiar conceivabilitypossibility link, the following principle follows: HD discontinuous : For any collection of properties C 1 and any collection of properties C 2 such that C 2 is discontinuous 29 with C 1 , if it is conceivable that C 1 is instantiated in some pattern P, then it is possible that C 1 is instantiated in P and no member of C 2 is instantiated.
This principle is also a version of HD because it denies that there are necessary connections of a certain sort between distinct properties that meet certain conditions. Again, these necessary connections would be that some patterns of property instantiation are necessarily connected with the instantiation of other, distinct properties.
To deny HD discontinuous , non-naturalists must break either the link between conceivability and possibility, or the link between discontinuity and conceivability. In my view, either move comes at a price. As I already said, there is no space here for an extensive discussion of the conceivability-possibility link. But it is worth considering the discontinuity-conceivability link, particularly because many non-naturalists will be inclined to deny it. The reason is that many non-naturalists will accept that supervenience failures are inconceivable. They hold that, for some (conceivable) patterns of instantiation of non-normative properties, it is inconceivable that there is that pattern of instantiation without there also being some pattern of instantiation of normative properties. But they hold that the normative is discontinuous with the non-normative. They therefore reject the link I have made between discontinuity and conceivability of absence. So where is the pressure to accept this link coming from?
The first thing worth noting is that the opponent of non-naturalism is free to accept the inconceivability of supervenience failures, but without rejecting the discontinuityconceivability link. The reason is that they reject the non-naturalist's discontinuity claim. The non-naturalist's counterexample to the discontinuity-conceivability link is therefore precisely the point of theoretical dispute. It appears dialectically inadmissible in this context as a data-point in evaluating the discontinuity-conceivability link.
But when we look at another area in which philosophers have made a discontinuity claim, it is striking that conceivability seems to pattern in conformance with the discontinuity-conceivability link. This case, which I mentioned before, concerns dualism in the philosophy of mind. Not all dualists need accept the zombie argument; some might, for example, deny the conceivability-possibility link. But what is striking is that, to my knowledge, no dualist has denied that zombies are conceivable. In the philosophy of mind, then, discontinuity does appear to be linked with conceivability of absence. This data provides at least some support for the discontinuity-conceivability link. 30 29 Discontinuity for collections of properties may be defined in terms of discontinuity between individual properties: C 2 is discontinuous with C 1 iff every member of C 2 is discontinuous with every member of C 1 . 30 One could object that this data is irrelevant since dualists about the mental deny the supervenience of the mental on the physical, whereas non-naturalists accept the supervenience of the normative on the non-normative. Yet it is unclear why exactly this renders the data irrelevant. Perhaps the suggestion is that our judgments about the conceivability of the relevant situations is simply being driven by whether we antecedently accept or deny the supervenience thesis. However, it seems to me that exactly the opposite is true. I suspect that what drives dualists to reject supervenience is that they find supervenience failures (e.g. zombies) conceivable whereas what drives non-naturalists to accept supervenience is that they find supervenience failures (e.g. normative zombies) inconceivable.
Let me quickly suggest some other considerations that support the discontinuityconceivability link. On a natural picture, the relevant kind of conceivability that links up with metaphysical possibility is constrained only by the real essences of the things conceived. Now as I noted when introducing non-naturalism, I aim to be non-committal about how non-naturalism should be understood in more theoretically-laden terms, such as the ideology of essence. However, it is plausible that any non-naturalism should uphold at least the following minimal claim about the essences of normative properties (if they are willing to countenance the ideology of essence at all) 31 : Essence Condition: Some normative properties have essences that do not entail purely non-normative sufficient conditions for their instantiation.
If Essence Condition is false, then all normative properties have essences that specify, in purely non-normative terms, sufficient conditions for their instantion. That would be a paradigm form of naturalism (or supernaturalism). Now consider one normative property, N, that satisfies Essence Condition. Take any arbitrary pattern of instantiation P of non-normative properties. We now ask: is it conceivable that (P & N is uninstantiated)? On the picture we are exploring, the answer should be 'yes' unless (P & N is uninstantiated) is ruled out by the essence of some entity involved in the conception. But the essences of the properties involved in P are silent on whether N is instantiated, since they are non-normative properties whose essences don't say anything normative. And by assumption, the essence of N does not say that P is a sufficient condition for N's instantiation. It therefore appears that the essences of the properties involved simply do not constrain the instantiation of N, and therefore (P & N is uninstantiated) should be conceivable. 32,33 31 For example, Essence Condition is entailed by Leary's (2021) essence-based definition of nonnaturalism. See condition (iii) of her definition of "essentially non-normative" properties. Note, however, that Corradini (2018) claims to be a non-naturalist view that denies Essence Condition. See Sects. 3.1.1-3.1.2 within for her explanation of why her view is still non-naturalist despite denying principles like Essence Condition. 32 One other possibility is that (P & N is uninstantiated) is ruled out not by the individual essence of N or the individual essences of any of properties in P, but rather by the "collective essence" of all of these properties taken together. Collective essences are said to be irreducibly collective, meaning that something can lie in the collective essence of a group of individuals without lying in the essence of any of those individuals (Kappes, 2020, p. 804). While this proposal is, therefore, not formally ruled out by anything said so far, it still seems implausible to me. For even if collective essences can include things not included in any of the individual essences, it appears that the collective essence is always in some way derivable from the individual essences. Take the classic example of the collective essence of Socrates and The Eiffel Tower (Kappes, 2020, p. 804). Supposedly, it lies in the collective essence of these taken together that Socrates is not identical to The Eiffel Tower, even though this fact does not lie in either of their individual essences. But this fact about the collective essence is derivable from facts about the individual essences, namely that the individual essences are distinct and that therefore what they are essences of must be distinct. However in our case, the alleged fact that would belong to the collective essence is not at all derivable from the individual essences, since by assumption the individual essences do not put any constraints on the instantiation of N, given P. I am therefore quite skeptical of this proposal involving collective essences. 33 The attentive reader will note that this doesn't by itself get us all the way to the discontinuityconceivability link. Rather it gets us such a link only for those normative properties that are discontinuous with non-normative properties in the sense of satisfying Essence Condition. But there is only a little gap here -one gets fully to a discontinuity-conceivability link (at least for the discontinuity relative to the It also seems to me that the notion of discontinuity, and in turn the just-too-different intuition that it is meant to capture, become opaque if the discontinuity-conceivability link is denied. If this pre-theoretical intuition is to matter at all for metaethical theorizing, it should have some theoretical implications. Yet the non-naturalist already denies that a discontinuity between two kinds of properties implies anything substantive about the modal relationship between them. If the discontinuity-conceivability link is severed, then a discontinuity fails to even imply anything about how the discontinuous kinds of properties may conceivably pattern. At this point one may begin to wonder whether the just-too-different intuition really says anything important at all. The denial of the discontinuity-conceivability link therefore burdens the non-naturalist to say more about what theoretical commitments follow from their views.
I hope to have said enough above to indicate why we might find the discontinuityconceivability link independently attractive, and why even the non-naturalist should hesitate to deny it. My remarks are undoubtedly somewhat speculative and deserve further consideration elsewhere, but my purpose in this section is again merely to illustrate how HD might be motivated on non-explanatory grounds, not to make these motivations airtight.
The argument for HD from conceivability presents another motivation for HD that is independent of the idea that violations of HD would be metaphysically inexplicable. This motivation rests instead on the overall plausibility of a conceivability-based approach to modal epistemology.
I conclude that there are at least two non-explanatory, big-picture considerations which can be used to derive limited versions of HD. My purpose here has not been to thoroughly defend these background considerations, but to show how we might come to versions of HD quite independently of issues of metaphysical explanation, thus challenging the explanatory picture.

The benefits of abandoning the explanatory picture
We've seen so far that prominent naturalists seeking to weaponize supervenience against non-naturalism have viewed HD as being motivated solely by explanatory considerations, a view I've called 'the explanatory picture'. This naturally gives rise to the view that the HD argument has no independent purchase from the explanatory challenge, since an otherwise reasonable explanation of a violation of HD would defeat the explanatory motivation for HD, leaving HD unmotivated. This result is problematic, since it puts a great deal of weight upon the explanatory challenge. Let me explain four related reasons for why there are advantages to not relying so heavily upon the explanatory challenge: Footnote 33 continued normative and non-normative) on the assumption that all the normative properties that are discontinuous with the non-normative satisfy Essence Condition. It is hard to make sense of a discontinuous normative property that doesn't satisfy Essence Condition; that would amount to saying that some normative properties are different in kind from non-normative properties even though their very essence specifies purely non-normative sufficient conditions for their instantiation. Anyway, even if this did make sense, we get enough of a discontinuity-conceivability link to threaten non-naturalism, provided that Essence Condition identifies a necessary consequence of non-naturalism.
There is currently no principled reason to think that the explanatory challenge cannot be met. 34 We can distinguish two versions of the explanatory argument, a weak version and a strong version. The weak version merely argues that the nonnaturalist does not in fact have a satisfactory explanation of supervenience, noting that this is a cost relative to naturalism. The strong version claims that the non-naturalist cannot explain supervenience at all. To advance the strong version of the explanatory argument, the naturalist would require some principled reason to think that the explanatory challenge to non-naturalism cannot be met. A natural line here would be to advance Schroeder's conjecture, on which all necessities are to be explained by the non-distinctness of the entities involved. But this thesis is currently just a conjecture, motivated primarily by looking at some paradigm examples of explained necessities. 35 Without a persuasive principled reason, naturalists must fall back on the weak version of the explanatory argument. But the dialectical force of this argument is limited, since it gives non-naturalists no reason to think that they won't eventually meet the explanatory challenge.
If HD is false, there cannot be such a principled reason to think that the explanatory challenge can't be met. Since the principled reason upon which the strong version of the explanatory argument rests must be motivated by explanatory principles like Schroeder's conjecture, the falsity of HD would doom the strong version of the explanatory argument. For there is a quick argument from the falsity of HD to the failure of explanatory principles like Schroeder's conjecture. Suppose that HD is false, so that some distinct entities are necessarily connected. Either these necessary connections have an explanation or not. If they do not, then Schoeder's conjecture is false because not all necessities are explained by non-distinctness. If they do have an explanation, then Schroeder's conjecture is false because, by assumption, some necessities are explained by something other than non-distinctness. Either way, if HD is false, then Schroeder's conjecture is false. Now if Schroeder's conjecture is false, then either some necessities are totally unexplained or some necessities are explained not by the non-distinctness of the entities involved. Neither result looks good for the strong version of the explanatory argument. So if we are to adopt the version of the explanatory argument which has strong dialectical force, we may already be committed to HD, and thus have some reason to move beyond the explanatory challenge anyway.
Without a principled reason to think that the explanatory challenge can't be met, the explanatory argument is a game of whack-a-mole. Without a principled reason, the explanatory argument must be run in its weak form. But, as I've said, the weak version gives non-naturalists no reason to think they won't eventually crack the explanatory challenge. This means that non-naturalists will interminably advance novel solutions to explain supervenience, since they have no reason, save perhaps inductive ones, to suspect their next attempt will fail. Proponents of the explanatory argument are left to play a game of whack-a-mole, trying to show that each succeeding version of non-naturalism fails to meet the explanatory challenge for different reasons. By contrast, moving beyond the explanatory picture and seeing HD as an independently plausible metaphysical principle gets us out of this game of whack-a-mole. In fact, HD itself could serve as the principled reason which validates the strong version of the explanatory argument.
Non-naturalists are making this game of whack-a-mole increasingly difficult. Nonnaturalist solutions to the explanatory challenge are rapidly increasing in number and sophistication. At least three broad kinds of non-naturalism have emerged in recent years in response to the explanatory challenge, each with different metaphysical commitments. The first kind appeals to fundamental pure normative laws which specify connections between non-normative and normative properties. 36 Another kind appeals to primitive reason-facts to connect the non-normative facts with the normative facts. 37 And the third kind posits hybrid properties whose essences specify both naturalistic sufficient conditions for their own instantiation and sufficient conditions for the instantiation of other pure normative properties. 38 Though debate over the success of these proposals is ongoing, 39 this indicates that it is unclear whether the explanatory challenge has in fact been met by non-naturalists. All the more reason to question the explanatory picture of HD, which would pin the naturalist's entire case on the explanatory challenge.
In Sect. 2, I've tried to show that a major reason for the neglect of the HD argument is the common adoption of the explanatory picture of HD. I've pushed back against the explanatory picture and argued that the explanatory picture distorts the dialectic between non-naturalists and their opponents in important ways. Moreover, abandoning the explanatory picture, which leaves the explanatory challenge as the only game in town, has several advantages for the opponent of non-naturalism. 40

HD arguments: an illustration
In the previous section I argued that the HD argument can be independent of the explanatory argument, and moreover that the HD argument enjoys some advantages over the explanatory argument. Having said that, it behooves me to show that the HD argument really is an argument. That is, I intend to show how HD arguments might work by showing that the versions of HD I derived in Sect. 2.2 can be employed in arguments from supervenience against non-naturalism. These arguments rely only on 36 Enoch (2011) and Scanlon (2014). 37 Moberger (2022). 38 Leary (2017). 39 For some references, see footnote 14. 40 To clarify, I argue for abandoning the explanatory picture, not necessarily the explanatory challenge or argument. While I have argued that the HD argument enjoys several advantages in comparison with the explanatory argument, this is compatible with thinking that the explanatory argument has other advantages over the HD argument. Moreover, the advantages I have described in this section do not settle the question of which argument is better overall. (I am actually skeptical of the idea that there is a fact of the matter about which is best overall, independently of all background beliefs, assumptions, etc.). a widely accepted thesis called 'Global Supervenience'. Since Global Supervenience does not by itself entail intensional equivalence, this premise is weaker than what is needed to run certain other forms of the supervenience challenge, including the direct argument. 41 Global supervenience concerns the modal relationship between world-wide patterns of instantiation of different collections of properties. Global supervenience can be formulated as follows: Global Supervenience: A-properties globally supervene on B-properties if and only if for any worlds W 1 and W 2 , if W 1 and W 2 have exactly the same world-wide pattern of instantiation of B-properties, then they have exactly the same world-wide pattern of instantiation of A-properties. 42 I will argue that the global supervenience of the normative properties on the nonnormative properties, together with either HD fundamental or HD discontinuous , rules out non-naturalism.
It is easiest to begin with HD fundamental . Recall that this principle says that any pattern of instantiation of some fundamental properties is possible without the instantiation of other distinct, fundamental properties. Now, if the normative is globally supervenient upon the non-normative, it follows that the total pattern of instantiation of non-normative properties at any world is necessarily accompanied by some total pattern of instantiation of normative properties. So consider some world W where some normative properties are instantiated. Call the total pattern of non-normative properties at that world P n . It follows that, necessarily, if P n , then some normative properties are instantiated.
Since the total pattern of non-normative properties globally supervenes on the total pattern of fundamental non-normative properties, there is some pattern of fundamental non-normative properties P fn , namely the one instantiated at W, such that necessarily, if P fn , then P n . Transitively, therefore, necessarily if P fn , then some normative properties are instantiated. Now for the non-naturalist, if some normative properties are instantiated, then some normative fundamental properties are instantiated. This conditional fails only if (1) all normative properties are metaphysically reducible to non-normative properties; or (2) all normative properties are metaphysically reducible to other normative properties. 43 (1) can be set aside because it contradicts non-naturalism. (2) comes in 41 For example, Jackson's (1998) direct argument relies on both Global Supervenience plus a local supervenience thesis, in addition to other substantive assumptions, c.f. Brown (2011, Sect. 2). (Note that Brown's (2011) version of the direct argument makes due only using Global Supervenience, but it does so by invoking an even more coarse-grained account of property individuation; see Brown (2011, pp. 212-213).) Some formulations of the explanatory argument appeal to a supervenience thesis called 'Strong Supervenience', e.g. Dreier (2019). Strong Supervenience entails Global Supervenience, but the reverse is true only under specific assumptions about the subvening and supervening properties, c.f. Bennett and Mclaughlin (2018, pp. 4.3.2-4.3.5) for discussion. 42 Bennett and McLaughlin (2018). I have changed their formulation only by exchanging 'pattern of instantiation' for 'pattern of distribution', for terminological consistency. 43 I make the simplifying assumption that all non-fundamental properties are in some way metaphysically reducible to fundamental properties. This assumption might be thought illegitimate in this context, since two distinct flavors; one possibility is that there is an infinite chain of increasingly fundamental normative properties that never terminates, and the other possibility is that there is a circle of reducibility relations among normative properties with none being irreducible and fundamental. However, I dismiss both of these flavors on several grounds. First, both proposals violate plausible, orthodox principles about the logic of metaphysical reduction and dependence. 44 Infinite non-terminating chains violate the 'well-foundedness' of metaphysical dependence, 45 and circles violate either the transitivity or the irreflexivity, asymmetry, and well-foundedness of metaphysical dependence. 46 Second, I know of no plausible examples of normative ontologies involving either infinite chains or circles of normative properties. 47 There are certainly different views about which normative properties are rock bottom (goodness? rightness? counting in favor of?…), but I have seen no substantive examples of views on which there is no rock bottom. Finally, I find the infinite chain suggestion especially unbecoming, as it strains the imagination to think that there might be a hitherto unknown, unconceptualized infinity of normative properties each derived from still others.
We are thus led to the conclusion that necessarily, if P fn , then some fundamental normative properties are instantiated. If we now add the characteristic non-naturalist assumption that normative properties are distinct from all non-normative ones, we get a violation of HD fundamental . For HD fundamental says that any pattern of instantiation of Footnote 43 continued one prominent version of non-naturalism, 'Bridge Law Non-Naturalism', claims that the fundamental normative entities are laws or principles, and that normative properties are grounded in non-normative properties together with these laws (Leary, 2021, p. 799). Personally, I have grave doubts about treating laws as more fundamental than the properties that figure in those laws. However, the more important point is that Bridge Law Non-Naturalism complicates the picture I am developing but does not overturn it.Humean theories of possibility allow for free recombination among fundamental entities, whatever type of entities those may be. I have focused on property recombination because I think it is a simple and natural setting in which to develop the recombinatorial motivation for HD. But if the non-naturalist insists that the fundamental entities include normative laws but not normative properties, we will simply have to employ recombination principles that apply to laws rather than properties. Gibbs (2019, pp 181-183) gives one natural implementation of this idea that appeals to Ramsified lawbooks. These recombination principles for laws may be motivated on the same grounds as the recombination principles for properties, namely that they provide the most theoretically virtuous account of the possibilities for the entities they concern. 44 The logic of metaphysical dependence tends to get discussed under one particular conception of metaphysical dependence, namely the grounding conception. Each of the principles below are considered standard, "orthodox" principles of the logic of grounding (Rabin, 2018, p. 38). Of course, however, for each of these principles, there are dissenters who claim that they admit of counterexamples. See the collection Bliss and Priest (2018) for discussion of each of these principles and some examples of dissent. 45 Roughly speaking, metaphysical dependence is well-founded just in case any dependence chain includes an element that is not dependent. 46 If dependence is transitive, then a circle of dependence would mean that each element is dependent upon itself, which violates irreflexivity. Also, each element would be dependence-related in both directions to each other element, violating asymmetry. In addition, it would mean that no element in the chain is not dependent, which violates well-foundedness. Thus, given circles, dependence either fails to be transitive, or fails to be irreflexive, asymmetric, and well-founded. 47 Roberts (2017) does argue that it is at least conceptually possible that the grounding chains of normative properties do not ultimately terminate in non-normative properties (or in other normative entities, like normative laws). And she rejects the view that normative properties could be ungrounded. This suggests that she must accept either infinite chains or circles. But Roberts does not supply any actual examples of a normative ontology that would instantiate these formal structures. That is the point I am making to cast doubt on their real possibility. some fundamental properties may possibly occur without the instantiation of other, distinct, fundamental properties. We may summarize this argument from HD fundamental as follows: 1F. If the normative is globally supervenient on the non-normative, then some patterns of instantiation of fundamental, non-normative properties are necessarily connected with the instantiation of some fundamental normative properties. 48 2F. If non-naturalism is true, these normative fundamental properties are distinct from all non-normative properties. 3F. (By 1F and 2F) If the normative is globally supervenient on the non-normative and non-naturalism is true, then some patterns of instantiation of fundamental, non-normative properties are necessarily connected with the instantiation of some distinct, fundamental normative properties. 4F. By HD fundamental , there are no such necessary connections. 5F. The normative is globally supervenient on the non-normative. 6F. (By 3F, 4F, and 5F) Therefore, non-naturalism is false. 49 I next consider how an argument from HD discontinuous and Global Supervenience against non-naturalism might go. Recall that HD discontinuous says that if some pattern of instantiation is conceivable, then it is possible at a world without the instantiation of any properties that are metaphysically discontinuous with each of the properties invoked in the pattern. Again, if the normative is globally supervenient upon the non-normative, it follows that the total pattern of instantiation of non-normative properties at any world is necessarily accompanied by some total pattern of instantiation of normative properties. For this argument we must consider a world such that some normative properties are instantiated at it and the total pattern of non-normative properties at it is conceivable. Surely there is such a world, even if it must have a fairly simply total pattern of non-normative property instantiation. Now let us call the total pattern of non-normative properties at this world, P cn . By global supervenience, it follows that necessarily, if P cn , then some normative properties are instantiated. By the characteristic claim of non-naturalism, each of these normative properties is discontinuous with each of the non-normative properties invoked in P cn . But since P cn is by assumption conceivable, by HD discontinuous , it follows that P cn is possible without the instantiation of any collection of properties discontinuous with those invoked in P cn . Therefore, non-naturalists are committed to a violation of HD discontinuous if they accept Global Supervenience. We may summarize this argument as follows: 48 Strictly speaking, this conditional is true only if there are some possibly instantiated normative properties. The non-naturalist obviously agrees with this assumption, so I have omitted it from the antecedent. An error theorist about normative properties would likely deny this assumption however, while holding that the normative trivially supervenes on the non-normative. The denial of this implicit assumption shows how the error theory is not the target of the HD argument. In what follows, I continue to implicitly assume that there are some possibly instantiated normative properties. 49 A similar argument from HD fundamental may rule out non-reductive naturalism as well. Assuming that the non-reductive naturalist takes normative properties to be fundamental, as they are not reducible to other natural properties, these fundamental properties will be necessarily connected with the instantiation of other fundamental natural properties, which violates HD fundamental . This result is not surprising, since non-reductive naturalism is structurally analogous to non-naturalism, it just rejects the claim that normative properties differ in kind from all non-normative properties.
1D. If the normative is globally supervenient on the non-normative, then some patterns of instantiation of non-normative properties are necessarily connected with the instantiation of some normative properties. 2D. Some such patterns of instantiation of non-normative properties are conceivable. 3D. If non-naturalism is true, then these normative properties are discontinuous with all non-normative properties. 4D. (By 1D, 2D, and 3D) If the normative is globally supervenient on the nonnormative and non-naturalism is true, then some conceivable patterns of instantiation of non-normative properties are necessarily connected with the instantiation of some discontinuous normative properties. 5D. By HD discontinuous , there are no such necessary connections. 6D. The normative is globally supervenient on the non-normative. 7D. (By 4D, 5D, 6D) Non-naturalism is false. I hope to have illustrated how relatively modest versions of HD, which can be motivated on broad metaphysical grounds independent of explanation, may be employed in arguments from supervenience against non-naturalism.

Miscellaneous objections considered
I said at the outset that my goal was to rescue the HD argument from its premature demise primarily by removing some barriers to its acceptance. I have already removed what I take to be the most operative barrier in Sect. 2, namely the explanatory picture on which the HD argument looks like an inferior form of the explanatory argument. For the remainder of the paper I consider a grab-bag of other objections to the HD argument.

The HD argument is not interestingly distinct from the direct argument
Recall that the direct argument attempts to establish, via supervenience, the intensional equivalence of normative properties with some non-normative ones, and then invokes the intensional account of property individuation, according to which intensionally equivalent properties are identical. Some authors seem to have thought that HD either entails, or just is, the intensional account of property individuation. 50 This could be problematic for the defender of the HD argument, since if HD commits us to the intensional account, then the HD argument may not be interestingly distinct from the direct argument. Or at least we would owe some account of why the former is preferable to the latter.
To sharpen this objection, I note that there is a very quick argument from the most generic versions of HD to the intensional account of property individuation. Suppose that 'necessary connections' include connections of necessary co-extensiveness. Then if there are no necessary connections between distinct properties, any necessarily coextensive properties are not distinct. Therefore, the intensional account follows from HD.
The response to this objection is that the HD argument may be run from considerably more modest and precise versions of HD, such as HD fundamental and HD discontinuous . Neither of these principles supports the argument above. HD fundamental , since it is limited to fundamental properties, is compatible with different individuation conditions for non-fundamental properties, and HD discontinuous invokes a notion of distinctness that is more demanding than mere non-identity. Therefore, the proponent of an HD argument is not thereby committed to the intensional account. These modest versions of HD carve out a distinct logical space from the direct account from which to develop interesting versions of the supervenience challenge.

Easy counterexamples to HD
There are a variety of quick counterexamples to HD which are often trotted out. This gives me occasion to respond to McPherson's other serious objection, as I had promised. The following passage from McPherson is along these lines: One might worry about our ability to type entities as distinct in a way that makes the dictum true but interesting. After all, a variety of seemingly non-identical properties are necessarily connected. For example, a surface's being scarlet is not identical to its being red, and yet seems to necessitate it. 51 The nice part for me about responding to this objection is that McPherson has already developed one powerful tool in responding to it, namely the criterion of discontinuity. Scarlet and red are paradigm examples of properties that are not metaphysically discontinuous. More generally, it is plausible that determinates and their determinables are metaphysically continuous. Therefore, though these kinds of examples may threaten very strong, generic versions of HD, they do not threaten HD discontinuous . Contra what McPherson seems to assume, there is nothing to prevent the proponent of the HD argument from adopting this criterion as a way of typing entities as distinct. As long as we do not restrict our version of HD to unexplained necessities as McPherson does, our argument is importantly distinct from the explanatory argument.
On the other hand, you may think the determinate/determinable case is more worrying for HD fundamental . Though scarlet and red are certainly not fundamental properties, why couldn't some fundamental properties exhibit this same determinate/determinable structure? If this were the case, then there would seem to be necessary connections between distinct, fundamental properties of the sort that HD fundamental says do not obtain.
This problem is in fact a very old one for proponents of property recombination principles like PATTERN. 52 I have nothing new to say about this issue here other than that it is a live issue about which much ink has been spilled. 53 One potential solution is a restriction of PATTERN to 'determinably-distinct' fundamental properties. 54 This restriction would yield a version of HD equally suitable to be employed against non-naturalism. Another solution proposes that only determinables and not their determinates are fundamental. 55 A potentially cheap solution is to use a stronger notion of distinctness in HD fundamental , like perhaps discontinuity. This would cleanly get around these counterexamples, but would leave the recombinatorial motivation behind HD fundamental in question. But even if we were forced to abandon one potential motivation for such a principle, it might still be plausible in its own right.

Necessarily instantiated properties
The final objection I will consider concerns necessarily instantiated properties. If there are such properties, such as being self-identical, they pose prima facie problems for various versions of HD.
I begin with a concession. If there are necessarily instantiated properties, then HD discontinuous is, as formulated, probably false. The problem is that if we consider some conceivable pattern of property instantiation that does not include the instantiation of some necessarily instantiated properties, HD discontinuous says that this pattern is possible without the instantiation of any other properties that are discontinuous with those involved in the pattern. But if the necessarily instantiated properties are discontinuous with those involved in the pattern, then this is not really possible. It is not possible that any collection of properties is instantiated in some pattern without being self-identical being instantiated.
That said, there is a simple fix. The fix is to amend HD discontinuous by saying that the conceivable pattern is possible without the instantiation of other, discontinuous properties except any necessarily instantiated ones. This isn't simply an ad hoc amendment. It naturally arises out of considerations about the conceivability principles that we used to motivate HD discontinuous . For in conceiving any pattern of property instantiation over some objects, I thereby also conceive those objects as being self-identical. Quite plausibly, in conceiving of anything I also conceive the instantiation of these necessarily 53 See Denby (2001), Hawthorne (2006), Weatherson (2006), Eddon (2007), Saucedo (2011), Wang (2013), andBricker (2017). 54 For this proposal, see Saucedo (2011, p. 246) and Wang (2013, p. 543). 55 Denby (2001), Hawthorne (2006), Weatherson (2006), and Bricker (2017). exemplified properties. 56 Therefore, the principles I used to derive HD discontinuous naturally should be qualified in this manner anyway. I've only neglected to do this for ease of exposition.
This amendment to HD discontinuous does open a back door for one somewhat eccentric variant of non-naturalism to escape the argument from HD discontinuous . This variant is known as 'Moral Platonism'. 57 It posits that the sui generis normative properties are instantiated not by particular token actions, objects, or events, but instead by either action-kinds (or event-kinds, etc.) or second-order properties. Add the assumption that these kinds or properties are platonic necessary existences which essentially instantiate the normative properties they do, and you get a view which posits necessarily instantiated normative properties. This would enable Moral Platonism to escape conflict with HD discontinuous through the backdoor amendment we just allowed for necessarily instantiated properties. Note that one needs to claim that all normative properties are necessarily instantiated in order to escape the argument from HD discontinuous via this backdoor. It is not enough to say merely that some normative properties are necessarily instantiated. 58 I don't see a reason to open this backdoor in HD fundamental , however. Even if there are necessarily exemplified properties, this is not inconsistent with HD fundamental so long as none of them are fundamental. Insofar as fundamental properties, in the sense invoked by the argument from HD fundamental , are meant to play such roles as explaining objective resemblances, forming a minimal supervenience base for the rest of the properties at a world, and figuring in the fundamental laws, trivially necessarily instantiated properties like being self-identical are surely not fit to serve as fundamental. HD fundamental is therefore only incompatible with the more controversial claims of substantive necessarily instantiated properties, like those that the Moral Platonist makes. HD fundamental is meant precisely to rule out such views. The Moral Platonist will have to contend, then, with the appeal of principles like HD fundamental .

Conclusions
Perhaps some progress has been made in undoing the premature burial of the HD argument against non-naturalism. The modern proponents of the explanatory argument have set aside this form of the supervenience challenge because they have, explicitly or implicitly, suggested that HD can only be motivated by explanatory principles. I argued against this view in Sect. 2, suggesting that it distorts the dialectic concerning non-naturalism in important ways, and also showing that once we are free of this explanatory picture, we can see that the HD argument enjoys some comparative advantages to the explanatory argument. In Sect. 3, I showed that the modest versions of HD which may be motivated on broad grounds independent of explanatory principles still cut against non-naturalism, demonstrating how an argument from Global Supervenience against non-naturalism might employ these principles. Finally, in Sect. 4, I removed various miscellaneous barriers to the acceptance of the HD argument as a promising form of the supervenience challenge. Some of these have actually been raised, others are entirely hypothetical, never having been considered since the HD argument lays nearly dormant. With the air thus cleared, the HD argument might yet survive arriving stillborn. 59 Funding Open access funding provided by SCELC, Statewide California Electronic Library Consortium.

Conflict of interest The author declares that there is no conflict of interest.
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