Why Lewis Would Have Rejected Grounding (2024)

Fraser MacBride,

Frederique Janssen-Lauret

1. Introduction

David Lewis is often credited with a leading role in the late twentieth-century revival of metaphysics.1 But, in the early twenty-first century, metaphysics has developed along lines Lewis did not himself anticipate. Now appeals to ‘metaphysical dependence’, ‘grounding’ and ‘ontological priority’ have become de rigueur. These weren’t terms in which Lewis ever sought to illuminate us or to put our philosophical perplexity to rest—they just weren’t expressions that belonged to his official philosophical vocabulary, not when he intended to talk seriously and precisely. The following question becomes pressing for us if we are to gauge how far the prevailing spirit of analytic philosophy has transformed since Lewis’s death in 2001. Do contemporary developments reflect an oversight, a failure to appreciate the strength and depth of the metaphysical tradition to which Lewis belonged, or are they an enrichment of a tradition that was otherwise stymied, or do they signal that the tradition to which Lewis belonged has just reached the end of the road?

In this chapter we argue that were Lewis to be writing today, he would—drawing upon the corpus of his established views—provide us with principled reasons for saying both (1) that we don’t need the notions of ‘metaphysical dependence’, ‘grounding’ or ‘ontological priority’ and (2) that they’re not intelligible notions anyway. They’re not needed because, he would have held, either there is work to be done but all the heavy lifting can be achieved by other means or there really isn’t any work that needs doing at all. They’re not intelligible, he would have continued, because they presuppose metaphysical modalities which Lewis had always held suspect. So much the worse, we conclude, for contemporary developments, whatever may be de rigueur. The tradition to which Lewis belonged cannot be consigned to the dustbin of history.

Here we use ‘grounding’ as a generic label for a range of theories which deem metaphysics stymied without metaphysical dependency, grounding or ontological priority. Although ‘metaphysical dependence’, ‘grounding’ and ‘ontological priority’ can be used differently, their proponents often share motivations and themes. We focus upon the alleged shortcomings of supervenience, the oft-cited need to invoke grounding to explain how a singleton relates to its sole member and the appeal to facts, essentialism or metaphysical necessity to explain grounding itself—all motivations and themes incompatible with Lewis’s philosophy.

2. Explaining Supervenience Without Grounding

Lewis, like Heracl*tus, saw order in nature, a pervasive cosmic stability which Lewis described in terms of supervenience. Lewis thought it a priori that every contingent truth supervenes upon the pattern of perfectly natural properties and relations (1994: 291). Impressed by the empirical success of physics to date, Lewis was also a committed materialist: based on physics’ extraordinary track record, he performed an optimistic induction, provisionally endorsing the a posteriori doctrine that as a matter of contingent fact all perfectly natural properties and relations that actually occur are fundamental physical ones. Putting these commitments together, he maintained an a posteriori supervenience of every contingent truth upon the pattern of fundamental physical properties and relations (1994: 293). Lewis conceived of this and other supervenience theses as ‘in a broad sense, reductionist’, but, as he put it, ‘unencumbered’ by ‘claims of ontological priority’ (1983a: 358). We will argue that Lewis would have seen no need to encumber his materialist supervenience with grounding.

Grounding is often called upon to remedy what is felt to be an explanatory deficiency of supervenience theses. The felt deficiency is that supervenience is not a ‘deep’ metaphysical relation but a ‘surface’ relation merely encoding a pattern of property co-variation, at best suggesting that ‘an interesting dependency relation’ might explain the pattern but not itself providing insight into why it obtains (Schiffer 1987: 153–4; Kim 1993: 167). To say, for example, that the mental supervenes upon the physical is to say that there is no mental difference without a physical difference but not why. Grounding is introduced to remedy this perceived explanatory deficit of supervenience (Schaffer 2009: 363–4; Rosen 2010: 113–14; Berker 2018: 735–7).

What this argument overlooks is the fact that Lewis never intended his supervenience theses to be taken in isolation, as the end of the story. The deeper, explanatory insight Lewis ultimately proffered was intended to arise from a combination of supervenience theses and analyses of the concepts required to state them. So unless scepticism about conceptual analysis is already presupposed, we cannot leap from the premise that supervenience theses merely report patterns of co-variation amongst properties to the conclusion that it is necessary to appeal to grounding. The explanation for why a pattern of supervenience obtains may arise from another source, the analysis of concepts.

Take Lewis’s materialism, a doctrine of global supervenience—that is, supervenience applied to whole possible worlds—favoured by Lewis because this allowed him to bypass questions about whether mental life is to some extent extrinsic to the subject (Lewis 1983a: 362). Holding that all natural properties of our actual world are physical ones based on his optimistic induction, Lewis proceeded to express his materialism thus: ‘if two worlds were physically isomorphic, and if no fundamental properties or relations alien to actuality occurred in either world, then these worlds would be exactly alike simpliciter’ (1994: 293). It follows that any two such worlds differing psychologically, ergo failing to be exactly alike, must differ physically. It also follows, Lewis reflected, that for anything mental there are physical conditions sufficient for its presence and physical conditions sufficient for its absence. For suppose, restricting our attention to worlds where no natural properties alien to our world are instantiated, that the physical condition P of one such world w₁ fails to be sufficient for the presence of a mental item M in w₁. Then there is another such world w₂ which satisfies P but doesn’t include M. But this contradicts Lewis’s materialism because then there are two such worlds which differ psychologically, i.e. with respect to the presence of M, without differing physically, i.e. with respect to the satisfaction of P. Similarly, if the physical condition P* of one such world w₃ from which M is absent fails to be sufficient for the absence of M in w₃, then there is another such world w₄ which satisfies P* and M is present, but this violates Lewis’s materialism too.

What features of the relationship between the mental and the physical did Lewis aim to explain in terms of his materialism? Lewis distinguished two features, which we label ‘placement’ and ‘tracking’. We distinguish a third, which we label ‘co-variance’, addressing the question why mental and physical properties co-vary. Although Lewis did not explicitly reflect upon it, we argue that his views about supervenience and conceptual analysis provide the wherewithal to explain co-variance if they explain placement and tracking. First we consider what Lewis said about placement and tracking.

Lewis distinguished the question how mental items ‘can find a place in the world of fundamental physics’ from the question how ‘Finite assemblies of particles—us—can track them’ (1994: 295, 297). The former Lewis took to be settled by his materialism. It follows from his materialism that a mental item can find a place in the world of fundamental physics in the sense that his materialism entails, as we have seen, that in a world where no natural properties alien to our world are instantiated there are physical conditions which determine the presence of any mental item and physical conditions which determines its absence. But Lewis’s materialism leaves open whether the physical conditions in question are finitely expressible, and graspable by us as finite creatures, or rather infinitely miscellaneous. So Lewis used ‘conceptual analysis’ as an additional resource to address the latter question how we are able to track the presence and absence of mental items (1994: 298).

His strategic idea was that an analysis of folk-psychological discourse would make available to us, in principle at least, finitely specifiable physical conditions which analytically suffice for the presence or absence of mental states. He conceived of terms for mental states as implicitly defined by the platitudes of folk psychology. These platitudes, according to Lewis, concern the manner in which mental states—states which typically belong to a system of states had by a person—are apt to cause behaviour and change under the impact of perceptual stimuli and other mental states which belong to the same system (1994: 299). When these implicit definitions are made explicit, they furnish analytic truths constraining the causal relations of the states (belonging to a system) to behaviour, perceptual stimuli, etc. So, according to Lewis, it is analytic that if someone has mental states, then she/he has states which, for the most part at least, exhibit the battery of causal relations described by the platitudes whereby mental terms are defined (1972: 250, 1974: 335). The analytic necessity whereby folk psychology constrains mental states is only ‘verbal’, not metaphysical (1994: 301); a mental state could have failed to exhibit the causal profile whereby the platitudes implicitly define mental terms. But in such circ*mstances it would no longer merit being described using mental terms.

The causal descriptions of states in which the conceptual analyses of mental terms terminate are ‘topic neutral’ (1994: 302): they say nothing about what sort of states exhibit the relevant battery of causal relations, nor whether they are mental or physical. They are bare descriptions of the causal profile analytically required of a mental state. But Lewis took physics, as a matter of contingent fact, to be ‘explanatorily adequate’ (1966: 23, 1994: 292). This gave him reason to suppose that states actually exhibiting the causal profiles analytically required of them to merit their being called mental are identical to physical states. Lewis was confident that physiology would reveal the states which exhibit these causal profiles to be neural ones (1966: 24, 1972: 249, 1994: 303). More generally, if Lewis’s materialism is true, there is nothing else for states which fulfil these analytic requirements to be except physical states. And because, according to Lewis, the analytic requirements are finitely encoded in tractable folk-psychological platitudes, they provide the ‘simple formula’ explaining our ability to track mental states in a world where every feature supervenes upon fundamental physics (1994: 305). We can track mental states because we are able to track whether there are physical states which exhibit the causal profiles analytically required of them to count as mental. This doesn’t mean that Lewis thought us capable—as a matter of course—of identifying individual neurological states which merit being called mental. But he did think we are able to track systems of neurological states satisfying the platitudes of folk psychology, systems belonging to persons whose behaviour is predicted and explained by folk psychology.

Much more could be said about Lewis’s account of how mental items find a place in a world of fundamental physics and how we are able to track them.2 But our focus here is: what about the complaint that Lewis’s account cannot explain why the mental supervenes upon the physical? It was this felt deficiency which motivated appeals to grounding. We call this the question of ‘co-variance’, the question why this pattern of co-variation obtains. In fact, we argue, the materials Lewis assembled to explain tracking and placement also explain co-variance.

Why, for Lewis, can there be no mental difference without a physical difference—not, at any rate, in worlds where no natural properties alien to our world are instantiated? Conceptual analysis reveals, according to Lewis, that as a matter of analytic necessity, a mental term denotes a state exhibiting a certain causal profile of relations to behaviour, perceptual stimuli, etc. The ‘explanatory adequacy’ of physics at our world then settles that the only states actually exhibiting these profiles are physical states. In light of Lewis’s conceptual analysis of folk psychology, for there to be a mental difference is for there to be a shift in the analytically relevant causal relations exhibited by the systems of states belonging to a person, a shift whereby a system ceases to satisfy the analytic requirements for including a certain mental state or begins to satisfy other requirements for previously absent mental states. Since the analytically relevant causal relations are, according to Lewis, causal relations physical states bear to other physical states, every difference in them is a physical difference. But not every difference in causal relations between physical states is a difference which is analytically relevant to whether a physical state merits being described as a mental state. Not just any causal difference in neural states matters to what we believe and desire, only the ones circ*mscribed by folk-psychological discourse. So, by Lewis’s lights, not every physical difference corresponds to a mental difference. The supervenience of the mental on the physical is asymmetric, not a brute modality but an intelligible consequence of conceptual analysis and the explanatory adequacy of physics.

This explanation of co-variance works only if Lewis can assume the explanatory adequacy of physics and the existence of stable folk-psychological platitudes as a basis for his conceptual analyses. There are well-known objections to these assumptions. It’s not a foregone conclusion that there’s a body of platitudes implicitly known by the folk such that the conceptual analysis of them terminates in purely causal descriptions without mental remainder; that is, ‘topic-neutral’ descriptions of causal profiles. In particular, it’s not obvious that a physical state is a conscious state solely because it exhibits a certain causal profile. It’s also often claimed that conscious states have phenomenal characters which cannot be explained in causal terms (though Lewis rebutted this claim (1988)). But to press such objections is to dispute the adequacy of Lewis’s assumptions. Our point here is that if his analyses are adequate and his materialism is true, then Lewis’s account does not suffer from a structural failure to explain co-variance which can only be remedied by introducing grounding. Lewis’s premises may be false, but taken together they do explain co-variance. Their explanation of co-variance does not require a further premise about grounding.

3. The Availability of Informative, Non-Trivial Conceptual Analyses

Lewis’s account of placement and tracking (hence co-variance) crucially relies upon the availability of informative conceptual analyses of mental terms. Proponents of grounding have objected that there aren’t informative conceptual analyses and that conceptual analyses are never non-trivial. According to the former challenge, conceptual analyses may only be framed in terms already grasped by fully competent speakers of the analysandum, but interesting, supposedly ‘analytic’ claims made by philosophers, including Lewis’s conceptual analyses of mental terms, are not available even to fully competent speakers. Ergo the ‘analytic’ claims, Lewis’s included, cannot be conceptual but must be understood as claims of grounding (Rosen 2015: 189). According to the latter challenge, it is implausible that when philosophers differ, one party to the dispute is conceptually confused or that they differ over linguistic or conceptual matters, rather than something substantive. If conceptual analyses are always trivial, then there cannot be, as Lewis claims, analyses of mental terms terminating in ‘topic-neutral’ descriptions without mental remainder—because that is not trivial. Hence ‘grounding’ is urged upon us to make up this explanatory shortfall, because whilst conceptual analyses can only be trivial or false, ‘grounding’ claims can be interesting and true (Berker 2018: 738–9). So here we have a different juncture: the availability of informative conceptual analyses—distinct from the necessity to explain mental–physical co-variation—at which the proponents of grounding find fault with Lewis’s account and press the need for introducing their favoured notions.

We argue Lewis would not have been moved by either challenge. The conceptual analyses of mental terms envisaged by Lewis are ‘topic-neutral’ descriptions of causal relations and behaviour which he conceived to be ‘available’ to the folk, because descriptions of causal relations and behaviour are already familiar parts of our ordinary conceptual repertoire (1994: 299–301). Remember, Lewis didn’t arrive at physicalism by conceptual analysis alone. To assure him that whatever satisfies these descriptions are physical states, Lewis relied on the explanatory adequacy of physics, which he considered an empirical matter. Lewis would have given short shrift to the first complaint because he thought the conceptual analyses of mental terms which he used actually are available to competent speakers.

The second challenge makes the assumption that conceptual analyses are invariably trivial if true—hence if a philosophical dispute appears substantial and difficult to resolve, it is correspondingly unlikely to rest upon a conceptual confusion by one or both parties, a confusion which should be obvious because it’s trivial. Lewis made clear that he did not share this assumption when reflecting upon his own analysis of value in terms of what we are disposed to value. He wrote, ‘The equivalence between value and what we are disposed to value is meant to be a piece of philosophical analysis, therefore analytic. But of course it is not obviously analytic; it is not even obviously true’ (Lewis 1989: 129). Lewis immediately added, ‘It is a philosophical problem how there can ever be unobvious analyticity’. But, he continued, ‘We need not solve that problem; suffice it to say that it is everybody’s problem, and it is not to be solved by denying the phenomenon. There are perfectly clear examples of it: the epsilon-delta analysis of an instantaneous rate of change, for one’. So, Lewis implied, once one unobvious analyticity has been admitted, such as the epsilon-delta analysis, the pro-grounding assumption that analyticities are invariably trivial or obvious has been overturned.3 We conclude that Lewis wouldn’t have been moved by the second challenge either because he denied that conceptual analyses, if true, are trivial.

Lewis thought of the question how there can be unobvious analyticities as a compulsory one on the philosophical examination paper. He did not return to it in his publications but his correspondence suggests how he would have answered. In one of his last letters, he compared conceptual analysis to inference to the best explanation, explaining how the former is no more trivial than the latter:

We interpret Lewis as making inter alia the following point. As ordinary-language speakers, we lack global oversight of how the different parts of our language fit together and relate to one another even though our practical mastery of individual expressions may be consummate. Once we appreciate this, we should no longer expect analytic connections between different parts of our language to be immediately transparent to us speakers, much less that they should appear trivial. The assumption that analyticities are trivial presupposes that language as a whole is perfectly tractable to speakers, but it isn’t. In short, Lewis’s appreciation of the fallible and speculative character of our engagement with language as a whole makes unobvious analyticities inevitable. That’s an important truth about us, not a reason for appealing to grounding.

4. Why Singletons Aren’t Grounded in Their Members

We turn from the accusation that supervenience claims are unsatisfactory because they merely register without explaining patterns of property co-variation to the quite different accusation that the notion of supervenience is expressively incapable of drawing the distinctions we need. Consider, for example, Socrates and his singleton—often invoked as a paradigm pair of one thing which ‘grounds’ another (Schaffer 2009: 375). Suppose it is necessary that if Socrates exists, he belongs to the singleton of Socrates, and necessarily that if that singleton exists, then Socrates exists.4 Then it follows that the existence of one supervenes on the existence of the other and vice versa. But according to proponents of grounding, the existence of Socrates grounds the existence of {Socrates}, not vice versa. They conclude that because supervenience goes both ways, but grounding only one way, supervenience is unable to capture the asymmetric sense in which {Socrates} depends upon its solitary member (Berker 2018: 736).

How would Lewis reply? One snappy response available to him would have been to refuse to accept that there is a metaphysical asymmetry between singletons and their members. This was the kind of response Lewis gave to one of Armstrong’s criticisms of natural-class nominalism, which analyses sameness of type in terms of membership of natural classes where the notion of naturalness is taken as primitive, and which Lewis considered a viable competitor to Armstrong’s theory of immanent universals (1983a: 347). In his Universals: An Opinionated Introduction (1989b), Armstrong sought to draw Lewis down a path unfavourable to class nominalism. Armstrong asked, ‘Is a thing the sort of thing that it is—an electron, say—because it is a member of the class of electrons? Or is it a member of the class because it is an electron?’ (1989b: 27–8). To decide upon an answer, Armstrong declared, ‘is a matter of deciding what is the direction of explanation’. To illuminate what he meant by ‘direction of explanation’, Armstrong invoked Socrates’ question in Plato’s Euthyphro: ‘Are pious acts pious because they are loved by the gods? That is, is being loved by the gods what constitutes their being pious? Or do the gods love these acts because of their piety?’ (1989b: 28). Armstrong’s favoured answer to his own question was, ‘it seems natural to say that a thing is a member of the class of electrons because of what it already is: an electron. It is unnatural to say that it is an electron because it is a member of the class’ (1989b: 28). His was an answer detrimental to class nominalism because if the type of a thing determines class membership, then class membership cannot be used to analyse the fact that different things are of the same type, as class nominalism maintains.

In 1988 Armstrong sent Lewis a draft of Universals: An Opinionated Introduction and Lewis replied with a twelve-page commentary.5 On the present point, Lewis’s snappy response to Armstrong was, ‘I think this saddles the natural class nominalist with a commitment he doesn’t have…He doesn’t say that they’re electrons because they’re members of the class—or vice versa’. Since, Lewis maintained, the class nominalist need not assume either thing, he, or she, need not be drawn by Armstrong’s leading question to confound his or her analysis. More generally, Lewis simply refused to be drawn by Armstrong’s Euthyphro question.

We consider Lewis’s response here to be indicative of how he would have responded to questions about whether X metaphysically depends upon Y or vice versa. The Euthyphro question is often employed as a technique for introducing the notion of grounding, to demonstrate how readily understood the notion supposedly is (Schaffer 2009: 375; Berker 2018: 730–1). Just ask: is an act pious because the gods approve or vice versa? Because then, it is claimed, interlocutors quickly appreciate that this is equivalent to asking: is the piety of an act grounded in the gods’ approval or vice versa? But evidently Lewis held that questions of this form aren’t always obligatory—witness his refusal to be drawn on whether electrons are members of the class of electrons because they are electrons or vice versa. Similarly here, with respect to singletons and their members, Lewis could well have refused to saddle the set theorist with a commitment to supposing that singletons are grounded in their members—or vice versa.

Although in our view refusing to be drawn on the question would already have been a legitimate move on Lewis’s part, we also argue Lewis had deeper reasons for holding that singletons neither depend upon their members nor vice versa. Drawing upon his evolving views about classes (Lewis 1991a, 1993), we argue on Lewis’s behalf that there is no two-way supervenience between Socrates and {Socrates}. During the 1980s Lewis had acknowledged the two-way supervenience, writing to Armstrong reluctantly to concede that, ‘I am committed to accept the mystery: I believe there are unit sets, they are wholly distinct from their members, and exist automatically if their members do…it’s just that I see no way around it’ (Letter 288. To Armstrong, 6 May 1987, in Beebee and Fisher 2020a: 578). But, we argue, Lewis’s mature philosophy of set theory gave him a way around having to acknowledge that singletons exist automatically if their members do. On Lewis’s 1990s view, Socrates and {Socrates} are entirely ‘loose and separate’. There is no necessary connexion between them. Hence, we conclude, Lewis would see no need to countenance grounding to explain how Socrates relates to {Socrates}. We begin with Parts of Classes (1991a) before considering its more radical successor, ‘Mathematics Is Megethology’ (1993).

In Parts of Classes Lewis argued that we can reconstruct set theory within mereology if we take the notion of singleton as primitive and take classes as fusions of singletons, as having singletons as their atomic parts, the members of a class being the members of the singletons of which the class is a fusion. Lewis considered the reconstruction illuminating because it allows us to isolate what’s distinctive, indeed peculiar, about classes. It’s not their combining many things to make one thing, a class from its members, as Cantor had supposed—because, Lewis argued, the combining is done by mereology, which is perfectly understood and unproblematic. What’s peculiar, according to Lewis, are the singletons themselves, because, he argued, the official axioms of set theory tell us ‘nothing about the nature of the singletons, and nothing about the nature of their relation to their element’ (1991a: 31). Nor, Lewis continued, are we helped much by what he called the ‘unofficial axioms’ of set theory, which he described as ‘passed along heedlessly from one author to another’, remarks such as ‘classes are outside of space and time’ or ‘classes have nothing much by way of intrinsic character’ or ‘singletons may not turn out to be among the atomic parts of ordinary things’ (1991a: 31, 33, 49). Lewis concluded that since singletons are the atomic parts of classes, our ignorance about the nature of singletons extends to the nature of classes generally. If we know nothing about singletons, then all we know about classes is that they are fusions of their atomic parts.

This already spells trouble for grounding. If Lewis is right that we don’t know anything about the nature of singletons, then eo ipso we don’t know that members ground singletons. We provide corroborative evidence for this outlook by reflecting that grounding does not figure in the official axioms of set theory. Whilst we grant that a word-search on the informal glosses of set theory will sometimes throw up words like ‘determines’, such occurrences typically serve merely to introduce axioms where it is the axioms themselves that are conceived as the proper carriers of the precise meaning. Consider, for example, the Axiom of Extensionality, originally dubbed by Zermelo ‘Axiom der Bestimmtheit’. Zermelo describes each set as being ‘determined through its members’ and Fraenkel et al. describe any set as being ‘completely determined by its members’. But what they meant by ‘determines’ (‘bestimmt’) wasn’t anything metaphysical but only an informal or shorter (‘kürzer’) gloss for what is precisely expressed by the Axiom of Extensionality itself: if X and Y have exactly the same members, then X =  Y (Zermelo 1908: 263; Fraenkel, Bar-Hillel and Levy 1973: 27).

We further reflect that when Lewis wrote, grounding did not figure in the ‘unofficial axioms’ of set theory either. Do proponents of grounding want to add a metaphysical update to the ‘unofficial axioms’—add that singletons are grounded in their members? It doesn’t take much to qualify as an unofficial axiom as Lewis characterised them: they’re only required to be ‘passed along heedlessly’ from one author to another. Nonetheless, we maintain Lewis would have found this unofficial axiom unhelpful too because it tells us nothing about the character of a singleton except that it is a ‘something we know not what’ grounded in its member.

There is, however, a deeper response to be made on Lewis’s behalf to conceiving singletons in terms of grounding. To explain it, we turn to his efforts to get past this unhappy situation, the mystery of singletons. In a certain sense what Lewis offered was a counsel of despair, analytic despair. He gave up on the conceptual analysis of set theory as a source of insight into the nature of singletons. The contrast between Lewis’s approach to set theory and his approach to folk psychology is striking. He held that conceptual analysis contributes to the vindication of folk psychology—because an analysis of folk psychology terminates in informative analyses of mental terms, ‘topic-neutral’ descriptions of causal profiles, which then facilitates an explanation of the placement and tracking of them. But he held conceptual analysis cannot perform a comparable role in vindicating set theory, because an analysis of set theory fails to terminate in anything informative. So Lewis sought to vindicate set theory along different lines, not by furnishing an analysis of it but by providing an explication instead. By ‘explication’ we mean what Quine meant: a substitute for a dubious expression or theory which fulfils whatever functions make the dubious expression or theory worth troubling about whilst avoiding its shortcomings (1960: 258–9).

Lewis described his explication for set theory as a kind of ‘structuralism’. He didn’t mean thereby that his substitute for set theory was a theory of some special entity, ‘an abstract structure’, because, he wrote, ‘I suspect such entities are trouble, but in any case, they’re an optional extra’ (1993: 16). The distinctive feature of Lewis’s structuralism is that it lays down purely ‘structural’ or ‘formal’ requirements on reality. It avoids committing to abstract structures or presupposing that the elements quantified over have a distinctive nature. So it circumvents the shortcoming of set theory Lewis had pressed in 1991, introducing sets as sui generis but neglecting to tell us what they are. Instead, according to his substitute theory, advanced in ‘Mathematics Is Megethology’, only atoms are singletons but an atom only counts as a singleton relative to a singleton function.6 What is an atom for Lewis? He defines the notion in solely mereological terms as anything that lacks proper parts (1991a: 15). What is a singleton function for Lewis? Any unary one–one function s which satisfies the following formal conditions: (1) the range of s consists of atoms (call them ‘s-singletons’, singletons relative to s); (2) its domain consists of all (small) fusions of s-singletons together with all things which don’t have s-singletons as parts (call them ‘s-individuals’, individuals relative to s); and (3) all things are generated by iterated application of s and mereological fusion (1993: 16).7

Let’s focus on s-singletons and s-individuals for some function s satisfying these conditions. An s-singleton x is the singleton of an s-individual y because x is an atom belonging to a collection of atoms which comprise the range of s, s maps y one–one onto x and s maps all the other s-individuals, fusions of s-singletons, fusions of s-individuals, etc. one–one onto s-singletons. What’s notable for present purposes is that in order for x to qualify as the singleton of y relative to s, x need not lie in any further relation to y. There need be no special sense in which y is ‘included’ in x. In fact, for any singleton function s there are many other singleton functions, also satisfying the formal requirements for being singleton functions, which have the same domain and range as s but map them together differently. Suppose that relative to s, x is the s-singleton of y and z is the s-singleton of w. Then there is another singleton function t relative to which x is the t-singleton of w and z is the t-singleton of y, because whereas s maps x onto y and z onto w, t maps x onto w and z onto y. More radically, x might have existed in circ*mstances in which y doesn’t exist but still figures in the range of another singleton function u, such that x is the u-singleton of something else that doesn’t actually exist. Indeed, x might have existed without being the singleton of anything because there might be no singleton functions whatsoever for want of a sufficient supply of atoms onto which the individuals and fusions of individuals etc. may be mapped one to one—because without an infinity of atoms that ‘transcends our commonplace alephs and beths in much the same way that any infinity transcends finitude’, set theory fails on Lewis’s explication (1993: 23).8

Bearing these points in mind, let us return to the claim made by proponents of grounding that there is two-way supervenience between the existence of Socrates and the existence of {Socrates}, but {Socrates} depends upon Socrates, not vice versa, hence the need to introduce grounding. It should now be evident that from Lewis’s point of view, this is all at best a mistake. At worst, it’s a flawed conception of set theory arising from a failure to appreciate that set theory is silent about its subject matter.9 What’s key is that according to Lewis’s structuralism, nothing is the singleton of Socrates per se but only relative to a singleton function. And relative to different singleton functions, different things qualify as Socrates’ singleton. In fact, whatever qualifies as Socrates’ singleton relative to one singleton function qualifies as Plato’s singleton relative to another—indeed qualifies as the singleton of any individual relative to some singleton function (assuming there are singleton functions). Moreover, whatever happens to actually qualify as a singleton of Socrates relative to some singleton function might have existed even though Socrates failed to exist and vice versa. Because singletons and their members are so loose and separate, grounding fails to gain purchase.

The relationship between singletons and their members is often presented as a paradigm case of grounding. But it follows from Lewis’s structuralism that there’s nothing paradigmatic about this case at all. From here we might indeed proceed case by putative case of grounding to consider what Lewis would have said negatively about each one. But in the next section we turn to more sweeping objections that flow from Lewis’s philosophy, objections to the very idea of grounding.

5. Against Facts

When we pass from presumed cases of grounding to what are held to be the theoretical principles underlying the classificatory practice, there is no consensus about what grounding is. This makes it difficult to engage en bloc with proponents of grounding. Nonetheless, it is possible to identify two principles such that many of its proponents will embrace one or other if not both of them, namely (1) that grounding is a relation between facts, either tout court or at least in a significant range of cases, and (2) that grounding holds of metaphysical necessity.10 We argue that if grounding is constrained by either of these principles, then Lewis would have denied its intelligibility. Ergo Lewis would have rejected most contemporary work on grounding.

According to (1), grounding is a relation which holds between facts. It is generally agreed that the facts in question are structured entities built up from worldly constituents including things, properties and relations. But whereas some proponents of grounding conceive of facts as a thing’s possessing a property or some things standing in a relation (Audi 2012: 686), others conceive of facts as true propositions whose constituents are the things, properties and relations that the propositions are about (Rosen 2010: 114). The two approaches come apart because a fact in the former sense, say a’s being F, could not have existed unless a had F, whereas a fact in the latter sense, a true proposition, say that a is F, could have existed if even if a didn’t have F—although then it wouldn’t have been a fact, just a false proposition. But this difference need not detain us. What they have in common is the assumption that facts are structured entities individuated by their worldly constituents and manner of composition. And it’s on the basis of this shared assumption that Lewis argued against the intelligibility of facts so conceived.

Facts in the former sense are Armstrong’s states of affairs by another name. Whilst states of affairs had made an appearance in Armstrong’s Universals and Scientific Realism (1978), they became increasingly central to Armstrong’s metaphysics, culminating in A World of States of Affairs (1997). Lewis set himself resolutely against this development, never faltering in his resistance.11 We reconstruct Lewis’s argument against states of affairs as proceeding in three steps.

Lewis endorsed a principle of ‘unrestricted composition’: it’s a sufficient condition for the existence of the whole a+F that its parts, a and F, exist (1986a: 212). But it isn’t a sufficient condition for the existence of the state of affairs a’s being F that its constituents exist. By contrast to a+F, a’s being F doesn’t automatically exist if a and F exist, but only if a also instantiates F. Lewis further endorsed a principle of ‘uniqueness of composition’: for given parts, only one whole is composed from them (1986b: 38). But from the given constituents a, b and non-symmetric R two distinct states affairs can be composed: a’s bearing R to a and b’s bearing R to a. For these two reasons constituency isn’t mereological. Since Lewis gave ‘the word “distinct” an exclusively mereological sense’, he concluded that a state of affairs and its constituents are distinct simpliciter (1999: 219).

Whilst Armstrong acknowledged that states of affairs aren’t mereological wholes, he maintained that they’re wholes in some ‘unmereological’ sense (1986: 85, 1989a: 41, 1997: 119–20). Earlier formulations of Lewis’s argument against states of affairs appealed to a further principle: that there is only one mode of composition, the mereological one. He swiftly concluded that ‘unmereological whole’ is a contradiction in terms (1986b: 92, 1992: 213). Unfortunately this left the Armstrong–Lewis dispute in a stand-off, with Lewis using the principle that there is only one mode of composition as a basis for denying that states of affairs are wholes, and Armstrong using states of affairs conceived as wholes as a counter-example to Lewis’s principle. In later formulations Lewis sought to break the impasse by granting Armstrong that the way states of affairs are built up from their constituents is in some ways analogous to the way wholes are composed from their parts whilst pointing out that the disanalogies outweigh the analogies (1999: 218–19, n.1).

It was in his correspondence that Lewis spelt out most fully what he had in mind. The analogy Lewis grants Armstrong is that just as the existence of a state of affairs entails the existence of its constituents, the existence of a whole entails the existence of its parts. But after that there are only disanalogies. The existence of a whole is entailed by the existence of its parts, but, as we have seen, the existence of a state of affairs isn’t entailed by the existence of its constituents. States of affairs are governed by a principle of instantiation: necessarily, if a instantiates F, then the state of affairs a’s being F exists. Mereology has no analogue of this principle. Lewis continued:

Lewis concluded the disanalogies are so weighty that ‘unmereological composition’ is ‘a straightforward contradiction in terms’.

If states of affairs aren’t (mereologically) composed of their constituents—in other words, if states of affairs and their constituents are (mereologically) distinct simpliciter—how else can we explain how states of affairs are constructed from their constituents? Constituency gives rise to necessary connexions which distinguish constituency from mereological composition—for example: necessarily, if one thing, a, instantiates another, F, then a third thing, a’s being F, exists, where neither the first thing nor the second thing are parts of the third thing. The only intelligible answer Lewis could fathom to the question how states of affairs are constructed from their constituents was to define their construction in terms of these necessary connexions. To be a constituent of a state of affairs is to be a thing which enters into the relevant necessary connexion to a state of affairs (Lewis 2001: 611).

Why could Lewis fathom no other answer? We speculate: because states of affairs are typically introduced and explained in terms of their necessary connexions—the state of affairs a’s being F is introduced just as the item which if it exists necessitates that a and F exist and a instantiates F, etc. It’s because states of affairs stand in these necessary connexions that they are posited; it’s what suits states of affairs to perform the truth-making role, their raison d’être so far as Armstrong was concerned (Armstrong 1989a: 41, 1997: 115–17), or what suits them to determine which things have which properties or stand in which relations. We’ve never been furnished with a description of states of affairs that, so to speak, operates at a deeper level and explains why these necessary connexions hold of states of affairs. Of course, describing states of affairs in terms of their ‘unmereological constituents’ may appear an initial step to remedy the lack, but Lewis’s point was that our only understanding of ‘unmereological constituents’ circles back to invoking necessary connexions between mereologically distinct existences, or more strongly, as he once wrote to Armstrong, all we have is ‘a metaphor inspired by the case of mereologically not distinct existences’ (Letter 288. To Armstrong, 6 May 1987, in Beebee and Fisher 2020a: 578).

In the same letter to Armstrong, Lewis voiced his concern thus: ‘In the case of soas [states of affairs] the question is: if Fa and a are mereologically distinct, how can it be necessary that the latter must exist if the former does?’. Lewis dubbed this ‘the mystery of states of affairs’ and likened it to what he took to be the ‘mystery of unit sets’: ‘the same problem is my main worry about sets: if a and its unit set are mereologically distinct how can it be necessary that the latter must exist if the former does?’ (Letter 288. To Armstrong, 6 May 1987, in Beebee and Fisher 2020a, 587). We’ve argued Lewis eventually resolved the latter mystery in 1993 by invoking structuralism—which allowed him to explain how singletons can be mathematically serviceable without being necessarily connected to their sole members. But if states of affairs are to perform the roles for which they are posited by Armstrong, they have to be necessarily connected to their (mereologically) distinct constituents. If the state of affairs a’s being F is to serve as a truthmaker, then it’d better be the case that necessarily, if it exists, then a has F. Since, according to Lewis, ‘it is the Humean prohibition of necessary connections that gives us our best handle on the question what possibilities there are’, he dismissed states of affairs as bad news for systematic metaphysics (2001: 611).12

We note that the second step of Lewis’s argument isn’t strictly required for his conclusion. One might, and Lewis sometimes did, proceed directly from the first to the third; that is, proceed from the acknowledgement that the composition of states of affairs is unmereological to the conclusion that the only understanding we have of their construction is in terms of necessary connections. He didn’t need to use the lemma that unmereological composition is more disanalogous than analogous to mereological composition. Our exegetical claim is that Lewis sometimes included the second step for its persuasive or therapeutic value, but sometimes left it out for reasons of brevity. What is its therapeutic value? Appreciating the extent of the disanalogies that obtain between mereological wholes and states of affairs helps dislodge any naïve confidence we may have that states of affairs are well understood or bona fide because of their similarity to mereological wholes and helps us appreciate that any understanding of states of affairs must draw upon a grasp of necessary connexions.

Our present focus is: what would Lewis have said about the principle (1) that grounding is a relation which holds between facts (at least in a significant range of cases)? If the facts in question are akin to Armstrong’s states of affairs, then Lewis would have rejected (1), because such facts violate the Humean prohibition on necessary connexions. But, as we noted earlier, some proponents of grounding conceive of facts differently, as true propositions with worldly constituents. What would Lewis have said about (1) if it’s facts so conceived that grounding relates? We maintain that Lewis would still have judged (1) to violate the Humean prohibition on necessary connexions. That’s because such facts (true propositions) are still conceived by proponents of grounding as structured items built up from things, properties and relations which aren’t composed mereologically. For example, the propositions that Lewis admired Armstrong and that Armstrong admired Lewis are both true, and they have the same constituents, but they’re not the same fact even though only one whole can be assembled from their constituents. Moreover, unexcused necessary connexions are nearby: the existence of the structured unit that a is F entails that a and F exist and a has F even though the proposition is (mereologically) distinct from a and F.13 We conclude Lewis would also have rejected grounding as a relation between true propositions conceived as structured but non-mereological wholes. If ‘unmereological whole’ is not a contradiction in terms, the only way to understand it is in terms of prohibited necessary connexions.

6. Against Metaphysical Necessity and Essentialism

Proponents of grounding who endorse principle (2), that grounding holds of metaphysical necessity, won’t be moved by Lewis’s appeal to a Humean prohibition of necessary connexions. This is because, in least in some cases, the dependencies they posit consist in relations holding of metaphysical necessity between distinct existences. Consider, for example, the avowedly ‘anti-Humean’ claim that regularities metaphysically depend upon nomic facts ‘as a matter of metaphysical necessity’ (Rosen 2010: 120). So we turn our attention to principle (2).

Proponents of grounding have not devoted themselves to demonstrating that the idea of a metaphysically necessary connexion is legitimate or intelligible. They have presupposed it, deploying the idea in the service of grounding. Whilst ‘grounding’ might not be strictly definable, it is conceived as belonging to a family of ideas and notions, which include ‘metaphysical necessity’ and ‘essence’, to which ‘grounding’ may be informatively related. Proponents of grounding have felt entitled to so proceed, describing this allegedly virtuous circle of notions, because of where they think they find themselves in history. They consider themselves to come in the latter stage of a revolution which Kripke began in the 1960s.

The first stage of this revolution was the project of rehabilitating the traditional notions of necessity and possibility. Rosen, for example, tells us that this is ‘a project now more or less complete, and whose value is beyond dispute’ (2010: 134). The second stage is now to complete that work by doing for ‘grounding’ what Kripke (and others) did for ‘metaphysical necessity’ and ‘essence’. What did Kripke and others do for the latter notions? According to Rosen, what they established was that we have ‘tolerably clear intuitions’ about whether, for example, this or that lectern could have been made of ice. Because of Kripke we now recognise that we have ‘moderately effective strategies’ for extending our modal knowledge by means of ‘argument and analogy’, which is enough to set the notions of metaphysical necessity and essence upon a secure footing (Rosen 2010: 134). With these achievements behind us, Rosen declares, now is the time to move to the second stage and to set ‘grounding’ on a similarly secure footing.

What would Lewis have said to this? He would have been unwilling to grant that Kripke and others had achieved so much. Lewis, contra Rosen, disputed the value of their achievements. Because Lewis would have held the first stage of the metaphysical revolution neither to have been completed nor in principle capable of completion, Lewis would have denied that now is the time to move to the second stage of the revolution. And he would have denied that that time will ever come.

Lewis is frequently recalled as the Übermensch of speculative metaphysics because of his doctrine that possible worlds are existent cosmoi—in this respect he is often viewed as an inspiration for the surge of metaphysics in the twenty-first century. But this overlooks how metaphysically circ*mspect Lewis was, which becomes evident when we turn to Lewis’s criticism of Kripke’s claim to have set ‘metaphysical necessity’ and ‘essence’ on a sure theoretical footing. Kripke had argued that misgivings about the intelligibility of de re modality can be refuted by pointing out that the distinction between essential and accidental properties is perfectly intuitive and that we are more certain of the claims we find intuitive than the soundness and validity of any philosophical argument brought against them (Kripke 1980: 41–2). What was key for Lewis—what he held Kripke to have overlooked—is that our intuitions about whether this or that lectern could have been made from ice or whether Kripke could have come from a different sperm and egg are not constant at all. Lewis observed, ‘Attend to the variety of what we say about modality and counterfactuals de re, and I think you will find abundant evidence that we do not have settled answers, fixed once and for all’ (1986a: 252). Rather, he explained, in different contexts ordinary speakers offer different answers, depending upon which features of a thing are conversationally salient. Sometimes it’s right to say that the lectern could have been made from ice, sometimes it isn’t. Suppose we are interested in the design of the wooden lectern rather than its constitution. Then we may vary its constitution in our counterfactual speculations but hold fixed its design—so it could have been made from ice.14 But if our interest is its constitution, then we may vary its design but hold fixed its constitution—so it couldn’t have been made from ice. Lewis labelled this the ‘inconstancy’ of modal discourse.15 Indeed, he continued, in the absence of a context to guide us, questions about whether this or that lectern could have been made from ice etc. have no determinate answers.

Lewis did not dispute Kripke’s claim that the distinction between essential and accidental properties has ‘intuitive content’ which means something to the ‘ordinary man’ (Kripke 1980: 41). Lewis granted that, at least in some contexts, ordinary speakers, not only men, may readily agree upon where to draw the boundary between essential and accidental properties. But Lewis also saw that more is needed to set ‘essentialism’ upon a sure intellectual footing. Essentialism requires that a thing’s essential properties are essential to it context-independently. But, Lewis reflected, when we consider the same thing in different contexts, we draw the boundary between its essential and accidental properties differently. That’s the inconstancy of modal discourse. Because of inconstancy, our ordinary ways of thinking and talking about how things might have been just don’t provide support for essentialism. Since Kripke sought to establish the legitimacy of essentialism based on nothing more than an appeal to ordinary usage, Lewis concluded that Kripke’s defence of essentialism was lacking. And there is no reason to think that the situation will ever change—no reason to suppose that the inconstancy of modal discourse will ever be resolved in favour of constancy.

In fact, Lewis went further and supplied an ‘error theory’ that explains the spurious credibility of essentialism. By the very act of deciding to defend the view, for example, that Kripke could not have come from a different sperm and egg, you ‘create a context’ in which that is the correct thing to say (Lewis 1986a: 251–2). So no wonder, Lewis would have said, that Rosen and his colleagues find themselves having ‘tolerably clear intuitions’ about essentialism (Rosen 2010: 134). From reading and re-reading Naming and Necessity over the years, teaching from it and spending time together in the seminar room discussing it, they create a context in which it’s correct to say that the lectern could not have been made from ice and Kripke could not have come from a different sperm and egg. But it makes just as much sense to institute a context in which it’s no less correct to say that constitution or origins are not essential.

Famously, Lewis had a distinctive proposal of his own for accommodating the inconstancy: counterpart theory (1986a: 254–5). According to counterpart theory, something is essentially thus-and-so if all and only its counterparts, distinct things from other possible worlds but relevantly similar, are thus-and-so. But what counts as a counterpart—that is, relevantly similar—depends upon which counterpart relation we have in mind. In different contexts our interests make salient different counterpart relations. So two things may be counterparts in one context but not another—a given lectern may have icy counterparts in one context but not another. The upshot is that it makes sense to describe something as having an essential property but only relative to a contextually variable choice of counterpart relation.16 Of course, Kripke had his well-known ‘Humphrey’ objection to counterpart theory. Kripke firmly held the intuition that what’s de re possible for something, say Humphrey, is what’s possible for him, not a counterpart of Humphrey (Kripke 1980: 45). Lewis replied, unmoved, that what’s possible for him is thanks to what happens to his counterparts (Lewis 1986a: 195–6). But the correctness of Lewis’s observation that the inconstancy of modal predications spells trouble for the notions of ‘metaphysical necessity’ and ‘essence’ that Kripke sought to vindicate is independent of counterpart theory.17 For now the significant and separate point is that Lewis showed that the notions of ‘metaphysical necessity’ and ‘essentialism’ remain to be set upon a sure footing. Hence, we conclude, Lewis would have denied that proponents of grounding can now legitimately take ‘metaphysical necessity’ and ‘essence’ for granted as part of a virtuous circle whereby ‘grounding’ is introduced or illuminated.

7. Conclusion

Since his early days in UCLA, Lewis had been haunted by the possibility that analytic philosophy should succumb to what he described as ‘a chaos of new beginnings’. For this reason, when he set about writing his book manuscript Confirmation Theory in 1969, Lewis concentrated upon Carnap’s approach to confirmation theory because, he wrote, it appropriated what was good in competing approaches rather than ripping up the rule book and starting again (1969/2022, preface).18 But by the time Lewis was installed in Princeton, he’d come to judge ‘a chaos of new beginnings’ not a threat but the reality. ‘Let me tell you of an unpopular idea of my own…Philosophers tend to be too open-minded to persevere on programs that seem to have promise of success and philosophy is a chaos of new beginnings’ (Letter 434. To Kissling, 5 February 1973, in Beebee and Fisher 2020b: 34). And Lewis’s judgement didn’t subsequently shift. Ten years later, he wrote, ‘Compare the sorry state of philosophy: we’re always eager to listen to someone who offers to revolutionise philosophy, with the result that one proposal after another goes out of fashion without very thorough examination—only to be revived another year. We could scarcely tell the natural scientists that they’d be better off if they followed our example!’ (Letter 480. To Ziolkowski, 24 May 1983, in Beebee and Fisher 2020b: 127). We have argued that Lewis would have seen the introduction of grounding as another unwelcome beginning, only adding to the chaos—unwelcome, first of all, because supervenience and conceptual analysis and structuralism about set theory still have the promise of success.19

We have also argued that Lewis would have found the introduction of grounding to be unwelcome for a second reason: that these notions conflict with his Humean prohibition on necessary connexions and his antipathy towards metaphysical necessity and essentialism. Rosen has described recent developments as ‘the recrudescence of premodern metaphysics in postmodern philosophy’ (2015: 189). The word ‘recrudescence’ has two meanings. One, which Rosen clearly intended, is, ‘A revival or rediscovery of something good or valuable’. But the other meaning of the word, its more usual one, is, ‘The action or fact of breaking out afresh; a recurrence of a disease or medical condition, or of an undesirable state of things, bad feelings, etc., esp. after a period of quiescence or remission’ (OED). We have argued that Lewis would have conceived of the re-emergence of grounding as a recrudescence in the latter sense. Lewis conceived of J.S. Mill, the great empiricist, as one of his historical ancestors (Lewis 1991b/2022). It’s fitting that Mill, in his account of the historical origins and development of positivism (1865: 24), should also have lamented ‘The recrudescence of a metaphysical Paganism’ amongst the philosophers of Alexandria.20

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