Relations

First published Tue Feb 9, 2016; substantive revision Fri Sep 5, 2025

The world we inhabit isn’t an undifferentiated bog. Everywherethere’s repetition and, importantly, we can even distinguishdifferent types of repetition. We see onecat and thenanothercat. But we can also see that our cat ison topof the mat and subsequently notice that the cat from next door ison top of the fence. Repetition in the first sense requiresonly one thing and then another. By contrast, repetition in the secondsense requires two (or more) things and then two (or more) otherthings. Properties are typically introduced to account for repetitionin the first sense, whereas relations are typically introduced toaccount for repetition in the second sense. There’s catrepetition because there is more than one thing that has a certainproperty, the property of being a cat. There’s above-and-belowrepetition because there is more than one thing that bears a certainrelation to something else, the relation that holds between two thingswhen one is on top of the other.

It’s doubtful whether the distinction between properties andrelations can be given in terms that do not ultimately presupposeit—the distinction is so basic. Nevertheless, there areelucidations on offer that may help us better appreciate thedistinction. Properties merely holdof the things that havethem, whereas relations aren’t relationsof anything,but holdbetween things, or, alternatively, relations areborne by one thingto other things, or, anotheralternative paraphrase, relations have a subject of inherencewhose relations they are and terminito which theyrelate the subject. More examples may help too. When we say that athing $A$ is black, or $A$ is long, then we are asserting thatthere is some property $A$ has. But when we say that $A$ is(wholly) inside $B$ then we are asserting that there is a relationin which $A$ stands to $B$.

Be careful though not to be misled by these examples. $A$ can onlybe (wholly) inside $B$ if $A$ is distinct from $B$. So therelation in which $A$ stands to $B$ if $A$ is (wholly) inside$B$ requires a distinct subject from terminus. By contrast, $A$can be black without prejudice to anything else. But we cannot inferstraightaway that every relation holds between more than one thingbecause there may be some relations that a thing bears to itself, if,for example, identity is a relation. So we cannot distinguishproperties from relations by appealing to the number of distinctthings required for their exhibition, since these may be the same,viz. one. Hence the plausibility of thinking that a relation differsfrom a property because a relation, unlike a property, proceedsfrom a subjectto a terminus, even if the subjectand terminus are identical.

Theories of relations add to, subtract from or qualify this basicpackage of ideas in various ways. But with the distinction betweenproperties and relations in hand we can already do some worthwhileconceptual geography, distinguishing four regions of logical space.(1) Rejection of both properties and relations. (2) Acceptance ofproperties but rejection of relations. (3) Acceptance of relations butrejection of properties. (4) Acceptance of both properties andrelations. (1) amounts to a form of thoroughgoing nominalism and istypically motivated by a blanket aversion to the doctrine that generalwords like “black” or “before” refer toworldly items, whether properties or relations. By contrast, (2), (3)and (4) amount to different forms of realism, whether about propertiesand/or relations. Typically they are motivated either by an especialantipathy toward relations (2) or, alternatively, an appreciation oftheir distinctive utility ((3) and (4)). Here we will focus upon whatit is about relations in particular that makes philosophers eitherlove them or hate them.

1. Preliminary Distinctions

To understand the contemporary debate about relations we will need tohave some logical and philosophical distinctions in place. Thesedistinctions aren’t to be taken for granted. Part of thedevelopment of the debate has consisted in the refinement of preciselythese distinctions. But you need to understand how, relativelyspeaking, things got started. (For introductions to the metaphysics ofrelations see Heil 2009, 2021 and MacBride 2024b).

To begin let’s distinguish between the “degree” or“adicity” or “arity” of relations (see, e.g.,Armstrong 1978b: 75). Properties are “one-place” or“monadic” or “unary” because properties areonly exhibited by particulars or other items, e.g., properties,individually or one by one. Relations are “many-place” or“$n$-adic” or “$n$-ary” (where $n\gt 1)$because they are exhibited by particulars only in relation to otherparticulars. A “2-place” or “dyadic” or“binary” relation is exhibited by one particular only inrelation to another. A “3-place” or “triadic”or “ternary” relation is exhibited by one particular onlyin relation to exactly two others. And so on. Some examples. Therelation that $x$ stands to $y$ whenever $x$ isadjacent to $y$ is binary. This is because it takes twothings to be adjacent to one another. The relation that $x$ standsto $y$ and $z$ whenever $x$ isbetween $y$ and $z$is ternary because it takes three things for one to fall between theother two.

In these terms, a relation can be distinguished from a“relational property”. Whereas a relation is borne by onething to another thing (possibly, in certain cases, to the samething), a relational property is the property of bearing a relation tosomething. Suppose $a$ bears a relation $R$ to $b$. Then $a$has the relational property of bearing $R$ to $b$ whilst $b$ hasthe property of $a$’s bearing $R$ to it. Here $R$ isbinary whilst the relational properties are unary because they are hadby single things, $a$ and $b$ respectively. For example, marriageis a relation between two people but being married to Cleopatra is arelational property had by Antony whilst being married to Antony is arelational property had by Cleopatra.

A “unigrade” relation $R$ is a relation that has adefinite degree or adicity: $R$ is either binary or ternary…or $n$-ary (for some unique $n$). By contrast a relation is“multigrade” if it fails to be unigrade (the expression isowed to Leonard & Goodman 1940: 50). Putative examples includecausal, biological, physical, geometrical, intentional and logicalrelations. Causation appears to be multigrade because a certain numberof events may be required to bring about an effect on one occasion,and a different number of events may be required to bring about aneffect on another. Similarly, entailment appears to be a multigraderelation because a certain number of premises may be required toentail one conclusion but a different number of premises to entailanother. Whether there are any multigrade relations and exactly whatit means to be multigrade is a fraught matter (see Armstrong 1978b:93–4, 1997: 85, 2010: 23–5 and Dixon 2019: 64–7 forarguments against multigrade relations and Morton 1975, Fine 2000: 22,MacBride 2005: 569–93 and Hossack 2007: 45–55 forarguments in favour of them).

Next, let’s draw a preliminary three-fold distinction betweenbinary relations in terms of their behaviour with respect to thethings they relate. A binary relation $R$ issymmetric iffwhenever $x$ bears $R$ to $y$, $y$ bears $R$ to $x$. Bycontrast, $R$ isnon-symmetric iff $R$ fails to besymmetric.Asymmetric relations are a special case ofnon-symmetric ones: R is asymmetric iff whenever $x$ bears $R$ to$y$ then $y$ does not bear $R$ to $x$. So whilst allasymmetric relations are non-symmetric, not all non-symmetricrelations are asymmetric. Some examples. Whenever $x$ is married to$y$, $y$ is married to $x$. So the relation in which $x$stands to $y$ when $x$ is married to $y$ is a symmetricrelation. But if $x$ loves $y$, it isn’t guaranteed that$y$ loves $x$. Because, alas, sometimes love is unrequited. So therelation in which $x$ stands to $y$ when $x$ loves $y$ is anon-symmetric relation. But love doesn’t have to go unrequitedotherwise far fewer people would get married, only cynics and golddiggers. So this relation isn’t asymmetric. By contrast, therelation in which $x$ stands to $y$ when $x$ is taller than$y$ is asymmetric because if $x$ is taller than $y$ then $y$isn’t taller than $x$. This doesn’t exhaust the logicalclassification of relations. It’s easy to see that furtherdistinctions will need be drawn. Consider, for example, thebetween relation. It’s a ternary relation that issymmetric in the sense that if $x$ is between $y$ and $z$ then$x$ is between $z$ and $y$. But thebetween relationisn’tcompletely orstrictly symmetric becauseit’s not the case that if $x$ is between $y$ and $z$ then$y$ is between $x$ and $z$. So the distinction between symmetricand non-symmetric relations for $n$-ary relations, where $n \gt2$, will need to be qualified. (In what follows we restrict ourselvesto the more straightforward cases discussed in the philosophicalliterature on relations. But see Russell 1919: 41–52 for a morewide-ranging discussion of the logical variety of relations.)

Nevertheless, equipped with this three-fold distinction for binarydistinction we can now appreciate the significance of one contemporarydebate about relations. Our ordinary and scientific views of the worldare replete with descriptions of things that are non-symmetrically orasymmetrically related. For this reason, Russell advocating admittingboth non-symmetric and asymmetric relations alongside symmetric ones(Russell 1903: §§212–6, 1914: 58–9, 1924: 339).But, more recently, it has been questioned whether it is necessary toadmit non-symmetric and asymmetric relations at all, i.e., it has beenargued that the world contains only symmetric relations by Armstrong1997: 143–4; Dorr 2004: 180–7; Simons 2010: 207–1,Button 2023; for counter-arguments see MacBride 2015:178–94.

Now non-symmetric relations, including asymmetric ones, are typicallydescribed as imposing an order upon the things they relate; they admitof what is also calleddifferential application (Fine 2000:8). This means that for any such non-symmetric $R$ there aremultiple different ways in which $R$ potentially applies to thethings it relates. Why? If the way thatloves applies to$a$ and $b$ when $a$ loves $b$ were no different from the waythatloves applies to them when $b$ loves $a$, then theloves relation couldn’t be non-symmetric, becauseotherwise $a$ couldn’t love $b$ without $b$’s loving$a$. So we have to distinguish between the two different ways thatloves is capable of applying to $a$ and $b$. Similarly wecan distinguish the six different ways that the ternarybetween relation potentially applies to three things. Bycontrast, we’re not required by the same reasoning todistinguish between the different ways in which a (strictly) symmetricrelation applies to the things it relates. There’s nocorresponding necessity to distinguish the way that theadjacency relation applies to $a$ and $b$ when $a$ isadjacent to $b$ from the way that the adjacency relation applies tothem when $b$ is adjacent to $a$ because $a$ cannot be adjacentto $b$ without $b$ being adjacent to $a$.

Putting the ideas of the last three paragraphs together, philosophersoften seek to capture what is distinctive about relations bydescribing them in the following manner. Unlike properties, binaryrelations aren’t exhibited by particulars but bypairsof particulars. Similarly ternary universals aren’t exhibited byparticulars but bytriples of particulars, etc. But, we haveseen, non-symmetric relations, including asymmetrical ones, areorder-sensitive. So the pairs, triples, etc., of objects that exhibitthese relations must be themselves ordered. It is only ordered pairs,ordered triples, more generally, $n$-tuples or sequences thatexhibit relations (Kim 1973: 222; Chisholm 1996: 53; Loux 1998:23).

There are a number of reasons we should be wary of characterisingrelations in such quasi-mathematical terms. First, sequencespresuppose order, the difference between a thing’s coming firstand thing’s coming second, so cannot be used to explain orderwithout our going around in a tight circle. Second, it’sproblematic to try to explain a relation between two things in termsof another thing, a sequence, which is those two things put togetherin an order (MacBride 2005: 590–2). If a sequence $s\ (= \langle x, y \rangle)$ is conceived as a “one” suchthat a relation $R$ is really only a monadic feature of it,$R(s)$, then it’s mysterious what $R$’s holding of$s$ has to do with $x$ and $y$ being arranged one way ratherthan another with respect to $R$. If you don’t see the mysterystraightaway, reflect that $s$ may have a monadic feature $F$,which is, so to speak, only to do with $s$ itself rather than itsmembers. But if $R$ and $F$ are conceived alike, as monadicfeatures of $s$, isn’t it mysterious that the possession by$s$ of $R$ has consequences for $x$ and $y$ whereas $F$doesn’t? If, alternatively, a sequence is conceived as a“many”, just $x$ and $y$, such that $R$ applies tothem directly, then it’s unclear what the appeal to sequenceshas achieved.

We cannot avoid the difficulties associated with sequences byappealing to the Kuratowski definition of sequences in terms ofunordered sets (where, e.g., $\langle x, y\rangle = \{\{x\}, \{x,y\}\}$) (or Bergmann’s Kuratowski inspired account of order interms of ‘diads’, see his 1981: 147). There areindefinitely many other set-theoretic constructions upon which we mayrely to model sequences, e.g., Wiener’s, hence the familiarobjection that we cannot legitimately fix upon any one suchconstruction as revelatory of the nature of sequences (see Kitcher(1978: 125–6) generalising upon a point originally made byBenacerraf (1965: 54–62)). But there is also the less familiarobjection that the Kuratowski definition cannot be used to analyse theorder inherent in the sequence $\langle x, y\rangle$. Compare: wecannot analyse personal identity in terms of memory because it’sbuilt into the what we mean by memory that you can only rememberyour own experiences, i.e., the very notion that we set outto explain, viz. personal identity, is presupposed by our analysans.Similarly $\{\{x\}, \{x, y\}\}$ only serves as a model of $\langlex, y\rangle$ relative to the assumption that the thing that belongsto a singleton member, namely $\{x\}$, of the first classcomesfirst in the pair $\langle x, y\rangle$, so the notion of orderis presupposed (Hochberg 1981, 2001: 176–7).

The next idea that needs to be introduced is the idea of aconverse relation. For any given binary relation $R$, theconverse of $R$ may be defined as the relation $R^*$ that $x$bears to $y$ whenever $y$ bears $R$ to $x$ (Fine 2000: 3; vanInwagen 2006: 459). Note that the relationship between a relation andits converse isn’t a matter of happenstance; there’s nopossibility of a converse of $R$ floating free from $R$ andholding between things independently of how $R$ arranges them.Rather it’s a more intimate relationship which, if it holds,holds of necessity. For $x$ to have one to $y$ is, so to speak,for $y$ to have the other to $x$; they neither exist nor can beobserved apart from one another but can only be distinguished inthought (Geach 1957: 33).Before andafter,above andbelow areprima facie examples ofmutually converse relations. Non-symmetric relations, includingasymmetric ones, are distinct from their converses (if they havethem). Suppose $R$ is a non-symmetric relation which $x$ bears to$y.$ Then the converse of $R$ is borne by $y$ to $x$. Butsuppose that $R$, as it may, fails to be borne by $y$ to $x$.Then something is true of the converse of $R$ that isn’t trueof $R$ itself,viz. that it is borne by $y$ to $x$.Hence, the converse of $R$ must be distinct from $R$. Moregenerally, whilst a binary non-symmetric relation has only oneconverse, a ternary one has five mutual, distinct converses, aquadratic relation has 23 converses, etc. The issue of whetherrelations have converses is another issue to which we will returnlater.

The next distinction we will need, or more accurately, family ofdistinctions, is betweeninternal and external relations.What makes them members of a single family is that a relation isinternal if its holding between things is somehow fixed by the thingsthemselves or how those things are; external relations are relationswhose holding between things isn’t fixed this way. Differentversions of the internal-external distinction correspond to differentexplanations of how internal relations are fixed (Ewing 1934:118–36; Dunn 1990: 188–192; Hakkarainen, Keinänen& Keskinen, 2019). We don’t need to describe every versionof the distinction but here are three that are essential tounderstanding the contemporary debate.

The first version of the distinction is owed to Moore. According toMoore’s (1919: 47), a binary relation $R$ is internal iff if$x$ bears $R$ to $y$ then $x$ does so necessarily. From theinternality of $R$, in Moore’s sense, it follows that if $x$exists at all then $x$ indeed bears $R$ to $y$. But ifit’s possible that $x$ exist whilst failing to bear $R$ to$y$, then $R$ is external. The doctrine of the necessity of originprovides one putative example of a relation that’s internal inMoore’s sense. If you essentially come from your biologicalparents then you could not have existed whilst failing to be theiroffspring, i.e., the relation ofbiologically originatingfrom must be internal in Moore’s sense.

The second version of the internal-external distinction is favoured byArmstrong (1978b: 84–5, 1989: 43, 1997: 87–9, van Inwagen1993: 33–4). According to Armstrong, a relation $R$ isinternal iff it’s holding between $x$ and $y$ isnecessitated by the intrinsic natures, i.e., non-relationalproperties, of $x$ and $y$; otherwise $R$ is external. The thirdversion is owed to Lewis. Lewis (1986: 62) advanced the view that aninternal relation is one that supervenes upon the intrinsic natures ofits relata. But Lewis’s definition of “external” ismore involved: $R$ is external iff (1) it fails to supervene uponthe nature of the relata taken separately, but (2) it does superveneon the nature of the composite of the relata taken together.

Suppose that $x$ is a cube and $y$ is also a cube. It follows that$x$ is the same shape as $y.$ So the relation that $x$ bears to$y$ when $x$ is the same shape as $y$ is internal inArmstrong’s sense. By contrast, spatio-temporal relations areexternal (in his sense) because the intrinsic characteristics of $x$and $y$ don’t necessitate how close or far apart $x$ and$y$ are. Lewis intends his distinction to classify the same way. If$x$ is the same shape as $y$, but $w$ is not the same shape asz, then there must be some difference in intrinsic nature eitherbetween $x$ and $w$ or else between $y$ and $z$. So therelation that $x$ bears to $y$ when $x$ is the same shape as$y$ is internal in Lewis’s sense. Lewis also claims that thedistance between $x$ and $y$ is an external relation because thefollowing conditions are met: (1) $x$ may be closer to $y$ than$w$ is to $z$ even though $x$ is an intrinsic duplicate of$w$, i.e., shares all and only the same intrinsic characteristics,and $y$ an intrinsic duplicate of $z$; (2) the distance between$x$ and $y$ does supervene upon the nature of the composite$x+y$. (We’ll return insection 3 to the question whether, as Lewis claims, distance relation superveneupon the nature of composites.) But the relation that $x$ bears to$y$ when $x$ is the same shape as $y$ isn’t internal inMoore’s sense because, let us suppose, $x$ might have beenspherical whilst $y$ remained a cube; so it doesn’t followfrom the mere fact that $x$ exists that $x$ is the same shape as$y$.

The final distinction, or rather, collection of distinctions, whichwe’ll introduce here and which needs to be borne in mind whenconsidering the recent debate on relations, concerns the existence andidentity conditions for relations. Since Lewis it has becomecommonplace to conceive relations as either “abundant” or“sparse”: according to an abundant conception there is arelation for every arbitrary pair (or triple, or ...) of thingswhereas according to a sparse conception there are only as manyrelations as are required to characterise the relational aspects oflikeness and difference (Lewis 1986: 61). But this binary distinctiondoesn’t do justice to the variety of different conceptions ofthe existence and identity of relations that have been put forward.Distinguish, on the one hand, ana posterori view accordingto which relations exist only if they are instantiated and whetherthey are instantiated is an empirical matter (Armstrong 1978a:76–77) from, on the other hand, ana priori viewaccording to which their existence is settled by purely logicalprinciples, for example strong comprehension principles whereby everyopen formula determines a corresponding property or relation (Zalta1988: 46–7; Williamson 2013: 226–7; for dissent toWilliamson’s view, see MacBride 2024a: 390–99). Whereasthe former view might admit space-time separation as a relation (orfamily of relations) whose instantiation is confirmed by physics, thelatter would affirm the existence of a relation corresponding to theopen formula “x is a much funnier anarchist thany”. Between these two extremes are views that allowsome relations, for example, relative velocities conceived asrelations between different things when one has a certain velocityrelative to another, to be uninstantiated for the purposes ofrounding out scientific theory (Bigelow and Pargetter 1990: 77) andviews that allow logically complex relations ($R \lor S$, $R \land S$etc.) insofar as they are built up by logical operations fromsimple relations ($R,$ $S,$etc.) found in nature (Bealer 1982:9–10).

There is a considerable variety of views concerning the identityconditions for relations. Distinguish, for example, a“coarse-grained” conception, according to which relationsare identical just in case they are necessarily equivalent,i.e. identical just in case, as a matter of necessity, theyare exhibited by the same pairs, triples... of things, from a“fine-grained” conception, according to which relationsare identical just in case they have the same analysis in terms ofsimpler relations (Bealer 1982: 2–3, Menzel 1993: 62–4).Another approach conceives relations as the same just in case theyconfer the same causal powers on the pairs, triples... of thingsrelated by them. A further approach to relation identity says that,for example, the dyadic relationsR andS areidentical if their corresponding relational properties (bearingR toa, bearingS toaetc.) are identical, where properties in general (which,recall, are unary) are identical just in case, according to this view,they are “encodings” of the same “abstractobject”. The sense of abstract object here is a specialised one:roughly speaking, an abstract object encodes a property byencapsulating what it is to exhibit the property in question (Zalta1988: 51–52). Even though these different conceptions of theexistence and identity conditions for relations yield differentresults in different cases it is important to also bear in mind thatthey need not invariably be seen as competing. It may be that theyprovide different notions of relation relevant to different purposes,for example, whether we are interested in relations because of theirrole in causation, semantics, intentionality and so on (Lewis 1986:55–59).

2. Eliminativism, External Relations and Bradley’s Regress

Some philosophers are wary of admitting relations because they aredifficult to locate. Glasgow is west of Edinburgh. This tells ussomething about the locations of these two cities. But where is therelation that holds between them in virtue of which Glasgow is west ofEdinburgh? The relation can’t be in one city at the expense ofthe other, nor in each of them taken separately, since then we losesight of the fact that the relation holds betweenthem(McTaggart 1920: §80). Rather the relation must somehow share thedivided locations of Glasgow and Edinburgh without itself beingdivided. This may sound peculiar if we assume that middle-sized thingslike cities, which have locations in a relatively straightforwardsense, set a standard for how entities in general ought to behave. Butwhy shouldn’t we admit other kinds of entity that have adifferent kind of relationship to space and time from such readilylocated things. Or even go down the route of conceiving relations asabstract, i.e., as “Nowhere and nowhen” (Russell 1912:55–6). As Lewis reflected, “if the price is right, wecould learn to tolerate it” (Lewis 1986: 68). But is the priceright? Do the theoretical benefits of admitting relations outweigh thecost of offending pre-theoretic intuition?

Other philosophers have been wary of admitting relations because, theycomplain, relations are unsatisfactorily characterised, “neitherfish nor fowl”, i.e., neither substances nor attributes (Heil2012: 141, 2016: 130) or “ontologically weird” (Lowe 2016:111). Now it’s certainly true that $n$-adic relations, where$n\gt 1$, cannot be characterised as either substances or (monadic)attributes. But it only follows that relations are unsatisfactorilycharacterised if we assume that substances and attributes provide thebenchmark for satisfactory characterisation. But we shouldn’tassume this ahead of investigating whether the theoretical benefits ofadmitting relations outweigh the costs. It’s also beencomplained that relations are suspect because theydependupon the existence of substances that bear them (Campbell 1990:108–9) or, as Lowe makes the point, “A ‘relationalaccident’, if there could be such a thing would not be‘in’, to at least not be wholly ‘in’, any ofits two or more ‘subjects’, nor even wholly‘in’ the totality of them. I consequently find it hard toconceive what such an entity could really be” (Lowe 2016:111–2). Of course it’s true that in order for a relationto be borne by one thing to another thing, then those things mustexist. But it doesn’t follow that relations don’t exist ifnothing bears them. Moreover, it doesn’t follow straightawayfrom the reflection that to be borne by something a relation needsthings to bear it, that relations are suspect. It doesn’t followany more than it follows from the reflection that to be had bysomething an attribute needs things to have it, that attributes aresuspect. If the price is right we should open our minds to thepossibility of things which are neither fish nor fowl but vegetables,i.e., neither substances nor attributes but relations.

In light of the internal/external distinction introduced above, we candistinguish two distinct questions here: (a) should we acknowledgeexternal relations? (b) should we acknowledge internal relations? Inthe present section our focus will be upon whether to accept or rejectexternal relations, before turning to internal relations in thenext.

F.H. Bradley regarded himself the nemesis of external relations, butnot only them. Famously, Bradley brought a vicious regress argumentagainst external relations. In his original version (1893:32–3), Bradley presented a dilemma to show that externalrelations are unintelligible. Here’s the dilemma. Either arelation $R$ isnothing to the things $a$ and $b$ itrelates, in which case it cannot relate them. Or, it issomething to them, in which case $R$ must be related tothem. But for $R$ to be related to $a$ and $b$ there must be notonly $R$ and the things it relates, but also a subsidiary relation$R'$ to relate $R$ to them. But now the same problem arises withregard to $R'$. It must be something to $R$ and the things itrelated in order for $R'$ to relate $R$ to them and this requiresa further subsidiary relation $R''$ between $R'$, $R$, $a$ and$b$. But positing more relations to fix up the problem is onlythrowing good money after bad. We fall into an infinite regressbecause the same reasoning applies equally to $R'$ and however manyother subsidiary relations we subsequently introduce—this iscommonly called “Bradley”s Regress’. As Bradleylater summed up,

“while we keep to our terms and relation as external, nointroduction of a third factor could help us to anything better thanan endless renewal of our failure” (Bradley 1935: 643; vanInwagen 1993: 35–6).

So whichever horn of Bradley’s dilemma we choose, whether arelation $R$ is nothing or something to the things it relates, wecannot make sense of $R$ relating them. Bradley concluded that weshould eliminate external relations from our ontology. (For acontemporary defence of Bradley’s argument, see Vallicella 2002,and or an argument according to which it is incoherent to conceive ofthe world in terms of things-in-relation,‘in the spirit ofBradley’ but invoking the notion of grounding, see Della Rocca2020: 59–82. For objections to Vallicella and Della Rocca, see,respectively Wieland & Betti 2008 and Paolini Paolletti 2025:104–114.)

It is often claimed that Bradley’s eliminativism may beaddressed by positing a special category of entities, either (1) facts(Hochberg 1978: 338–40; Armstrong 1989: 109–110, 1997:118–9; Hossack 2007: 41–5) or (2) tropes (Maurin 2010:321–3; Simons 2010: 201–3). The strategic idea they shareis that we can embrace the second horn of Bradley’s dilemmawithout being stymied by his regress argument. Bradley assumedexternal relations to begeneral entities, i.e., universals,capable of relating different things from whatever things they happento actually relate. It follows from this assumption that the mereexistence of $R$ and the things it relates cannot suffice for $R$to relate them, because $R$ might have been engaged elsewhererelating other things, and this problem will persist however manysubsidiary (universal) $R$s we introduce. But, it is argued, thisproblem can be avoided by either positing facts or tropes, which areparticular rather than general entities. Both of these purportedsolutions are open to question. It seems that they presuppose whatthey set out to show: the capacity of relations to relate (MacBride2011: 168–72). Why is this?

First, take (1): Hochberg, Armstrong and Hossack argue that the factthat $a\Rel b$ does indeed suffice for $R$ to hold between $a$and $b$, because the fact that $a\Rel b$ cannot exist unless $R$does hold between $a$ and $b$. So once the fact that $a\Rel b$has been admitted into our ontology, the later stages ofBradley’s regress either don’t arise or they harmlesslysupervene upon the admission of this fact. But, MacBride counters,this presupposes that Bradley’s argument has already beendisarmed. Ask yourself: what is a fact? What distinguishes the factthat $a\Rel b$ from the mere plurality of its constituents: $R$,$a$, $b$? Just this: that the constituents of the fact arerelated. But Bradley’s argument is intended toestablish that we cannot understand how it is possible for things toberelated. So unless we have already established whereBradley’s argument goes wrong, we cannot appeal to the existenceof facts because facts are cast into doubt by his argument too.

Next, take (2). Maurin (2010, 2012: 802–4, 2024: 255–8)and Simons (2010) argue that if we reject Bradley’s assumptionthat relations are general entities, i.e. universals, and hold insteadthat relations are a special class of particular entities, i.e.,non-transferable tropes which are specific to their relata, then wecan avoid Bradley’s Regress. What is a relation conceived as a“non-transferable trope”? It’s a relation $r_1$that’s borne essentially by the things it relates and whichcould not have been borne by anything else (Simons 2002/3: 6). Suppose$r_1$ is borne by $a$ to $b$. Then $r_1$ is a trope in thesense that it is non-repeatable and so is not a “one-overmany”, not a universal. And $r_1$ is non-transferable in thesense that it could not be transferred from one pair of particulars toanother, i.e., $r_1$ could not be borne by $c$ to $d$ where $c\ne a$ and $d \ne b$. So, necessarily, if $r_1$ exists then (i)$a$ and $b$ exist and (ii) $a\mathbin{r_1} b$. So in virtue ofthe manner in which $r_1$ depends upon the things it relates itfollows that the mere existence of $r_1$ accounts for $a$ and$b$’s being related without having to go down the route ofappealing to subsidiary relations to glue them together. But again,this proposal assumes that relations really do relate, and thereforethat Bradley’s regress has already been disarmed. By way ofanalogy, consider the Problem of Evil: the difficulty of theoreticallysquaring God’s omnipotence, omniscience, omnibenevolence, etc.,with the (apparent) fact of unnecessary evil in the world. Of courseif God exists then there cannot be any unnecessary evil, i.e., theremust be a solution to the Problem of Evil. But we don’t providea theoretical solution to the Problem by simply positing God. Thatdoesn’t make any easier the theoretical task of squaring whatappears to be unnecessary evil with the nature of an omnipotent,omniscient, omnibenevolent deity. Similarly we don’t provide aneffective theoretical response to Bradley’s argument by simplypositing relational tropes. Positing them doesn’t provide anexplanation of how a relation can succeed in relating some thingswithout requiring to be related by a further relation to them. (Forfurther criticisms of the relata specific response to Bradley’sregress, see Hakkarainen & Keinänen, 2022)

Bradley’s problem with relations would be resolved if we couldachieve a perspective whereby we appreciated that the bearing of arelation isn’t itself a relation or require a further relation,whatever grammar may suggest (Armstrong 1978a: 106–111; Grossman1983: 167–8; Lewis 2002: 6–7). But we won’t achievethat perspective until we can provide a diagnosis of Bradley’sreasoning that relieves us of the temptation to think that the bearingof a relation is itself a relation.

Bertrand Russell dismissed Bradley’s argument on the groundsthat philosophers who disbelieve in the reality of external relationscannot possibly interpret those numerous parts of sciences whichappear to be committed to external relations (Russell 1924: 339).Russell’s argument bears comparison to Quine’s famousindispensability argument for mathematics: just as we cannot makesense of scientific discourse without taking seriously those portionsirredeemably committed to, e.g., numbers, we cannot make sense ofscientific discourse without taking seriously those portions committedto, e.g., spatio-temporal relations (Quine 1981: 149–50). From amethodological point of view, Russell deemed it more likely that anerror is concealed somewhere in Bradley’s argument than thatmodern science should have incorporated so fundamental a falsehood,i.e., portrayed a world of spatio-temporal relations when really thereare none. Of course even if we have good or even overwhelming reasonto think that Bradley’s argument must be wrong, thisdoesn’t relieve us of the philosophical responsibility ofdetermining where exactly Bradley slipped up.

3. Reductionism about Internal Relations

What about internal relations? Do we need to acknowledge theirexistence? We distinguished 3 senses of “internalrelation” in our preliminary discussion: internal relations aredetermined by the mere existence of the things they relate, orinternal relations are determined by the intrinsic characters of thethings they relate, or internal relations supervene upon the intrinsiccharacters of the things they relate. So, in one sense or another, itsuffices for the obtaining of an internal relation that either thethings it relates exist or that the things it relates havesuch-and-such intrinsic characters. On this basis, some philosophershave concluded that there is no need to admit internal relations asadditional pieces of furniture in the Universe (Fisk 1972:146–9; Armstrong 1978b: 86; Campbell 1990: 99–101; Simons2010: 204–5; 2014: 314–5; Lowe 2013: 242, 2016:105–6; Heil 2009: 316–7, 2012: 144–6, 2016: 130,2024: 35–44, Lowe 2016: 102–11, Keinänen 2020). Whydo they think this? Truths about internal relations aren’tnecessarily equivalent to truths about the things they relate or theintrinsic characters of the things they relate. Nevertheless, all thetruths about internal relations are determined by the existence or theintrinsic characters of the things they relate, or supervene uponthem. So internal relations are variously declared to be “noaddition of being” (Armstrong) or “absent from thefundamental ontology” (Heil) or, more clearly, “there areno such things” (Simons).

Consider an example:

Ben Vorlich, a Scottish mountain, is taller than Ben Vane, itsneighbour.
Ben Vorlich is 3094 ft high whilst Ben Vane is 3002 ft.

(1) and (2) aren’t necessarily equivalent because Ben Vorlichmight have been taller than Ben Vane even if Ben Vorlich and Ben Vanehad had different heights from the ones they actually have, i.e., (2)doesn’t follow from (1). Nevertheless, Ben Vorlich’s beingtaller than Ben Vane is determined by its having the height it does(3094 ft) and Ben Vane’s having the height it does (3002 ft); itisn’t possible for these mountains to have the heights they dowhilst Ben Vorlich fail to be taller than Ben Vane. So thebeingtaller than relation is internal in Armstrong’s sense. Thisrelation is also internal in Lewis’s sense which, recall, reliesupon comparison between pairs of a relation’s relata (Lewis1986: 62). Ben Lomond doesn’t stand to Ben More as Ben Vorlichdoes to Ben Vane, because Ben Lomond is 3196 ft, Ben More, 3852 ft.Hence Ben Vorlich’s being taller than Ben Vane supervenes, inLewis’s sense, upon the heights of Ben Vorlich and Ben Vane. Soonce Ben Vorlich and Ben Vane have been bestowed with their respectiveheights there is no need to admit a further (internal) relation toaccount for Ben Vorlich’s being taller than Ben Vane (Campbell1990: 100, 103; Lewis 1994: 294).

This, broadly speaking, reductionist argument is open to question in avariety of respects. Some philosophers have maintained that it cannotbe effective because we literally perceive proportions and otherinternal relations (Mulligan 1991; Hochberg 2013: 232). Theserelations must exist, otherwise we couldn’t perceive them. So,they conclude, something must have gone wrong with this argumentbecause it leads to the mistaken conclusion that there aren’tany internal relations out there to be perceived.

More generally, this reductionist argument assumes that if an internalrelation $R$ isn’t required to perform the role of determiningthat $R$ holds between the things it relates, because that role hasalready been discharged by the intrinsic characteristics of the thingsit relates, then $R$ doesn’t exist at all or doesn’tconstitute “an addition to the world’s furniture”(Armstrong 1997: 87). But from the fact that $R$ doesn’tfulfill one role it doesn’t follow that it isn’t requiredto fulfill another. It doesn’t follow because there may still beother good reasons for us to believe that internal relations exist, orconstitute “an addition to the world’sfurniture”.

To bring the issues here into focus we will need to take on board twofurther distinctions. First, the distinction betweentruth-making andontological commitment. Atruth-maker for a statement $S$ is (at least) something theexistence of which determines (necessitates) that $S$ is true(Armstrong 2004: 5–7). By contrast, something $x$ is anontological commitment of $S$ if, roughly speaking, $S$ could notbe true unless $x$ existed (Quine 169: 95). Typically, a statement$S$ incurs ontological commitment to an entity or some entitiesbecause we refer to it or quantify over them when we make thestatement. It’s easy to runtruth-making andontological commitment together but it’s important tokeep them separate. A statement $S$ cannot be true unless theentities to which $S$ is ontologically committed exist. Soit’s anecessary condition of the truth of $S$ thatthe entities to which $S$ is ontologically committed exist. Bycontrast, it’s asufficient condition for the truth of$S$ that a truth-maker for $S$ exist because a truth-maker for$S$ determines that $S$ is true.

Armed with the distinction between truth-making and ontologicalcommitment, we are now able to state a very general shortcoming of theargument. Let it be agreed that Ben Vorlich’s being the heightit is and Ben Vane’s being the height it is together make truethe claim that Ben Vorlich is taller than Ben Vane; so no furthertruth-makers are required for this claim to be true. But itdoesn’t follow that we aren’t ontologically committed tothe existence of an internal relation between Ben Vorlich and Ben Vanewhen we make this claim. We surely are ontologically committed to BenVorlich and Ben Vane when we make the claim that one is taller thanthe other even though they’re not themselves truth-makers forthis claim. They’re not truth-makers for this claim because theymight have existed even though Ben Vane was taller than Ben Vorlich orthe same height. But recognising that the class of ontologicalcommitments and the class of truth-makers for a given claim may bedifferent classes opens up the possibility that we may beontologically committed to an internal relation between Ben Vorlichand Ben Vane even though this relation isn’t a truthmaker forthis claim either (MacBride 2011: 162–6, Heil 2021: 29–31,44–5, MacBride 2024c: 737–8).

The second distinction we need is betweenthick andthin relations (Mulligan 1998: 342–7). Some examples.Thick relations:loves,kills,gives. Thinrelations:identity,resemblance,greaterthan. Thick relations have more of a “material”content and are less “formal” than thin relations. Itcorrespondingly makes more sense to think of thick relations as“topic-neutral”. Thin relations, by contrast to thickrelations, are typically internal (see Clementz 2014 and Johansson2014 for further discussion of the thick/thin distinction).

We are now equipped state another, more specific gap in thereductionist argument. Whilst Ben Vorlich and Ben Vane’s havingthe heights they do avoids the need to admit the existence of a thick(internal) relation that Ben Vorlich bears to Ben Vane, thisdoesn’t avoid the need to admit the existence of a thin(internal) relation. This is because we need to explain why BenVorlich’s having the height it does (3094 ft) and BenVane’s having the height it does (3004 ft) entails that BenVorlich is taller than Ben Vane. The answer is that the thin relationbeing greater than holds between the number of feet in heightthat Ben Vorlich has and the number of feet in height that Ben Vanehas (Russell 1903: §214; Bigelow & Pargetter 1990:55–6. So internal relations may not (in general) be required toperform the role of truth-makers for comparative claims. Nonethelessthin internal relations may perform an indispensable role inexplaining why the truth-makers for comparative claims are equipped tomake such claims true, rather than leaving this a matter of bruteinexplicable necessity—because, in this case, the heights of themountains in question lie in a certain relation of proportion to oneanother. Similarly Ben Vorlich’s being taller than BenVane’s supervenes upon their heights but this is onlyintelligible because the height of Ben Vorlich is larger than theheight of Ben Vane.

This leaves us a choice. Either we can rewind the argument and admitinternal relations, whether thick or thin, as additions to theworld’s furniture (Bigelow & Pargetter 1990: 56–60).Or we can adopt the more austere view that only admits thin (internal)relations to hold amongst intrinsic characteristics (Mulligan 1998:347).

The situation may be summed up in schematic terms. There are truecomparative claims of the form $a\Rel b$ which are either determinedto be true by, or supervene upon, the intrinsic natures of the thingsthey relate, $Fa \amp Gb$. Philosophers of a reductionist persuasionhave argued that this shows there is no need to admit an internalrelation $(R)$ over and above the intrinsic characteristics ormonadic foundations of the things related $(F, G)$. But theintrinsic characteristics of the things related only determine thetruth of the claim that $a\Rel b$, or the truth of the claim that$a\Rel b$ only supervenes upon $Fa \amp Gb$, because thesecharacteristics stand in an internal relation themselves, $F \fR G$.Because the monadic foundations, $Fa$ and $Gb$, are only empoweredto determine that $a\Rel b$ because $F$ and $G$ lie in thisinternal relation, $\fR$, it follows that internal relations stillperform an indispensable role in our theorising about the world. Oneline of response to this argument is to deny that the ontologicalcommitments and the truth-makers for a claim might come apart bydefining ontological commitment in terms of truth making, by roughlyspeaking, maintaining that to be is to be a truth-maker, but then analternative theory of truth will need to be provided for ordinary andformal languages if quantificational structure is never ontologicallycommitting and it unclear whether this can be done (For furtherdiscussion, see Heil 2016: 129, 2021: 29–36, 44–6;MacBride 2018: 157).

The case against reductionism about internal relations has proceededso far from a relativelya priori basis. But there’salso a case to be made against it ona posteriori grounds.Grant for the sake of argument the claim that if a relation isinternal then it doesn’t count as an ontological addition. Thesignificance of this claim, when it comes to providing an inventory ofthe world’s furniture, depends upon the extent to whichrelational truths have monadic foundations. But relativity theory,quantum theory and even classical mechanics provide anaposteriori basis for doubting that there are very many, if any,$F$s and $G$s capable of serving as monadic foundations for target$R$s, i.e., intrinsic properties to which a significant range ofrelations may be reduced. Heil goes even further and suggests thatmodern physics sits comfortably with a version of Bradley’smonism (Heil 2021: 48–57).

It is has often be taken for granted by metaphysicians, at least untilrecently, that the fundamental quantities of physics provide a readysource of monadic foundations because physical quantities areintrinsic characteristics. The vision of reality metaphysicianstypically ascribe to physics is akin to Lewis’s doctrine of“Humean Supervenience”, according to which there areintrinsic properties which need nothing bigger than a point to beinstantiated and there’s no difference without a difference inthe arrangement of intrinsic properties over points (Lewis 1986:ix–xvi). And certainly some metaphysicians have endeavoured tomake good upon this vision, arguing, for example, that velocity is anintrinsic property (Tooley 1988; Bigelow & Pargetter 1990:62–82). Heil has gone so far as to claim that HumeanSupervenience is compatible with the conception of the cosmos as a‘seamless substance’ (Heil 2021: 42–4). But not onlyare these views contentious, there are many other notions in classicalmechanics, such as force, stress, strain and elasticity, representedusing vectors and tensors, that belie the conception of fundamentalphysical quantities as intrinsic (Butterfield 2006, 2011).

When we turn from classical mechanics to relativity theory and quantumtheory, thea posteriori case against conceiving physicalquantities as intrinsic is even stronger. Consider mass and charge,most metaphysicians’ favourite and commonly relied upon examplesof intrinsic properties of a piece of matter (in particular a pointparticle). In relativity theory, mass is identified with energy(“mass-energy”). In a relativistic field theory, such asMaxwell’s theory of electromagnetism, the all-pervadingelectromagnetic field has mass-energy. It also has momentum, stressand other traditionally mechanical properties. So mass is no longerconceived as an intrinsic property of localized lumps of stuff, letalone of point particles. Furthermore, the attribution of mass,momentum, stress, etc., to the electromagnetic field, or to a matterfield such as a continuous fluid, depends upon the metrical structureof spacetime. So these quantities are not intrinsic, but, rather,relational properties. They are properties that the radiation ormatter field in question has in virtue of its relation to spacetimestructure (Lehmkuhl 2011). For further contributions to the debateabout whether quantities are intrinsic, see Dasgupta 2013, Eddon 2013,Wolf 2020 and Baker 2021.

In quantum theory there are even fewer candidates for intrinsicproperties. Even in elementary quantum mechanics, the closest analogueof a point particle as classically understood (“a quantumparticle”) is an all-pervading field,viz. anassignment, for each instant of time, of a (complex) number to everypoint of space. Here, the mass of the quantum particle cannot beattributed to any single point of space, but must be associated withthe whole field, although not in the simple manner of “spreadingthe mass out” as a classical mass density distribution. (Thesame remark applies to the quantum particle’s charge.) Turningto quantum field theory: here again, the notion of particle isradically changed from the idea of a classical point particleand from the idea of a quantum particle. There are notvarious quantum particles, each of them represented by acomplex-valued field as above, and each of them, say, electrons.Rather there is a single all-pervadingelectron field andeach electron, as treated in elementary quantum mechanics, is replacedby a unit of energetic excitation of the electron field. The same goesfor other types of matter, such as quarks. According to quantum fieldtheory, quarks are really excitations of an all-pervading quark field.Besides, quantum field theory exhibits a second, and fundamentallydifferent, way in which mass and charge fail to be intrinsicproperties. This goes under the label of“renormalization”. In short, the mass and charge of aparticleà la quantum field theory, i.e., a unit ofexcitation of an all-pervading field, depends upon the length-scale atwhich you choose to describe, or to experimentally probe, the field(Butterfield 2014, 2015).

So, to sum up this case against reductionism about relations: thereseem to be far fewer (if any) monadic physical properties than thereductionist view needs. But this is not the only case that has beenmade. It has also been argued on the basis of quantum entanglement,the fact that the quantum states of entangled particles cannot bedescribed independently of one another, that physics is committed torelations that don’t supervene upon intrinsic properties (Teller1986). More radically, it has been argued that a consideration ofquantum entanglement pushes us towards the metaphysical outlook thatthere are no intrinsic properties (Esfeld 2004, 2016) or even thatthere are no objects but only relations (French & Ladyman 2003;Ladyman & Ross 2007: 130–89, Ladyman 2016). For a range ofdifferent objections to these views, see Psillos 2001, 2006,Briceño & Mumford 2016, and Ioannidis, Psillos &Pechlivanidi 2022. For debate about the intelligibility of conceivingthe world in wholly relational terms, see Dipert 1997, Barker 2009,Odeberg 2011, 2012 and Shackel 2011.

Consider next the related suggestion that spatio-temporal relationsare internal relations between, variously, space-time points, regionsor events (Simons 2010: 207–8, 2014: 312–4; Simons 2016:2010: 207–8; Heil 2012: 147; Lowe 2016: 109–111). Generalrelativity makes serious trouble for this suggestion. The thinkingbehind the suggestion is that it belongs to the essential natures ofspace-time points, etc., that they bear the catalogue ofspatio-temporal relations they do. In philosophy of physics, the viewis called “metrical essentialism”, the doctrine thatpoints have their metrical properties and relations essentially(Maudlin 1988). From metrical essentialism it follows that the mereexistence of a point, etc., already suffices to determine that itinhabits this network of spatio-temporal relations. Hence, it isargued, there is no need to admit spatio-temporal relations, or atleast consider spatio-temporal relations as ontological additions tothe furniture of the Universe. This kind of view is open to the hea priori concerns already aired – for furtherobjections see Orilia & Paolini Paoletti 2023: 396–401, But,moreover, this argument for refusing to admit spatio-temporalrelations as ontological additions is encumbered by theaposteriori difficulties of the metrical essentialism itpresupposes.

These difficulties for metrical essentialism arise from the fact thatin general relativity space-time structure varies from one model,i.e., possible world, to another (Butterfield 1989: 16–27). Wecannot leave points or regions to themselves to settle their internalrelations, because general relativity denies the idea of aspatial-temporal framework whose make-up is settled independently ofthe matter or radiation. So there isn’t a stableonce-and-for-all class of points or regions which once posited sufficefor the purposes of modern physics, hence no stable class oftruth-makers that suffice for making all the distance-duration claimstrue.

It may help to understand this debate about space-time points if weconsider the following line of response on behalf of metricalessentialism. According to metrical essentialism, points have theirmetrical properties and relations, i.e., space-time framework,essentially. So if the space-time framework differs between modelsthen the points must differ between models too. But this viewisn’t compromised by the fact, as Butterfield (1989) points out,that in general relativity the space-time framework varies from onemodel to another; it only follows that the points must vary betweenthese models too. What, however, this line of response fails to takeinto account is how extremely fragile spatio-temporal relations mustbe conceived to be when metrical essentialism is combined with generalrelativity.

Take a possible world which is truly described by a dynamical theoryof space-time (i.e., described by general relativity and any likelysuccessor to it). Now take a second possible world which differs fromthe first only with respect to a tiny change, on a millimeter scale asregards length, and on a milligram scale as regards mass. Thisinvolves altering spatio-temporal relations throughout the entireuniverse. Hence it’s a consequence of metrical essentialism thatin the second possible world, all space-time points are non-identicalto those in the first. But surely it’s implausible that such asmall adjustment should add up to such a world-altering difference(thanks here to Jeremy Butterfield).

We are now in a position to evaluate David Lewis’s claim thatwhilst distance relations fail to supervene on the intrinsiccharacters of the things they relate, there is nevertheless adifferent way in which relations of distance do supervene on intrinsiccharacter:

If, instead of taking a duplicate of the electron and a duplicate ofthe proton, we take a duplicate of the whole atom, then it willexhibit the same electron-proton distance as the original atom.Although distance fails to supervene on the intrinsic nature of therelata taken separately, it does supervene on the intrinsic nature ofthe composite of the relata taken together – in this case thecomposite hydrogen atom. (1986: 62)

This remark has suggested to some philosophers that we can do awaywith relations generally because we need only posit the compositesupon whose intrinsic nature the relation supervenes (Parsons 2009:223–5). But Lewis’s view and the more general form ofreductionism inspired by it encounter a dilemma. Either (1) thecomposite is just the proton and electron taken together. But then thedistance between them fails to supervene upon the intrinsic characterof the composite, because there are duplicates of the proton andduplicates of the electron that vary in distance. Or (2) the compositeis more than just the proton and electron, i.e., the proton andelectron related together in some (external) way. But then wehaven’t gotten rid of external relations in favour of internalones but only presupposed them.

Can this dilemma for Lewis be sidestepped by the following manoeuvre?Suppose the composite is the mereological fusion of the proton and theelectron. Then can’t we say that the fusion has its intrinsiccharacter, and the distance relation between the parts of the fusionsupervenes on this character?

But this can’t be enough to say. We don’t really have anyidea of what the intrinsic properties of fusions are because theaxioms of mereology are silent about the intrinsic characters offusions. So we don’t really have any idea of whether thedistance relations of their parts supervene upon their properties.Perhaps fusions don’t have much, if any, intrinsic character!Since we don’t know much about these things, we’re not ina position to rule out the possibility of a duplicate of Lewis’sproton-electron fusion being a fusion that has parts which are adifferent distance apart than the original proton and electron Lewisdescribed.

Of course, there’s no logical bar to putting forward a theorythat posits a rich supply of fusions whose intrinsic natures are suchthat the distance relations that hold between the parts of thesefusions supervenes upon their intrinsic characters. But why believesuch a theory? It’s difficult to see how the supervenience ofdistance relations upon intrinsic characters of fusions could ever beexplained (’superdupervenience’), because it’sdifficult to see how an intrinsic property $F$ of one thing $X$can intelligibly give rise to an external relation $R$ between twoother things $x$ and $y$, even if $x$ and $y$ are parts of$X$.

Finally, in the light of the foregoing discussion, Lewis’s viewis open to an objection from physics: there is no absolute fact of thematter about the distance between the electron and protonindependently of facts about the physical fields more generally, e.g.,the electromagnetic field and the electron field, i.e., factsconcerning what is happening elsewhere. So the distance relation ofthe electron and fusion cannot supervene upon the intrinsic characterof the proton-electron fusion. It follows that distance relationscannot be external in Lewis’s sense.

4. The Nature of Relations: Order and Direction

Supposing they exist, what is the nature of relations, whetherinternal or external? It is the crucial feature of binarynon-symmetric relations, which distinguishes them from binarysymmetric relations, or, more generally, distinguishes relations whichfail to be strictly symmetric from strictly symmetric ones, that theybestow order upon the things they relate. Non-symmetric relations doso, because they admit ofdifferential application, i.e., forany such relation there are multiple ways in which it potentiallyapplies to the things it relates. So, in particular, for every binarynon-symmetric $R$, $a\Rel b \ne b\Rel a$. But can we say anythingmore about how it is possible for non-symmetric relations to admit ofdifferential application? More generally, can we say anythingdiscursive about the origins of order?

According to what is sometimes called the “standard”account of non-symmetric relations, and now sometimes“directionalism”, we can (Russell 1903: §94; Grossman1983: 164; Paul 2012: 251–2). A non-symmetrical relation $R$has adirection whereby it travelsfrom one thingto another. What distinguishes $a\Rel b$ from $b\Rel a$is that $R$ runs from $a$ to $b$ in the former case and $b$ to$a$ in the latter. So differential application is possible becausenon-symmetric relations have direction, i.e., because non-symmetricrelations run one way, rather than another, amongst the things theyrelate. (See Johansson 2011 for an argument that‘direction’ is really a cluster term covering differentlogical and intentional notions.)

Directionalism faces the serious objection that it is liable toover-generate relations, specifically converse ones, givingrise to anover-abundance of states that results from theholding of converse relations (Fine 2000: 2–5; cf. Russell 1913:85–7; Castañeda 1975: 239–40; Armstrong 1978b: 42,94; for related puzzles arising out of the commitments of Armstrong1997, see Hochberg 1999: 157–60, Cross 2000 and Newman 2002, andfor further objections to directionalim see Paolini Paoletti 2025:164–72). Employing the notion ofdirection that thestandard account uses to explain order, we can define the converse ofa non-symmetric $R$ as the relation $R^*$ that runs from $b$ to$a$ whenever $R$ runs from $a$ to $b$. But once given thisnotion of a (distinct) converse, it is difficult to deny that $R$has a converse. This is because it would be arbitrary to admit theexistence of $R$ whilst denying its converse (orviceversa). Hence, the directionalism is committed to the generalprinciple,

(Converses) Every non-symmetric relation has a distinctconverse.

But the admission of converse relations threatens to conflict with twoplausible sounding principles about the number of states that arisefrom the holding of non-symmetric relations and their converses.

(Identity) Any state that arises from the holding of arelation $R$ is identical to a state that arises from the holding ofits converse $R^*$.

(Uniqueness) No one state arises from the holding of morethan one relation.

Support forIdentity is garnered from intuitions: that thecat’s beingon top of the mat is the very same state asthe mat’s beingunderneath the cat, that Obama’sbeingtaller than Putin is the very same state asPutin’s beingshorter than Obama, and so on.Uniqueness is based upon more overtly theoretical grounds.States are often conceived as complexes of things, properties andrelations. They are, so to speak, metaphysical molecules built up fromtheir constituents, so states built up from different things orproperties or relations cannot be identical. Hence it cannot be thecase that the holding of two distinct relations give rise to the samestate.

Nowone problem with directionalism is thatConverses,Identity andUniqueness form aninconsistent triad, assuming there are non-symmetric relations. Takethe cat’s being on top of the mat.Identity dictatesthat this state is the very same state as the mat’s being underthe cat. But if there is only one state in play here,Uniqueness tells us that this state can only arise from theholding of one relation. So the relation that holds between the catand the mat in virtue of which the cat is on top of the mat must beidentical to the relation that holds between the cat and the mat invirtue of which the mat is underneath the cat. But this conflicts withConverses which tells us these relations are distinct.

In the face of this inconsistency a number of theoretical options areopen to us. If we wish to maintain the standard view (directionalism)that non-symmetric relations have direction then something has togive. (1) Either we can give upIdentity,Uniqueness, or both, or we can ditchConverses.Indeed it has been argued thatConverses should be ditchedanyway because it gives rise to a special kind of semanticindeterminacy: we cannot establish by our use of a sign for a binarynon-symmetric relation $R$ that we mean to pickit outrather than its converse $R^*$. This is because we can shift thereferent of a predicate from a given relation to its converse whilstleaving undisturbed the truth conditions of the statement in which thepredicate occurs by making compensating adjustments to the manner inwhich the names flanking the predicate are interpreted (Williamson1985: 252–55, 1987; van Inwagen 2006: 458–68 but seeLiebesman 2013 for a reply to Williamson). But whetherConverses is ditched to avoid semantic indeterminacy, orbecauseConverses,Identity andUniquenessform an inconsistent triad, or for some other reason, the link betweenConverses and the standard view will have to be broken. (2)Alternatively we can jettisonConverses but throw out thestandard view with it. But if that’s the option we choose, thenwe will have to go back to the drawing board to provide a freshaccount of differential application that doesn’t appeal todirection.

There is certainly room for development if we go down route (1). Theintuitions that provide evidence forIdentity are, likeintuitions generally, considered by many philosophers to be relativelyweak sources of evidence, because there may be all variety ofpsychological, historical and sociological forces responsible for ourholding this or that view to be intuitive which have little to do withwhether the view is true or credible. There’s also a traditionof conceiving states or facts as capable of being analysed into morethan one collection of constituents, i.e., a tradition that deniesUniqueness (Frege 1884: §64; Hodes 1982; see Macbride2007: 54–55 and Liebesman 2014 for counterarguments toFine’s assumption; but see Leo 2013 for a view which deniesuniqueness for states).

Alternatively we may leaveIdentity andUniquenessin place but question whether the standard view really does give risetoConverses. According to one line of thought (Russell 1913:86–7), it would be arbitrary for us to choose from the inside,so to speak, one binary non-symmetric relation at the expense of itsconverse. But, it may be argued, this doesn’t preclude naturechoosing for us from the outside, providing an externalist rather thanan internalist foundation for admitting a non-symmetric relation butnot its converse. Nevertheless, even ifConverses is ditchedthere is another obstacle to the standard view, namely that it appearsto give rise to questions about direction for which there are nosensible answers, even if doesn’t over-generate converserelations and states to house them (MacBride 2014: 5–7,9–10). Consider the states, (a) Antony is married to Fulvia and(b) Antony is to the left of Fulvia. If these states arise from theirrespective relations travelling from one thing they relate to another,then there must be a fact of the matter about which things proceedfrom and which to. So, we may ask, do they both proceed from the samething or do they proceed from different things? It doesn’t seemthat there could be a sensible answer to this question and so weshould avoid an account of differential application that commits us tothere being such recherché and undetectable facts of thematter. (But note that whilst many philosophers have felt the pull ofthese kind of intuitive considerations, others have argued that is isnecessary to introduce ordinal properties or relations to imposestructure upon relational facts. For example, Hochberg posits“ordering relations” (Hochberg 1987, 1999: 180–81)whilst Tegtmeier posits “ordinators”, such as“coming first” and “coming second” in a fact(Tegtmeier 1990, 2004, 2016).)

What are the options if we go down route (2)? Some philosophers havesuggested that if we ditchConverses then we will have togive up non-symmetric relations altogether (van Inwagen 2006: 453).But this presupposes that the only way to account for the distinctivefeature of non-symmetric relations,viz. differentialapplication, is to appeal to directionalism. However, philosophersthat go down this second route endeavour to account for differentialapplication by different means thandirection and therebyearn the right to hold onto non-symmetric relations. We candistinguish three broad strategies for explaining differentialapplication without direction, namely (1)positionalism, (2)anti-positionalism, and (3)primitivism, althoughthis does not exhaust the range of possibilities considered in theliterature (consider, for example, the view that relations areproperties of sequences discussed above (sec. 1).

First,positionalism may be roughly sketched as the view thatan $n$-ary relation has $n$ argument positions, where argumentpositions are conceived as themselves entities, and, crucially, thereis no intrinsic order to the argument places of a relation; neitherargument position of a binary relation is the first or the second,etc. (Castañeda 1972, 1975; Williamson 1985: 257; Grossman1992: 57; Armstrong 1997: 90–1; Cross 2002: 219–223, Fine2000: 10–16; Orilia 2011; Gilmore 2013). In the case of a binarynon-symmetric relation $R$, this amounts to the idea that $R$ isassociated with two further entities, % and #, its argument positions,and $R$ holds between two given relata $a$ and $b$ relative toan assignment of objects to % and #. The difference between $a\Relb$ and $b\Rel a$ (differential application) is then explained interms of the different relative assignments of $a$ and $b$ to %and #: $a$ to % and $b$ to # in one case, and $b$ to % and $a$to # in the other. But since the argument positions do not themselvesexhibit any significant order or direction, there is no basis here forintroducing converses, i.e., relations that differ only with respectto their direction of travel.

Different versions ofpositionalism vary with regard to$(i)$ the nature of argument positions, crucially upon whether theycan be shared by different relations, and $(ii)$ the variety ofargument positions, whether there are different kinds of positionspecifically suited to different kinds of relata, and $(iii)$ thecomplexity of the relation whereby things are assigned to positions.According to Orilia’sonto-thematic role view,positions are capable of recurring in different relations and suitedby their nature to different sorts of filling (Orilia 2011, 2014,2019, 2020 Paolini Paoletti 2016, 2021a sec. 2, 2025: 192–241)and for an earlier version inspired by Leibniz, Orilia 2008). Forexample, theloves relation has agent and patient roles,roles which also recur in thehates relation. Other morecomplex relations involve more and different roles, for example,theme, location, instrument, goaletc. Orilia’s idea isthat the schedule of ontological roles belonging to relations shouldbe informed by the linguistic programme of identifying thematic rolesin natural language because this helps us to provide informativegeneralisations about what different relations have in common (seeDavis 2011 for a helpful introduction to the thematic role literaturein linguistics).

By contrast, other versions ofpositionalism conceive ofpositions as unique to the relations in which they figure. Somepositionalists of this stripe conceives of each position as beingassigned a unique occupant (Gilmore 2013). But others whilst sharingthe assumption that positions are unique to the relations in whichthey figure, also maintain that it is a mistake to assume that thereis a 1–1 correspondence between positions and their occupants.So this version of positonalism conceives relations more liberally ashaving positions capable of being assigned several distinct occupantsat once (see, for example, Armstrong 1997: 91, Yi 1999: 169; Fine2000: 17, MacBride 2005: 588). Further sophistication is added by theidea that positions can be occupied by a number of things varyingbetween fixed limits (Hazen & Taylor 1992: 376, 390, Dixon 2018:207) and by the suggestion that the assignment of things to positionsshould also be relative to a number to allow for one and the samething to simultaneously occur more than once in the same position(Dixon 2018: 213–6). For further problems and alternativerefinements and developments of slot theory see Effingham 2020.

A different line is taken by Donnelly’srelativepositionalism, according to which positions are a kind of“relative property” (Donnelly 2016, 2021). Donnelly takesbeing north as an example of a relative property: a propertybecause it is had by one thing but only relative to something else, inthis case a location. Her basic idea is that corresponding to eachn-ary relationR is a package of relative properties(as many asn factorial properties) such that whennthings instantiate R they instantiate these properties relative to oneanother (Donnelly 2016: 91). It is a distinctive feature of the viewthat by positing for each relation a bespoke package of relativeproperties which models the logical character of the relation inquestion,relative positonalism is able to handle relationsof arbitrary complexity, including symmetric and partially symmetricrelations (Donnelly 2016: 88–96, Dixon 2019: 56–67). Bycontrast, theonto-thematic role approach doesn’tprovide a once and for all recipe for positing roles but takes this tobe a matter to be settled by systematic “ontologicalinvestigation” into the character of relations as we find them(Orilia 2014: 296–301 but see objections from MacBride 2014:11–12, Ostertag 2019: 1492–5).

There are two immediate challenges faced by the different versions ofpositionalism. Can the account be generalized to cover allrelations? If, for example, the binary symmetric relation that holdsbetween two things $a$ and $b$ whenever $a$ is next to $b$ hastwo argument positions, £ and $, then there should be twopossible ways of assigning $a$ and $b$ to £ and $. But thismeans (absurdly) that there are two different ways for $a$ and $b$to be next to one another,i.e., symmetric relations admit ofdifferential application too! Orilia’sonto-thematicrole approach handles this kind of case by conceiving of thematicroles as “intra-repeatable” (Orilia 2014: 294), so thereis only one role or position in play in this relation (£ = $)and so the possibility of different assignments to different roles orpositions doesn’t arise. Similarly, Dixon’s“pocket” theory handles this case by conceiving thenext to relation as having only one pocket. But, as Fine(2000: 17–18, 2007: 58–9) and MacBride (2007: 36–44)argue, and as Dixon (2018: 31, n.21) acknowledges, this kind ofstrategy doesn’t straightforwardly extend to more complexpartially symmetric relations such asbeing arranged clockwise ina circle in that very order, or,playing-tug-of-war.Accordingly Orilia (2011: 6–9, 2014: 296–301) proposes aprogramme of analysis whereby more complex relations are reduced tosimpler ones that the onto-thematic role approach can accommodate. Bycontrast,relative positionalism can handle these kinds ofpartially symmetric relations without any analysis (Donnelly 2016:88–96, Dixon 2019: 56–67), but not multigrade relationswhich lack a fixed arityn, hence cannot be modelled by an! package of relative properties (Donnelly 2016: 94, Dixon2019: 59) (For further criticisms ofpositionalism seeHochberg 1999: 153; 2000: 40–1).

The more general challenge that faces the different versions ofpositionalism is whether we have a better understanding of positionsand the assignment relation that the different versions ofpositionalism propose, than we do of the relations that positions andassignment are used to explain (Fine 2000: 16; MacBride 2014:11–12; Orilia 2014: 302–3). Consider, for example,Donnelly’s claim that the application ofn-aryrelations needs to be explained in terms of packages ofn!relative properties. One problem, acknowledged by Donnelly (2016: 98),is the ontological cost of having to trade in each relation for somany properties. Another problem concerns why relative properties of agiven package cannot be instantiated independently of one another butonly relative to other properties belonging to that package, eventhough each relative property is meant to be unary. The obviousexplanation is that such properties aren’t unary at all but tobe explained in terms of an underlying relation (Bigelow &Pargetter 1990: 55, MacBride: 2000: 83). So saying that one thingx has a property relative to something elsey havinganother relative propertyetc is more perspicuously renderedby saying thatx bears a relation toy. (For furtherobjections to positonialism and relative positionalism, see PaoliniPaoletti 2025: 174–182).

Both the standard view and the different versions ofpositionalism assume that the differential application of anon-symmetric relation $R$ with respect to $a$ and $b$ is to beexplained “locally” in terms of $R$, whether itsdirection or argument positions, and the things it relates, $a$ and$b$.Anti-positionalism is the doctrine, due to Kit Fine,which eschews direction and argument positions and explainsdifferential application by abandoning the assumption thatdifferential application is to be explained locally (Fine 2000:20–32). According to anti-positionalism, the difference between$a\Rel b$ and $b\Rel a$ arises from how these states areinter-connected with other states that arise from $R$ holding ofother relata. So order is not a feature of given state capable ofbeing isolated, but only emerges over a totality of states to which$R$ gives rise. What distinguishes $a\Rel b$ from $b\Rel a$ isthat inter-connectedness with other states such as $c\Rel d$. Payspecial attention to the relative arrangement of the schematic namesin the following sentence. Whereas the state $a\Rel b$ is acompletion of $R$ by $a, b$ in the same manner as $c\Rel d$ is acompletion of $R$ by $c, d$, the state $b\Rel a$ isn’t.The state $b\Rel a$ is a completion of $R$ by $b$, $a$ in thesame manner as $c\Rel d$ is a completion of $R$ by $c, d$ (Fine2000: 21). However, Fine doesn’t take the notion of two statesbeing put together “in the same manner” as primitive, butexplains this in terms of a substitution relation. To say that a state$s$ is a completion of relation $R$ by $a_1, a_2 \ldots, a_n$ inthe same manner as a state $t$ is completion of $R$ by $b_1, b_2\ldots, b_n$ is to say that $s$ is completion of $R$ by $a_1,a_2 \ldots, a_n$ that results from simultaneously substituting $a_1,a_2 \ldots, a_n$ for $b_1$, b$_2 \ldots, b_n$ in $t$ (andvice versa) (Fine 2000: 25–6). Fine goes on to suggestthat the notion of simultaneous substitution of many objects may bedefined, under certain conditions, in terms of the single substitutionof one object (Fine 2000: 26 n. 15).Converses is avoidedbecause the notion of direction, in terms of which the distinctionbetween converse relations is drawn, is eschewed. This is becausesubstitution is doing all the work.

Fine’s version ofanti-positionalism has certainlimitations, including the assumption that states arising from asingle relation have a unique composition and that substitutionoperates on objects rather than occurrences of objects. Leo (2013)provides a more fully developed version ofanti-positonalismthat addresses these limitations from a more formal point of view.According to Leo, relational complexes are structured perspectives onstates ‘out there’ in reality, relational complexes haveoccurrences of objects, and different complexes of the same relationmay correspond to the same state. It is key to this approach thatsubstitution is a primitive operation taking a complex and anassignment of objects to the occurrences of objects in the complex asinput and returning another complex as output. For example, for thecomplex of Anthony’s loving Cleopatra, substituting Abelard forthe occurrence of Anthony and Eloise for the occurrence of Cleopatraresults in the complex of Abelard’s loving Eloise. Substitutionis identified in terms of basic principles governing substitutions, inparticular a composition principle, which forms the heart of anantipositional theory of relations (Leo 2013: 363; for furtherdevelopments and defence of this version ofanti-positionalism, see Leo 2008a, 2008b, 2010, 2014,2016).

By abandoning the assumption that differential application is to beexplained locally,anti-positionalism in effect tells us thatwe’re misconceiving the logical form of our ordinary relationaljudgements. Contrary to their surface form, relational judgementsalways have a suppressed argument position for other states, by whichmeans the state we are making a judgement about is implicitly comparedto another. The following two objections arise (MacBride 2007:47–53). First of all,anti-positionalism presupposes anon-symmetric relation of substitution to explain differentialapplication. But this suggests that the explanation anti-positonalismprovides involves a circularity because it uses a non-symmetricrelation, substitution, whose differential application is take forgranted. Take the state $a\Rel b$. The result of substituting $a$for $b$ is the state $a\Rel a$; where the result of substituting$b$ for $a$ is the state $b\Rel b$. The outcomes are differentso the substitution relation must be order sensitive. Secondly,anti-positionalism entails that there cannot be lonely binaryrelations, i.e., relations which only hold of exactly two things. Butprima facie we should be able to distinguish $a\Rel b$ from$b\Rel a$ even if $R$ holds between nothing except $a$ and$b$, Antony’s loving Cleopatra from Cleopatra’s lovingAntony even if the world is otherwise loveless (see, for replies tothese objections, Fine 2007: 59–62, Wieland 2010: 487–93,Leo 2014: 279–81).

The third strategy,primitivism, seeks to avoid thedifficulties that threatenpositionalism andanti-positionalism by taking the radical step of asking us toreconfigure our explanatory settings. It’s a familiar thoughtthat we cannot account for the fact that one thing bears a relation$R$ to another by appealing to a further relation $R'$ relating$R$ to them—that way Bradley’s regress beckons. To avoidthe regress we must recognise that a relation is not related to thethings it relates, however language may mislead us to think otherwise.We simply have to accept as primitive, in the sense that it cannot befurther explained, the fact that one thing bears a relation toanother. But, according to primitivism, it is not only the factthat one thing bears a (non-symmetric) relation $R$ toanother that needs to be recognised as ultimate and irreducible.How $R$ applies, whether the $a\Rel b$ way or the $b\Rela$ way, needs to be taken as primitive too (MacBride 2014:14–15). The difficulties encountered by attempts to provide ananalysis of differential application provide corroborative evidencefor primitivism. Does refraining from providing an analysis ofdifferential application constitute shirking an obligatory task forsystematic philosophy, hence that we should always favour a theorythat ‘solves’ the problem of differential application(Armstrong 1980: 440–1, Paolini Paoletti 2025: 174)? Hardly.Let’s not forget thatevery theory has its primitives,hence does not provide a discursive analysis of everything itpresupposes. And remember what Lewis correctly said about predication:“Not everyaccount is ananalysis”(Lewis 1983: 352). More fully, predication is basic to the conceptualrepertoire we draw upon to describe the world that we cannot analyseit in terms of more basic ingredients. The same moral applies here: wecannot get underneath differential application to something more basicwhich doesn’t presuppose it. What we need, according to thisthird strategy, primitivism, is a form of realism committed to theexistence of relations but which denies that the manner of theirapplication, their differential application, can be further analysedor explained.

One of the most intriguing features of the debate about relations anddifferential application is that there is no single sense in which thedifferent views we have considered seek to “explain” or“account” for differential application. They all run ondifferent explanatory settings. Some think there is a genuine problemto do with the capacity of relations to differentially apply whichneeds to be explained by any successful theory of relations. Hence,because he thinks there is a genuine problem here, Hochberg claimsthat the question which any adequate theory of relations must answer“is about how the order is ontologically grounded infacts” (Hochberg 2001: 178). But others think that thereisn’t a problem of differential application at all, much lessone that requires an ontological analysis of relations or facts. Thismeans that the debate cannot be reduced to simply an exchange ofopinions about which view explains the most at the least cost.

For example, positionalists typically maintain that their positingroles, slots, pockets and relative properties enables them to answerwhat they take to be obligatory questions about why each relationapplies the way it can and why some relations are capable of applyingdifferently from the ways others can (Dixon 2019: 68). So thedifferent ways that the relationsbetween andadjacency are capable of applying is explained bypositionalists in terms of the different ontological structuresattributed to these relations,i.e. their number of roles orpositions and the nature of the relation that assigns things to rolesor positions. By contrast, anti-positionalists and primitivists denythat such questions about why relations have the capacity todifferentially apply need answering at all. As Fine describes his ownversion ofanti-positionalism, “It is a fundamentalfact for [the anti-positionalist] that relations are capable of givingrise to a diversity of completions in application to any given relataand there is no explanation of this diversity in terms of a differencein the way the completions are formed from the relation and itsrelata” (Fine 2000: 19).

However, the explanatory differences between the parties to the debaterun deeper than merely whether they think the capacity of relations todifferentially apply is open to informative explanation. Whilst Finetakes the capacity of relations to differentially apply as primitive,Fine also holds that it is necessary to provide an account of whereinthe difference consists between two states which arise from the sameconstituents: “Yet clearly the state of Anthony’s lovingCleopatra and the state of Cleopatra’s loving Anthony aredistinguishable; they are not merely two indiscernible”atoms“ within the space of states. But if these statesare not to be distinguished by how they derive from the given relationand its relata, then how are they to be distinguished?”(Fine2000: 20). Fine’s answer, as we have seen, is spelled out interms of substitution. But Leo’s version of anti-positionalismhas different explanatory settings again. He doesn’t engage withmetaphysical questions about wherein the difference between statesconsists and instead provides principles of substitution intended toexplicate rather than explain the differential application ofrelations (Leo 2013: 361–8). And primitivism, we have seen,takes the fact that relations admit of differential application asunanalysable too but also takes the difference between states arisingfrom the same constituents as primitive (MacBride 2014: 8,cf. van Fraassen 1982). The dispute about relations anddifferential application continues because it remains to beestablished which questions about relations are genuine and which arepseudo-problems.

In conclusion, it should be emphasised that nothing written here isintended to imply that there is just one problem of order(differential application). It’s plausible that there areseveral problems in the vicinity, whether logical, metaphysical orepistemological and they will require different if co-ordinatedsolutions.

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Acknowledgments

I am grateful to Stephen Barnett and especially Jeremy Butterfield forclose advice concerning the physics and philosophy of physics dealtwith in this entry. For further discussion of the material in thisentry, I am grateful to Helen Beebee, Chris Daly, Bill Demopoulos,Scott Dixon, Cian Dorr, Kit Fine, Berta Grimau, Herbert Hochberg,Frederique Janssen-Lauret, Sam Lebens, Joop Leo, Stephan Leuenberger,Anna-Sofia Maurin, Kevin Mulligan, Francesco Orilia, Gary Ostertag,Laurie Paul, Jonathan Schaffer, Stewart Shapiro, Peter Simons, TimWilliamson, and Ed Zalta. The material in this entry formed the basisfor a course on relations at Glasgow University and I’d like tothank the students for their many helpful comments andsuggestions.

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