HUME’S DICTUM AND METAETHICS

This paper explores the metaethical ramiﬁcations of a coarse-grained criterion of property identity, sometimes referred to as Hume’s dictum . According to Hume’s dictum, properties are identical if and only if they are necessarily co-extensive. Assuming the supervenience of the normative on the natural, this criterion threatens the non-naturalist view that there are instantiable normative properties which are distinct from natural properties. In response, non-naturalists typically point to various counterintuitive implications of Hume’s dictum. The paper clariﬁes this strategy and defends it against objections by Bart Streumer and Ralf Bader. In addition, it is argued that proponents of naturalist and supernaturalist views, along with proponents of a certain kind of nihilism, should also reject Hume’s dictum. This shows that non-naturalists can also attack the criterion indirectly, by pointing to partners in guilt. Also, it shows that not just any opponent of non-naturalism can appeal to Hume’s dictum. Only certain nihilists can.


I. INTRODUCTION
In this paper I explore the metaethical ramifications of a coarse-grained criterion of property identity, sometimes referred to as Hume's dictum.According to Hume's dictum, properties are identical if and only if they are necessarily coextensive.As I explain in Section II, this criterion threatens the non-naturalist view that there are instantiable normative properties which are distinct from natural properties.This is because of the widely accepted supervenience of the normative on the natural, which implies (given some further plausible assumptions) that for every normative property, there is a natural property that is necessarily co-extensive with it.
In response, non-naturalists typically attack Hume's dictum head-on, by pointing to various counterintuitive implications.In Section III, I clarify this strategy and defend it against objections made by Bart Streumer in his recent book Unbelievable Errors (2017).I also argue that the strategy, if successful, applies C The Author 2019.Published by Oxford University Press on behalf of The Scots Philosophical Association and the University of St Andrews.This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/),which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
with almost equal force to a variant of Hume's dictum, proposed by Ralf Bader  (2017).
In Section IV, I argue that proponents of naturalist and supernaturalist views, along with proponents of a certain kind of nihilism, should also reject Hume's dictum.This is interesting for two reasons.Firstly, it shows that non-naturalists are not confined to the head-on strategy.They can also attack the criterion indirectly, by pointing to partners in guilt.Secondly, it shows that not just any opponent of non-naturalism can appeal to Hume's dictum.Only certain nihilists can.I end the paper in Section V with a brief conclusion.

II.1. Non-naturalism
Let me begin by clarifying the non-naturalist view threatened by Hume's dictum.Although the term 'non-naturalism' is sometimes used for the semantic thesis that normative sentences ascribe non-natural properties, or for the epistemological thesis that normative knowledge or justification is attainable only through a special kind of intuition, here the view is to be thought of as a purely metaphysical thesis.It is a view about the existence and nature of normative properties, not a view about normative language or justification.I will understand non-naturalism as follows: Non-naturalism: There are instantiable normative properties which are distinct from natural properties.
The term 'normative' can here be taken in a broad sense to include moral, prudential, epistemic and aesthetic properties, and perhaps others as well.It can also be taken to include evaluative as well as deontic properties.
The properties also have to be instantiable, since the view that there are normative properties (whether natural or not) none of which can be instantiated is a form of nihilism (I will return to this point later on).
The distinctness that non-naturalists typically have in mind is stronger than mere numerical distinctness.1But since Hume's dictum threatens even the weaker claim that normative properties are numerically distinct from natural properties, I will understand non-naturalism accordingly.
Finally, how should we understand the notion of a 'natural property' in this context?I assume that the class of natural properties includes both relational properties and (perhaps infinitely) disjunctive/conjunctive properties.Beyond that it is difficult to say anything very precise.One suggestion is that VICTOR MOBERGER we understand natural properties as empirically accessible properties. 2 But this would make non-naturalism too broad, since non-naturalists think of normative properties as distinct also from supernatural properties-such as being created by God-and various 'philosophical' properties-such as having free will or having psychological persistence conditions-which are hardly empirically accessible in any interesting sense.Another proposal is that we understand natural properties as properties of the same kind as empirically accessible properties. 3But this characterization is too porous to be of much use; it secures extensional adequacy only through what we read into it.Yet another suggestion is that natural properties be understood as properties that can be ascribed using only non-normative terms. 4There are worries about extensional adequacy here, however. 5Another worry is that the suggestion relies too heavily on linguistic facts, which do not seem to get to the metaphysical heart of the matter. 6 will not try to settle this issue here; however, I trust that what has been said makes non-naturalism clear enough to sustain the following discussion.7 I now turn to the conflict between non-naturalism and Hume's dictum.

II.2. Hume's dictum and supervenience
Hume's dictum individuates properties by necessary co-extension. 8Crucially, the modality in question is metaphysical.Anything weaker would make the criterion false.This is because it is metaphysically impossible for a property and itself to go separate ways, and so nothing weaker than metaphysically necessary co-extension can be sufficient for identity.(Put another way, identity rules something out that is not ruled out by anything weaker than metaphysically necessary co-extension, namely the metaphysical possibility of divergence.)And anything stronger than metaphysical modality would make the criterion ineffective against non-naturalism.This is because non-naturalists are not committed to normative and natural properties being co-extensive as a matter of conceptual or logical necessity.Firstly, non-naturalism rules out there being logical or analytic connections between specific normative and 2 E.g.Moore (1993: 92) and Shafer-Landau (2003: 59). 3 Enoch (2011: 103). 4E.g.Jackson (1998: 121) and Streumer (2017:  §1).Both use the term 'descriptive property'. 5See Dancy (2006: 126) and Sturgeon (2009: 76-7). 6See Schroeder (2005: 10) and Enoch (2011: 103). 7I discuss the issue at greater length in Moberger (2018: §3.1). 8I follow Olson (2014: §5.1) in using the label 'Hume's dictum' specifically for this view of property identity.Note that the label is sometimes used for a different thesis, namely that there are no metaphysically necessary connections between distinct entities (in some sense of 'distinct').The thesis I have in mind applies only to properties, it rules out only necessary co-extension, and 'distinct' specifically means 'numerically distinct'.See Wilson (2010) for a critical discussion of the other thesis.
natural properties (e.g. between rightness and happiness-maximization).Secondly, while the supervenience of the normative on the natural may be a conceptual truth, 9 this does not suffice for conceptually necessary co-extension.If there are no non-natural normative properties, then supervenience will hold trivially, even though necessary co-extension fails.And the existence of non-natural normative properties is presumably not a conceptual truth.
Hume's dictum should thus be understood as follows: Hume's dictum: Properties are identical if and only if they are co-extensive as a matter of metaphysical necessity.
Since Hume's dictum is formulated in terms of metaphysical necessity, it threatens non-naturalism only on the assumption that normative properties supervene on natural properties with (at least) metaphysical necessity.What is required is thus the following thesis, which I will call simply Supervenience.
Supervenience: It is metaphysically impossible that x and y differ in some normative respect, but not in any natural respect. 10 Supervenience is both plausible and widely accepted, not least among nonnaturalists. 11I will assume it in what follows. 12on-naturalism and Supervenience, conjoined with some further plausible assumptions, imply that there are distinct natural and normative properties which are co-extensive as a matter of metaphysical necessity.This is incompatible with Hume's dictum, and so non-naturalists must reject it. 139 See Olson (2014: 89-90) for an argument to this effect. 10This formulation is a bit simplified.Here is a more precise formulation: For any metaphysically possible worlds w and w * and for any objects x in w and y in w * , if x in w and y in w * are identical in all natural respects (including relational and disjunctive/ conjunctive respects), then x in w and y in w * are identical in all normative respects.
Note that Supervenience can be accepted also by naturalists (including supernaturalists) and nihilists.According to naturalists, normative respects just are natural respects, and so for them the thesis is trivial.According to nihilists, it is metaphysically impossible for normative properties (if such there are) to be instantiated, and so for them the thesis is trivial as well.If nothing can have normative properties, then nothing can differ from anything else in normative respects, which of course includes things that are identical in natural respects.Cf.McPherson (2012:  §2).See McLaughlin and Bennett (2018:  §4) for various other notions of supervenience.

III. DOUBTS ABOUT HUME'S DICTUM
My aim in this section is to clarify the non-naturalist's head-on strategy and the dialectic surrounding it (III.1).I will also defend the strategy against the objections due to Streumer (III.2) and Bader (III.3).

III.1. The head-on strategy
The basic idea of the head-on strategy is to build a cumulative prima facie case against Hume's dictum, by pointing to various examples where the criterion has counterintuitive implications.The underlying assumption is that we have a reasonably firm grip on the notion of a property, which allows us to have discerning intuitions about cases. 14The upshot is that properties are intuitively more fine-grained than Hume's dictum allows.
It has not been noted, however, that the examples pointed to in the literature are of two importantly different kinds. 15First, there are examples where the necessary co-extension of the properties is common ground between proponents of the head-on strategy and adherents of Hume's dictum, but where the identity of the properties appears counterintuitive.Secondly, there are examples where, again, the identity of the properties appears counterintuitive, but where the necessary co-extension of the properties is a matter of contention.This latter kind of example is dialectically less forceful, or so I will argue.Let me begin by considering two examples of this kind: (i) Identity/necessary co-extension Russ Shafer-Landau points out that if Hume's dictum is correct, then 'the relational properties of being necessarily coextensive with and being identical to [. . .]  are necessarily coextensive, and [thus] identical.[. . .] But this seems false.It seems that we are referring to different features when we assert the existence of an identity relation, as opposed to one of necessary coextension.' (2003: 91)  Although Shafer-Landau does not distinguish here between relations and relational properties, we can safely assume that if the relations being necessarily coextensive with and being identical to are distinct, then so are the relational properties of being necessarily co-extensive with x and being identical to x.
(ii) Hume's dictum/the correct dictum Jonas Olson (2014: 94) suggests that being Hume's dictum is intuitively a different property from being the correct dictum about whether there are relations of necessary co-extension between distinct properties.If Hume's dictum is correct, however, these 14 Contrast Lewis (1986: 55-6). 15I am grateful to Erik Carlson for first drawing my attention to this distinction.
properties are necessarily co-extensive (assuming Hume's dictum is not a contingent proposition), and thus identical. 16he problem with these examples is that the step from necessary coextension to identity is not an open question to begin with.Concerning (i), if necessary co-extension and identity are indeed necessarily co-extensive, then it follows that necessary co-extension is both necessary and sufficient for identity.But that is precisely what Hume's dictum says, and so the there is nothing left to debate.Similarly, if being Hume's dictum and being the correct dictum. . .are necessarily co-extensive, then Hume's dictum is the correct dictum.Again, there is nothing left to debate.
Thus, we know beforehand, on independent and conclusive grounds, that the properties pointed to by Shafer-Landau and Olson, respectively, could not be necessarily co-extensive and distinct.What is controversial is whether necessary co-extension obtains in the first place.But this matter cannot be decided independently of the present debate, since whether the respective properties are necessarily co-extensive is the very issue at hand.To deny that they are necessarily co-extensive (and thus identical) is tantamount to simply denying Hume's dictum, and so the examples have limited, if any, dialectical force.
More effective examples have been proposed, however: (iii) Mathematical properties Derek Parfit (2011: 297) suggests that being the only even prime number and being the positive square root of four are distinct properties, despite being necessarily coextensive.Both properties attach uniquely, and with metaphysical necessity, to the number two, and yet they seem to be two distinct aspects of this number.
One property has to do with a particular relation between the number two and the number four.The other property has to do with the unique evenness of the number two in a specific series of numbers.Intuitively these are different ways for the number two to be.(Similar examples could be constructed using other mathematical properties.) (iv) Impossible properties Olson (2014: 93) suggests that at least some properties that necessarily lack extension (and thus are necessarily co-extensive), like being a round square and being an even prime larger than two, are intuitively distinct.Here we can do one 16 Following Olson (2014: 94), we might try to reinforce this example by noting that the properties in question are not equally interesting.Hume's dictum is trivially Hume's dictum, but interestingly the correct dictum . . .(if it is).This argument does not work, however.Both properties are equally interesting/trivial, depending on how we put things.Being Hume's dictum is certainly a trivial property of Hume's dictum, but not of the correct dictum. . . .Likewise, being the correct dictum . . . is an interesting property of Hume's dictum, but not of the correct dictum. . . .The interesting/trivial-distinction thus resides in language and thought, not in the world.
VICTOR MOBERGER better by pointing to contrary properties, such as being a round square and being a triangular square, whose distinctness seems even more obvious.
These examples have more dialectical force, since denying the identity of the respective properties is not tantamount to simply denying Hume's dictum.
Let me give one more example, inspired by example (ii) above: (v) View X/the correct view of subject Y Hume's dictum will imply counterintuitive identity-claims about any domain of inquiry, where the correct answers are plausibly taken to be necessarily true if true at all.For example, whatever the correct view of personal identity turns out to be, being that view (animalism, say) and being the correct view of personal identity will be one and the same property.I believe this example captures the intent behind example (ii).The problem was just that Olson chose to focus on Hume's dictum and not some other philosophical thesis, rendering the example needlessly vulnerable. 17 Proponents of Hume's dictum might respond in three different ways.Firstly, they might try to mitigate the intuitive case, either by denying that the criterion has one or more of the alleged implications, or by denying that these implications are counterintuitive.Secondly, they might try to undercut the intuitive case by debunking the relevant intuitions.Third, they might try to override the intuitive case by providing independent support for Hume's dictum that is strong enough to outweigh the intuitive case.Streumer (2017) does a bit of all three, and I will consider his responses in the following section. 18(In Section III.3I turn to Bader's more indirect response.)
Although I have argued that examples (i) and (ii) have limited dialectical force against Hume's dictum, it is still worth considering Streumer's responses to these examples.This is because his response to example (ii) bears also on example (v), which is more forceful, and because example (i) will become relevant later on (in Section III.3).
In response to example (i), Streumer suggests that the intuitive verdict is irrelevant, since Hume's dictum will not render the predicates necessarily co-extensive in the first place.He writes: [T]hese predicates are not necessarily coextensive, since 'is identical to' applies to properties and 'is necessarily coextensive with' applies to predicates.(2017: 19, n. 25)   This is too quick, however.Firstly, I don't see why we could not, as I have done above, talk of properties being necessarily co-extensive.I take it that properties have extensions, just like predicates and concepts, the extension of a property being the set or class of objects that exemplify it.More importantly, however, if we insist that only predicates can be co-extensive, then we can talk instead of necessary correlation.Shafer-Landau's point is then that if Hume's dictum is correct, the predicates 'is necessarily correlated with' and 'is identical to'-both of which do apply to properties-will be necessarily co-extensive, and so the relations being necessarily correlated with and being identical to will be identical.
In response to example (ii), Streumer again suggests that Hume's dictum does not imply necessary co-extension.He writes: [T]hese predicates are not necessarily coextensive, since the claim Olson calls 'Hume's dictum' could have been someone else's dictum instead: in the (admittedly not very close) possible world in which Derrida rather than Hume made this claim, it is Derrida's dictum.(2017: 19, n. 25) But this misunderstands the example.The predicate 'is Hume's dictum' should not be read as 'is the dictum formulated by David Hume', but rather as 'is the thesis Hume's dictum' (or 'is the thesis that all and only necessarily co-extensive properties are identical'). 19Olson's point is thus that if Hume's dictum is correct, there is a unique proposition which necessarily has the properties of being the thesis Hume's dictum and being the correct dictum about whether there are relations of necessary co-extension (or correlation) between distinct properties.(As I mentioned above, Streumer's response bears also on example (v), focusing perhaps on the properties of being Goldbach's conjecture and being the correct conjecture about whether every even integer greater than 2 can be expressed as the sum of two primes).
In response to Parfit's mathematical example (example (iii) above), Streumer first asks us to consider the sentence 'Two is the positive square root of four'.He goes on to say that this sentence 'does not seem to ascribe a property to VICTOR MOBERGER the number two.Instead, it seems to say that the number two is identical to the positive square root of four.' (2017: 16)  Thus, what may look like an ascription of a property is in fact a statement of identity.Streumer then points out that Parfit himself 'admits that the phrases 'the positive square root of four' and 'the only even prime number' both refer to the number two.This makes it hard to see how he can deny that the predicates 'is the positive square root of four' and 'is the only even prime number' ascribe a single property: the property of being the number two.' (2017: 17)  So the argument seems to be this: Since the definite descriptions 'the positive square root of four' and 'the only even prime number' both refer to a single object-the number two-the predicates 'is the positive square root of four' and 'is the only even prime number' both ascribe a single property-being the number two-and not two distinct properties as Parfit claims.
In response, note first that it is not generally the case that when two definite descriptions both refer to a single object, then the predicates generated by adding 'is' to those definite descriptions will ascribe a single property, namely the property of being that object.For example, the definite descriptions 'the person who wrote Convention' and 'the person who wrote Counterfactuals' both refer to David Lewis, but that does not suggest that the predicates 'is the person who wrote Convention' and 'is the person who wrote Counterfactuals' ascribe a single property, namely being David Lewis.Instead they ascribe two distinct properties.Thus, the import of Streumer's point about the reference of the definite descriptions is unclear.
In any case, Streumer's response can be circumvented by slightly modifying the example, switching from the predicates 'is the positive square root of four' and 'is the only even prime number', to the predicates 'is a positive square root of four' and 'is an even prime number'. 20Since these predicates are not built from definite descriptions in the first place, in this case there is no temptation to think that the predicates ascribe the single property of being the object to which the definite descriptions refer.
In response to Olson's impossible-properties example (example (iv) above), Streumer argues that the example is not available to non-naturalists in the first place.Drawing on Jackson's (1998) discussion, Streumer (2017: 12-3) maintains that the debate between non-naturalists and naturalists is premised on a certain conception of properties, on which properties are ways objects can be.On this conception, 'what it is for an object to have a certain property is that this object itself is a certain way ' (2017: 12).On an alternative conception, properties are merely shadows of concepts.On this conception, 'what it is for an object to have a certain property is that this object falls under a certain concept' (2017: 12).Assuming, then, that properties are ways objects can be, Streumer suggests that there are no impossible properties.He writes: [I]f properties are ways objects can be, the predicates 'is a round square' and 'is an even [prime] number larger than two' ascribe non-existent properties.Do these predicates ascribe a single non-existent property or different non-existent properties?If properties are ways objects can be, this question does not make sense.It only makes sense if properties are shadows of concepts.(2017: 17-8) Thus, Olson's example is intelligible only if we assume a conception of properties which is not on the cards to begin with.
I will not dispute the claim that non-naturalists must assume that properties are ways objects can be.And I agree, of course, that it makes no sense to ask whether two predicates ascribe the same or different non-existent properties.What I don't see, however, is why the ways-objects-can-be conception would imply that the properties in question are non-existent rather than just uninstantiable.Perhaps Streumer takes this to follow from the 'can' in 'ways objects can be'.But such a reading of 'can' would undermine the claim that non-naturalists are committed to the ways-objects-can-be conception in the first place.Nothing, or at least nothing obvious, prevents non-naturalists from adopting a broadly speaking Platonic account of properties, which does allow for the existence of uninstantiable properties.For example, they might follow Peter van Inwagen (2004) in construing properties as abstract, proposition-like entities ('unsaturated assertibles'), which, unlike propositions (or 'saturated assertibles'), cannot be asserted (or be true or false) simpliciter, but which can be asserted of (or be true or false of) things.On this account, there is such a property as being a round square, since that it is round and square is something that one can assert (falsely) of something.21This is not the place to decide whether van Inwagen's account of properties is right, of course.The point is just that the ways-objects-can-be conception of properties does not by itself, as Streumer suggests, rule out Olson's impossible-properties example.

III.2.2. Undercutting
Another possibility is that the intuitions elicited by the above examples are due to an implicit conflation of properties with concepts.Streumer writes: [I]f properties are ways objects can be, (N) is the correct criterion of property identity.[. . .] Why does not everyone agree with this conclusion?One reason for this may be the hold that a mistaken idea has on us: the idea that an object's name somehow reflects its essence.(2017: 24) In my experience, most [non-naturalists] remain unmoved [. . .].What explains this?One explanation may be that they tacitly conflate properties with concepts.(2017: 40) 22 Thus, when we intuitively judge for example that being a positive square root of four and being an even prime number are distinct properties, perhaps we are really tracking a difference between the related concepts.If so, the intuitions can be explained away as unreliable.
Although this explanation has the right ingredients to be debunking-it would explain the occurrence of the relevant intuitions without assuming that their content is true-it remains to be shown that the explanation is credible.It may be that properties are sometimes more coarse-grained than our conceptual lenses make them appear, but this cannot just be assumed whenever it is convenient.Unless something is said about why, in specific cases, our concepts would be more fine-grained than the properties, we have been given no independent reason to distrust our intuitions about the properties.
It is useful here to consider another influential debunking strategy for comparison.According to the error theory about moral discourse, moral judgments qua moral judgments ascribe non-existent (or at least uninstantiable) properties, rendering all moral judgments false. 23An important objection to the error theory points to our first-order moral intuitions, for example the intuition that it is wrong to set a cat on fire for fun, and argues in Moorean fashion that it is far-fetched to suppose that a controversial philosophical position such as the error theory could get enough leverage to overturn them. 24rror theorists typically respond by giving a projectivist account of our moral intuitions: When we intuit that setting a cat on fire for fun is wrong, we are really just projecting an emotional response onto the situation.This projection gives rise to the sense that actions of the type in question instantiate not-to-bedoneness.This account is then backed up by an evolutionary story of why such projection was useful to our ancestors in various ways. 25ut suppose error theorists were to simply omit this last part, merely asserting that our moral intuitions are due to emotional projection; or, to use Streumer's (2017: 40) words, that '[o]ne explanation may be that' our moral intuitions are due to emotional projection.Clearly, that would seriously compromise the force of the strategy.It is precisely in virtue of the independent evolutionary evidence (assuming it stands up) in favour of the projectivist account that the error theorist's debunking strategy threatens to undercut the Moorean objection.Analogously, what is needed in the present case is some sort of independent evidence to the effect that we would have a fine-grained conception of the relevant properties independently of the truth of the matter.

III.2.3. Overriding
A further possibility is that the intuitive case can be overridden by compelling independent considerations in favour of Hume's dictum.One potential source of support (which Streumer does not appeal to) is intuitive. 26Assuming that properties are distinct, why should it not be possible to pull them apart, as it were?What kind of queer metaphysical glue might be holding them together?
This question may seem pressing, but its relevance is doubtful.Consider the properties of being a circle and having a diameter.These properties are distinct, and although they are not necessarily co-extensive (other objects also have diameters), there is at least one-way necessitation: It is metaphysically necessary that any circle has a diameter.But since the properties are distinct, why shouldn't it be possible to have circles without diameters?What kind of queer metaphysical glue is holding diameters within circles?Although there may be a mystery here, it is not one that should make us doubt that one-way necessitation between distinct properties is possible.And once we realize this, it is not clear why we should doubt that two-way necessitation between distinct properties is also possible. 27nother potential source of support is theoretical.It might be argued that Hume's dictum is superior to even the most defensible criterion of property identity available to non-naturalists in terms of simplicity, elegance and ontological parsimony.This is difficult to tell at this point, since it is not obvious which criterion non-naturalists should adopt.However, in the remainder of this section, I will discuss a candidate criterion of property identity that is congenial with non-naturalism (even though it may not be congenial with the views of some non-naturalists).The criterion is also very simple and elegant.And although it is in one respect less parsimonious than Hume's dictum, it is in another respect more parsimonious.
If we adopt van Inwagen's above-mentioned account of properties, then it is not obvious that we should maintain a distinction between properties and concepts.If properties are abstract, proposition-like entities which can be asserted of (or be true or false of) things, then they seem well suited to fill the role of concepts as well.Let us call the resulting picture-i.e. the conjunction of van Inwagen's account of properties and the identification of properties and concepts-the Inwagian picture.Note that the term 'concept' should not here be understood as 'idea', 'representation', or the like.Concepts/properties on the Inwagian picture are not things in our heads, but rather things that, like propositions, form the content of things in our heads.
Assuming the Inwagian picture, we get the following criterion of property identity: The conceptual criterion: Properties are identical if and only if they cannot be conceptually distinguished.
I am not claiming that non-naturalists should adopt the conceptual criterion, but it is at least a possibility proof concerning simplicity, elegance and even parsimony.Admittedly, the criterion does lead to a more fine-grained individuation of properties than Hume's dictum, and, other things being equal, this implies a more expansive ontology.But other things are not equal.By identifying properties and concepts, the conceptual criterion in one respect leads to a more parsimonious ontology. 28ne might worry, however, that the conceptual criterion leads to a reckless multiplication of properties.As Streumer puts it: If the predicates 'is a closed figure that has three sides' and 'is a closed figure that has three angles' ascribed two different properties, why would the predicate 'is a triangle' not ascribe a third property?And [. . .] why would the predicate 'is a closed figure with six half-sides and six half-angles' not ascribe a fourth property?If properties are ways objects can be, this multiplication of properties has to stop somewhere.(2017: 14) 29 We should all agree that the multiplication of properties has to stop somewhere.But it is not as if the conceptual criterion gives carte blanche to any multiplication of properties.What is required is a genuine conceptual distinction, and it is not clear that all four of Streumer's examples are conceptually distinct.For example, the predicates 'is a triangle' and 'is a closed figure that has three angles' do not appear conceptually distinct. 30ore importantly, however, since properties on the Inwagian picture are abstract objects, it is not clear that the number of properties is an important factor from the point of view of parsimony.If there are abstract objects at all, then presumably integers will be among them.And since the number of integers is infinite, there will in any case be infinitely many abstract objects.
But perhaps this just goes to show that the real problem from the point of view of parsimony is countenancing abstract objects in the first place, at least if they are taken to be mind-independent, immutable entities.Note, however, that even if this consideration does favour Hume's dictum over the conceptual criterion in terms of overall parsimony, it should still be comforting for nonnaturalists to have this issue tied to the more general metaphysical debate over the existence and nature of abstract objects.
Before I move on, let me briefly address two further worries about the conceptual criterion.Firstly, I agreed above that non-naturalists are committed to the ways-objects-can-be conception of properties, but Streumer (2017: 11-2)  suggests that the conceptual criterion presupposes that properties are 'shadows of concepts'. 31If this is right, then non-naturalist cannot accept the conceptual criterion.I don't think it is right, however.On Streumer's explication of the ways-objects-can-be conception, an object having a property is a matter of the 'object itself [being] a certain way' (2017: 12).I don't see why the conceptual criterion would rule this out.On the Inwagian picture, an object having a property is a matter of the object standing in a relation of instantiation (or exemplification) to a certain abstract object, and I don't know why that wouldn't qualify as the object itself being a certain way.(Perhaps the right thing to say is that the Inwagian picture collapses the distinction between the two conceptions.) Finally, one might object that there are forceful counterexamples to the conceptual criterion.For example, haven't we learned that being water is the same property as being H 2 O, despite a conceptual difference?32No, we have not.As Olson (2016: 468) points out, it is not clear why being water and being H 2 O could not instead be thought of as distinct properties which are necessarily and uniquely instantiated by one and the same stuff, namely water (or H 2 O).(One possibility is that being water is the rigidified de dicto property of being the stuff, whatever it is, that in the actual world is causally responsible for our 'watery experience'.)33

III.3. Bader's dictum
Even if the non-naturalist head-on strategy should successfully refute Hume's dictum, non-naturalists may still not be in the clear.This is because there may be other criteria of property identity which also rule out non-naturalism, but which are not (equally) vulnerable to the intuitive counterexamples. 34ader (2017) proposes a variant of Hume's dictum which individuates properties not by necessary co-extension, but by necessary co-grounding: Bader's dictum: Properties are identical if and only if they are co-extensive as a matter of metaphysical necessity and have the same grounds. 35e phrase 'same grounds' should here be read as 'same full grounds'.Thus, the criterion does not say that being necessarily co-extensive and having the same partial grounds is sufficient for identity (Bader 2017: 117).
By thus adding a conjunct to the sufficient condition for property identity, Bader's dictum is in this respect rendered logically weaker than Hume's dictum.The upshot is that, as Bader (2017: 115) puts it, '[r]ejecting necessary co-extension as the criterion for property identity [. . .] does not suffice for defending [non-naturalism]'. 36o evaluate the threat to non-naturalism posed by Bader's dictum, we need to consider two issues: (1) whether the criterion is indeed incompatible with non-naturalism, and (2) whether it fares any better than Hume's dictum with respect to the non-naturalist's head-on strategy.Concerning (1), I will not dispute Bader's claim that his dictum rules out non-naturalism. 37Instead, I will focus on (2).In what follows I argue that Bader's dictum is almost equally vulnerable to the intuitive counterexamples.The reason why I say 'almost' here is that my modified version of Olson's first example does not seem to work against Bader's dictum.This is because the properties being the view X and being the correct view of subject Y intuitively have distinct metaphysical grounds.The property of being a certain view is presumably fully grounded by the nature of a certain proposition, whereas the property of being the correct view of a certain subject matter is at least partly grounded by a certain aspect of the world.Thus, Bader's dictum does have the resources to distinguish them.Beyond that, however, the intuitive counterexamples remain equally forceful.
Consider first my slightly modified version of Parfit's mathematical example: The properties being an even prime number and being a positive square root of four are necessarily co-extensive and yet intuitively distinct.What grounds these properties?In virtue of what, metaphysically speaking, does the number 2 have these properties?I would suppose that they are grounded in the nature of the number 2. If this is right, then the properties turn out to have the same metaphysical ground, in which case the example is equally forceful against Bader's dictum.To give his dictum an advantage over Hume's dictum with respect to the present example, Bader would have to argue that the properties have distinct grounds.But it is not clear what those might be.
It is worth considering here what Bader says about another mathematical example: properties] can be understood along the same lines.The fact that triangularity and trilaterality are distinct properties can, however, be explained in terms of these properties having different grounds.The former property is had in virtue of having three angles, whereas the latter is had in virtue of having three sides.(2017: 119) Thus, Bader thinks his dictum escapes the counterexample since triangularity and trilaterality have different metaphysical grounds.But do they?The triangularity of closed figures cannot be grounded in their having three angles, as Bader suggests, since grounding is plausibly asymmetric, whereas being triangular and having three angles are presumably identical on any account of properties.The same goes for being trilateral and having three sides.What grounds trilaterality and triangularity is more plausibly the nature of certain objects, namely triangles (or trilaterals).Again, insofar as the present example works against Hume's dictum, it works equally against Bader's dictum.
Next, consider Olson's impossible-properties example: The properties being a round square and being a triangular square are necessarily co-extensive but intuitively distinct.Since these properties cannot be instantiated, they do not have any grounds. 38This raises the question what Bader's dictum implies about necessarily co-extensive ungrounded properties.Do they count as having the same grounds?If they do, then the example is equally effective against Bader's dictum.But if they do not, and since they obviously do not have distinct grounds, Bader's dictum is importantly incomplete.There will be properties about which the dictum is silent, thus undermining its status as a criterion of property identity.Like Streumer, Bader might of course deny that there are any uninstantiable properties.But that manoeuvre would merely serve to underwrite the parity of the two criteria with respect to the present counterexample.
Finally, consider Shafer-Landau's example, appealing to the intuitive distinctness of the properties being necessarily co-extensive with x and being identical to x.The analogous example vis à vis Bader's dictum would appeal to the properties being necessarily co-extensive and co-grounded with x and being identical to x.I suggested that Shafer-Landau's example has limited dialectical force vis à vis Hume's dictum, since denying the identity of the properties is tantamount to simply denying Hume's dictum.The analogous point applies here too.For my purposes, however, the important point is that, again, there is parity.

IV. HUME'S DICTUM, NATURALISM, SUPERNATURALISM AND NIHILISM
In this section, I argue that proponents of other views should also reject Hume's dictum. 39This includes naturalists, supernaturalists and even certain nihilists.I begin with naturalism and supernaturalism.

IV.1. Naturalism and supernaturalism
There are important differences between naturalism and supernaturalism, but for present purposes the two views can be treated as one.This is because, as I mentioned in Section II.1, supernatural properties, such as being in accordance with God's will, are still natural properties in the technical sense that is relevant here, which excludes only normative properties as non-naturalists see them.Thus, both naturalists and supernaturalists accept the following view, which I will refer to simply as 'naturalism'.
Naturalism: There are instantiable normative properties, all of which are identical to natural properties.
There are two basic reasons why naturalists should reject Hume's dictum.The first has to do with the nature of normative explanation, i.e. the kind of explanation that we seek in normative ethics.Normative explanation is plausibly fine-grained in that necessarily co-extensive properties are not (always) interchangeable salva veritate.To illustrate, even if it should turn out that being happiness-maximizing is necessarily co-extensive with being divinely commanded, hedonistic utilitarianism and divine command theory would still be competing views about what makes actions right. 40But Hume's dictum implies that this is not so, since the views would appeal to one and the same right-making property, albeit under different conceptual guises.They would be no more incompatible than the claim that glaciers are made of water and the claim that glaciers are made of H 2 O.
Thus, if there are instantiable normative properties, then either Hume's dictum is false, or normative explanation is not fine-grained.But normative explanation is plausibly fine-grained.Hence, if there are instantiable normative properties, Hume's dictum is false.Since naturalists accept the antecedent, they too are committed to rejecting Hume's dictum.
Naturalists might object that the fine-grainedness of normative explanation can be secured in a different way.Assuming that being happiness-maximizing and being divinely commanded are identical, normative explanations invoking this property can be rendered incompatible if we assume that the mode of representation is built into the explanans.On this picture, then, actions are not made right or wrong by properties alone, but by properties represented in a certain way. 41owever, I submit that this is just false.What is normatively relevant is what happens out there in the world, independently of our modes of representation.Suffering, for example, is normatively important (if it is) regardless of how we happen to represent it.Thus, normative explanation is not plausibly construed in the way suggested by the objection. 40Cf.Plantinga (2010) and FitzPatrick (2011: §3). 41I am grateful to Niklas Möller for pressing this objection.
The second reason why naturalists should reject Hume's dictum is that the criterion will not allow them to differentiate between what makes something have some normative property, and what it is for something to have that property.Rightness, for example, will be necessarily co-extensive with, and thus identical to, the (possibly disjunctive/conjunctive) natural property that makes actions right.Thus, by giving a constitutive account of rightness, naturalists will automatically specify the right-making property, and vice versa.But there is significant pressure to keep these apart.First, as McNaughton and Rawling  (2003:  §IV) point out, it is hard to see how identification of these properties leaves room for the explanatory asymmetry that plausibly obtains between normative properties and the relevant natural ones.But I want to highlight an additional problem: Very different constraints of plausibility apply to the metaethical project of specifying what it is for an act to be right, as opposed to the normative ethical project of specifying what makes acts right.A constitutive account of rightness and other normative properties needs to capture the normativity of the normative.If an action is right, then, as J. L. Mackie puts it, this is not just some further 'inert' fact about it, but 'something that involves a call for action ' (1977: 33).In order to do justice to this feature, it seems naturalists have no choice but to give a broadly speaking subjectivist account of normative properties.What it is for something to have a normative property must be specified in terms of the desire-like attitudes of some actual or (naturalistically) idealized person toward that thing.This may still not be sufficient, but I don't see how any other kind of naturalistic account might hope to accommodate the call-for-action aspect of the normative.
Problems arise, however, when Hume's dictum imposes this metaethical (or metanormative) plausibility constraint on the first order project of specifying the normatively relevant properties.It is implausible to suppose that the desire-like attitudes of some actual or idealized person are what make things have normative properties.First, any such subjectivist criterion will run up against competitors in the form of extant normative theories (in ethics, and perhaps also in other fields such as aesthetics and epistemology).Not only will the naturalist's constitutive account of normative properties be held hostage to the outcome of first order normative inquiry; it is also prima facie unlikely that this outcome will be favourable to the naturalist.Secondly, and more importantly, any subjectivist criterion will face a Euthyphro-problem.To avoid complete arbitrariness, the attitudes appealed to by the criterion would have to be responses to features of the bearers of normative properties.But if so, those features themselves are more plausible candidates for being the normatively relevant ones than the responses they prompt (or would prompt).To illustrate, suppose God responds with a favourable attitude to all and only acts which maximize happiness.It is hard to see how those responses could be morally relevant, unless the feature guiding them-happiness-maximization-already is.
Thus, I conclude that naturalists (including supernaturalists) should join non-naturalists in rejecting Hume's dictum.42I now move on to nihilism.

Nihilists claim the following:
Nihilism: There are no instantiable normative properties. 43 can distinguish between two different kinds of nihilism, corresponding to two different ways of arriving at the view: Sparse nihilism: There are no normative properties.
Abundant nihilism: There are normative properties, none of which is instantiable.
Among abundant nihilists, we can further distinguish between those who agree with naturalists and those who agree with non-naturalists about the nature of normative properties.We thus get: Naturalist abundant nihilism: There are natural normative properties, none of which is instantiable.

Non-naturalist abundant nihilism:
There are non-natural normative properties, none of which is instantiable. 44take it that most abundant nihilists are of the non-naturalist type.Indeed, often the very reason for holding that normative properties cannot be instantiated is precisely that they are distinct from natural properties.45I'm not sure that there are any abundant nihilists of the naturalist type, however.One way of arriving at such a view would be to combine metaphysical naturalism with a supernaturalist view of normative properties, but that is not a particularly attractive package. 46ume's dictum rules out both kinds of abundant nihilism.To see this, note first that Hume's dictum implies that there can be at most one uninstantiable property.Thus, abundant nihilists who accept Hume's dictum will have to say, for example, that rightness is identical to round-squareness.This implication straightforwardly rules out non-naturalist abundant nihilism, since round-squareness is clearly a natural property by non-naturalist lights, whereas rightness is supposed to be a non-natural one.And naturalist abundant nihilism is ruled out as well, since it would in any case be absurd to identify rightness with round-squareness.Even if both are natural, one is a normative property while the other is not.
In response, abundant nihilists who want to retain Hume's dictum might claim that there is no intuitively natural or non-normative uninstantiable property, such as round-squareness, for rightness to be identical to.This response does not work, however.To begin with it seems objectionably ad hoc.But more importantly, the class of normative properties on its own will still include several contrary properties, such as rightness and wrongness, virtue and vice.Abundant nihilists will have to say, absurdly, that these are all one and the same. 47hus, the only view left standing with Hume's dictum is sparse nihilism.This gives non-naturalists leverage in the metaethical dialectic.Firstly, the disjunction of non-naturalism, naturalism, supernaturalism and abundant nihilism has more credibility than non-naturalism on its own, and so non-naturalists can cast further doubt on Hume's dictum by appealing to these other views as (temporary) partners in guilt.Secondly, since Hume's dictum is available only to sparse nihilists, all other opponents of non-naturalism are robbed of a potentially powerful argument.

V. CONCLUSION
In this paper, I have explored the metaethical dialectic surrounding Hume's dictum.Apart from clarifying the non-naturalist head-on strategy and the different kinds of responses available, I have defended the strategy against Streumer's and Bader's critiques.Also, I have argued that only sparse nihilists are in a position to invoke Hume's dictum. 48