The Metaphysics of Opacity

1. Introduction

Leibniz’s Law is the principle that a true identity claim involving genuine singular terms licenses substitution of the term on one side of the identity sign for the term on the other side salva veritate. It is prima facie a fundamental logical principle, present in all classical first- and second-order systems with an identity symbol.¹ Recently, however, two sophisticated challenges to Leibniz’s Law have emerged — one due to Michael Caie, Jeremy Goodman, and Harvey Lederman (2020) (henceforth CGL), the other due to Andrew Bacon and Jeffrey Russell (2019) (henceforth BR). Motivated by putative ordinary-language counterexamples, BR and CGL engage in logical revision: BR restrict quantifier rules, whereas CGL develop a system in which identity is coarser-grained than other definable equivalence relations. In this paper, we argue that their proposals are incompatible with the widely held assumption that the world contains individuals; moreover, if identity and a finest-grained equivalence relation are both present, there is strong theoretical pressure for them to coincide. The first part of our paper sets out these arguments; the second proposes a metaphysical picture — a version of stuff ontology — on which the rejection of individualism makes sense; finally, we provide a model theory that demonstrates the coherence of this picture.

That there is a connection among identity, individualism, and Leibniz’s Law should be unsurprising: Leibniz’s Law appears to capture the core theoretical role of identity precisely because it accounts for a thing’s being one thing, regardless of how many names refer to it. Moreover, even in languages without an identity symbol, an implicit mastery of the notion of identity is required in order to make sense of quantification: as Hawthorne (2003) remarks, to grasp the difference between the truth conditions of ∃x∃y(Fx ∧ Gy) and those of ∃x(Fx ∧ Gx), one must grasp the difference between one thing and two things.

For our purposes, we shall take the following to be the canonical form of Leibniz’s Law:

(LL)

where t₁ and t₂ are terms of type e (names, variables, expressions for the value of functions, or the like) and ϕ(t₁) and ϕ(t₂), as usual, represent the result of uniform capture-free substitution by the displayed term.

Nonetheless, violations of (LL) appear to abound. The most prominent cases of such putative failures include belief and attitude reports: it seems, for instance, that we cannot infer from

• (1)

  George Sand = Amantine Dupin

and

• (2)

  George Sand is believed by Madeline to be the author of the novel she is reading.

to

• (3)

  Amantine Dupin is believed by Madeline to be the author of the novel she is reading.

Representing (1) as s = d, (2) as Bs, and (3) as Bd, in a case where the inference fails, we have

• (4)

  s = d ∧ Bs ∧ ¬Bd,

a violation of (LL). We term putative counterexamples to (LL) such as (4) cases of opacity.

Philosophers have also argued that (LL) fails in the case of simple sentence pairs in a wide range of instances. For one example, consider the pair

• (5)

  Clark Kent walked into the phone booth and Superman walked out

and

• (6)

  Superman walked into the phone booth and Clark Kent walked out.

Prima facie, when applied to a single situation, these appear to differ in truth value (Saul, 2007), even though Superman is identical to Clark Kent — thus yielding a candidate counterexample to (LL).

Leibniz’s Law also comes under pressure from the ease with which it can be used to proliferate objects in metaphysics. It is frequently employed to argue for the non-identity of things that naive intuition might take to be identical (Magidor, 2011). Consider, for example, the following case from Fine (2003):

• (7)

  That statue is Romanesque

and

• (8)

  That clay is not Romanesque,

where the statue and clay occupy the same spacetime region, which is ostended to introduce the demonstratives. Fine urges us to conclude from this and similar arguments that the statue is not the clay. Those who eschew multiply occupied regions of spacetime must object to this use of (LL). But even statue/clay dualists may baulk at proliferating entities in cases of role-relative predication, which arise because a single entity can occupy two different roles. For instance, the Chief Justice of the United States is empowered to issue writs of habeas corpus, but the Chancellor of the Smithsonian Institution is not empowered to issue writs of habeas corpus. Now, it happens that the same man — John Roberts — plays the role both of Chief Justice and of Chancellor of the Smithsonian. Thus, it seems that:

• (9)

  Justice Roberts is empowered to issue writs of habeas corpus

and

• (10)

  Chancellor Roberts is not empowered to issue writs of habeas corpus.

Nevertheless, it would certainly be ontologically profligate to conclude that Roberts-the-Justice is not identical to Roberts-the-Chancellor.²

One might worry that the expressions in (7)–(8) and (9)–(10) are not genuine singular terms. Although demonstratives, such as ‘this statue’ or ‘this clay’ are often taken to be directly referential, with their reference fixed by ostension (Kaplan, 1989), this is not universally accepted. Moreover, titles, such as ‘Justice’ and ‘Chancellor’ seem to occupy an intermediate place between ordinary bare proper names and descriptions. But, at the cost of introducing slightly more backstory, it would be straightforward to modify these examples to use ordinary language proper names. We could introduce proper names for the statue and clay by ostension,³ and we can imagine a swearing-in ceremony’s including baptism with a new first name for each role.⁴ Once these names are established, sentence pairs with names corresponding to (7)–(8) and (9)–(10) would display the same behaviour as our examples.⁵

The tension between the theoretical importance of (LL) and the pervasiveness of intuitively compelling counterexamples to it has divided responses into two camps.⁶

Members of the first camp employ a variety of strategies to explain away the apparent counterexamples. For instance, Fregeans about proper names will argue that, in certain contexts, names do not denote their referents but rather their senses. Fregeans then use this to explain away apparent failures of substitutivity in such contexts: when the underlying logical form, rather than the surface structure, of the sentence is spelled out, there is no true failure (Frege, 1892; Church, 1951a, 1951b). Similarly, on Bertrand Russell’s view, ordinary-language proper names are not logically proper names at all but rather disguised descriptions, which are in turn abbreviations of quantified expressions, so natural-language apparent counterexamples do not threaten Leibniz’s Law (Russell, 1918, 1919a, 1919b, 1919c). Others, such as Nathan Salmon (1986) and Scott Soames (1989), argue that our initial judgments about truth values are mistaken and result from confusing the claim with a nearby but quasi-quotational statement.

Those in the second camp, by contrast, suggest modifying Leibniz’s Law, either by restricting the range of contexts to which it applies or jettisoning it entirely. For instance, Carnap (1956) restricts Leibniz’s Law to extensional contexts (those not containing modal operators) and introduces a separate principle for intensional formulas: □t₁ = t₂ → (ϕ(t₁) ↔ ϕ(t₂)).⁷ But, until recently, there have been few precedents for systems allowing failures of Leibniz’s Law for singular terms in formulas of arbitrary structure (as appears to occur in natural-language cases such as (1)–(10)).

Recently, however, CGL and BR have sought to address this lacuna by providing sophisticated formal discussions of systems in which (LL) fails. Both approaches are situated within a higher-order language in which classical identity — that is, identity that obeys (LL) — can be formally defined as Leibniz equivalence, which we will denote by ≈:

(LEquiv)

The central difference between BR’s and CGL’s approaches is that BR accept that (LEquiv) characterises genuine identity, while CGL dispute this. In order to maintain the equation of identity with Leibniz equivalence, BR modify both first- and higher-order principles of Universal Instantiation,

(UI1)

(UI2)

(where t is a first-order and T a higher-order term). Doing so permits violations of (LL) without allowing any instance of t₁ = t₂ ∧ ∃X(Xt₁ ∧ ¬Xt₂).⁸ In effect, their strategy is an implementation of positive free logic, with the quantifier restricted from the perspective of the metalanguage. Because they do not allow unrestricted second-order generalization, they can accept a failure of the inference in (1)–(3) as a true violation of (LL) and regiment it using (4) without accepting that there is a property by which Sand and Dupin differ.

CGL adopt a different strategy: they permit cases in which t₁ and t₂ flank the identity sign but nonetheless fail to be Leibniz-equivalent. The distinctive commitment of this view is what we term SUBMAXIMAL FINENESS:

(SUBMAXIMAL FINENESS) There exists a unique finest-grained equivalence relation, but identity is not it: genuine identity is a coarser-grained relation.

This allows them to retain full (UI1–2). They call their view Classical Opacity, because it preserves the core of classical higher-order logic, while allowing violations of (LL) (Caie et al., 2020, 532). For clarity, we term the coarser-grained relation denoted by the identity sign on their view weak associative equivalence (WA-equivalence), denoted by ≡: we can thus express the distinctive claim of the Classical Opacicist as the thesis that ≡, not ≈, expresses genuine identity.

BR and CGL offer contrasting justifications for their views concerning whether genuine identity must track Leibniz equivalence. BR argue that, if we allow genuine identity to float free of Leibniz equivalence, we will lose our grip on the subject matter (Bacon and Russell, 2019, p. 84).⁹ CGL (Caie et al., 2020, p. 540) respond by claiming that BR err in according too much evidential force to a theoretical desideratum—the requirement that identity be the finest-grained equivalence relation—rather than the common-sense data (which they take to include, for example, the claim that Hesperus is identical to Phosphorus and the claim that they vary in properties, in that one was known by the ancients to rise in the evening and the other was not). Because of these different desiderata, CGL and BR offer contrasting modifications to the classical picture: CGL argue that the identity relation is coarser-grained than Leibniz equivalence, whereas BR restrict the quantifier (relative to the ‘external’ realm of objects for which there are singular terms).

Neither BR nor CGL, however, offer any account of the underlying metaphysics of their systems for opacity—the way the world would have to be for their accounts to be right. They take the apparent linguistic counterexamples as data and develop candidate logics to accommodate them, building models to demonstrate consistency, but they do not explain the intended interpretations of their systems in metaphysical terms.

In this paper, we show that there is a tension between these systems incorporating opacity and widely held, intuitively compelling metaphysical principles. In particular, we show that the core commitments of a metaphysics of individuals conflict with accepting a case of (LL) violation, such as (4). First, in §2, we address the objection that (1)–(3) and similar cases are merely superficial and can be explained away metalinguistically. We then raise the objection that acceptance of (4) conflicts with adherence to a metaphysics of individuals. In order to avoid this result, one must adopt a highly revisionary ontology: either an aspect-based metaphysics, such as that proposed by Donald Baxter (2018) or an ontology on which individuals do not play a primary role. We then show that Baxter’s view cannot be used to solve the problem.

We next raise a metalogical objection to CGL’s attempt to pry apart Leibniz equivalence and identity: we argue that SUBMAXIMAL FINENESS is incompatible with standard accounts of counting and the logicality of identity. We concentrate on CGL here, because their denial that identity is Leibniz equivalence is a particularly novel and radical view and because their acceptance of classical quantifier rules makes their position particularly attractive to those who take logical conservatism as a definitive advantage. After concluding our argument against existing ways of denying (LL), we develop in §4 an alternative, positive proposal that accommodates violations of Leibniz’s Law within a metaphysically well-motivated system. On our view, in order to deny (LL) intelligibly, one must both downgrade the status of individuals and reject SUBMAXIMAL FINENESS by positing an infinite hierarchy of ever-finer-grained equivalence relations.

We propose an ontology of stuff that would motivate this combination of views. We do not mount a thoroughgoing case for this position, but we show that it provides a coherent metaphysical story. Although we do not develop the idea here, it is worth noting that this metaphysics has a claim to capture the intuition behind at least two historically prominent views — Anaxagoras’s “theory of extreme mixture” (Marmodoro, 2017) and Leibniz’s account of the phenomenal world as an infinitely variegated, infinitely divisible multitude.¹⁰

Finally, in the Appendix, we give a model theory that corresponds heuristically to a stuff metaphysics of this type.

2. Metalinguistic Leibniz’s Law

Our motivating cases, such as (1)–(3), might seem merely superficial and thus unworthy of extensive logical and metaphysical attention. After all, in the absence of any restrictions on allowable values of ϕ, t₁, and t₂, natural-language (LL) can fail in explicitly or implicitly quotational contexts, but these appear to be tangential, cheap cases. In this section, we show that a metalinguistic version of (LL), developed by Timothy Williamson (2002) to exclude such cases, does not provide an easy way to avoid CGL’s and BR’s challenges. CGL, in particular, are committed to genuinely identical objects differing in properties in the only sense of ‘property’ that they countenance (that is, as the value of the term generated by λ-abstraction from any open formula).

Let’s consider Williamson’s (2002, pp. 288–89) formulation, which uses sameness of semantic value on minimal pairs of variable assignments in place of intersubstitutability salva veritate. A (first-order) assignment for a language is a function mapping the language’s variables to objects; we use α, β, and variants for assignments. We say that β is an a₂/a₁-variant of α just in case α and β differ only in that there exists one variable x such that α(x) = a₁ and β(x) = a₂.

Williamson’s principle (slightly paraphrased) is:

• (CLL)

  For all a₁, a₂, if there exist assignments α and β such that β is an a₁/a₂-variant of α and a formula ϕ is true on α but not true on β, then a₁ ≠ a₂.

Quotational cases do not provide even arguable counterexamples to (CLL). For instance, ‘“George Sand” was chosen to trade on the gender expectations of nineteenth-century readers’ is true but the result of substituting ‘Amantine Dupin’ for ‘George Sand’ in this sentence is false. But there is no way to turn the sentence ‘“George Sand” was chosen to trade on the gender expectations of nineteenth-century readers’ into an open formula ‘x was chosen to trade on the gender expectations of nineteenth-century readers’ whose truth-value differs on a minimal assignment pair where one assignment takes x to George Sand and the other takes x to Amantine Dupin. (Of course, the truth-value would differ on a minimal assignment pair involving the expressions ‘George Sand’ and ‘Amantine Dupin’, but no one has ever claimed that the ten-letter expression ‘George Sand’ is identical to the fourteen-letter expression ‘Amantine Dupin’.)¹¹

As we will now show, CGL are committed to the existence of violations of (CLL) as well as (LL). To see this, note that they accept violations of the quantified principle

(ULL)

For instance, they take (4) to be a witness to the negation of (ULL). But any such case will provide a counterexample to (CLL) as well, for the truth of the universally quantified formula is defined in terms of the satisfaction of its instances on all assignments. Let α and β differ (if at all) only in that α assigns Dupin to x and Sand to y, whereas β assigns Sand to both. Consider the open formula ‘x is believed by Madeline to be the author of the novel she is reading’; on CGL’s view, this is false on α and true on β. It is unavailing to object that intensional predicates cannot be treated by the usual apparatus of assignments: CGL also are forced to accept that x ≈ y, i.e. ∀X(Xx ↔ Xy), which is free from all intensional vocabulary, is false on α and true on β.

This argument shows that Classical Opacicism violates even the metalinguistic (CLL). This creates a problem for the Classical Opacicist, for (CLL) appears to be what Williamson terms “a special case of a trivial theorem of classical mathematics”:

• (CMT)

  Let f and f∗ be functions on a domain D such that for some d ∈ D, f(e) = f∗(e) whenever d ≠ e. Suppose that some object x has R to f but not to f∗. Then, f(d) ≠ f∗(d). (Williamson, 2002, p. 289).

This theorem is indeed classically trivial, but it can only be derived by reasoning that CGL would reject. For it is proved from an extensionality assumption for functions:

• (11)

  If, for all e ∈ D, f(e) = f∗(e), then f = f∗,

and the principle

• (12)

  If there exists an object x that has R to f but not to f∗, then f ≠ f∗.

But (12) is itself a contraposed instance of Leibniz’s Law that CGL are obliged to reject, when = is understood as ≡, for f ≡ f′ → (Raf ↔ Raf′), and more generally f ≡ f′ → (Ff ↔ Ff′), has false instances in their system. Thus, CGL must reject certain statements of classical mathematics in full generality — and they have the resources, on their own terms, to motivate that rejection. But they can offer restricted versions of these statements that will hold when R is transparent — that is, when it does not induce a case of opacity. They suggest some principles that would allow canonical logical vocabulary to be transparent and they might likewise provide a restricted form of (CMT) that would hold for mathematical — i.e., purely extensional — functions (Caie et al., 2020, pp. 534, 536–40).

As for BR, the situation is more complicated because they accept (ULL) — it is a trivial variation of what they term LL(xy) — even though they countenance what seem to be counterexamples to it (Bacon and Russell, 2019, p. 101). (Rejecting unrestricted universal generalization allows them to hold these two positions consistently.) But they will still view (1)–(3) as providing a counterexample to a schematic version of (CLL),

• (CLL′)

  If β is a t₁/t₂-variant of α and a formula ϕ is true on α but not true on β, then t₁ ≠ t₂,

since the open formula Bx will be true on an assignment taking x to George Sand but not on one taking x to Dupin. They will thus have to reject the corresponding schematic instance of (CMT).

Since ordinary mathematics assumes full classical logic, and thus unrestricted universal instantiation, no distinction is standardly made between (CMT) and its schematic analogue. If BR wish to avoid restricting (CMT) in the way that CGL do, they will need to introduce a distinction exogenous to mathematical practice.

The preceding discussion shows that neither CGL’s nor BR’s systems can be accommodated within a conciliatory strategy on which their motivating cases merely involve violations of the letter but not the spirit of Leibniz’s Law. The metaphysical questions cannot be evaded.

3. Two Challenges for Metaphysical Opacity

We will now develop our two main negative arguments: the first is directed against the combination of (LL) violations with an intuitively appealing metaphysical picture, the second against views that claim that there is a finest-grained equivalence relation but identity is not it. In the first argument, we set out four principles about the relationship between language (on the assumption of reasonably perspicuous logical form) and the world that, we claim, are unavoidable consequences of the individualist picture of objects and properties. We challenge both CGL and BR to specify an alternative metaphysical picture on which the rejection of one of these principles makes sense.¹² The second argument applies to CGL in particular: we show that any system on which there is a finest-grained equivalence relation that is not identity must reject the logicality of identity and of the cardinality quantifiers on the standard criteria. We thus conclude that the only reasonable choice for the individualist is to accept a finest-grained equivalence relation that is identity and with it (LL); the alternative, as we discuss later, requires rejecting the individualist picture and the existence of any finest-grained equivalence relation.

4. Stuff Ontology

In order to avoid these two objections, the opponent of (LL) must reject both individualism and SUBMAXIMAL FINENESS. Is there any metaphysical picture on which these commitments make sense? We argue that a stuff ontology provides an attractive approach for the opponent of (LL), because the principles governing the division of stuff motivate both rejecting (S1) and denying the existence of a finest-grained equivalence relation. Our purpose is not to defend such a stuff ontology: we merely wish to elucidate it and show that it motivates a coherent anti-(LL) posture.

There have been a number of recent discussions of stuff ontologies (Zimmerman, 1997; Markosian, 2004; Kleinschmidt, 2007; McKay, 2015), which focus primarily on applying the notion of stuffs to solve coincidence problems.²³ Here, in contrast, we try to motivate a particular large-scale metaphysical picture, concentrating on the hierarchical property structure of the stuff-universe; it is this, on our view, that explains why stuffs can help the foe of (LL).

The thing/stuff distinction is motivated by the observation that natural languages contain not only count nouns, such as ‘table’, ‘cat’, and ‘electron’, but mass nouns, such as ‘water’, ‘wine’, and ‘cheese’. Count nouns admit indefinite and numerical determiners (‘a’/‘an’, ‘one’, ‘six’, ‘many’); mass nouns admit only mass determiners (‘some’, ‘much’). Clearly, in not every case does the count/mass distinction track something of metaphysical significance, and it is open to the orthodox individualist to provide paraphrases in a counting idiom at the level of logical form. For instance, she could explain reference to ‘some cheese’ in terms of a set of cheesy things, or a mereological sum of cheesy things, or a maximal portion of cheese, or some such account.²⁴ The stuff ontologist, however, thinks that these strategies do not succeed in all cases; at least sometimes, she maintains, that which mass nouns pick out has unique structural properties that no object-based approach can capture. It is not our purpose here to convince the dedicated defender of orthodoxy, but we think there is value in exploring the metaphysics that results from taking our pervasive non-individualistic natural language commitments as a guide to underlying ontological structure.

The key distinctive property of stuff is that it cannot be counted. We can say that there are six chairs in the room, but we cannot literally say that there are four waters or five woods. At most, this is elliptical for ‘four units of water’ or ‘five kinds of wood’: in such cases, we count not stuff itself but portions or kinds of stuff.

We can leave open whether, in these ordinary examples, a reduction to an enumerable object (in the sense that the stuff is nothing over and above that object) is possible. With some mass nouns it clearly is: although we cannot ask how many ‘furnitures’ there are in the room, ‘furniture’ seems to refer to a derivative entity that can be exhaustively ontologically accounted for through such things as chairs and tables. With water and wine the case is less clear: perhaps the water is nothing over and above the water molecules, or perhaps it is composed of them without being ontologically reducible to them.

But, crucially, the serious stuff ontologist maintains that at least for some stuffs—fundamental stuffs—there is no reduction to things. And the very serious stuff ontologist maintains that these fundamental stuffs are what ultimately make up the world.

It is this sort of very serious stuff ontologist who we think is in a position to reject (LL). She starts from the observation that fundamental stuff requires a new quantifier. The existential quantifier, ∃, is informally taken to stand for ‘there is a …’, which results in a conflict with the grammar of mass nouns: while we can say ‘there is a table’ and symbolise this as ∃x Tx, we cannot symbolise ‘there is cheese’ as ∃x Cx, for this would be to say that there is a discrete individual x that is a certain way, but ‘there is cheese’ is different from ‘there is a portion of cheese’ or ‘there is a kind of cheese’, or, indeed, any formulation that would be committed to one thing that is cheese.

This argument has not been unopposed: Ned Markosian 2015, p. 685 has claimed that the grammatical feature is superficial: we can use the existential quantifier indifferently for both stuff and things, allowing the natural language paraphrase to vary. On our view, however, Markosian fails to take into account the link between quantification and counting. If we accept that ‘there is both hard cheese and non-hard cheese’ can be perspicuously regimented as ∃x∃y(Cx ∧ Cy ∧ Hx ∧ ¬Hy), then, on the standard account of quantifiers and counting, ‘how many cheeses?’ questions will clearly be coherent, and it is precisely this which the stuff ontologist must reject.

We can think of the serious stuff ontologist as speaking a language that includes, in lieu of ordinary existential and universal quantifiers, primitive stuff quantifiers, which form sentences from terms for stuff-kinds without using bindable variables: read ${\displaystyle \sum W}$ as ‘there is some wine’ and ∏ C as ‘everything is wine’. Optional location parameters can be added to express claims like ‘there is some wine at Karla’s’ $\left({\displaystyle \sum }^{\alpha }W\right)$; we can also introduce measure quantifiers, which correspond to numerical quantifiers for objects. Thus, the existence of at least two litres of wine at Karla’s entails the existence of at least one litre of wine at Karla’s: ${\displaystyle {\sum }_{\ge 2\text{L}}^{\alpha }\text{W}}⊢{\displaystyle {\sum }_{\ge 1\text{L}}^{\alpha }\text{W}}$. (The fact that stuff ontology allows for a natural treatment of these inferences is one of its theoretical advantages and provides a motivation for distinguishing between portions of stuff and genuine objects.)

We have used mundane examples — wine, cheese, and the like — to explain the framework, but the very serious stuff ontologist’s fundamental stuffs may look quite different: they may be the fields postulated by physics or perhaps a neutral monist stuff with both a physical and mental aspect or a stuff of some other sort. (A positive defence of the view would have to say something about this question; for our purposes, however, we can remain neutral.)

In principle, a variety of views about kinds of stuff and their interdependence are compatible with very serious stuff ontology. But there is a general theoretical impetus, on grounds of elegance and simplicity, towards reducing fundamental ontological commitments and hence towards priority monism (Schaffer, 2010) that applies particularly strongly to stuff ontology. The thing ontologist faces countervailing pressure from the existence of clear boundaries between discrete objects; it is characteristic of fundamental stuff that it has no such boundaries. Once we have moved from thinking of discrete things to some stuff, it is a short step to the view that all unbounded stuff is derived from a single, variegated basic world-stuff.

If the basic world-stuff is variegated, then there will be subkinds of stuff, corresponding to each of the ways in which it varies. (As we use the term here, a ‘subkind’ of stuff is all the stuff of that particular type — some but not all of the variegated stuff — not a universal or other abstract object used to categorise the stuff.²⁵)

The subkinds might be related to the world-stuff in a number of ways. There could be two levels: a fundamental world-stuff and then a plethora of subkinds, all on a par. But this would make the world-stuff very unlike our familiar stuffs, which come in hierarchies of determination. For instance, wine is a subkind of liquid, claret a subkind of wine, Graves a subkind of claret, and so on; important structure would be lost if all of these divisions were posited at once. It thus seems better motivated to assume that at the first stage the world-stuff would be partitioned into a few, relatively fundamental kinds; these kinds would then be further divided in turn, and so on. The process might culminate in atomic kinds — kinds that admit no subkinds — or it might go on forever; the serious stuff ontologist endorses the latter view.

The serious stuff ontologist thus claims that stuff is never perfectly homogeneous; however finely we separate out kinds of stuff, what remains is still variegated. In effect, she adopts Leibniz’s picture of an infinitely divisible universe of limitless qualitative diversity (though without the underlying atomic realm of monads he takes as its metaphysical basis). This picture has two consequences: first, stuffs are infinitely numerous; second, the relations of stuff to substuff always go infinitely deep: there is no final level of substuff. Were there a terminal subkind, it would parcel out homogeneous stuff. Because she views stuff as ineradicably mixed, the serious stuff ontologist has a natural affinity for a conception of the hierarchy as ‘indefinitely extensible’ (Dummett, 1963; 1991) rather than given all at once. The serious stuff ontologist views the stuffs in the hierarchy as very different from individual substances as conceived on the pegboard model of ontological structure: whereas substances would play the role of pegs, stuffs are constituted by their qualitative properties.²⁶

In addition to the primary hierarchy of kinds, stuff is secondarily divided into portions as well: for instance, stuffs derivatively form portions such as a glass of water or a decanter of wine.²⁷ For the serious stuff ontologist, mere portions of stuff are not metaphysically privileged in the way that kinds of stuff are: kinds are explanatorily fundamental and account for the constitutive structure of the world, but portions are merely accidental divisions.

The kind hierarchy and the portion hierarchy cut across one another: a portion of stuff may contain many kinds of stuff, and a kind of stuff may comprise numerous scattered portions. For any carving of the world-stuff into kinds, each kind can be portioned out, and, correspondingly, for each portion of world-stuff, we can separate out different kinds within it.

These portions have the status of quasi-things: we can refer to them using singular terms, but unlike the substance ontologist’s notion of a genuine thing, which is always no more and no less than one thing, portions come in amounts. For the very serious stuff ontologist, all singular reference must ultimately be explained through these portions: she does not eschew the ordinary ∃ and ∀, but she views facts expressible using them as derivative on those facts we capture with ∑ and ∏.

The infinitely descending hierarchy of qualitative properties is the only fundamental posit for the very serious stuff ontologist. As a consequence, all relations must be constructed out of monadic, qualitative properties. In particular, every equivalence relation is exactly as fine as the available configurations of monadic properties on some level. So there can be no finest-grained equivalence relation, since there is no ultimate level in the hierarchy of stuffs and substuffs. But for the very serious stuff ontologist, who admits no individuals prior in nature to the stuffs, identity just is the equivalence relation on portions induced by a partitioning of stuff at a certain level.²⁸ So, for every relation that can serve as a candidate for identity, there will be a stronger one, corresponding to a level further down in the hierarchy.

This picture thus meets the two desiderata set out at the beginning: the very serious stuff ontologist can reject (S1), because she is not committed to individuals figuring in states of affairs. Nevertheless, she is free to use singular terms to designate quasi-individuals, portions of stuff. Using the qualitative divisions available at a level to form equivalence classes, she can then introduce an identity symbol to express that portion a is within the same level-relative equivalence class as portion b. Thus, Leibniz’s Law is not a mere vacuity. These identity statements, however, will always be evaluated relative to some bounded depth in the hierarchy. For this reason, violations of (LL) may arise: at any level ℓ, a portion a of stuff may be ℓ-identical to a portion b of stuff (i.e., they stand in the finest-grained equivalence relation existing at ℓ), even though a and b can be differentiated by properties that arise only at a level beyond ℓ.

The very serious stuff ontologist will give something like the following story to explain how singular reference comes about: I ostend a particular portion of stuff, and pick it out by one of its kinds, for instance as ‘this watery stuff.’ I then baptise the portion so picked out by saying: ‘This watery stuff is named “Lake Ontario.”’ The content of subsequent uses of ‘Lake Ontario’ will then be dthat (this watery stuff). In something like this manner, singular reference can come about through apportionments of stuff.

This account of singular reference explains how the singular terms a and b come to denote portions of stuff that are level ℓ-identical but perhaps distinguishable at a level beyond ℓ. For instance, consider a particular portion of liquid that is both winey and watery. (We use these in our toy example as placeholders for ultimate stuffs; real wine and water, of course, are better thought of as a mixture.) Applying the account above, we can imagine that we have come to name this portion of liquid in two ways: first, we might use the definite description ‘this watery stuff’ and form a singular term dthat (this watery stuff); then, we might use ‘this winey stuff’ and form the singular term dthat (this winey stuff). We now have two singular terms referring to this portion of liquid, derived from the conceptual distinction between watery and winey stuff. If, however, this conceptual distinction does not correlate to a distinction among qualitative properties present at level ℓ, then dthat (this watery stuff) will be ℓ-identical to dthat (this winey stuff), but they will be distinct at the first level at which watery stuff is distinguished from winey stuff.

Although it is not our goal to give a positive defence of stuff ontology, it is worth discussing a key potential objection. It might be claimed that successful physical theories, such as the standard model of particle physics, involve prima facie commitment to individual particles and hence an individualistic ontology; thus, the very serious stuff ontologist runs afoul of the naturalistic constraint that metaphysics should reflect the deliverances of the natural sciences.²⁹

We first note a methodological problem with the objection: the quantum field theory in which the standard model is couched is an effective field theory and does not purport to track fundamental structure; in fact, since a high-energy cutoff is built into QFT, it would violate the theory’s own presumptions to try to apply it at sub-Planckian scales. And, as David Wallace (2011, p. 211) has noted:

Whatever our sub-Planckian physics looks like (string theory? twistor theory? loop quantum gravity? non-commutative geometry? causal set theory? something as-yet-undreamed-of?) there are pretty powerful reasons not to expect it to look like quantum field theory on a classical background spacetime. As such, what QFT (of any variety) says about the nature of the world on length scales below ∼ 10⁻⁴³ m […] doesn’t actually tell us anything about reality.

It should be emphasised that the problem is not merely that the standard model is not final physics. Rather, the standard model itself says that we cannot coherently extend it to arbitrarily small scales: unlike the theory of Newtonian gravity (for example), it is not even a complete picture of a world with different laws from our own.

Nonetheless, we do not find the objection compelling even as applied to the standard model. Following David Baker (2016), we can distinguish two broad families of interpretations of QFT — ‘particle-theoretic’ and ‘field-theoretic’. Particle-theoretic interpretations vary in the extent to which the core commitments of the classical notion of ‘particle’ remain intact.

Even the most stringently particle-theoretic interpretations, however, preserve little of the intuitive conception of particles as localised individuals that can be counted. Localisation is ruled out by no-go theorems (Malament, 1996; Halvorson and Clifton, 2002). As for counting, given a specific Fock space representation of a system in QFT — the most hospitable framework for particle interpretations — there will, in general, be a well-defined ‘particle number observable’, but the system may be in a superposition of this observable. This already rules out any direct equation between the particle ‘number’ operator and a literal count of individuals as specified using existential quantification.

Moreover, this hospitable framework is often unavailable. Even in the toy case of a free (noninteracting) boson field, Minkowski and Rindler representations — corresponding to inertial and accelerated observers, respectively — have different particle number observables, and each representation interprets the other’s observable as yielding an expectation value distinct from its own (Clifton and Halvorson, 2001; Ruetsche, 2011, pp. 191–219). As Baker (2016, p. 7) remarks,

Since there is nothing physically privileged about either observer’s definition, it must be that the number of particles is a perspective-dependent fact. If particles belonged to QFT’s fundamental ontology, the number of particles would not be dependent on one’s perspective — here we assume, plausibly, that the number of fundamental entities in the universe is an objective fact. So if this argument succeeds, it rules out fundamental particle interpretations.

Still worse, it is not always the case that there are Fock space representations at all: Doreen Fraser (2008) has shown that none exist for a large class of interacting theories, a result which, as Baker (2016, pp. 8–9) notes, undermines even very weak interpretations that attempt to find a place for particles in non-fundamental, much less fundamental, ontology.

In contrast, field-theoretic interpretations abandon any commitment to particles, even in a minimal sense, and instead take the physical realm to consist in properties of spacetime points or regions (Baker, 2016, p. 9). There are several proposals to implement this idea: the simplest (‘wavefunctional’) field-theoretic interpretations treat the quantum field as (roughly) composed of configurations of classical fields, represented in a complex-valued Hilbert space; as such, their ontological commitments are not qualitatively different from those of classical field theories. While standardly formulated using spacetime points, there are alternative, empirically equivalent formulations of classical field theory using gunky spacetime (Arntzenius, 2003, 2008). Purged of commitment to spacetime points, classical field theories would provide a hospitable setting for very serious stuff ontology. Baker (2009) does argue that wavefunctional interpretations confront a difficulty parallel to that faced by particle interpretations, but his preferred alternative approach, which uses algebras of operators, remains fundamentally field-theoretic (in that it assigns properties to spacetime regions) and thus does not present any new challenges to very serious stuff ontology.

Of course, this is not a demonstration that stuff ontology is compatible with physics; such a result must await the development of a theory integrating gravity and quantum field theory. But at present we have no reason to suspect that our basic scientific commitments rule out a stuff ontology.³⁰

This concludes our sketch of a principled metaphysical stance on which violations of (LL) would be countenanced without falling afoul of the arguments developed in §3. At this point, however, it is worth asking whether a consistent theory can be developed that would enable us to reason about stuffs without reintroducing a finest-grained equivalence relation. In the next section, we show that this is possible using a toy theory: we describe a class of hierarchical models and provide a semantics, complete with level-relative notions of evaluation, for a second-order language with identity over those models. These models are not fully faithful to the very serious stuff ontologist’s metaphysics, for they are (as is standard) constructed in set theory, and thus built out of sets of individuals. But they nonetheless share the stuff ontologist’s decisive commitment: there is no way to single out individuals from within the object language. The toy semantics thus provides a mathematically tractable way to demonstrate the internal coherence of the stuff ontologist’s account.

5. Model Theory for Stuff

The central idea behind our toy model for stuffs is to employ infinite sets to simulate stuff. Nothing in the very serious stuff ontologist’s fundamental ontology corresponds to the elements of this set: they are merely an artefact of the model theory, but they give us a structure that captures the infinite division of stuff.

For convenience, we build the model over the natural numbers, but any infinite collection could be taken as a first-order domain.³¹ The fundamental requirements are that the second-order domain (standing in for the hierarchy of kinds) contain not every subset of the first-order domain, but only carefully chosen sets: every set in the second-order domain is infinite (or empty), sets subdivide as we progress down the levels, and any two distinct elements are ultimately separated on some level. In this way, we obtain a progressive winnowing of sets corresponding to the infinite divisibility of stuff: at a given level, only those sets are available which correspond to the kinds produced at that level.

We start from the set of all natural numbers, standing in for the world-stuff, and we partition at each level by separating out the even-numbered from the odd-numbered elements (under the obvious ordering). The zeroth level contains just {0, 1, 2, 3, …}; the first, {1, 3, 5, 7, …} and {0, 2, 4, 6, …}. The second level contains four sets, {1, 5, 9, 13, …}, {3, 7, 11, 15, …}, {0, 4, 8, 12, …}, and {2, 6, 10, 14, …}. The third contains eight infinite sets, the fourth sixteen, and so on ad infinitum.

In order to turn the hierarchy of levels into a domain containing the values of monadic predicate expressions (modelling kinds of stuff, whether these are cheese, wood, and wine, and so on, or something more esoteric), we take the union of all the levels and close under complements and finite unions and intersections. We do not close under countable unions and intersections, for this would lead to singletons: we do not wish the model to be able to discriminate out any nonempty sets with finitely many members. (The very serious stuff ontologist admits no divisions that cannot be further divided.)

Thus we have a first-order domain, D₀, which is just $\omega$, and a second-order singulary domain, $D_1^1$, formed of finite unions and intersections and complements of the subsets of $\omega$ constructed at various levels by the splitting procedure. We now need relational domains $D_1^n$ for n > 1; crucially, $D_1^2$ contains all the two-place relations we accept, and thus all the equivalence relations: these provide the only acceptable candidates for the identity relation. Here the idea is that relations should cut no finer than properties: everything in $D_1^n$ can be constructed from a collection of sets in $D_1^1$, one for each argument place, and no two individuals which the sets do not distinguish can be distinguished by the relation. Crucially, each equivalence relation will be constructed from a finite number of sets in $D_1^1$. As a result, there will always be a level ℓ such that the relation cuts no finer than the partition on the natural numbers induced at that level. Since we do not close $D_1^1$ under infinite intersections, there is no way to construct a ‘level-$\omega$’ equivalence relation that would capture all of the divisions: we are limited to an infinite chain of equivalence relations, each of finite level.

The collection of domains forms a fixed frame. We use the frame to provide a semantics for a second-order language $\mathcal{L}\text{ : }\mathcal{L}$ has firstand second-order variables, possibly constants, the usual connectives and quantifiers, and equality. A model for $\mathcal{L}$ is constructed from the frame by adding an interpretation: the interpretation assigns to each first-order constant an element of D₀ and to each second-order constant an element of the appropriate $D_1^n$. Similarly, a variable assignment on the model maps first-order variables to elements of D₀ and second-order variables to elements of the corresponding $D_1^n$. The values of second-order variables on an assignment may occur at any level in the hierarchy.

We interpret formulas in L relative to two level parameters: a formula holds or does not hold in a model, on an assignment, at an equality-level ℓ₁ and at a quantifier-level ℓ₂. Specifically, the formula ⌜t₁ = t₂⌝ holds on a given assignment at equality-level ℓ₁ just in case the values of t₁ and t₂ on that assignment are not separated by the finest-grained equivalence relation constructible at level ℓ₁. Similarly, ⌜∀X ϕ⌝, with X a second-order variable, holds at quantifier-level ℓ₂ just in case ϕ holds on every relevant assignment mapping X to a value located at or above level ℓ₂. (First-order quantifiers are treated standardly. In contrast to bound second-order variables, there are no level restrictions on free second-order variables.)

Evaluating identity and second-order quantified formulas relative to a level captures the serious stuff ontologist’s commitment to the indefinite extensibility of the stuff hierarchy. For her, it makes no sense to evaluate a formula ‘from the outside’, independently of levels, for the hierarchy is not given all at once: it develops as the variegated stuff is separated out into kinds and subkinds.

The two level parameters give us a variety of well-behaved notions of truth-in-a-model and validity. (And logical consequence, although for simplicity we consider only the single-formula case.) The notions collapse for identity-free first-order sentences, but the differences are crucial for sentences with identity or second-order variables. We introduce six such notions in the appendix; here we explain the four most important (holding simpliciter, holding on pairs, holding on pairs pointwise in the limit, and holding on pairs uniformly in the limit).

A formula holds simpliciter in a model if it holds on all assignments at every equality-level and every quantifier-level; it is valid simpliciter if it holds simpliciter on all models.

This is the most demanding notion, and Quantified Leibniz’s Law,

(QLL)

fails to be valid simpliciter. Consider a model and assignment on which t₁ denotes 4, t₂ denotes 6, and the formula is evaluated at quantifier-level 2 and equality-level 1. The finest-grained equivalence relation extant at level 1 treats {0, 2, 4, 6, …} as a partition cell; but 4 and 6 are separated by the set {0, 4, 8, …}, which is available at level 2 to be the value of the quantified variable X.

Since (QLL) is not valid simpliciter, the schematic version of Leibniz’s Law,

(LL)

is also not valid simpliciter.

In addition, second-order existential generalisation (and its dual, universal instantiation),

(EG2)

also fails: consider a model and assignment where T has as its semantic value a set at level 10 of $D_1^1$, but the formula is evaluated at a quantifier-level of 5, and no set at level 5 or above of $D_1^1$ satisfies the open formula Φ.

Another natural, but less demanding, semantic notion is holding on pairs: a formula holds on pairs in a model if it holds on every assignment in every circumstance where the equality-level is the same as the quantifier-level; a formula is valid on pairs if it holds on pairs in every model.

Since the identity-level was irrelevant to the counterexample, (EG2) still fails to be valid on pairs. (LL) is also not valid on pairs: if we continue to let t₁ denote 4 and t₂ denote 6, and we evaluate at equality-depth and quantifier-depth 1, the instance t₁ = t₂ → (Xt₁ ↔ Xt₂) will fail when the free variable X is assigned a set below level 1 that splits 4 and 6.

But (QLL) is valid on pairs. A proof is given in the appendix, but the reasoning is clear: when the quantifier and equality levels move in tandem, no set that is available to be the value of a bound second-order variable can separate a pair unless an equivalence relation available at that level already separates them.

A formula holds on pairs uniformly in the limit in a model if there exists some ℓ such that for every ℓ′ > ℓ, for every assignment, the formula holds in the model on that assignment at identity- and quantifier-level ℓ′. A formula holds pointwise on pairs in the limit in a model if, for every assignment α, there exists some ℓ such that for every ℓ′ > ℓ, the formula holds on α in the model at identity- and quantifier-level ℓ′. A formula is pointwise (resp. uniformly) valid on pairs in the limit if it holds pointwise (resp. uniformly) on pairs in the limit in all models.

Since validity on pairs requires that a formula holds on all pairs, whereas both limit notions require only that the formula holds on certain pairs, any formula that is valid on pairs will be both pointwise and uniformly valid on pairs. Thus (QLL) is both pointwise and uniformly valid on pairs in the limit. Neither (LL) nor (EG2) is uniformly valid on pairs in the limit, since for any candidate ℓ there will be some assignment α falsifying the conditional using a value below ℓ.

But (LL) and (EG2) do hold pointwise on pairs in the limit: for every assignment α we can find an equivalence relation that distinguishes everything that can be distinguished by finitely many values of variables on that assignment: when this equivalence relation is taken to be identity, (LL) holds. Similarly, when the quantifier is evaluated at this level, (EG2) holds.

It is important to be clear about the limitations of these notions: they all represent ‘external’ perspectives on the totality of levels, and thus — despite their heuristic utility — none of them represents what an inhabitant of the serious stuff ontologist’s world can directly access. Nonetheless, even as representational aids, there is a sense in which the notions of holding simpliciter and holding on pairs are more faithful to the programme of stuff ontology than the limit notions. All notions of truth-in-a-model require quantification over all levels (or pairs of levels), but holding simpliciter and holding on pairs permit truth-at-a-level (or pair-of-levels) to be evaluated purely locally at the first step. In contrast, the limit notions require a potentially unbounded search through finer and finer levels before even this step can be completed.

Hence, (LL) (and, on one variant, (QLL)) fail on our toy models when we use the semantic notions that make the most sense for the system. Unlike CGL’s framework, however, our models do not deliver the existence of a finest-grained equivalence relation. These models demonstrate the consistency of an approach that denies (LL), while endorsing the conditional claim that if there is a finest-grained equivalence relation, it must be identity.

The model provides a guide to the proper interpretation of identity claims about portions of stuff from the perspective of the very serious stuff ontologist. In general, from the claim that this portion of stuff is identical to that portion of stuff, it will not follow that if this portion of stuff is some way, then that portion of stuff is that way as well. Consider, for instance, the case of a portion of liquid, composed of a mix of wine and water, which we pick out twice by direct reference (through rigidified descriptions, the first based on its winey nature and the second on its watery nature): ‘this stuff is that stuff’ is true at an identity-level above that on which water and wine are separated and false below that level. Even at a level on which the identity claim does hold, this stuff will still be distinguished from that stuff by a property: whether that property is available to be quantified over will depend on the quantifier-level.

We do not claim that the model theory provides a royal road to understanding stuff ontology: there will always be a gap between what we can represent using set-theoretical constructions and what the very serious stuff ontologist is committed to. But we think that the presentation given here fleshes out some of the formal behaviour that the very serious stuff ontologist should expect of her identity connective.

6. Conclusion

Our purpose in this paper has been twofold: we have offered both a critique of current work that endorses genuine Leibniz’s Law violations and a positive suggestion for a metaphysical picture on which such violations make sense.

The work of BR and CGL represents the best-developed recent attempt to allow for Leibniz’s Law violations in a systematic way. In particular, CGL’s classical opacicism promises an appealing combination of orthodox logic and nontrivial cases of opacity. But, as we argued in §3, classical opacicism and its variants are besieged by problems: they lack a reasonable response to the metaphysical challenge of the S-argument and they leave their proponents in the untenable position of simultaneously accepting a finest-grained equivalence relation and denying that it is identity.

The proponent of the stuff ontology we sketch in §§4–5 faces none of these problems: she motivates Leibniz’s Law violations by pointing to the infinite qualitative heterogeneity of stuff. On her ontological picture, the world is infinitely qualitatively heterogeneous. Our identity statements give rise to (LL) violations because our attributions of identity only hold relative to a differentiation into properties that is always subject to finer-grained division. Stuff ontology represents the best alternative of which we are aware to a fully classical, Leibniz’s Law-endorsing treatment of identity.