Since Timothy Williamson’s (1985) and Kit Fine’s (2000) critiques of Bertrand Russell’s (1903) view about the nature of relations, directionalism, according to which relations are understood as having a direction, applying to their relata in an order, philosophers have largely turned away from it.¹ They have turned toward views according to which relations are adirectional, or neutral. One popular sort of theory of neutral relations is absolute positionalism, according to which relations have positions or roles associated with them which their relata occupy or have, respectively Dixon (2018).² As Fine (2000) argues, however, absolute positionalist views face the problem of symmetric relations; they are unable to properly treat relations with certain symmetries. That is, they are unable to deliver the correct possible completions of such a relation, where a completion of a relation is anything which results from that relation applying to some things in a certain way, e.g., a fact, a state of affairs, or a proposition.³ I will characterize a way a relation can apply formally in what follows, but for now, a couple of examples will serve to elucidate the idea. The binary relation loving, for example, seems able to apply to two objects in two ways. Goethe’s loving Charlotte Buff is a different state of affairs from Buff’s loving Goethe. The binary relation being next to, on the other hand, seems able to apply to two objects in only one way. Goethe’s being next to Buff is the same state of affairs as Buff’s being next to Goethe.

The difficulty absolute positionalism has with symmetric relations has led to the development of other, neutral, views of relations which can solve this problem, including Fine’s (2000) antipositionalism, Fraser MacBride’s (2014) relational primitivism, and Maureen Donnelly’s (2016, 2021) relative positionalism. These views properly treat any fixed arity relation, no matter its particular symmetry structure. Donnelly has recently argued that directionalism is, like absolute positionalism, also unable to properly treat symmetric relations. I begin, in the remainder of this section, by explaining the difficulty Donnelly alleges directionalism has with symmetric relations, which emerges clearly even in the case of binary relations, and state how my reply on behalf of directionalism goes in that case. I then remind the reader of fixed arity relations of arity greater than two, which can have more complex symmetries, and which any account of relations, including directionalism, ought to be able to to treat properly.

In section 1, I develop a way of formally representing the symmetry structure of any fixed arity relation, similar to Donnelly’s (2016), and a way of formally modeling the ways a fixed arity relation can apply. Along the way, I discuss several relations with various symmetry structures, some of which are known to cause problems for absolute positionalism. In section 2, I explain how Donnelly takes her objection to generalize to \(n\)-ary relations for all \(n \geq 2\), and I develop my reply to this generalized criticism by showing how the directionalist can link the application conditions of any fixed arity relation, no matter its arity or symmetry structure, and its converse(s) in such a way that directionalism yields the correct manners in which it can apply.⁴ I thus establish that directionalism possesses the same advantage Donnelly’s, Fine’s, and MacBride’s accounts of relations have over absolute positionalism, which, it is well known, cannot handle all such relations.

In section 3, I turn to the task of evaluating directionalism, with my previous results in mind, in relation to other accounts of relations that avoid the problem of symmetric relations, viz., Fine’s, MacBride’s, and Donnelly’s. I argue that directionalism has advantages over each of these views. In the case of Donnelly’s relative positionalism, directionalism’s advantage is that it is not, like relative positionalism, committed to the involvement of relative properties in every irreducibly relational claim (i.e., in every relational claim which cannot be construed as a claim involving the instantiation of only ordinary non-relative properties). I close by conceding, in section 4, that, as Donnelly notes, directionalism is committed to the primitive relation of ordered relational application. But I don’t find this notion as mysterious as Donnelly does. I conclude that, even if one construes this feature of directionalism as a drawback, the two views are at worst at a draw, other things being equal, since this drawback is mitigated by the advantage directionalism has over relative positionalism. Unfortunately, I won’t have the space to properly address all of the objections that have been leveled against directionalism over the years, including Williamson’s and Fine’s, and instead leave replies to these objections for another occasion.

Directionalism is usually formulated in terms of binary relations only.⁵ It is typically taken to consist of three central theses.

The relation loving, for example, is understood by the directionalist as applying first to Goethe then to Buff when Goethe loves Buff, or, alternatively, proceeding from Goethe to Buff.⁶

The converse of loving, for example, is being loved by. It applies first to Buff and second to Goethe when Goethe loves Buff—in the opposite order or direction to that in which loving applies to them under the same condition.

where a (binary) relation \(R\) is necessarily symmetric if and only if, necessarily, \(Rxy\) if and only if \(Ryx\), and is non-symmetric otherwise. So while loving is distinct from its converse being loved by, a symmetric binary relation, like being next to, is its own converse.

Donnelly’s criticism of directionalism emerges clearly even in the case of binary symmetric relations. She says,

If the different ways \(R\) can hold among \(x_{1}, \ldots, x_{\text{n}}\) amount to just different orders of application of \(R\) to \(x_{1} \ldots, x_{\text{n}}\), then any difference in the order of \(x_{1}, \ldots, x_{\text{n}}\) should correspond to a different way for \(R\) to hold among \(x_{1}, \ldots, x_{\text{n}}\). (2021, 6, ital. orig.) ⁷

Donnelly is concerned that, because the directionalist imparts a direction to every (binary) relation, not just non-symmetric ones, she will be forced to say that, just as a non-symmetric binary relation like loving can apply to two objects in two ways, a symmetric binary relation like being next to will have to too. Note, however, that D2 saves the directionalist from this consequence. Since being next to is necessarily symmetric, by D3, it is its own converse, and so D2 demands that, when it applies to two objects like Goethe and Buff in that order, it must also apply to them in the opposite order. So there is only one way for it to apply to Goethe and Buff: the way in which it applies to Goethe and Buff both in that order and the opposite order. Contrast that with how directionalism treats loving. Since it is non-symmetric, by D3, it is distinct from its converse being loved by. D2 demands that, when loving applies to Goethe and Buff in that order, being loved by must apply to them in the opposite order (and vice versa). But this yields two ways for loving (and being loved by) to apply to Goethe and Buff: the way in which loving applies to Goethe and Buff in that order and being loved by does so in the opposite order, and the way in which loving applies to Buff and Goethe in that order and being loved by does so in the opposite order. Of course many countenance relations of arity greater than two, and such relations exhibit a variety of different symmetry structures, and as I will discuss later, Donnelly takes her concern to generalize to many of these structures. So the directionalist’s response can’t be as simple as this. To understand Donnelly’s criticism in full, and the directionalist’s response to it, we first need to see the full picture of the possible symmetry structures relations can have. This task I undertake in the next section.

1 Relations of Arbitrary Symmetry

Following Donnelly (2016), I represent a relation’s symmetry (structure) by its symmetry group. A group is a set of objects that is closed under an associative operation \(\cdot\), the group operation, which has a unique identity element \(e\) such that \(x\), \(e \cdot x = x \cdot e = x\) and, for each element \(x\), a unique inverse element \(x^{-1}\) such that \(x \cdot x^{-1} = x^{-1} \cdot x = e\). A symmetry group of an \(n\)-ary relation is a group of permutations of \(\{1, 2, \ldots, n\}\) (i) whose group operation is function composition, \(\circ\), (ii) whose identity element is the identity permutation (i.e., the permutation that maps 1 to 1, 2 to 2, …, and \(n\) to \(n\)), and (iii) for which the inverse of each element is that element’s inverse permutation. In particular,

As Donnelly notes (2016, 83, incl. fn.10), the symmetry group of any \(n\)-ary relation will be a subgroup of the group of all possible permutations of \(\{1, \ldots, n\}\), i.e, of the symmetric group of degree \(n\), or \(S_{n}\).⁹

A question arises at this point, for each \(n\)-ary relation \(R\), whether the fact that, necessarily, for all \(x_{1}, \ldots, x_{n}\), \(R x_{1} \ldots\, x_{n}\) iff \(R x_{p(1)} \ldots\, x_{p(n)}\) really is sufficient for \(p\) to be in \(R\)’s symmetry group, as the above definition of symmetry groups stipulates, or whether instead \([R x_{1} \ldots\, x_{n}]\) must be identical to \([R x_{p(1)} \ldots\, x_{p(n)}]\) to guarantee this to be the case, where \([R x_{1} \ldots\, x_{n}]\) and \([R x_{p(1)} \ldots\, x_{p(n)}]\) are completions of the same type (viz., facts, states of affairs, or propositions). But since “\(R\)” appears on both sides of the biconditional in the definition, there will presumably be no cases in which \([R x_{1} \ldots\, x_{n}]\) is distinct from \([R x_{p(1)} \ldots\, x_{p(n)}]\). It is plausible that, for any relations \(R\) and \(R'\), when \(R = R'\), if necessarily, \(R x_{1} \ldots\, x_{n}\) iff \(R' x_{p(1)} \ldots\, x_{p(n)}\), then \([R x_{1} \ldots\, x_{n}] = [R' x_{p(1)} \ldots\, x_{p(n)}]\), even if this is implausible when \(R \neq R'\). So an intensional definition of symmetry groups should be adequate. For this reason, I’ll allow myself to move back and forth between talk of (non-)identity of completions and (non-)equivalence of relational claims in what follows.

The discussion of relations’ symmetry groups has been pretty abstract so far, so I’ll consider some examples. I’ll begin with the symmetry groups of the binary relations being next to and loving. Since, necessarily, for any \(x_{1}\) and \(x_{2}\), \(x_{1}\) is next to \(x_{2}\)

• iff \(x_{1}\) is next to \(x_{2}\) (equivalently: \(x_{[1\, 2](1)}\) is next to \(x_{[1\, 2](2)}\)),

and

• iff \(x_{2}\) is next to \(x_{1}\) (equivalently: \(x_{[2\, 1](1)}\) is next to \(x_{[2\, 1](2)}\)),

where \(\ulcorner [i_{1}\, i_{2} \ldots\, i_{n}]\urcorner\) denotes the permutation of \(\{1, 2, \ldots, n\}\) that maps \(1\) to \(i_{1}\), \(2\) to \(i_{2}\), …, and \(n\) to \(i_{n}\), the symmetry group of being next to,

In other words, every permutation of \(x_{1}\) and \(x_{2}\) results in an equivalent claim. But since (i) necessarily, for any \(x_{1}\) and \(x_{2}\), \(x_{1}\) loves \(x_{2}\) iff

• \(x_{1}\) loves \(x_{2}\) (equivalently: \(x_{[1\, 2](1)}\) loves \(x_{[1\, 2](2)}\))

but (ii) it is not the case that, necessarily, for any \(x_{1}\) and \(x_{2}\), \(x_{1}\) loves \(x_{2}\) iff

• \(x_{2}\) loves \(x_{1}\) (equivalently: \(x_{[2\, 1](1)}\) loves \(x_{[2\, 1](2)}\)),

the symmetry group of loving,

In other words, the only permutation of \(x_{1}\) and \(x_{2}\) that results in an equivalent claim is the identity permutation, i.e., the permutation that leaves the two terms where they are.

An \(n\)-ary relation such that, necessarily, for all \(x_{1}, \ldots, x_{n}\), \(R x_{1} \ldots\, x_{n}\) iff \(R x_{p(1)} \ldots\, x_{p(n)}\) for every permutation \(p \in S_{n}\), is completely symmetric, while one that is such that this is true only when \(p\) is the identity permutation of \(S_{n}\), \([1\, 2\, \ldots \, n]\), is completely non-symmetric. Being next to is an example of the former, and loving the latter. Indeed, any binary relation can only be either completely symmetric or completely non-symmetric, since there are only two subgroups of the group of \(S_{2}\), viz., \(S_{2}\) itself, and the group that consists of just the identity permutation of \(S_{2}\), i.e., \(\{[1\, 2]\}\). There are, of course, also completely symmetric and completely non-symmetric \(n\)-ary relations for \(n > 2\) as well, though I will not consider any here.

Fine (2000, 17–18, incl. fn.10) argues that absolute positionalism is unable to properly treat fixed arity relations with certain symmetries (see also Donnelly 2016, sec. 5.3). According to absolute positionalism, relations are neutral (directionless), but feature positions, which have been interpreted as worldly correlates of thematic roles in linguistics that their relata fill Orilia (2014), or as entities akin to holes which their relata occupy Dixon (2018). Such views properly treat relations with some symmetries just fine. But there are relations with other symmetries that they cannot properly treat. They can properly treat any completely symmetric or completely non-symmetric relation one might throw at them.

For a theory of relations to properly treat a given \(n\)-ary relation, I mean that the theory has the resources to ensure that that relation can apply in the ways that we think it should be able to apply. But what is a way for an \(n\)-ary relation to apply? And, for a given \(n\)-ary relation, what are the ways that it should be able to apply? The ways such a relation can apply can be identified with the left cosets of that relation’s symmetry group. For a given ordering of \(n\) objects, yielding a certain completion of an \(n\)-ary relation \(R\), \(Sym_{R}\) includes exactly those permutations of that ordering that yield the same completion of \(R\), which of course include the identity permutation. This amounts to one way the relation can apply. For some relations (any relation that is not completely symmetric), there will be non-identity permutations of that initial ordering (in \(S_{n}\) but not in \(Sym_{R}\)) that yield distinct completions of a given sort (facts, states of affairs, or propositions). Consider such a relation \(R\) and such a non-identity permutation \(q\). Then \([R x_{1} \ldots\, x_{n}] = [R x_{[1\, \ldots\, n](1)} \ldots\, x_{[1\, \ldots\, n](n)}] \neq [R x_{q(1)} \ldots\, x_{q(n)}]\). And \([R x_{q(1)} \ldots\, x_{q(n)}]\) will be identical to every other completion (of the same sort) that results from permuting the arguments of \(R\) in \([R x_{q(1)} \ldots\, x_{q(n)}]\) by some permutation in \(Sym_{R}\). The sets of permutations identified by considering every \(q \in S_{n}\) form the left cosets of \(Sym_{R}\) in \(S_{n}\), and represent the ways \(R\) can apply to \(n\) fixed objects. More formally,

The left cosets of the symmetry group of an \(n\)-ary relation \(R\) partition \(S_{n}\) into between \(1\) and \(n!\) equally-sized sets of permutations, depending on \(R\)’s symmetry group. And by Lagrange’s theorem, which implies that the number of left cosets of a subgroup \(H\) of a group \(G\) equals \(|G| \div |H|\), the number of left cosets of \(Sym_{R} = |S_{n}| \div |Sym_{R}|\). So there are \(|S_{n}| \div |Sym_{R}|\) ways for \(R\) to apply to \(n\) objects.^10,11

A completely symmetric \(n\)-ary relation \(R\) will therefore be able to apply to \(n\) objects in only \(|S_{n}| \div |Sym_{R}| = |S_{n}| \div |S_{n}| = 1\) way, corresponding to the single coset of \(Sym_{R}\) in \(S_{n}\). The single way being next to can apply to two objects, for example, corresponds to the single left coset of \(Sym_{\textit{being next to}} = \{[1\, 2], [2\, 1]\}\) in \(S_{2} = \{[1\, 2], [2\, 1]\}\), viz., \(\{[1\, 2], [2\, 1]\}\) itself.¹² (\(|S_{2}| \div |Sym_{\textit{being next to}}| = 2 \div 2 = 1\).) A completely non-symmetric \(n\)-ary relation, on the other hand, will be able to apply in \(|S_{n}| \div |Sym_{R}| = |S_{n}| \div |\{[1\, 2\, \ldots\, n]\}| = |S_{n}| \div 1 = n!\) ways to \(n\) objects, corresponding to the \(n!\) cosets of \(Sym_{R}\) in \(S_{n}\). The two ways loving can apply to two objects, for example, correspond to the two left cosets of \(Sym_{\textit{loving}} = \{[1\, 2]\}\) in \(S_{2} = \{[1\, 2], [2\, 1]\}\), viz., \(\{[1\, 2]\}\) and \(\{[2\, 1]\}\). (\(|S_{2}| \div |Sym_{\textit{loving}}| = 2 \div 1 = 2\).)

The absolute positionalist can say that a completely symmetric relation has just one position which can take up to \(n\) arguments. This results in there being just one way for such a relation to apply to \(n\) objects: that constituted by each of those objects being assigned to that single position. So, for example, the absolute positionalist would say that being next to has one position, \(p_{1}\), which can take up to two arguments, and so there is only one way for it to apply to two objects like Goethe and Buff. Goethe and Buff can only both be assigned to \(p_{1}\). And, as mentioned, there is indeed only one way for being next to to apply to two objects like Goethe and Buff. Goethe’s being next to Buff is the same state of affairs as Buff’s being next to Goethe. The absolute positionalist can say that a complete non-symmetric \(n\)-ary relation has \(n\) positions, each of which can take just a single argument. This results in there being \(n!\) ways for such a relation to apply to \(n\) objects, each corresponding to a different assignment of those \(n\) objects to those \(n\) positions. For example, the absolute positionalist would say that loving has two positions, \(p_{2}\) and \(p_{3}\), each of which can take just a single argument, and so there are two ways for it to apply to two objects, such as Goethe and Buff. Goethe can be assigned to \(p_{2}\) and Buff to \(p_{3}\), or Buff can be assigned to \(p_{2}\) and Goethe to \(p_{3}\). And, as mentioned, there are indeed two ways for loving to apply to two objects like Goethe and Buff: one in which Goethe is doing the loving, and Buff is being loved, and one in which Buff is doing the loving, and Goethe is being loved.

In addition to completely symmetric and non-symmetric \(n\)-ary relations for \(n > 2\), however, there are also partially (non-)symmetric such relations. The symmetry group of a partially symmetric \(n\)-ary relation is a proper non-trivial subgroup of \(S_{n}\). That is, it will contain some, though not all, non-identity permutations of \(\{1, \ldots, n\}\). The ternary relation being between is an example of such a relation. Since (i) necessarily, for any \(x_{1}\), \(x_{2}\), and \(x_{3}\), \(x_{1}\) is between \(x_{2}\) and \(x_{3}\)

• iff \(x_{1}\) is between \(x_{2}\) and \(x_{3}\) (equivalently: \(x_{[1\, 2\, 3](1)}\) is between \(x_{[1\, 2\, 3](2)}\) and \(x_{[1\, 2\, 3](3)}\)),

and

• iff \(x_{1}\) is between \(x_{3}\) and \(x_{2}\) (equivalently: \(x_{[1\, 3\, 2](1)}\) is between \(x_{[1\, 3\, 2](2)}\) and \(x_{[1\, 3\, 2](3)}\)),

but (ii) this is false of every other permutation of \(\{1, 2, 3\}\), the symmetry group of being between,

Absolute positionalist views can properly treat some partially symmetric relations, like this one. The absolute positionalist can say that such a relation, while ternary, has only two positions, \(p_{4}\) and \(p_{5}\), the first of which can take only a single argument, while the other can take up to two (see Dixon 2018, 208). This results in there being three ways for such a relation to apply to three objects, such as Larry, Curly, and Moe. Larry can be assigned to \(p_{4}\) and the other two to \(p_{5}\), or Curly can be assigned to \(p_{4}\), and the other two to \(p_{5}\), or Moe can be assigned to \(p_{4}\), and the other two to \(p_{5}\). And there are, indeed, three ways for such a relation to apply to Larry, Curly, and Moe. Larry could be between the other two, or Curly could be, or Moe could be. These three ways correspond to the three left cosets of \(Sym_{\textit{being between}} = \{[1\, 2\, 3], [1\, 3\, 2]\}\) in \(S_{3} = \{[1\, 2\, 3], [1\, 3\, 2], [2\, 1\, 3], [2\, 3\, 1], [3\, 1\, 2], [3\, 2\, 1]\}\), viz., \(\{[1\, 2\, 3], [1\, 3\, 2]\}\), \(\{[2\, 1\, 3], [2\, 3\, 1]\}\), and \(\{[3\, 1\, 2], [3\, 2\, 1]\}\). (\(|S_{3}| \div |Sym_{\textit{being between}}| = 6 \div 2 = 3\).)

But absolute positionalist views cannot handle all partially symmetric relations. The ternary relation being arranged clockwise in that order is such a relation.¹³ Since (i) necessarily, for any \(x_{1}\), \(x_{2}\), and \(x_{3}\), \(x_{1}\), \(x_{2}\), and \(x_{3}\) are arranged clockwise in that order

and

but (ii) this is false of every other permutation of \(\{1, 2, 3\}\), the symmetry group of being arranged clockwise in that order,

The absolute positionalist appears to have only four options for treating being arranged clockwise. But none of these options yields the correct number of possible ways for it to apply to three objects. If it has one position, then it can apply in only one way. If it has two positions, one which can take only a single argument while the other can take up to two, it can apply in three ways (as was the case with in the previous example). If it has two positions, either of which can take up to two arguments, then it can apply in six ways. And if it has three positions, it can apply in six ways. But there are two ways for such a relation to apply to three objects, like Larry, Curly, and Moe. Larry, Curly, and Moe could be arranged clockwise in that order. Or Larry, Moe, and Curly could be arranged in that order instead. These two ways correspond to the two left cosets of \(Sym_{\textit{being arranged clockwise in that order}}\) in \(S_{3} = \{[1\, 2\, 3], [1\, 3\, 2], [2\, 1\, 3], [2\, 3\, 1], [3\, 1\, 2], [3\, 2\, 1]\}\), viz., \(\{[1\, 2\, 3], [2\, 3\, 1], [3\, 1\, 2]\}\) and \(\{[1\, 3\, 2], [3\, 2\, 1], [2\, 1\, 3]\}\). (\(|S_{3}| \div |Sym_{\textit{being arranged clockwise in that order}}| = 6 \div 3 = 2\).)

2 Generalizing Directionalism

It is the shortcoming of absolute positionalism just related which has motivated others to develop alternative accounts of relations. This includes Donnelly, who develops relative positionalism, which provably yields the correct possible completions of any fixed arity relation. She recognizes that the problem of symmetric relations is at its heart an algebra problem, and uses this to draw insights about what relations would have to be like to avoid the problem. But she thinks that directionalism is unable to do the same. Donnelly (2016, 83–85; and 2021, 6) takes her concern about directionalism’s ability to deal with symmetric relations, which I explicated above, to generalize to any relation that is anything but completely non-symmetric. Stated generally, her concern is that, because each \(n\)-ary relation \(R\) applies to \(n\) relata in a total order, there will always be \(n!\) ways for \(R\) to apply to \(n\) relata, clashing with our intuitive judgement about \(n\)-ary relations that are anything but completely non-symmetric that they apply in \(m\) ways where \(m < n!\). In this section, I show how directionalism can properly treat relations of any fixed arity relation.

It is clear that directionalism, as formulated earlier in the text, like absolute positionalism, cannot properly treat the relation being arranged clockwise in that order. This is for the simple reason that directionalism was formulated there in terms of binary relations only, and a relation expressed by the predicate “…, …, and \(\ldots\) are arranged clockwise in that order” is presumably ternary. (For the same reason, directionalism, as formulated above, can’t even handle being between—something that I noted absolute positionalism can do.) But all the directionalist needs to do is construe D1 as allowing for some relations to take more than two relata. The direction of such a relation can be understood as the ordering of those relata, proceeding from the first relatum to the second, to the third, …, to the \(n\)th.

Then, once a couple more adjustments are made to the original formulation of directionalism, it becomes clear that directionalism can treat these relations, and indeed relations of any fixed arity, and that it can do so properly, no matter these relations’ symmetries. First, for any \(n\)-ary relation \(R\) and each possible ordering of \(n\) relata, \(x_{1}, \ldots, x_{n}\), the directionalist posits a unique converse for \(R\) which applies to \(x_{1}, \ldots, x_{n}\) in that ordering of them exactly when \(R\) applies to \(x_{1}, \ldots, x_{n}\) in that order. More precisely,

where

\(p\)-Converse Existence effectively replaces D2.

I will not define the notion of a converse of a relation, as it appears in clause (i) of the above definition. A straightforward way to do so—in terms of a \(p\)-converse of that relation (given a definition of \(p\)-converses which omits clause (i) of the above definition)—is as follows:

But if one thinks that there are distinct though intensionally equivalent relations, this definition would be too permissive. For example, it would seem that, necessarily, for any \(x_{1}\) and \(x_{2}\), \(x_{1}\) is triangular and taller than \(x_{2}\) iff \(x_{2}\) is shorter than \(x_{1}\) and \(x_{1}\) is trilateral. But presumably being a \(y\) and \(z\) such that \(y\) is shorter than \(z\) and \(z\) is triangular—and not being a \(y\) and \(z\) such that \(y\) is shorter than \(z\) and \(z\) is trilateral—is the single distinct converse of the completely symmetry binary relation being triangular and larger than. Such cases would also prevent one from supposing that the \(p\)-converse of a relation is unique, as I stipulate in the above definition of \(p\)-converses.

I will prevent such cases from causing problems by instead taking the notion of a converse as primitive, regarding facts about which relations are which relations’ converses as brute, and adopting the following principle:

I assume that every relation is (one of) its own converse(s), so that \(R = R_{[1\, \ldots\, n]}\) for every \(n\)-ary relation \(R\). Thus the notion of a converse I have in mind, and which I will employ in what follows (mainly to simplify the discussion), is different than that given by D3—the claim that every necessarily symmetric binary relation is identical to its converse, while every other binary relation is distinct from its converse. But even if revised according to this new terminology, D3 will still entail a difference between necessarily symmetric relations and all other fixed arity relations; each of the former is its own only converse, while each of the latter has at least one converse distinct from itself.

To be able to properly treat any \(n\)-ary relation \(R\), no matter its symmetry, all the directionalist needs to do is identify those \(p\)-converses of \(R\) whose orderings of relata \(x_{1}, \ldots, x_{n}\), when \(R x_{1} \ldots\, x_{n}\), “can be transformed into one another by a permutation in the symmetry group” of \(R\) (Donnelly 2016, 94). More precisely,

\(p\)-Converse Identity effectively replaces D3.¹⁴ I assume that the symmetry structure of any \(n\)-ary relation \(R\) is represented by some subgroup of \(S_{n}\).¹⁵ Whatever subgroup of \(S_{n}\) \(Sym_{R}\) turns out to be, \(p\)-Converse Identity guarantees that \(R = R_{p}\) iff \(p \in Sym_{R}\). (\(R = R_{[1\, \ldots\, n]}\), and so \(p\)-Converse Identity implies that \(R = R_{q^{*}}\) iff there is some \(p \in Sym_{R}\) such that \(q^{*} = p \,\circ\, [1\, \ldots\, n] = p\).) This ensures that the ways \(R\) can apply to \(n\) objects correspond to the left cosets of \(Sym_{R}\), as they should. This is because, by the definition of \(p\)-converses, \(R = R_{p}\) iff, necessarily, for any \(x_{1}, \ldots, x_{n}\), \(R x_{1} \ldots\, x_{n}\) iff \(R x_{p(1)} \ldots\, x_{p(n)}\), and so the directionalist has ensured that \(p \in Sym_{R}\) iff, necessarily, for any \(x_{1}, \ldots, x_{n}\), \(R x_{1} \ldots\, x_{n}\) iff \(R x_{p(1)} \ldots\, x_{p(n)}\), which is in agreement with the definition of symmetry groups. And I explained in the previous section why the ways an \(n\)-ary relation \(R\) can apply correspond to the left cosets of \(Sym_{R}\) in \(S_{n}\). This in turn ensures, of course, that the number of ways \(R\) can apply to \(n\) objects equals \(|S_{n}| \div |Sym_{R}|\), as it should.

But directionalism must also imply that the symmetry group of any converse of an \(n\)-ary relation \(R\) is isomorphic to that of \(R\). What I’ve just said establishes that any \(n\)-ary relation will, according to directionalism, be able to apply to \(n\) objects in the ways we think it should. But we also expect \(R\)’s (non-identical) converses (if it has any) to apply to \(n\) objects in the same ways as \(R\) (or, at least, in ways that are structurally the same). To show that this isomorphism holds, consider any \(n\)-ary relation \(R\) and any permutation \(p\) of \(\{1, 2, \ldots\}\). There is bijective function \(f_{p}\) from \(Sym_{R}\) to \(Sym_{R_{p}}\) such that, for any \(i, j \in Sym_{R}\), \(f_{p}(i \circ j) = f_{p}(i) \circ f_{p}(j)\). \(f_{p}(q) = p \circ q \circ p^{-1}\) fits this bill. (Recall that \(p^{-1}\) is the inverse of \(p\). See section 1 above.) In other words, \(f_{p}(q)\) is the permutation in \(S_{n}\) that maps \(a\) to \(b\) iff \(q\) maps \(p^{-1}(a)\) to \(p^{-1}(b)\), where \(a, b \in \{1, \ldots, n\}\), i.e., \(f_{p}(q)(a) = b\) iff \(q(p^{-1}(a)) = p^{-1}(b)\).¹⁶ In general, \(Sym_{R_{p}} = \{p \circ q \circ p^{-1} : q \in Sym_{R}\}\).¹⁷

This discussion has been very abstract, so to provide the reader with a better idea of how directionalism handles relations of different arities and symmetries, and to highlight some interesting differences between directionalism, understood as applying to relations of any fixed arity, as compared to the binary formulation of it that I gave on pages four and five, I’ll show how directionalism, as formulated above, treats the examples I discussed in section 1. I’ve already noted that, according to directionalism, a (completely) symmetric binary relation is its own only converse, while a (completely) non-symmetric binary relation has a single converse distinct from it. (Though now even a non-symmetric binary relation is a converse of itself.) But it will be instructive to see how \(p\)-Converse Existence and \(p\)-Converse Identity result in these treatments. Consider first the binary (completely) symmetric relation being next to. \(p\)-Converse Existence implies that being next to has \(p\)-converses \(\textit{being next to}_{[1\, 2]}\) and \(\textit{being next to}_{[2\, 1]}\). Since \([1\, 2], [2\, 1] \in Sym_{R} = \{[1\, 2], [2\, 1]\}\), by \(p\)-Converse Identity, \(\textit{being next to} = \textit{being next to}_{[1\, 2]} = \textit{being next to}_{[2\, 1]}\). So by the definition of \(p\)-converses, being next to has a single converse, viz., itself. This means that, according to directionalism, being next to can apply to two things, such as Goethe and Buff, in only one way. If being next to applies to Goethe and Buff in that order, then being next to’s converse must apply to them in the opposite order. And if being next to’s converse applies to Goethe and Buff in that order, then being next to must apply to them in the opposite order. But since being next to is its own converse, there is no difference between these two possibilities, which are depicted in figure 1.

By assigning Goethe to \(1\) and Buff to \(2\), it is clear that this single manner of application corresponds to the single left coset \(\{[1\, 2], [2\, 1]\}\) of \(\text{SYM}_{\textit{being next to}}\) in \(S_{2}\).

Things go the same in the case of a non-symmetric relation like loving, except that, because \([2\, 1] \notin Sym_{\textit{loving}} = \{[1\, 2]\}\), it follows by \(p\)-Converse Identity that \(\textit{loving} = \textit{loving}_{[1\, 2]} \neq \textit{loving}_{[2\, 1]}\). So by the definition of \(p\)-converses, loving has two converses, one of which is itself, the other, presumably, being being loved by. This means loving can apply to two things, such as Goethe and Buff, in two ways. If loving applies to Goethe and Buff in that order, then loving’s distinct converse, being loved by, must apply to them in the opposite order. And if being loved by applies to Goethe and Buff in that order, then loving must apply to them in the opposite order. But since loving \(\neq\) being loved by, these are two different possibilities, which are depicted in figure 2.

By assigning Goethe to \(1\) and Buff to \(2\), it is clear that these two manners of application correspond to the two left cosets \(\{[1\, 2]\}\) and \(\{[2\, 1]\}\) of \(\text{SYM}_{\textit{loving}}\) in \(S_{2}\).

Things become more complicated for ternary relations. Consider being between. Recall that

By \(p\)-Converse Existence, the directionalist would say that being between (\(R\) for now) has \(p\)-converses \(R_{[1\, 2\, 3]}\) (\(= R\)), \(R_{[1\, 3\, 2]}\), \(R_{[2\, 1\, 3]}\), \(R_{[2\, 3\, 1]}\), \(R_{[3\, 1\, 2]}\), and \(R_{[3\, 2\, 1]}\). And because

and

the directionalist would say, by \(p\)-Converse Identity, that (i) \(R_{[1\, 2\, 3]} = R_{[1\, 3\, 2]}\), (ii) \(R_{[2\, 1\, 3]} = R_{[3\, 1\, 2]}\), and (iii) \(R_{[2\, 3\, 1]} = R_{[3\, 2\, 1]}\). But because

and

the directionalist would say, by \(p\)-Converse Identity, that \(R_{[1\, 2\, 3]} \neq R_{[2\, 1\, 3]}\), \(R_{[1\, 2\, 3]} \neq R_{[2\, 3\, 1]}\), and \(R_{[2\, 1\, 3]} \neq R_{[2\, 3\, 1]}\) (and so \(R_{[1\, 3\, 2]} \neq R_{[2\, 3\, 1]}\) and \(R_{[3\, 1\, 2]} \neq R_{[3\, 2\, 1]}\)).

By the definition of \(p\)-converses, this means that being between, according to the directionalist, has three converses, one of which is itself. To identify plausible interpretations of the two converses distinct from being between, suppose Larry is between Curly and Moe, and consider the following diagram.

applies to Larry, Curly, and Moe in that order, and also to Larry, Moe, and Curly in that order (see (i) above). But what relation applies to Curly, Larry, and Moe in that order and to Moe, Larry, and Curly in that order (as (ii) above demands)? A plausible interpretation of this relation is being on the far side of from the perspective of. Curly is on the far side of Larry from the perspective of Moe, and Moe is on the far side of Larry from the perspective of Curly. And what relation applies to Curly, Moe, and Larry in that order and to Moe, Curly, and Larry in that order (as (iii) above demands)? A plausible interpretation of this relation is being on the opposite side as from the perspective of. Curly is on the opposite side as Moe from the perspective of Larry, and Moe is on the opposite side as Curly from the perspective of Larry.¹⁸

Given the way the application conditions of these three relations are connected to one another, there are, according to directionalism, three possible ways for each of them to apply to three objects, like Larry, Curly, and Moe. These three manners of application are depicted in figure 4.¹⁹

The reader can check, by assigning Larry to \(1\), Curly to \(2\), and Moe to \(3\), that these three manners of application correspond to the three left cosets \(\{[1\, 2\, 3], [1\, 3\, 2]\}\), \(\{[2\, 1\, 3], [2\, 3\, 1]\}\), and \(\{[3\, 1\, 2], [3\, 2\, 1]\}\) of \(Sym_{\textit{being between}}\) in \(S_{3}\).

Being between is noteworthy because it has more than one converse distinct from it, which undermines the idea, expressed in D2, that the order in which a relation \(R\)’s converse applies to its relata is opposite to that in which \(R\) does; they are merely different. When Larry is between Curly and Moe, being between applies in the orders \([l\, c\, m]\) and \([l\, m\, c]\), being on the far side of from the perspective of applies in the orders \([c\, l\, m]\) and \([m\, l\, c]\), and being on the opposite side as from the perspective of applies in the orders \([c\, m\, l]\) and \([m\, c\, l]\). But neither of these two pairs of orders seem to be opposite the two orders in which being between applies; they appear only to be different. It is somewhat plausible that the first and third of these relations apply in opposite orders. It is, after all, Larry who is the one who is privileged in the scenario under consideration (i.e., when Larry is between Curly and Moe). And Larry is at opposite ends of the light grey and dark grey arrows in the leftmost column of figure 4, which depicts this scenario. But neither of the first and third relations could plausibly be understood to apply in orders opposite to those in which the second relation (depicted by the black arrow) applies, and the second relation is nonetheless a converse of each of the other two. This, in conjunction with my choice to count every relation—even every completely non-symmetric relation—as its own converse means that the most we can hang onto as far as D2 goes is that, except in cases of completely non-symmetric relations, a relation \(R\)’s converse (even in the case when it is its own converse) applies to its relata in an order that is different, not opposite, from the order in which \(R\) applies to them.

Our discussion of being between also helps to illustrate why D3, which covers only (completely) symmetric and (completely) non-symmetric binary relations, needs to be replaced with something, like \(p\)-Converse Identity, that can accommodate complete non-symmetries and partial symmetries which arise in relations of higher arities. According to D3, a symmetric binary relation is identical to its converse, while a non-symmetric one is distinct from its converse. But a completely non-symmetric \(n\)-ary relation, where \(n \in \{3, 4, \ldots\}\), will have more than one converse distinct from it, \(n! - 1\), to be exact. And a partially symmetric relation will have more than one converse (some factor of \(n!\) between \(1\) and \(n!\)), though one of those converses will be identical to it. Of completely symmetric relations of any arity, the directionalist can say that it has a single converse, viz., itself.

Consider last the ternary relation being arranged clockwise in that order—the relation with a symmetry structure that causes problems for the absolute positionalist. Recall that

By \(p\)-Converse Existence, the directionalist would say that being arranged clockwise in that order (\(R\) for now) has \(p\)-converses \(R_{[1\, 2\, 3]}\) (\(= R\)), \(R_{[1\, 3\, 2]}\), \(R_{[2\, 1\, 3]}\), \(R_{[2\, 3\, 1]}\), \(R_{[3\, 1\, 2]}\), and \(R_{[3\, 2\, 1]}\). And because

and

the directionalist would say, by \(p\)-Converse Identity, that (i) \(R_{[1\, 2\, 3]} = R_{[2\, 3\, 1]}\) and (ii) \(R_{[1\, 2\, 3]} = R_{[3\, 1\, 2]}\) (and so \(R_{[2\, 3\, 1]} = R_{[3\, 1\, 2]}\)). Similarly, because

and

the directionalist would say, by \(p\)-Converse Identity, that (i) \(R_{[1\, 3\, 2]} = R_{[2\, 1\, 3]}\) and (ii) \(R_{[1\, 3\, 2]} = R_{[3\, 2\, 1]}\) (and so \(R_{[2\, 1\, 3]} = R_{[3\, 2\, 1]}\)). And finally, because

we know by \(p\)-Converse Identity that \(R_{[1\, 2\, 3]} \neq R_{[1\, 3\, 2]}\) (and so \(R_{[1\, 2\, 3]} \neq R_{[2\, 1\, 3]}\), \(R_{[1\, 2\, 3]} \neq R_{[3\, 2\, 1]}\), \(R_{[1\, 3\, 2]} \neq R_{[2\, 3\, 1]}\), and \(R_{[1\, 3\, 2]} \neq R_{[3\, 1\, 2]}\)).

By the definition of \(p\)-converses, this means that being arranged clockwise in that order has two converses, one of which is itself. A plausible interpretation of the converse of being arranged clockwise in that order distinct from it is being arranged counterclockwise in that order.²⁰ Given the way the application conditions of these two relations are coordinated, there are, according to directionalism, two possible ways for each of them to apply to three objects, like Larry, Curly, and Moe. These three manners of application are depicted in figure 5.²¹

The reader can check, by assigning Larry to \(1\), Curly to \(2\), and Moe to \(3\), that these two manners of application correspond to the two left cosets \(\{[1\, 2\, 3], [2\, 3\, 1], [3\, 1\, 2]\}\) and \(\{[1\, 3\, 2], [3\, 2\, 1], [2\, 1\, 3]\}\) of \(Sym_{\textit{being arranged clockwise in that order}}\) in \(S_{3}\).

3 Directionalism’s Advantages Over Its Closest Competitors

I’ve shown how directionalism avoids Donnelly’s charge, in that it is able to properly treat any fixed arity relation with any symmetry such a relation can have. As such, it possesses the same advantage over absolute positionalist theories that is enjoyed by Donnelly’s relative positionalism, Fine’s (2000) antipositionalism, and MacBride’s (2014) relational primitivism. In this section, I describe some advantages that directionalism has over each of these three accounts of relations. First, directionalism, unlike primitivism, supplies an explanation of why a given relation can apply in the ways it can. Second, directionalism, unlike antipositionalism and primitivism, supplies an explanation of why two relations can apply in the same or different ways (as the case may be). And third, directionalism, unlike relative positionalism, isn’t committed to the involvement of relative properties in every irreducibly relational claim (i.e., in every relational claim which cannot be captured by a claim involving the instantiation of only ordinary non-relative properties). I’ll describe each of these advantages in that order, explaining the views along the way as necessary.

Relations can apply in a variety of ways. But why is a given relation able to apply in the ways in can? Not all accounts of relations answer this question. Directionalism does. For example, directionalism explains why the binary relation being next to can apply to two objects in the single way it can. This is because it can apply to up to two objects (i.e., it is a binary relation), it is its own unique converse, and necessarily, for any \(x_{1}\) and \(x_{2}\), if it applies to \(x_{1}\) and \(x_{2}\) in that order (i.e., if \(x_{1}\) is next to \(x_{2}\)), then its converse applies to \(x_{2}\) and \(x_{1}\) in that order (i.e., \(x_{2}\) is next to \(x_{1}\)). The binary relation loving, on the other hand, can apply to two objects in the two ways it can, according to the directionalist, because (i) it can apply to up to two objects, (ii) it has a single converse distinct from it, viz., being loved by, and (iii) necessarily, for any \(x_{1}\) and \(x_{2}\), if loving applies to \(x_{1}\) and \(x_{2}\) in that order (i.e., if \(x_{1}\) loves \(x_{2}\)), then its distinct converse applies to \(x_{2}\) and \(x_{1}\) in that order (i.e., \(x_{2}\) is loved by \(x_{1}\)). This ensures that it is possible for loving to apply to \(x_{2}\) and \(x_{1}\) in that order whether or not it applies to \(x_{1}\) and \(x_{2}\) in that order, and vice versa, yielding two ways in which it can apply to two objects. In general, for any \(n\)-ary relation \(R\), \(R\) can apply to \(n\) objects in the ways it can because \(R\) can take up to the number of relata it can, it has the number of converses it does, and the application conditions of it and its \(p\)-converses are necessarily connected in the ways that they are.

Contrast this with MacBride’s relational primitivism, which is the view that, in general, there is no explanation for why any given relation can apply in the ways it can. It is, according to the primitivist, a matter of brute fact, for example, that being next to can apply in the single way it can, and that loving can apply in the two ways it can. By refusing to explain such facts, the primitivist avoids postulating any machinery that might treat a relation improperly (as does the absolute positionalist’s machinery). Thus primitivism can properly treat any relation directionalism can properly treat. But primitivism has a pro tanto disadvantage compared to directionalism, in that it does not supply an explanation of the behavior of each relation it can properly treat, whereas directionalism does.

Now to directionalism’s second advantage. Some relations seem able to apply in the same ways as one another, while others seem able to apply in different ways from one another. Consider loving and hating. Each of these relations can apply to two objects in two ways. Moreover, they seem to be applicable in the same two ways. \(Sym_{\textit{loving}} = Sym_{\textit{hating}} = \{[1\, 2]\}\), and so the two left cosets of each these relations’ symmetry groups are the same; they are the two left cosets of \(\{[1\, 2]\}\) in \(S_{2}\), viz., \(\{[1\, 2]\}\) and \(\{[2\, 1]\}\). The single way in which the binary relation being next to can apply to two objects is distinct from each of the two ways in which loving or hating can do so. That way is represented by the single left coset of \(Sym_{\textit{being next to}} = \{[1\, 2], [2\, 1]\}\) in \(S_{2}\), viz., \(\{[1\, 2], [2\, 1]\}\) itself. Directionalism supplies explanations of the identities and distinctions between the ways any two fixed arity relations can apply to appropriate numbers of objects in terms of the relation’s arity, the number of converses it has, and how the application conditions of it and its \(p\)-converses are necessarily connected.

In the case of loving and hating, the directionalist says that these relations can apply in the same two ways because each has arity two, each has two converses, one of which is itself and the other distinct from it, and each is such that, necessarily, for any \(x_{1}\) and \(x_{2}\), if it applies to \(x_{1}\) and \(x_{2}\) in that order, then its distinct converse applies to \(x_{2}\) and \(x_{1}\) in that order. The way in which being next to can apply to two objects is different from the two ways in which loving (or hating) can do so, according to directionalism, because, while these relations have the same arity, the former relation is its own only converse, while the latter relation is distinct from one of its converses. As a result, while the latter can apply to two objects in two ways, the former, whenever it applies to \(x_{1}\) and \(x_{2}\) in that order, it must, as its own only converse, apply to \(x_{2}\) and \(x_{1}\) in that order as well, yielding only a single way in which it can apply.

Some relations, while able to apply in the same number of ways, can nonetheless apply in different ways from one another. The ternary relation being arranged clockwise in that order, for example, can apply to three objects in two ways. But these two ways are different than the two in which loving or hating can apply. An intuitive explanation for this is that the first two ways can involve up to three objects, while the latter two can’t. In terms of cosets, this can be explained by the fact that the two cosets of \(Sym_{\textit{being arranged clockwise in that order}}\) in \(S_{3}\), viz., \(\{[1\, 2\, 3], [2\, 3\, 1], [3\, 1\, 2]\}\) and \(\{[1\, 3\, 2], [3\, 2\, 1], [2\, 1\, 3]\}\) are pairwise distinct from the two left cosets that represent the ways loving or hating can apply. The directionalist can explain these differences by appealing to the fact that being arranged clockwise in that order has a different arity than each of loving and hating.

Some relations have the same arity, apply in the same number of ways, but nonetheless apply in different ways. Such relations, though of the same arity, still have non-isomorphic symmetry groups, and thus the way such relations can apply are still represented by different left cosets. For example, the six ways in which the quaternary being arranged clockwise in that order\(^{4}\) (as in Alice, Bob, Carol, and Diane are arranged clockwise in that order) can apply to four objects are pairwise distinct from the six ways in which being closer together than (as in Alice and Bob are closer together than Carol and Diane) can apply to them.²² I will not go to the trouble of listing these cosets, but instead just briefly explain why the symmetry groups of these two relations, viz.,

and

are not isomorphic. This is illustrated by the fact that the latter relation yields the same completion if certain pairs of relata are transposed in its application to them, while the former relation does not. For example,

but

(see Dixon 2019, 68–69 for discussion of this point). These relations have the same number of converses (six). The directionalist will explain these differences in possible applications by appealing to differences in the ways the application conditions of these relations are necessarily connected.²³

In contrast, neither Fine’s antipositionalism nor MacBride’s relational primitivism supplies explanations of the identities or differences in the ways distinct relations can apply. The primitivist supplies no explanation for why any given relation can apply in the ways it can, and so, ipso facto, can supply no explanation for why two relations can apply in the same or different ways as the case may be.²⁴ According to antipositionalism, relations do not have positions. What determines the ways in which a given relation can apply to some things—its manners of completion—are not facts about the internal structure of the completions that result from its application. Instead, the ways a relation can apply are determined by identity and distinctness relationships that hold between completions of that relation by different sets of objects. In previous work, I say,

the manner in which Goethe and Buff complete loving\(\ldots\) in Goethe’s loving Buff is the same, on Fine’s view, as exactly one of the two manners in which W. B. Yeats and Maud Gonne complete that relation in Yeats’s loving Gonne and Gonne’s loving Yeats, and it is distinct from the other. Which identity and distinctness relationships hold of these two possible but mutually exclusive sets of possibilities is, according to the antipositionalist, a matter of brute fact. (Dixon 2019, 65)

Antipositionalism can properly treat any relation that directionalism (and relative positionalism) can treat, as long as any such relation is instantiated by enough distinct sets of objects. But, as I note (see Dixon 2019, 70, fn.17), because Fine defines the identity of manners of completions of relations \(R\) and \(R'\) only when \(R = R'\), the antipositionalist is left without a way to compare manners of completions of distinct relations. Here is Fine’s statement of the definition:

to say that \(s\) is a completion of a relation \(R\) by \(a_{1}, a_{2} \ldots, a_{\text{m}}\), in the same manner as \(t\) is a completion of \(R\) by \(b_{1}, b_{2}, \ldots, b_{\text{m}}\) is simply to say that \(s\) is a completion of \(R\) by \(a_{1}, a_{2} \ldots, a_{\text{m}}\) that results from simultaneously substituting \(a_{1}, a_{2} \ldots, a_{\text{m}}\) for \(b_{1}, b_{2}, \ldots, b_{\text{m}}\) in \(t\) (and vice versa). (2000, 25–26)

Moreover, I also note, it is not clear that Fine’s definition could be modified in such a way that it could apply when \(R \neq R'\). There will be no principled way to identify the manner in which a non-symmetric relation \(R\) applies to some things with any one of the manners in which a distinct non-symmetric relation applies to some other things rather than any of the other ways \(R'\) applies to those other things. Why, for example, should Goethe’s loving Buff result from simultaneously substituting Goethe and Buff for Yeats and Gonne (and loving for hating) in Yeats’s hating Gonne rather than in Gonne’s hating Yeats? Only if this question has an answer will the antipositionalist have a way to explain why the way in which loving applies to Goethe and Buff in Goethe’s loving Buff is identical to the way in which, say, hating applies to Yeats and Gonne in Yeats’s hating Gonne and distinct from the way in which hating applies to them in Gonne’s hating Yeats rather than vice versa. There does not seem to be a non-ad hoc way to answer questions like this, and so the antipositionalist seems to be left unable to compare the manners of completions of distinct relations.^25,26

Directionalism has an advantage over relative positionalism too. Relative positionalism is the view that, when a relation applies to some things, its doing so consists in those things occupying positions of the relation relative to one another. But the positions of a relation are not understood, on relative positionalism, as roles that objects fill, or holes that they occupy, as they are understood on absolute positionalist views. Instead, they are construed as unary relative properties, which relata instantiate relative to one another. A relative property is a property that can be instantiated by a thing only relative to a thing or some things, while a non-relative property is a property that can be instantiated by a thing full stop. If being north is a property, rather than a binary relation, it is presumably a relative property, since something can be north, it would seem, only relative to something or some things. It makes no sense, for example, to say that Washington, D.C. is north. Washington, D.C. is north relative to something, such as Kingston, Jamaica. In contrast, many would take a property like being spherical to be non-relative. Exceptions to even the latter sort of case are certain endurantists, who regard putative non-relative properties as relative properties that can be instantiated only relative to a time. More on this below.

Structurally, relative positionalism and directionalism are quite similar. The directionalist sees the application of each relation as being order-sensitive, and involving attendant order-sensitive applications of its converse(s). And while the relative positionalist regards each relation as neutral (directionless), she also regards each as having one or more relative properties—equal in number to the number of converses a relation has according to directionalism—which are instantiated by \(x_{p(1)}\) relative to \(x_{p(2)}\), …, relative to \(x_{p(n)}\) in exactly those orders that the directionalist would have her relation and its distinct converse apply to \(x_{1}, \ldots, x_{n}\). So according to the relative positionalist, being next to has one relative property, which one might interpret as being adjacent, that is instantiated by \(x_{1}\) relative to \(x_{2}\) and by \(x_{2}\) relative to \(x_{1}\) whenever \(x_{1}\) is next to \(x_{2}\), yielding only a single way for being next to to apply two objects. Loving, on the other hand, has two relative properties, being a lover and being beloved, the first of which is instantiated by \(x_{1}\) relative to \(x_{2}\) when \(x_{1}\) loves \(x_{2}\) and the second of which is instantiated by \(x_{2}\) relative to \(x_{1}\), and vice versa when \(x_{2}\) loves \(x_{1}\), yielding two ways for loving to apply to two objects. The ternary relation being between has three relative properties, resulting in it being able to apply in the three ways discussed in section 1, while the ternary relation being arranged clockwise in that order has two, resulting in it being able to apply in the two ways discussed in section 1.²⁷ Like directionalism, relative positionalism can provably properly treat any fixed arity relation with any symmetry such a relation can have (see Donnelly 2016, 94–96).

Directionalism possesses an advantage over relative positionalism in that it is not, while relative positionalism is, committed to the involvement of relative properties in every irreducibly relational claim. An irreducibly relational claim is a claim which cannot be captured by a claim involving the instantiation of only ordinary non-relative properties. For example, the claim that Goethe and Buff are mortal can be captured by the claim that Goethe is mortal and Buff is mortal, which, if it involves the instantiation of properties at all, is most plausibly understood as involving the instantiation of ordinary non-relative properties, viz., the property being mortal. I’m putting aside some endurantists’ view, mentioned above, that anything that we might have thought is a non-relative property is actually a relative property which can be instantiated only relative to a time. But even the claims I’ve been discussing at length, like “Goethe is next to Buff” and “Goethe loves Buff,” could as easily be regarded as irreducibly relational by such endurantists as by others, since people across that divide think that such claims cannot be adequately paraphrased as claims that involve the instantiation of only non-relative properties.

Relative positionalism’s commitment to both relations and relative properties is problematic for the simple reason that it makes that view ontologically less parsimonious than directionalism, as the latter view is committed to only one type of entity, viz., relations. In answer to a different objection, Donnelly (2016, 98–99) considers a version of relative positionalism according to which there are no relations, just relative properties; relational predicates are associated immediately with a certain number of relative properties.²⁸ Adopting this relationless relative positionalism would enable the relative positionalist to do away with relations altogether, and be committed to the same number of types of entities as the directionalist. But directionalism possesses an advantage over relationless relative positionalism as well. Directionalism is a theory of non-relative relations only, and makes claims only about their application. It explains why a given non-relative relation \(R\) can apply in the ways it can in terms of the fact that it has a certain number of converses, all of whose application conditions are necessarily connected in a certain way. It says nothing about relative properties. It does not explain the application of non-relative relations in terms of relative properties, and it does not posit relative properties anywhere else. But it is compatible with their existence. Directionalism is perfectly compatible with the existence of relational claims that involve the instantiation of relative properties rather than the application of relations; it just won’t say anything about why these relative properties can be instantiated in the ways they can. That is the job of a theory of relative properties—something which directionalism does not purport to be. Relationless relative positionalism, on the other hand, is committed to the claim that any irreducibly relational claim involves the instantiation of relative properties and not the application of relations.

Thus relationless relative positionalism is compatible with a narrower range of epistemic possibilities than directionalism, and is therefore methodologically inferior in this respect. It is incompatible with the existence of relations, while directionalism is not similarly incompatible with the existence of relative properties. In addition to this, however, there is reason to think that, while some irreducibly relational claims are best understood in terms of the instantiation of relative properties, others are best understood in terms of the application of relations. Jack Spencer (2016) argues that this is the case. Spencer is interested in relativity, the phenomenon of something’s being a certain way relative to a thing or some things.²⁹ One of Donnelly’s examples of a relative property, which I mentioned above, is that something is north only relative to a location (or an object in a location). Each example that Spencer has in mind is something that, at least on its face, seems like it can be appropriately construed as the instantiation of a relative property, like being north, as Donnelly conceives of it, or of a relative relation like being closer than, as in San Francisco’s being closer than (i.e., as compared to) Seattle relative to (i.e., from the perspective of) Los Angeles.

Spencer argues that there are at least two ways to cash out talk about relativity, only one of which invokes genuine relative properties (or relative relations). According to the first, relationalism, a putative relative property or relation is actually a non-relative relation of greater arity. Instead of there being a genuine relative \(n\)-ary property or relation that is instantiated relative to a thing, there is in fact a non-relative \(n + 1\)-ary relation. Being north, on this view, is not a unary property, instantiated relative to a location, but is instead a binary relation, which takes a location as its second argument. According to the second way to cash out talk of relativity, variabilism, a relative \(n\)-ary property or relation is understood as being genuinely \(n\)-ary, and its relativity is captured by the fact that the extension of that property or relation can change when the value of a parameter associated with that property or relation (an index) changes. Being north, on this view, is a genuine unary property. But its extension function has a location parameter, and can yield different extensions when that parameter takes different values. So, for example, when the location parameter is Lima, Peru, the extension of being north includes Kingston, Jamaica, whereas when the location parameter is Washington, D.C., it does not.

On Spencer’s account, the difference between relationalism and variabilism, and thus between relative and non-relative properties and relations, is substantive. Relative properties’ and relations’ extensions vary across parameters, which can take different values, while non-relative properties’ and relations’ extensions do not, since they don’t have such parameters. This means that a relative property or relation is always instantiated relative to at least one thing whenever it is instantiated at all, while a non-relative property or relation is never so instantiated. If instantiation is itself non-relative, then the instantiation relation that \(n\) objects stand in to a non-relative \(n\)-ary property or relation \(R\) can be at most \(n + 1\)-ary (due to the fact that it will take \(R\) as an argument in addition to up to \(n\) other arguments). But the instantiation relation that \(n\) objects stand in to an \(n\)-ary relative property or relation \(R'\) can be up to \(n + k + 1\)-ary, where \(k\) is the number of parameters relative to which \(R'\) may be instantiated. If, on the other hand, the instantiation relation is relative for relative relations and non-relative for non-relative relations, then \(R\) and \(R'\) will stand in different instantiation relations altogether.

Spencer (2016, 440–444) notes that there are certain tests to which we can subject a putative relative property or relation that can tell us whether it is a genuine example of such an entity, or whether it is in fact a non-relative relation with a higher-than-expected arity. Moreover, these tests deliver examples of both sorts of entity—both genuine relative properties and relations and non-relative relations.³⁰ The first test Spencer discusses is the switch-the-index test. (For simplicity, I’ll explain Spencer’s tests in terms of relative properties and non-relative binary relations only.) Suppose that \(x\) instantiates a property \(F\) relative to some putative index \(i\). Now pick a property \(G\) that is incompatible with \(F\) (i.e., \(x\) can’t instantiate both \(F\) and \(G\) relative to the same putative index), and let \(x\) instantiate \(G\) relative to a different parameter \(j\). If, intuitively, a change in \(x\) has taken place, then \(F\) and \(G\) are genuine relative properties. If, on the other hand, intuitively, no change in \(x\) has taken place, then each is a non-relative binary relation.

Consider the examples Spencer uses to illustrate how this test works. David Lewis (1986, 202–204) argues that the endurantist faces a challenge because they are apparently committed to the idea that the very same object is both bent and straight, since they are committed to the view that objects persist by being wholly present at each moment at which they exist. One way of responding to this challenge is to claim that properties like shape are relative, instantiated relative to times (as in Haslanger 1989, 123). That they are in fact relative properties and not disguised binary relations between objects and times can be shown by applying the switch-the-index test. Suppose Lewis instantiates being bent at \(t_{1}\) and being straight at \(t_{2}\). (These two properties are incompatible.) Has Lewis undergone a change in properties between these times? Intuitively, yes. So being bent and being straight are indeed relative properties. Contrast the case of shape with that of size. A big mouse, Remy, is big compared to other mice. But when the average-sized mice surrounding him are replaced with larger animals, say dogs, Remy is no longer big. Has Remy undergone a change through this replacement? Intuitively, no. So being large is not instantiated relative to anything, but instead “\(\ldots\) is large” expresses a binary relation—presumably something like being large compared to—which holds between an object and the objects in certain groups. A difference in properties over time implies genuine change; a thing’s simply being related by different relations to different things at different times doesn’t. This is what the switch-the-index test is supposed to capture. And it delivers results that imply that both genuine relative properties and non-relative relations exist, assuming that these claims involving shape and size are true and irreducibly relational.

Now to Spencer’s second test, the real similarity test. Suppose that \(x\) instantiates a property \(F\) relative to some putative index \(i\). Now switch \(x\) out with a different object \(y\), and switch the value of \(i\) to a new acceptable value \(j\). If, intuitively, \(x\) is exactly similar to \(y\) with respect to \(F\), then \(F\) is a genuine relative property. If, on the other hand, intuitively, \(x\) is not exactly similar to \(y\) with respect to \(F\), then \(F\) is a non-relative binary relation. The rationale for these conclusions is, roughly, that similarity is a matter of sharing properties, not of instantiating relations to different objects (see Spencer 2016, 443). Consider what this test says about the two examples discussed above. Begin by supposing that Lewis instantiates being bent at \(t_{1}\), and then switch Lewis with Haslanger and \(t_{1}\) with \(t_{2}\) to yield the result that Haslanger instantiates being bent at \(t_{2}\). Intuitively, Haslanger is exactly similar to Lewis with respect to being bent, and therefore being bent is a genuine relative property. Being large, on the other hand, is a non-relative relation according to the real similarity test. Remy instantiates being large relative to mice. Now replace Remy with Jupiter and replace mice with planets of the solar system. Intuitively, Remy is not exactly similar to Jupiter with respect to being large. Jupiter is, after all, much larger than Remy.

According to Spencer’s account of relativity, there is a real difference between relative properties and relations on the one hand and non-relative properties and relations on the other. And in light of the deliverances of Spencer’s tests, I’m happy to grant that relative properties exist. But the relationless relative positionalist is committed to an analysis of every instance of relativity (i.e., every irreducibly relational claim) in terms of relative properties. Indeed, even the relative positionalist who countenances relations is so committed. Directionalism, on the other hand, can provide an account of relativity in exactly those cases which we have reason to believe involve relations only, and is simply silent in those cases which we have reason to believe involve relative properties only, if any such cases exist.³¹

Consider one of the examples of relativity I have been discussing all along—that connected with someone loving someone. Suppose that Buff is beloved relative to Goethe. Is there a genuine relative property being beloved? Or does this case of relativity involve a non-relative relation, loving, instead? According to the switch-the-index test, it is the latter that is the case. First, consider the fact that Buff is beloved relative to Goethe. Next, consider the fact that she is not beloved relative to Joseph II. (Being beloved and not being beloved are incompatible.) However, intuitively, Buff has not undergone a change. To briefly summarize the advantage I have argued directionalism has over relative positionalism: if the relative positionalist countenances relations as well as relative properties, her ontology is more profligate than that of the directionalist. But if she dispenses with relations, then she is forced to posit the involvement of relative properties in relational claims which we have reason to believe involve relations only, while the directionalist is not forced to do the reverse.³²

4 Where Directionalism Stands Now

I’ve shown how directionalism can rise to Donnelly’s challenge and properly treat fixed arity relations with any symmetry such a relation can have. Consequently, directionalism has a distinct advantage over absolute positionalist views of relations. Granted, other views, like Fine’s antipositionalism, MacBride’s relational primitivism, and Donnelly’s own relative positionalism can solve this problem as well. But unlike primitivism, directionalism supplies an explanation of why each relation can apply in the ways it can. And unlike both primitivism and antipositionalism, it supplies explanations of why distinct relations can apply in the same or only different ways (as the case may be). Directionalism has an advantage over relative positionalism as well, in that it is not, like relative positionalism, committed to the involvement of relative properties in every irreducibly relational claim.

Still, more remains to be said before we can conclude that directionalism wins the day. I’ve dealt with only one objection—the problem concerning symmetric relations that Donnelly poses for relative positionalism. But Donnelly gives another objection to directionalism; she charges the directionalist’s primitive notion of order-sensitive relational application with being obscure (2016, 82 and 97–98; and 2021, 5–6), since the ordering of a relations’ relata by it can’t be understood to be “a process which unfolds over time or across space” (2016, 82). She adds,

[I]t is hard to see how the idea of an order of relational application could be filled out. It is not as though relata are somehow fed into a relation as paper is fed into a printer or wood into a chipper. Relations are not the kinds of things that can “pick up” their relata in a temporal or spatial succession. Perhaps there is some other way for relations to apply to their relata in an order, but no one has tried to explain what this is supposed to be. (2021, 6)

I won’t try to explain what order-sensitive relational application is supposed to be, but I’m not as concerned about this as Donnelly is. It’s not clear to me that the directionalist is on the hook to provide a general account of this notion, given that relational predicates are themselves order-sensitive. Of course, I doubt Donnelly would be satisfied by this. But this problem strikes me as being no worse than the problem I identified for relative positionalism in the previous section, concerning its commitment to the involvement of relative properties in every irreducibly relational claim. So, other things being equal, the two views are at worst on a par. Of course, everything depends on whether other things really are equal between the two views. As I’ve mentioned, there are other objections to directionalism that still warrant replies, notably Fine’s and Williamson’s, mentioned in the introduction. There are also important concerns raised by MacBride (e.g., 2014, 5–6) and others. I must leave replies to these objections for another occasion.^33,