1 A Trilemma

Assuming that one goes in for talk of states of affairs (as I shall), the following may be considered a non-negotiable datum (cf., e.g., MacBride 2007, 27):

It also seems prima facie hard to deny that

But this last statement might give rise to linguistic qualms; for, given that ‘is loved by’ is not even a complete phrase, it does not look like an appropriate target for the attribution of a semantic value. We can get around this by adopting the notational expedient of \(\lambda\)-expressions. Instead of ‘loves’ and ‘is loved by,’ we might speak of ‘\(\lambda x,y\ (x\text{ loves }y)\)’ and ‘\(\lambda x,y\ (x\text{ is loved by }y)\),’ and lay down a semantics of \(\lambda\)-expressions under which \(\ulcorner\lambda x,y\ (x\text{ $\varphi$s }y)\urcorner\) denotes whatever dyadic relation is such that the instantiation of that relation by any entities \(x\) and \(y\), in this order, is just the state of affairs that \(x\) \(\varphi\)s \(y\).² Under such a semantics, ‘\(\lambda x,y\ (x\text{ loves }y)\)’ denotes the dyadic relation whose instantiation by any entities \(x\) and \(y\) (in this order) is the state of affairs that \(x\) loves \(y\). Analogously for ‘\(\lambda x,y\ (x\text{ is loved by }y)\),’ which may also be said to denote the converse of \(\lambda x,y\ (x\text{ loves }y)\).

Using \(\lambda\)-expressions as names for relations, (D2) becomes:

And this is hard to deny. As the argument is both straightforward and tedious, I delegate it to a footnote.³ (D2) closely reflects what Bertrand Russell implies when he, in his Principles of Mathematics (1903), speaks of an “indubitable distinction between greater and less,” adding that

These two words have certainly each a meaning, even when no terms are mentioned as related by them. And they certainly have different meanings, and [what they mean] are certainly relations. (1903, 228)

So far, no problem. (D1) and (D2\('\)) can both be maintained without giving rise to any obvious contradiction. But a problem does arise once we adopt a further assumption, to the effect that

In other words, nothing is an instantiation of two relations. In Kit Fine’s seminal “Neutral Relations” (2000), this assumption (formulated using somewhat different terminology) is referred to as ‘Uniqueness.’ And now—at least assuming that there exists an instantiation of \(\lambda x,y\ (x\text{ loves }y)\) by Abelard and Héloïse (in this order) as well as an instantiation of \(\lambda x,y\ (x\text{ is loved by }y)\) by Héloïse and Abelard—we have a problem. For, by the semantics of \(\lambda\)-expressions suggested above, the former instantiation is the state of affairs that Abelard loves Héloïse, just as the latter instantiation is the state of affairs that Héloïse is loved by Abelard. By (D1), these ‘two’ states of affairs are one and the same. So, by (D2\('\)), we have here a single state of affairs that is an instantiation of two distinct relations. So we have a counter-example to (U). But, at least at first blush, (U) may seem an attractive thesis. For instance, the above-quoted passage from Russell’s Principles continues as follows:

Hence if we are to hold that “\(a\) is greater than \(b\)” and “\(b\) is less than \(a\)” are the same proposition, we shall have to maintain that both greater and less enter into each of these propositions, which seems obviously false; or else we shall have to hold that what really occurs is neither of the two […]. (1903, 228, boldface emphasis added)

What seems to bother Russell here is (i) the thought that the relation less should “enter into” an instantiation of the distinct relation greater and (ii) the analogous thought that greater should enter into an instantiation of less. According to MacBride (2020, sec. 4), adherents of (U) may offer the following motivation (cf. also Fine 2000, 4):

States are often conceived as complexes of things, properties and relations. They are, so to speak, metaphysical molecules built up from their constituents, so states built up from different things or properties or relations cannot be identical. Hence it cannot be the case that the holding of two distinct relations give rise to the same state. (MacBride 2020, sec. 4)

However, the picture of a relational state (i.e., of an instantiation of a relation) as a “metaphysical molecule,” admitting only a single way in which such a state is put together from its constituents, can seem slightly naïve or at least under-motivated. A possible way to motivate it may be to hold, on the one hand, that, if one and the same relational state is an instantiation of two relations, then there needs to be some explanation of how this can be (cf. Fine 2000, 15; MacBride 2007, 55; 2014, 4; Ostertag 2019, 1482), and, on the other hand, that it is not easy to see what such an explanation might look like. But this argument will be persuasive only as long as no plausible candidate explanation has been produced. So it seems appropriate to take a skeptical attitude towards (U), as MacBride does at the end of his (2007). More recently, David Liebesman notes that prima facie “the motivation for Uniqueness looks suspect” (2014, 412) and that “the intuitions elicited by Fine fail to establish Uniqueness” (2014, 413).

Given that the case for (U) looks fairly weak, and given how blatantly this thesis conflicts with (D1) and (D2\('\)), one may naturally expect that the literature on relations would have come down rather strongly against (U). However, this is not what we find.

In the Principles, Russell’s way out of the conflict between (U) on the one hand and (D1) and (D2\('\)) on the other was in effect to opt for the denial of (D1). Using Peirce’s notation for the converse of a relation, he concluded that “\(R\) and \(\breve{R}\) must be distinct, and ‘\(aRb\) implies \(b\breve{R}a\)’ must be a genuine inference” (1903, 229).⁴ This last remark suggests that the state of affairs that Abelard loves Héloïse would on Russell’s view be distinct from the state of affairs that Héloïse is loved by Abelard. A decade later, however, we find him endorsing the existence of entities that, following Fine, have become known as neutral relations. The text in question is his manuscript on the Theory of Knowledge (1984), which is worth quoting from at some length:

The subject of “sense” in relations is rendered difficult by the fact that the words or symbols by which we express a dual complex always have a time-order or a space-order, and that this order is an essential element in their meaning. When we point out, for example, that “\(x\) precedes \(y\)” is different from “\(y\) precedes \(x\)”, we are making use of the order of \(x\) and \(y\) in the two complex symbols by which we symbolize our two complexes. […] Nevertheless, we decided that there are not two different relations, one called before and the other called after, but only one relation, for which two words are required because it gives rise to two possible complexes with the same terms. (1984, 86)

A few paragraphs further down, the terms ‘before’ and ‘after’ are recycled for the purpose of naming two special relations that Russell refers to as positions:

Let us suppose an \(a\) and a \(b\) given, and let us suppose it known that \(a\) is before \(b\). Of the two possible complexes, one is realized in this case. Given another case of sequence, between \(x\) and \(y\), how are we to know whether \(x\) and \(y\) have the same time-order as \(a\) and \(b\), or the opposite time-order?

To solve this problem, we require the notion of position in a complex with respect to the relating relation. With respect to time-sequence, for example, two terms which have the relation of sequence have recognizably two different positions, in the way that makes us call one of them before and the other after. Thus if, starting from a given sequence, we have recognized the two positions, we can recognize them again in another case of sequence, and say again that the term in one position is before while the term in the other position is after. That is, generalizing, if we are given any relation \(R\), there are two relations, both functions of \(R\), such that, if \(x\) and \(y\) are terms in a dual complex whose relating relation is \(R\), \(x\) will have one of these relations to the complex, while \(y\) will have the other. The other complex with the same constituents reverses these relations. (1984, 87–88)

In this relatively brief passage, Russell introduces a member of what has become one of the most prominent families of views on the metaphysics of relations, namely positionalism. (The term is due to Fine, who coined it in his “Neutral Relations”; but I here use it in a slightly relaxed sense, on which a form of positionalism need not involve a commitment to what Fine calls ‘neutral relations.’) It has received more or less tacit endorsements by Segelberg (1947, 190), Armstrong (1978, 1997), Williamson (1985), Svenonius (1987, sec. 4), Barwise (1989, 180–181), Grossmann (1992, 57), Paul (2012, 251), Gilmore (2013), and Dixon (2018), among others. Where Russell speaks of ‘positions,’ these other authors speak in related senses of ‘sides,’ ‘relation places,’ ‘gaps,’ ‘empty places,’ ‘argument places,’ ‘slots,’ ‘ends,’ or ‘pockets’ of, or in, a relation.⁵ Castañeda (1972, 1975, 1982) attributes a form of positionalism to both Plato and Leibniz.⁶ More recently, Francesco Orilia (2008, 2011, 2014, 2019a, 2019b) has defended a form of positionalism under which positions, referred to as ‘onto-thematic roles,’ are widely shared among relations. These ‘roles’ are thought of as ontological counterparts of the thematic roles known from linguistics.

2 Positionalism

Most of the positionalists just cited conceive of relations as unordered or—using Fine’s term—‘neutral,’ i.e., as not imposing any order on the positions with which the respective relations are associated. (The only clear exceptions seem to be Gilmore and Dixon.) Nor has the appeal of unordered relations been limited to positionalists. The so-called antipositionalist views defended by Fine (2000, 2007) and Leo (2008a, 2008b, 2010, 2013, 2014, 2016) also conceive of relations as unordered, as does the ‘primitivist’ view proposed by MacBride (2014).⁷

Let us now look back at (D2\('\)). What would a proponent of unordered relations make of that thesis?

According to Williamson (1985), any relation \(R\) is identical with its converse, so that we have the equation ‘\(R=\breve{R}\).’⁸ But, he says, in this equation ‘\(R\)’ functions as a singular term, whereas, in ‘\(Rxy\),’ it instead functions as a relational expression, and this is supposed to block the inference from ‘\(Rxy\)’ to ‘\(\breve{R}xy\)’ which one might otherwise have felt entitled to on the strength of ‘\(R=\breve{R}\).’ Crucially, while ‘\(R\)’ “stands for the relation \(\mathrm{R}\), this does not exhaust its semantic significance: it stands for \(\mathrm{R}\) with a particular convention as to which flanking name corresponds to which gap in R” (italics in the original). He adds that “‘\(\breve{\mathrm{R}}\)’ as a relational expression uses the opposite convention” (1985, 257). On a certain flat-footed way of applying this treatment to the case of \(\lambda x,y\ (x\text{ loves }y)\), one would say that this relation is in fact identical with its converse \(\lambda x,y\ (x\text{ is loved by }y)\) and that (D2\('\)) is therefore false. But this would be to ignore the stipulatively specified semantics of \(\lambda\)-expressions on which that thesis was based (and with the help of which it was justified in footnote 3). What the Williamsonian positionalist should really say is that (D2\('\)) is not false but meaningless, due to a crippling mistake in the underlying semantics of \(\lambda\)-expressions. For under that semantics, “‘\(\lambda x,y\,(x\text{ loves }y)\)’ denotes the dyadic relation whose instantiation by any entities \(x\) and \(y\) (in this order) is the state of affairs that \(x\) loves \(y\).” To the Williamsonian positionalist, this talk of instantiation can make no sense, because it can make no sense, by his lights, to speak of a relation as having an instantiation by some entities \(x\) and \(y\) in a given order. After all, the Williamsonian positionalist conceives of relations as unordered. Mention to someone a certain unordered relation \(R\), together with some entities \(x\) and \(y\) and an ordering of \(x\) and \(y\): the receiver of this information cannot possibly deduce which of the two positions of \(R\) (or ‘gaps,’ in Williamson’s terminology) is supposed to be filled with \(x\) and which with \(y\). Any information about an ordering of \(x\) and \(y\) is simply irrelevant. What is needed is not a function from some set of ordinals to \(x\) and \(y\), but rather a function from the set of \(R\)’s positions to \(x\) and \(y\).⁹

We have now encountered one way in which the conflict between (D1), (D2\('\)), and (U) might be resolved while holding onto (U): namely, to treat (D2\('\)) as meaningless. Another option, which does not require the positing of unordered relations, would be to deny that relations have converses, so that, e.g., there only exists the relation \(\lambda x,y\,(x\text{ loves }y)\) or the relation \(\lambda x,y\,(x\text{ is loved by }y)\), but not both.¹⁰ There is also a third way, which requires that ‘relation’ may be said in at least two ways. Thus it might be thought that, in one of its senses, the term ‘relation’ applies to unordered relations while, in another sense, it applies to what one might call ‘ordered’ or (using another phrase coined by Fine) ‘biased’ relations. One might then go on to suggest that this latter sense is operative in (D2\('\)) and the former in (U). In this way the conflict between the three theses would be resolved through the power of equivocation, as it were, without having to abandon any of the three. But now there arises a question: How exactly should the believer in unordered relations conceive of ordered relations? We might be content with thinking of unordered relations as unanalyzable metaphysical whatnots, but the question of how ordered relations come by their peculiar directedness still deserves an answer.

According to one such answer, suggested by Fine, the positionalist might

think of each biased relation as the result of imposing an order on the argument-places [i.e. positions] of an unbiased relation. Thus, each biased relation may be identified with an ordered pair \((R, O)\) consisting of an unbiased relation \(R\) and an ordering \(O\) of its argument-places. Loves, for example, might be identified with the ordered pair of the neutral amatory relation and the ordering of its argument-places in which Lover comes first and Beloved second; and similarly for is loved by, though with the argument-places reversed. (2000, 11, original italics)

If we let \(\mathscr{A}\) be the “neutral amatory relation” and understand an “ordering of its argument-places in which Lover comes first and Beloved second” to be the ordered pair \((\textit{Lover}, \textit{Beloved})\), then this amounts to the suggestion that the ordered relation loves is the ordered pair \((\mathscr{A}, (\textit{Lover}, \textit{Beloved}))\) while its converse is the ordered pair \((\mathscr{A}, (\textit{Beloved}, \textit{Lover}))\). On a common construal of ordered triples, one might also put this by saying that loves is the ordered triple \((\mathscr{A}, \textit{Lover}, \textit{Beloved})\) while its converse is the ordered triple \((\mathscr{A}, \textit{Beloved}, \textit{Lover})\).

On this proposal, then, ordered relations are certain set-theoretic constructions. Such a proposal is apt to provoke resistance in anyone who is used to conceiving of ordered relations as the objectively determined semantic values of such verbs as ‘loves’ or ‘stabs,’ which these latter verbs stand for “without need of philosophical stipulation” (Williamson 1985, 254). It is also apt to provoke resistance in anyone who conceives of relations as “fundamental entities, not mere projections onto the world of idiosyncratic facts about human language” [Dorr (2004), 187; emphasis in the original]. However, the thesis that transitive verbs have determinate semantic values, outside of any more or less arbitrary assignment scheme, is a strong assumption that it is not a priori easy to see how to defend. And the idea that relations, whatever they are, can only be “fundamental” entities looks far from incontrovertible in light of the fact that it was once not unusual to conceive of relations as mere entia rationis (see, e.g., Brower 2024, sec. 5.2).

Once we have reached a point at which we are prepared to take seriously the identification of loves with \(( \mathscr{A}, \textit{Lover}, \textit{Beloved})\), it becomes natural to ask whether we might not, in the interest of both ontological and ideological parsimony, get rid of unordered relations altogether and take ordered \(n\)-adic relations to be simply ordered \(n\)-tuples of positions. On this view, loves would be the ordered pair \((\textit{Lover}, \textit{Beloved})\) and its converse would be \((\textit{Beloved}, \textit{Lover})\). In the case of certain symmetric relations, one might even make do with a single position. Thus the dyadic relation of adjacency might be construed as the ordered pair \((\textit{Next}, \textit{Next})\).¹¹ A great advantage of this construction lies in the fact that it immediately reveals this relation to be identical with its converse and thereby offers a satisfying explanation of why adjacency is symmetric.

However, presumably not every ordered pair of positions should count as a relation; and it might be argued that here is where unordered relations earn their keep. For instance, it might be thought that the pair \((\textit{Lover}, \textit{Giver})\) should not count as an ordered relation because there are no states of affairs in which both Lover and Giver are occupied; and the non-existence of such states may in turn be thought to be due to the putative fact that Lover and Giver do not belong to the same unordered relation.¹² Thus, more generally, unordered relations may be thought of as organizing positions into groups such that only members of the same group can have occupants in the same states of affairs. But again one might wonder why the work that is thus ascribed to unordered relations cannot be done more cheaply. After all, together with the category of unordered relations, we would need to have in our conceptual inventory a non-symmetric relational notion of ‘belonging’ that applies to unordered relations and their respective positions. Yet if unordered relations merely serve to ‘collect together’ certain sets of positions, then why not adopt instead a symmetric notion of connectedness that holds directly between positions? Rather than to say that Lover and Beloved are the only two positions that ‘belong’ to a certain unordered relation, we might then, for example, say that Lover and Beloved form a maximal clique of connected positions. Some other options will be mentioned in section 4.

3 The Instantiation Problem

Whether one keeps unordered relations in the picture or not, the task of working out the details of a positionalist theory of relations is not trivial. Above all, the positionalist will have to specify what exactly is required for a given ordered relation to be instantiated by some entities \(x_1,\dotsc,x_n\), in this order. While it may in principle be open to the positionalist to leave the concept of being instantiated by \(\dotsc\) (in this order) unanalyzed, this would be profoundly unsatisfactory. After all, on the positionalist view, at least of the sort now under discussion, ordered relations are fairly artificial set-theoretic constructs, and one would not expect that any metaphysically fundamental notion, other than the ‘formal’ notions of set-membership and identity (and perhaps mereological notions, if one follows Lewis (1991) in thinking of sets as fusions of singletons), would apply directly to ordered relations, any more than one would expect a set to have mass or charge other than in a derivative sense.¹³ Consequently the notion of instantiation, given that it does apply directly to ordered relations, would not plausibly be thought of as metaphysically fundamental. What we would like to have, then, is an account of what it takes for a given ordered relation to be instantiated by such-and-such entities in a given order.¹⁴

Can this instantiation problem, to give it a name, be avoided by abjuring (with Williamson, for example) all talk of ordered relations and acknowledging only unordered ones? Strictly speaking, yes. But the believer in unordered relations will then still be faced with the problem—which I shall call the contribution problem—of explaining what metaphysical work those unordered relations are supposed to do; and since their only reasonably clear hope for employment lies in contributing to the truth-conditions of relational predications, our theorist will thus be confronted with the task of specifying just what that contribution consists in. For example, someone who posits a ‘neutral amatory relation’ will need to tell some story, in the terms of her favored metaphysic, of what it takes for it to be the case that Abelard loves Héloïse; and that amatory relation had better play a prominent part in that story. (Or at least, so one may argue.¹⁵) Moreover, since for it to be the case that Abelard loves Héloïse is patently not the same as for it to be the case that Héloïse loves Abelard, the unordered-relations theorist will need to be able to tell a different story of what it takes for it to be the case that Héloïse loves Abelard, or at the very least allow that the relational state of Abelard’s loving Héloïse is distinct from that of Héloïse’s loving Abelard.

Arguably, however, mere numerical distinctness is not quite sufficient. Consider a ‘minimalist’ view that takes any two states \(Rab\) and \(Rba\) (for distinct \(a\) and \(b\)) to be merely numerically distinct ‘completions’ of some unordered relation \(R\): “two indiscernible ‘atoms’ within the space of states,” in Fine’s memorable phrase. If such a view were correct, it would be more perspicuous to write ‘\((R\{a,b\})_1\)’ and ‘\((R\{a,b\})_2\)’ instead of ‘\(Rab\)’ and ‘\(Rba\),’ using the subscripts ‘\(1\)’ and ‘\(2\)’ as nothing more than arbitrary tags. With the help of this amended notation, the minimalist view can be seen to suffer from the following difficulty: Suppose we have three particulars \(a\), \(b\), and \(c\), giving rise to six possible instantiations of \(R\), namely \((R\{a,b\})_1\), \((R\{a,b\})_2\), \((R\{b,c\})_1\), \((R\{b,c\})_2\), \((R\{a,c\})_1\), and \((R\{a,c\})_2\). Suppose further that, of these six states, only the following three obtain: \((R\{a,b\})_1\), \((R\{b,c\})_1\), and \((R\{a,c\})_2\). Question: Is \(R\) transitive on the set \(\{a,b,c\}\)? There appears to be no fact of the matter, or maybe one should say that the question is ill-posed. In either case, the minimalist has no ready way of capturing the distinction between transitive and non-transitive relations.¹⁶

How might the Finean antipositionalist address the contribution problem? A crucial feature of antipositionalism, as developed towards the end of “Neutral Relations,” is that it conceives of the ‘completions’ of neutral relations as interrelated by substitution, where the relevant notion of substitution is taken as primitive. Positions and ordered relations do not enter the picture at the ground level (as it were) but are rather conceived of as abstractions and set-theoretic constructions. While the antipositionalist is able—unlike the minimalist—to distinguish between transitive and non-transitive relations, she is unable to characterize the difference between, say, Abelard’s loving Héloïse and Héloïse’s loving Abelard without appeal to a reference state, such as as that of Antony’s loving Cleopatra (cf. Fine 2000, 29–30). As a result, the antipositionalist is unable to say what it takes for it to be the case that Abelard loves Héloïse independently of who else loves whom. This need not by itself constitute a problem. The antipositionalist might maintain that in fact there is nothing interesting to be said in response to the question of what it takes for Abelard to love Héloïse: she might regard Abelard’s loving Héloïse as a “basic relational fact (at least in the relevant respect),” as Fine (2007, 62) puts it. However, this view still leaves us in a curious position: plausibly there exist precisely two completions (or possible completions) of the neutral amatory relation in which Abelard and Héloïse function as relata. But antipositionalism offers no explanation as to why there should be exactly two such completions, rather than only one (as in the case of the adjacency relation), or three, or a hundred. Under antipositionalism, the fact that, for any given pair of distinct entities, there are exactly two completions of the amatory relation with those two entities as relata appears to be effectively treated as brute.¹⁷

While there is certainly more to be said about antipositionalism, I will have to leave the matter here.

4 Positionalism Developed

Let us now return to the positionalist’s instantiation problem, which (as may be recalled) was to provide “an account of what it takes for a given ordered relation to be instantiated by such-and-such entities in a given order.” This problem is inseparable from the question of how facts concerning positions—and, where applicable, unordered relations—determine what ordered relations there are. In addition, it is inextricably linked to the positionalist’s selection of basic notions and to the question of what role positions play in the individuation of relational states (where a relational state is just an instantiation of a relation). The menu of available options is marked by at least five noteworthy choice points.

Choice point #1: The occupation predicate. Arguably the central notion in the positionalist’s ideology is that of occupation, which in its simplest form applies to an entity, a position, and a relational state. While more complicated notions of occupation are conceivable, in the following we will only be discussing forms of positionalism that operate with this simple triadic concept, expressed by the predicate ‘occupies \(\dotsc\) in \(\dotsc\)’.

Choice point #2: Unordered relations. As already noted, positionalists have traditionally assumed that there are such things as unordered or ‘neutral’ relations with which positions are in some sense associated. However, at least in those forms of positionalism that (unlike the view put forward by Orilia) do not allow for positions to be shared among relations, the only theoretically significant work performed by unordered relations seems to lie in organizing positions into different ‘groups,’ where the theoretical role of these groups in turn lies in determining what relational states there are. Thus it might be said that it is because Lover does not ‘belong’ to the same unordered relation as Giver that there does not exist a state in which Antony occupies Lover and Cleopatra occupies Giver. To the positionalist who rejects unordered relations, by contrast, it is open to dispense with the concept of an unordered relation as well as with that of ‘belonging,’ and to work instead with a concept of connectedness that applies directly to positions (cf. section 2 above). She will then be able to say that it is simply because Lover is not connected to Giver that there does not exist a state in which Antony occupies Lover and Cleopatra occupies Giver.¹⁸

In following this route, the positionalist can further choose among several options. For example, she might assume that connectedness is transitive. But likewise she might hold that it isn’t, and allow that there are positions \(p\), \(q\), and \(r\) such that \(p\) is connected to \(q\) and \(q\) to \(r\), but \(p\) is not connected to \(r\), and that, correspondingly, there exist relational states in which both \(p\) and \(q\) are occupied, and also states in which both \(q\) and \(r\) are occupied, but no states in which both \(p\) and \(r\) are occupied. Another possibility would be to hold that what matters for the question of whether there exists a state in which two given positions \(p\) and \(q\) are occupied is not whether \(p\) and \(q\) are directly connected but rather whether they are directly or indirectly connected, i.e., whether there exist any positions \(p_1,\dotsc,p_n\) such that (i) \(p=p_1\), (ii) \(q=p_n\), and (iii) for each \(i\) with \(1\leq i<n\), \(p_i\) is connected to \(p_{i+1}\). Or again, she might hold that what matters is whether \(p\) and \(q\) are both members of the same maximal clique of connected positions.

Another interesting option would be to understand being connected as a multigrade notion, i.e., as a relational concept that can apply to different numbers of arguments. Equipped with such a concept, the positionalist might propose that the question of whether there exists a relational state in which some given positions \(p_1,p_2,\dotsc\), and no others, are occupied depends on whether \(p_1,p_2,\dotsc\) are connected, where this is not analyzable in terms of whether any two of them are connected.

Choice point #3: Non-obtaining states. The third choice point we have to consider concerns the question of whether to allow for non-obtaining relational states. Let us use the term state-positivism for the view that every state of affairs obtains (or in other words: for the view that every state of affairs is a fact).¹⁹ According to the state-positivist, there is no distinction to be drawn between obtainment and existence: Abelard loves Héloïse if and only if the state of Abelard’s loving Héloïse exists. The state-antipositivist, by contrast, will allow that this latter state exists even if Abelard does not love Héloïse.

Choice point #4: Multiply occupiable positions. To see how the positionalist might address the instantiation problem, let us focus on that form of positionalism that (i) employs a simple triadic notion of occupation, (ii) dispenses with unordered relations in favor of a multigrade notion of connectedness, and (iii) rejects state-positivism. On such a view, the question of how facts about positions determine what relations there are may be answered as follows:

It may further be natural to adopt the following uniqueness claim for relational states:

However, if the positionalist wishes to allow for positions to be multiply occupiable, a weaker claim is needed:

Finally, the instantiation problem may be addressed in two steps. In the first and main step, the positionalist may adopt a thesis that characterizes instantiations of ordered relations:

Note that, together with (R) and (US\('\)), it follows from this that any ordered relation has only at most one instantiation by a given sequence of entities. One can now specify what it takes for a given ordered relation to be instantiated by some such sequence:

This solves the instantiation problem for the form of positionalism that we have here been considering.

Choice point #5: The place of relations in the world. So far it has been left largely implicit what thesis positionalism amounts to: just what it is that positionalists want us to believe about the world. To remedy this situation, one could employ the concept of a relational phenomenon. For present purposes, a relational phenomenon may be understood to be simply any state of affairs that can be felicitously expressed with the help of ‘relational’ vocabulary—notably, transitive verbs and prepositions, as in ‘the cat is on the mat’ or ‘Abelard loves Héloïse.’ Unlike the concept of a relational state (i.e., of an instantiation of a relation), the concept of a relational phenomenon is not directly tied to that of a relation. Once we settle on a specific conception of relations, and also clarify the notion of an instantiation of a relation, we will have specified what a relational state is; but we will not thereby have specified how relational states relate to relational phenomena. Among the options that the positionalist is presented with in this regard, we can usefully identify two extremes, which might be called the strong and the weak thesis, respectively:

Of course, neither (ST) nor (WT) by itself amounts to a form of positionalism. However, we obtain a form of positionalism if we combine either (ST) or (WT) with a positionalistic conception of relations and relational states; and one such conception is given by (R) and (I1) above. A form of positionalism that entails (ST) may be called ‘strong positionalism,’ while a theory that entails only (WT) may be called ‘weak positionalism.’ Unlike the strong positionalist, the weak positionalist may well deny that the sentence ‘Abelard loves Héloïse’ expresses a relational state (although she will presumably agree that it expresses a relational phenomenon) and, correspondingly, that there exists such a thing as the relation \(\lambda x\,(x\text{ loves }y)\). For the sake of the example, however, I will in the following continue to assume that there is such a relation.

On the background of the above solution to the instantiation problem, let us now return one last time to the conflict observed in section 1 between (D1), (D2\('\)), and (U). To recapitulate, (D2\('\)) states that the (ordered) relation \(\lambda x,y\,(x\text{ loves }y)\) is distinct from \(\lambda x,y\,(x\text{ is loved by }y)\). The positionalist who wishes to analyze relational states like that of Abelard’s loving Héloïse in terms of the occupation of two positions Lover and Beloved will, if she also accepts (R), identify the relations \(\lambda x,y\,(x\text{ loves }y)\) and \(\lambda x,y\,(x\text{ is loved by }y)\) with, respectively, the ordered pairs \((\textit{Lover},\textit{Beloved})\) and \((\textit{Beloved},\textit{Lover})\). That these are distinct follows straightforwardly from the assumed distinctness of Lover and Beloved. So (D2\('\)) holds true. By contrast, (U)—the thesis that nothing is an instantiation of two relations—looks now more questionable than ever. For if one thinks of an ordered relation as an ordered tuple of positions, one will hardly be inclined to think of its instantiations as ‘metaphysical molecules’ in which it figures as a constituent. But then it becomes difficult to see the intuitive appeal of (U). With (U) accordingly given up, nothing prevents us from accepting (D1), i.e., the thesis that Abelard’s loving Héloïse is the same state as that of Héloïse’s being loved by Abelard. And indeed, if one identifies \(\lambda x,y\,(x\text{ loves }y)\) with \((\textit{Lover},\textit{Beloved})\) and \(\lambda x,y\,(x\text{ is loved by }y)\) with \((\textit{Beloved},\textit{Lover})\), then (D1) can be seen to follow from (US\('\)) and (I1).²³

5 Potential Objections

Still, it is not all smooth sailing for the positionalist. A first worry is akin to ‘Bradley’s regress.’ As we have seen, the positionalist (at least of the sort considered in this essay) characterizes relational states in terms of what positions are occupied in them by what entities. If now \(s\) is the state of Abelard’s loving Héloïse, shouldn’t there also be a further state of affairs to the effect that, in \(s\), the position Lover is occupied by Abelard—as well as a state of affairs to the effect that the position Beloved is in \(s\) occupied by Héloïse? If the positionalist is to apply her approach to these further states, she has to introduce three additional positions, of State, Occupant, and Position.²⁴ With their help the state of Abelard’s occupying Lover in \(s\)—call it \(s'\)—can be characterized as a state in which \(s\) occupies the position of State, Lover occupies Position, and Abelard occupies Occupant. (See figure 1.) But now we seem to have three further states on our hands, one of which may be characterized by saying that \(s'\) occupies in it the position of State, \(s\) the position of Occupant, and State the position of Position. And so the regress takes its course.²⁵ It is not obvious, however, that this regress is vicious. For it is not as if the state of Abelard’s loving Héloïse is in any sense grounded in (or ‘explained by’) the fact that Abelard occupies in it the role of Lover; rather, the former state is merely (in some suitable sense) “characterized” by the latter. We thus have a “regress of characterization,” not of grounding or explanation.

To be sure, the positionalist should presumably allow that

is in a certain sense a more perspicuous representation of Abelard’s loving Héloïse than the simpler and more familiar ‘Abelard loves Héloïse’: because (1), but not ‘Abelard loves Héloïse,’ lets us know about the existence of the two positions of Lover and Beloved. By the same token, a positionalist who posits the aforementioned positions of State, Occupant, and Position should presumably allow that

is more perspicuous than (1); but this is only because from (2)—and not from (1)—we can infer the existence of those three positions. Hence it is not the case that the positionalist has now embarked on some infinite ‘regress of perspicuity.’ Nor has she embarked on an infinite regress of analysis, in the form of some incompletable attempt at providing a metaphysical analysis of the ‘occupies \(\dots\) in \(\dots\)’ locution. To think that she has would be to presuppose that (2) is put forward as an attempt at such an analysis; but this would be highly uncharitable, given that (2) itself is rife with instances of that locution. The positionalist, at least of the stripe considered here, is ‘stuck’ with that locution in the same way in which a more traditional proponent of universals is stuck with ‘instantiates’ or ‘is an instantiation of \(\dotsc\) by \(\dotsc\).’ But this in itself is not an objection.

So much for potential worries about a vicious regress. In his “Neutral Relations,” Fine has raised a number of additional concerns about positionalism. According to one of his objections, positionalism is guilty of “ontological excesses” (2000, 16–17). This objection, however, appears to rest largely on the claim that “surely we would not […] wish to be committed to the existence of argument-places [a.k.a. positions] as the intermediaries through which the exemplification of the relations was effected” (2000, 16–17).

Fine has also maintained that positionalism is unable to accommodate strictly symmetric or multigrade (‘variably polyadic’) unordered relations (2000, 17, 22), where ” [a]n unbiased binary relation \(R\) is said to be strictly symmetric if its completion by the objects \(a\) and \(b\) is always the same regardless of the argument-places to which they are assigned” (2000, 17). This claim relies on a special feature of the particular form of positionalism discussed by Fine, namely that no position is ever occupied by more than one entity in the same state. There seems to be nothing incoherent, however, in embracing an alternative form of positionalism that does allow for multiple occupancy.²⁶

Admittedly, a positionalist who, contrary to the form of positionalism discussed by Fine, does not admit any unordered relations will a fortiori not be able to accommodate unordered relations that are strictly symmetric or multigrade. However, the idea that there are strictly symmetric or multigrade unordered relations is less of a datum than a metaphysical hypothesis. A theorist might be drawn to the idea that there are strictly symmetric unordered relations because it helps to accommodate certain intuitive identities between relational phenomena, such as the identity of \(a\)’s being next to \(b\) with \(b\)’s being next to \(a\). And a theorist might be drawn to the idea that there are multigrade unordered relations because it helps to accommodate certain analogies between relational phenomena, such as the analogy between, on the one hand, the state of affairs that \(a\) and \(b\) jointly support \(c\) and, on the other hand, the state of affairs that \(a\), \(b\), and \(c\) jointly support \(d\). But neither of these considerations constitutes a compelling argument for invoking unordered relations. The first intuition—that \(a\)’s being next to \(b\) is the same state of affairs as \(b\)’s being next to \(a\)—can be accommodated by adopting a form of positionalism under which \(a\)’s being next to \(b\) and \(b\)’s being next to \(a\) are ‘both’ characterized as a state in which a certain position Next is occupied by both \(a\) and \(b\). And the intuitive analogy between the state of affairs that \(a\) and \(b\) jointly support \(c\) and the state of affairs that \(a\), \(b\), and \(c\) jointly support \(d\) can be accommodated by positing two connected positions, Supporter and Supportee, of which at least the first is multiply occupiable (cf. Marmodoro 2021, 173).

6 Symmetries

Nonetheless, at least under a sufficiently ‘abundant’ view as to what (ordered) relations there are, some of them—in particular ones that exhibit a ‘cyclical’ symmetry—do not easily lend themselves to the positionalist approach.²⁷ To elaborate this point, we first have to go over some technical preliminaries.

Let us say that a function \(f\) is a symmetry of an \(n\)-adic ordered relation \(R\) iff \(f\) is a permutation of the set \(\{1,\dotsc,n\}\) such that, for any sequence of entities \(x_1,\dots,x_n\) and any \(y\): \(y\) is an instantiation of \(R\) by \(x_1,\dotsc,x_n\), in this order, iff \(y\) is an instantiation of \(R\) by \(x_{f(1)},\dotsc,x_{f(n)}\), in this order.²⁸ It is easy to verify that, for any \(n\)-adic unigrade ordered relation \(R\), the symmetries of \(R\) form a group with respect to function composition. That is to say, where \(S_R\) is the set of \(R\)’s symmetries, the following three conditions are satisfied:

This set \(S_R\) is also called the symmetry group of \(R\).²⁹ Further, for any group \(G\) of functions defined on a common set, let us say that the latter is the domain of \(G\). For example, if a given group consists of permutations of the set \(\{1,\dotsc,n\}\) (for some \(n>0\)), then this set is the domain of that group.

Consider now an \(n\)-adic ordered relation \(R\) (for some \(n>2\)) whose symmetry group satisfies the following condition:

A well-known example of such a relation is due to Fine (2000, 17, n.10): “the relation \(R\) that holds of \(a\), \(b\), \(c\), \(d\) when \(a\), \(b\), \(c\), \(d\) are arranged in a circle (in that very order)”. Fine goes on to say that “the following represent the very same state \(s\): (i) \(Rabcd\); (ii) \(Rbcda\); (iii) \(Rcdab\); (iv) \(Rdabc\).” If this list is supposed to be exhaustive, then the relation in question will have to be understood as a relation of circular arrangement that is either clockwise or counter-clockwise relative to some vantage point; for otherwise the state \(s\) may also be represented as (v) \(Rdcba\), (vi) \(Rcbad\), (vii) \(Rbadc\), and (viii) \(Radcb\).³⁰ Given that Fine specifies neither a vantage point nor a direction (clockwise or counter-clockwise), let us take \(R\) to be ‘direction invariant’ in this latter sense, i.e., so that the state \(Rabcd\) is identical not only with \(Rbcda\) (etc.), but also with \(Rdcba\). \(R\)’s symmetry group will then have eight members, which may be respectively represented as (i) \(id_4\), (ii) \((1\,4\,3\,2)\), (iii) \((1\,3)(2\,4)\), (iv) \((1\,2\,3\,4)\), (v) \((1\,4)(2\,3)\), (vi) \((1\,3)(2)(4)\), (vii) \((1\,2)(3\,4)\), and (viii) \((1)(2\,4)(3)\).³¹

This set is also known as a ‘dihedral group of order eight.’ To verify that it satisfies (C), it is enough to note that it, on the one hand, contains the permutation \((1\,4\,3\,2)\), which for instance maps \(1\) to \(4\), but on the other hand does not contain the permutation \((1\,4)(2)(3)\) that merely transposes \(1\) and \(4\). As Maureen Donnelly (2016, 88–89) points out, relations whose symmetry groups are of this kind—i.e., such as to satisfy (C)—tend to pose a problem for positionalism. More specifically, they pose a problem for the sort of positionalism that operates with a simple triadic occupation predicate and individuates relational states exclusively in terms of what entities occupy in them which positions. To see this, let us focus on the particular form of positionalism that conceives of relations in accordance with the statement (R) in section 4 above, and which conceives of instantiations of relations in accordance with the statements (US\('\)) and (I1) in the same section.

To begin with, we can note that the question of what position(s) an entity \(a\) occupies in the instantiation of \(R\) by some given sequence of entities \(x_1,\dotsc,x_4\) (at least one of which is \(a\) itself) depends, apart from \(R\), only on where \(a\) appears in this sequence.³² From this it follows that \(a\) has to occupy exactly the same position(s) in \(Radbc\) as it does in \(Rabcd\). Further, since the former state is identical with \(Rdbca\) (as is reflected in the fact that \(R\)’s symmetry group contains the permutation \((1\,2\,3\,4)\)), it follows that \(a\) occupies exactly the same position(s) in \(Rdbca\) as it does in \(Radbc\). Putting the previous two statements together, we have that \(a\) occupies the same position(s) in \(Rdbca\) as it does in \(Rabcd\). By analogous reasoning, it can be shown that \(d\) occupies the same position(s) in \(Rdbca\) as it does in \(Rabcd\). Hence, the two states \(Rabcd\) and \(Rdbca\) cannot differ with respect to which positions are in them respectively occupied by \(a\) and \(d\). And clearly they cannot differ, either, with respect to which positions are in them respectively occupied by \(b\) and \(c\). Accordingly, since, under the form of positionalism now in question, relational states are characterizable up to uniqueness in terms of what entities occupy in them which positions, it follows that the two states are identical. But they aren’t, as is reflected in the fact that \(R\)’s symmetry group fails to contain the permutation \((1\,4)(2)(3)\). So we have a contradiction.

To have a name for this difficulty, let us refer to it as the symmetry problem. How might a positionalist respond to it? The first thing to note is that it is not obviously a problem for what has above (in section 4) been called weak positionalism. This is because—as has in essence already been pointed out by MacBride (2007, 41)—it is open to the weak positionalist to deny the existence of relations whose symmetry groups satisfy (C).³³ In the particular case of Fine’s example, the weak positionalist may maintain that, for any entities \(a\), \(b\), \(c\), and \(d\), the state of affairs that \(a\), \(b\), \(c\), and \(d\), in this order, are arranged in a circle is only a relational phenomenon rather than a relational state: in other words, that it is not an instantiation of a relation. (It is compatible with this claim that the state of affairs in question is grounded in, or analyzable in terms of, states of affairs that are relational states.) Thus the positionalist may hope to obviate the symmetry problem by retreating to some form of weak positionalism and, with it, to a ‘sparse’ ontology of relations. Admittedly, however, this move is not likely to appeal to a theorist who is unwilling to give up the advantages of an abundant ontology of intensional entities.³⁴

Alternatively, the positionalist might opt for giving up the assumption that relational states are characterizable up to uniqueness in terms of what entities occupy in them which positions. She might then for instance allow that the states \(Rabcd\) and \(Rdbca\), although distinct, are both such that \(a\), \(b\), \(c\), and \(d\) occupy in them one and the same position \(p\). The idea that all four relata thus occupy the same position can be readily motivated by the symmetry of \(R\). This line of thought is not available, however, in the case of Leo’s (2008a, 2008b, 2010) example of a triadic relation \(S\) whose instantiation by any entities \(x\), \(y\), and \(z\) (in this order) is the state of affairs that \(x\) loves \(y\) and \(y\) loves \(z\). Given that this relation is thoroughly non-symmetric—its symmetry group contains only the identity permutation—the positionalist should find it hard to avoid positing three positions \(p_1\), \(p_2\), and \(p_3\) such that, for any \(x\), \(y\), and \(z\), the instantiation of \(S\) by \(x\), \(y\), and \(z\) (in this order) is a state in which \(p_1\) is occupied only by \(x\), \(p_2\) only by \(y\), and \(p_3\) only by \(z\). But if she follows this approach, she will not be able to accommodate the idea that, for any \(x\) and \(y\), the state \(Sxyx\) is identical with \(Syxy\). Plausibly \(Sxyx\) and \(Syxy\) are ‘both’ the state of affairs that \(x\) and \(y\) love each other, yet on the approach in question, \(p_2\) is in \(Sxyx\) occupied only by \(y\), while, in \(Syxy\), \(p_2\) is occupied only by \(x\).³⁵

A very different view has recently been proposed by Donnelly (2016). According to her relative positionalism, there exist unordered relations, associated with which there are ‘relative properties.’ At least from a formal point of view, these relative properties behave much like ordered relations: just as an ordered relation may be instantiated by some entities \(x_1,\dotsc,x_n\) (in this order), so a relative property may be instantiated by an entity \(x_1\) “relative to” an entity \(x_2\), \(\dotsc\), “relative to” an entity \(x_n\).³⁶ Relatedly, Donnelly’s view is not limited with regard to the symmetry groups it can accommodate; but this flexibility comes at a steep price in ontological commitment. Suppose \(R\) is a tetradic ordered relation whose symmetry group contains only \(id_4\). In place of \(R\), the relative positionalist would posit \(4!=24\) different relative properties. A non-relative positionalist, by contrast, would only posit four different positions \(p_1,\dotsc,p_4\). It is true that, given standard set theory, there would then also exist \(24\) different tuples \((p_i, p_j, p_k, p_l)\) for pairwise distinct \(i,j,k,l\in\{1,\dotsc,4\}\); and, as proposed above, these tuples could play the role of ordered relations. But the ontological commitment to these tuples would be a consequence of set theory, given the existence of \(p_1,\dotsc,p_4\). They would be ‘derivative’ entities. By contrast, the \(24\) relative properties posited by the relative positionalist would presumably have to be regarded as ontologically fundamental; for it is not easy to see (and Donnelly doesn’t specify) how they might be derived from anything more basic.³⁷

7 The Contributions to this Special Issue

Four of the papers of this Special Issue have first been presented at a workshop on “Properties, Relations, and Relational States” that has taken place in Lugano in October 2020.

Scott Dixon presents an extensive defense of what is often called the ‘standard view’ of relations, or ‘directionalism,’ against objections recently raised by Maureen Donnelly. A central thesis of directionalism is to the effect that a relation “applies to its relata in an order, proceeding from one to another.” Donnelly (2021, 3592) has criticized this conception as “obscure” and as failing “to connect with ordinary thinking about” the semantic difference between such statements as ‘Abelard loves Héloïse’ and ‘Héloïse loves Abelard.’ She also argues that directionalism “does not have the right structure to explain the differential application of partly symmetric relations like between or stand clockwise in a circle” (2021, 3592). Dixon responds to these criticisms and moreover argues that directionalism has advantages over a number of competing views, including Donnelly’s own.

Joop Leo describes a new form of positionalism, dubbed ‘thin positionalism,’ which can be regarded as a middle ground between traditional forms of positionalism on the one hand and antipositionalism on the other.³⁸ Thin positionalism, like its more traditional counterparts, accords a central place to the notion of a position. But positions are here conceived of as “substitutable places in a structure or form.” The substitution of entities for such positions yields relational complexes, which are also related among each other by substitution relationships. As in Fine’s antipositionalism, the relevant notion of substitution is taken as primitive. And, like Fine’s antipositionalism, thin positionalism is immune to the symmetry problem discussed in the previous section.

Fraser MacBride argues that quantification into predicate position, as one finds it in second-order logic, cannot be understood as quantification over “relations conveived of as the referents of predicates.” He argues for this thesis by constructing a dilemma. On the one hand, if converse predicates—understood as open sentences, such as ‘\(\xi\) is on top of \(\zeta\)’ and ‘\(\xi\) is underneath \(\zeta\)’—co-refer, then we fail to understand the higher-order predicates that are involved in quantification into relational predicate position: predicates (understood, again, as open sentences) such as ‘Alexander \(\Phi\) Bucephalus.’ On the other hand, if converse predicates do not co-refer, then we can still not make sense of those higher-order predicates unless we “impute implausible readings to lower-order constructions.” For instance, even a symmetric predicate, such as ‘\(\xi\) differs from \(\zeta\),’ would have to be read as applying to its relata in a given order, which, MacBride argues, would be implausible.

Francesco Orilia offers a sophisticated form of positionalism, dubbed dualist role positionalism, that on the one hand embraces very finely individuated ‘biased’ relations (and their abundant converses) at the ‘semantic’ level while, on the other hand, rejecting them “at the truthmaker or ontological level of sparse attributes.” At this more fundamental level, Orilia allows only neutral relations, whose exemplification he conceives of as being mediated through ‘roles’ such as agent and patient or inferior and superior. For instance, where \(V\) is a neutral relation of vertical alignment with respect to the Earth’s surface, Orilia would write (in boldface) ‘\(V(\mathrm{superior}(a), \mathrm{inferior}(b))\)’ to represent the state of affairs of a plane \(a\)’s being above a bird \(b\).

MacBride and Orilia, in their joint contribution, respond to van Inwagen’s (2006) argument for the conclusion that we do not have any “formal and systematic” names for non-symmetric relations. They concede the plausibility of supposing that, if non-symmetric relations had distinct converses, then it would be impossible to introduce such names for them. But they do not follow van Inwagen in holding that non-symmetric relations do have distinct converses. They point out that there are alternative conceptions of non-symmetric relations under which the existence of distinct converses—and hence the conclusion of van Inwagen’s argument—can be avoided. And they moreover argue, contra van Inwagen, that it is possible (either in English or a modest extension of English) to introduce names for non-symmetric relations of an adicity greater than \(2\).

Finally, Edward Zalta replies to two papers by MacBride. More specifically, he replies (i) to MacBride’s argument, in his contribution to the present issue, for the conclusion that second-order quantifiers cannot be interpreted as ranging over relations and (ii) to the argument in MacBride (2014) for the conclusion that (as Zalta puts it) “unwelcome consequences arise if relations and relatedness are analyzed rather than taken as primitive” (emphases in the original). Both arguments are examined in the light of Zalta’s theory of relations, as developed in the context of his object theory.³⁹ The resources of this theory are brought to bear on the individuation of states of affairs, an issue which Zalta identifies as central to both of MacBride’s arguments.

As I hope can be seen from this brief overview, the metaphysics of relations and relational states continues to be a fertile field of inquiry.