Maureen Donnelly’s (

Some relations can apply to their relata in different ways while some cannot.

I begin, in section 2, by introducing relative positionalism, providing all the information the reader will need to understand the discussion that follows. In section 3, I explain why relative positionalism is unable to handle variable arity relations. This involves considering and rejecting a way Donnelly proposes to amend relative positionalism to handle them. I begin section 4 by explaining why relative positionalism’s inability to handle variable arity relations means that the relative positionalist cannot comfortably accommodate their existence (e.g., by providing a non-uniform account of fixed and variable arity relations) by introducing relative positionalism’s closest competitors and showing that they can provide uniform accounts of both sorts of relation. I then turn to my defense of relative positionalism from the problem of variable arity relations. I explicate a way the relative positionalist can explain the satisfaction of any predicate which putatively expresses a variable arity relation in terms of the application of fixed arity relations, opening the door for her to deny the existence of variable arity relations altogether. I note that whether the relative positionalist may successfully deploy this strategy depends both on the strength of the reasons we have to believe in variable arity relations and on whether relative positionalism has sufficient theoretical advantages over its closest competitors to offset those reasons. But I argue that the reasons we have to believe in variable arity relations are at best defeasible in the remainder of section 4. And in section 5, I argue that relative positionalism has an important virtue that its closest competitors lack, which is sufficient to offset its deficiency in connection with variable arity relations. In particular, I argue that it supplies an explanation of relational application of the sort that a theory of relations ought to supply, while none of its closest competitors does so. I conclude, in section 6, by explaining why this means we ought, all else being equal, to prefer relative positionalism to its closest competitors, at least to the extent that the explanation it supplies of relational application is plausible.

Relative positionalism is the view that, when some things stand in a relation, they do so by occupying certain positions of the relation relative to one another. Positions are understood, not as holes or slots in a relation that relata occupy, as they are on absolute positionalist views (such as those defended in

Donnelly’s relative positionalism is able to correctly handle any fixed arity relation regardless of its symmetry, which provides her view with an important advantage over absolute positionalist theories, which cannot handle relations with certain symmetries, such as cyclic symmetries. Donnelly achieves this by using group theory to represent the symmetries (or symmetry structures, in Donnelly’s parlance) of relations. She begins by imposing the following basic constraints on the behavior of the relative properties she takes to be involved in the application of relations.

Every _{1}, …, _{n}

each of _{1}, …, _{n}

and

every ordering of _{1}, …, _{n}

(Adapted from

So, for example, if Goethe and Charlotte Buff stand in the binary relation

Donnelly represents the symmetry of each relation by its

For any [order-determined] _{R}_{1}’, …, ‘_{n}’ referring to objects in the domain of

(*)
_{1} … _{n}

is equivalent to

(*
_{P})_{P(1)} … _{P(n)}.

(

As Donnelly notes (_{n}

(∇) | _{Q} = _{Q}_{R}^{*}_{n}, _{Q} is the property which (*) entails that _{Q(1)} has relative to _{Q(2)}, …, relative to _{Q(n)}. ( |

That is, such relatives are identical just in case ‘they can be transformed into one another by a permutation in the symmetry group’ of the predicate/relation (

The relative positionalist identifies the way(s) a relation _{R}^{2} is an example of such a relation. (Henceforth, I indicate the arity of a fixed arity relation denoted by a term with a superscripted positive integer in the way I have just done.) This can be seen by considering the order-determined predicate ‘… is next to …’, which expresses it. This predicate has the symmetry group

SYM_{… is next to …} = {[1 2], [2 1]},

where ^{⌜}[_{1} _{2} … _{n}^{⌝} denotes the permutation of {1, 2, …_{1}, 2 to _{2}, …, and _{n}

(*) | _{1} is next to _{2} |

(*_{[2 1]}) |
_{2} is next to _{1} |

By the Base Thesis, when ^{2} applies to two things, one of its relative properties (for all we currently know, not necessarily the same one) is instantiated by each of them relative to the other in each of the two possible orderings of them. And because

[2 1] ∈ SYM… _{is next to} … and [2 1] ◦ [1 2] = [2 1],

we know by (▽) that the relative property that _{1} instantiates relative to _{2} is the same as that which _{2} instantiates relative to _{1}.^{2} has a single relative property, _{1}, and hence that there is, according to relative positionalism, only one way it can apply to two objects, such as Goethe and Buff (Figure ^{2} can apply to two objects; there is only one completion of a given sort (fact, state of affairs, or proposition) that can result from it applying to two objects.

The fixed arity relation ^{2}.

In this diagram and the ones to follow, a relative property instantiated by one thing relative to another is represented by an arrow going from the first thing to the second.

An ^{2} is an example of such a relation. This can be seen by considering the order-determined predicate ‘… loves …’, which expresses it. This predicate has the symmetry group

SYM_{… loves …} = {[1 2]}.

This can be seen by noting that corresponding instances of the following schemas are not equivalent to one another.

(*) | _{1}loves _{2} |

(*_{[2 1]}) |
_{2} loves _{1} |

By the Base Thesis, when ^{2} applies to two things, one of its relative properties (for all we currently know, not necessarily the same one) is instantiated by each of them relative to the other in each of the two possible orderings of them. And because

there is no P ∈ SYM_{… loves …} such that P ◦ [1 2] = [2 1],

we know by (▽) that the relative property that _{1} instantiates relative to _{2} is distinct from that which _{2} instantiates relative to _{1}. So we know that ^{2} has two relative properties, _{2} and _{3}, and hence that there are, according to relative positionalism, two ways ^{2} can apply to two objects (Figure ^{2} can apply to two objects; there are two completions of a given sort that can result from it applying to two objects.

The fixed arity relation ^{2}.

Donnelly (_{2} be the relative property that Goethe instantiates relative to Buff and _{3} be the one that Buff instantiates relative to Goethe when Goethe loves Buff and not vice versa? In this case, Donnelly interprets _{2} and _{3} as ^{2}, she makes a remark (

Many of relative positionalism’s competitors, including some absolute positionalist theories, are able to correctly handle relations with complete symmetry and complete non-symmetry just as well as relative positionalism. Relative positionalism’s primary advantage over such views is its ability to handle any relation with a ^{3}, for example, permits of transpositions between two of its arguments but not of any permutations involving the other argument, assuming those arguments are pairwise distinct. And while some absolute positionalist theories can handle relations with this sort of symmetry, they cannot handle relations with other sorts of partial symmetries, most notably, those with cyclic symmetries, like ^{3}. I will not discuss why they cannot, since it is not relevant to my purpose here. But see

In general, an ^{3}, such as ‘…, …, and … are arranged clockwise’, has the symmetry group

SYM_{…, …, and … are arranged clockwise in that order} = {[1 2 3], [2 3 1], [3 1 2]}.

This can be seen by noting that corresponding instances of the following schemas are equivalent to one another.

(*) | _{1}, _{2}, and _{3} are arranged clockwise in that order |

(*_{[2 3 1]}) |
_{2}, _{3}, and _{1} are arranged clockwise in that order |

(*_{[3 1 2]}) |
_{3}, _{1}, and _{2} are arranged clockwise in that order |

But none is equivalent to any instance which results from any other permutation of _{1}, _{2}, and _{3} with respect to the predicate. By the Base Thesis, when ^{3} applies to three things, some relative property of that relation (for all we currently know, not necessarily the same one) is instantiated by each of them relative to the others in each of the six possible orderings of them. And because

[2 3 1] ∈ SYM_{…, …, and …are arranged clockwise in that order} and [2 3 1] ◦ [1 2 3] = [2 3 1] and

[3 1 2] ∈ SYM_{…, …, and …are arranged clockwise in that order} and [3 1 2] ◦ [1 2 3] = [3 1 2],

we know by (▽) that the relative property that _{1} instantiates relative to _{2}, relative to _{3} is the same as (i) that which _{2} instantiates relative to _{3}, relative to _{1}, and (ii) that which _{3} instantiates relative to _{1}, relative to _{2}. Furthermore, since

[2 3 1] ∈ SYM_{…, …, and …are arranged clockwise in that order} and [2 3 1] ◦ [1 3 2] = [2 1 3] and

[3 1 2] ∈ SYM_{…, …, and …are arranged clockwise in that order} and [3 1 2] ◦ [1 3 2] = [3 2 1],

we know by (▽) that the relative property _{1} instantiates relative to _{3}, relative to _{2} is the same as (i) that which _{2} instantiates relative to _{1}, relative to _{3}, and (ii) that which _{3} instantiates relative to _{2}, relative to _{1}. Moreover, because

there is no permutation P ∈ SYM_{…, …, and … are arranged clockwise in that order} such that, e.g., P ◦ [1 3 2] = [1 2 3],

we know by (▽) that the relative property that _{1} instantiates relative to _{2}, relative to _{3} must be distinct from that which _{1} instantiates relative to _{3}, relative to _{2}. So we know that the relation has two relative properties, _{4} and _{5}, which Donnelly interprets, respectively, as

This means that the relative positionalist will say that there are two ways in which ^{3} can apply to three objects, such as Larry, Curly, and Moe (Figure

• | _{4} relative to |
• | _{4} relative to |

• | _{4} relative to |
• | _{4} relative to |

• | _{4} relative to |
• | _{4} relative to |

• | _{5} relative to |
• | _{5} relative to |

• | _{5} relative to |
• | _{5} relative to |

• | _{5} relative to |
• | _{5} relative to |

The fixed arity relation ^{3}.

In this diagram and the ones to follow, a relative property instantiated by one thing relative to another, relative to another is represented by an arrow going from the first thing to the second, then to the third thing. In this diagram, red arrows depict assignments of τ_{4}, while blue arrows depict assignments of τ_{5}.

Again, relative positionalism gets things right. There are, intuitively, two ways for Larry, Curly, and Moe, to complete this relation. First, they may do so in the order just specified. Or Larry, Moe, and Curly can do it in ^{3} can apply to three objects; there are two completions of a given sort that can result form it applying to two objects.

The remarkable thing about relative positionalism, as opposed to absolute positionalist theories in particular, is that, for

While a fixed arity relation can take only a single number of arguments, a variable arity (or multigrade) relation can take more than one number.

Donnelly considers a modification of relative positionalism in an attempt to extend it to cover variable arity relations. But while this modification can handle

… and … are meeting for lunch

…, …, and … are meeting for lunch

…, …, …, and … are meeting for lunch

⋮

The relative positionalist would then proceed as she normally would with any other set of fixed arity predicates. Applying the Definition of Symmetry Groups to each would yield the result that the _{6}, which, when the relation applies to two objects, is instantiated by each relative to the other (left side of Figure _{7}, which, when the relation applies to three objects, each instantiates relative to each of the others, relative to the remaining one (right side of Figure

The variable arity relation

Donnelly’s amendment to relative positionalism would then become operative. Effectively, her proposal is to allow relative properties to be able to be instantiated by something relative to more than one number of things. This would allow the relative positionalist to identify the relative properties associated with these predicates with one another (so that _{6} = _{7}). This, in turn, would enable the relative positionalist to identify the relations expressed by them. (Absent Donnelly’s amendment, one might take the fact that some relatives are instantiated relative to different numbers of things to be enough to distinguish them.) The relative positionalist would then be able to countenance the variable arity relation

One might attempt to undermine Donnelly’s proposal by arguing in various ways that the relations expressed by the predicates in the above list must be pairwise distinct. In the interest of space, however, I will ignore such arguments, since there is a more pressing problem with Donnelly’s proposed modification. While nothing may stand in the way of the relative positionalist identifying the relatives associated with the various predicates in the list above, and thus identifying the relations expressed by them, this is an artifact of the example; each of these predicates and relations is completely symmetric. Completely symmetric predicates and relations stand apart from their counterparts with incomplete symmetries (i.e., symmetries that are anything other than complete) in that each has a single relative property, but, more importantly, in that the number of relatives each has is the same no matter its arity. Relations with incomplete symmetries which differ in arity may differ in the number of relative properties they have. Thus a variable arity relation which has incomplete symmetry in some of its manifestations of arity may cause problems for relative positionalism.

A plausible candidate for such a relation, and one that causes problems for relative positionalism, is the variable arity counterpart of the fixed arity relation which Donnelly uses to motivate her view over absolute positionalist theories, viz.,

In her attempt to accommodate the allegedly variable arity relation

… and … are arranged clockwise in that order

…, …, and … are arranged clockwise in that order

…, …, …, and … are arranged clockwise in that order

⋮

The relative positionalist would then apply the Definition of Symmetry Groups to each predicate in the list. Applying it to the first would yield the result that the relation it expresses has the same symmetry group as ^{2}. Its symmetry group contains all of the possible orderings of its relata, since every permutation of terms flanking the predicate results in an equivalent claim. Thus that relation, like ^{2}, will have one relative property, _{8} (left side of Figure ^{3} (right side of Figure _{4} and _{5}).

The variable arity relation

As before, red arrows depict assignments of τ_{4}, while blue arrows depict assignments of τ_{5}.

But Donnelly’s suggestion that relative properties can apply relative to more than one number of things is of no help in this case, since it is not just the number of things relative to which a relative property of the putative variable arity relation can be instantiated that differs when the relation applies to different numbers of relata. The

The relative positionalist might reply by insisting that there is a single variable arity relation expressed by each of these predicates, which has all of the relative properties for each of the arities the relation can manifest. In general, a variable arity relation which can apply to _{2} + _{3}, + … relative properties, where _{i}

To address this concern, Donnelly proposes that the relative positionalist abandons relations altogether, and supposes instead that relational predicates are associated directly with a certain number of relative properties. This ‘relationless’ relative positionalism is committed only to relative instantiation. There are no relations, and hence there is nothing that stands in the non-relative instantiation relation. So there is no reason for the relative positionalist to posit this relation at all. Indeed, doing so would be needlessly extravagant. Answering the ‘two forms of instantiation’ concern in this way, however, prevents the relative positionalist from making use of the ‘one plow, many yokes’ reply to the variable arity problem, according to which the relative positionalist says that each variable arity relation has all of the relative properties needed for each of the arities the relation can exhibit. First of all, according to relationless relative positionalism, there are no relations, and so there is no

The simple fact that relative positionalism cannot handle variable arity relations doesn’t mean that the relative positionalist must reject their existence. She might treat them in an alternate way. But this would yield a non-uniform account of relations, which would be awkward at best, and at worst will put relative positionalism at a decided theoretical disadvantage relative to its closest competitors. The only two views out there that can properly handle relations with all of the symmetries that relative positionalism can handle (including relations with cyclic symmetries), viz., Fine’s (

Antipositionalism, as its name suggests, does not posit positions in relations. The ways (or ^{2} 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 distinct from the other. Which identity and distinctness relationships hold of these two possible but mutually exclusive sets is, according to the antipositionalist, a matter of brute fact.

For various reasons, MacBride rejects antipositionalism and endorses a view that he calls ‘ostrich realism.’ According to his view, there is no explanation whatsoever for why any relation can apply in the way(s) that it can. Each such fact is taken as primitive. Ostrich realism has even more flexibility than Fine’s antipositionalism when it comes to correctly handling relations with any possible symmetry, including variable arity relations. There need be no objects to which a given relation applies (such objects are required on Fine’s account) for MacBride to explain why it can apply in the way(s) that it can, since he supplies no explanation of this fact at all.

Relative positionalism’s deficiency concerning variable arity relations is a particularly tempting basis on which the antipositionalist or ostrich realist might build an argument against their rival. Because antipositionalism and ostrich realism can provide uniform accounts of fixed and variable arity relations, they enjoy a theoretical advantage over relative positionalism. If the relative positionalist is going to have a fighting chance against these two competitors, she would be better off simply denying the existence of variable arity relations. But she must find an alternative explanation for the truth of relational claims that allegedly involve commitment to them. There must be

It is good news for the relative positionalist that it seems that this strategy can always be implemented, no matter the predicates allegedly expressing a variable arity relation with which one is confronted. For any such predicates one happens upon, one can always posit a class of fixed arity relations in terms of which one can analyze their satisfaction, and reject the existence of the alleged variable arity relation. But if the relative positionalist is going to be justified in making use of the strategy, the following two conditions must be met.

(C1) | Any reasons we have for believing in the existence of variable arity relations must be defeasible. |

(C2) | Relative positionalism must have advantages over its closest competitors, viz., antipositionalism and ostrich realism, that are sufficient to defeat the reasons we have for believing in variable arity relations, all else being equal (i.e., assuming that relative positionalism does not have any disadvantages relative to its competitors that would tip the balance back in their favor). |

If (C1) were not met, then the strategy just outlined to do away with variable arity relations would be decidedly unacceptable, since it implies that variable arity relations do not exist. And if (C2) were not met, then the strategy would be unmotivated and thus

What reasons do we have for believing in the existence of variable arity relations? The strongest (and only) one that I know of is discussed by MacBride (

Thus MacBride thinks that our use of relational predicates, in general, commits us to the existence of corresponding relations. So, MacBride concludes, our use of collective variable arity relational predicates, in general, commits us to the existence of corresponding variable arity relations. I grant that MacBride’s argument is strong enough to establish that, absent the presence of countervailing considerations, one should recognize the existence of variable arity relations (if one recognizes the existence of relations at all). But I do not grant that it

So the fact that the relative positionalist supplies a revisionary account of an entire class of predicate, viz., collective variable arity predicates, is not in principle objectionable in light of MacBride’s argument. If it were, then widely accepted views like eliminativism and the identity theory would have to be rejected. But of course, the relative positionalist must have good reasons to deny the existence of variable arity relations, just as the eliminativist must provide good reasons to deny that relational mental predicates express relations, and just as the identity theorist must provide good reasons to hold that they express physical relations rather than mental ones. Fortunately for relative positionalism, it has an important theoretical virtue which both antipositionalism and ostrich realism lack, which is substantial enough to override MacBride’s argument for the existence of variable arity relations. I turn now to a discussion of this virtue.

I now set out to establish that (C2) is met — that relative positionalism has an advantage over antipositionalism and ostrich realism that is, other things being equal, sufficient to defeat the reason we have to believe in variable arity relations. That advantage consists of the fact that relative positionalism supplies the sort of explanation of relational application that a theory of relations ought to supply, while neither antipositionalism nor ostrich realism does so. This advantage is quite substantial, since supplying such an explanation is plausibly one of the primary goals of a theory of relations and their application. I begin by spelling out this explanatory target.

Ideally, a theory of relations will supply not only an account of their nature, but also an account of their application to things. The latter involves accomplishing at least two things. First, such a theory will ideally correctly handle the application of any relation we take to exist. That is, it must say, for any given relation in our ontology, that it can apply to some things in the ways that we think it can. Thus the theory should say, for example, that the binary relation ^{2} can apply to two objects in only one way, and that ^{2} can apply to two objects in two ways. As we have seen, antipositionalism and ostrich realism fare just as well as (and, if we include variable arity relations, better than) relative positionalism with respect to this issue. The second thing a theory of relations must do if it is to provide an (adequate) account of relational application is to supply answers to the following two questions.

(Q1) | Why can each relation apply in the way(s) it can? |

(Q2) | Why are the ways in which some relations can apply to their relata the same as one another, and those in which others can apply to their relata different from one another? |

As far as (Q1) goes, a theory of relations will ideally explain why, for example, ^{2} can apply to two things in only one way. It will also explain why ^{2} can apply to two things in two ways. I expect little skepticism about the importance of the role (Q1) has played in the literature on relations. It is, after all, one of the central questions which preoccupies Fine (

(Q2) deserves more discussion, and not just in its defense as an essential part of developing a theory of relations. It is important to get clear first on exactly what the question is. Examples will help. According to (Q2), a theory of relations will ideally explain why, for example, the two ways in which ^{2} can apply to two things are the same as the two ways in which ^{2} can do so, and why they are different from the way in which ^{2} can apply to two things.

The quaternary relations ^{4} and ^{2-2} can apply in the same number of ways as one another to four objects. (Here the superscript ‘^{2-2}’ indicates that the relation relates the distances between two pairs of objects, as in ‘Alice and Bob are closer together than Carol and Diane’.) But, I will argue, they are not the same ways, as are, for example, the ways in which ^{4} and ^{4} can apply to four objects. The reader can use the same method from section 2 to confirm that the symmetry groups of these relations are

SYM_{…, …, …, and … are arranged clockwise in that order} = {[1 2 3 4], [2 3 4 1], [3 4 1 2], [4 1 2 3]}

SYM_{… and … are closer together than … and …} = {[1 2 3 4], [1 2 4 3], [2 1 3 4], [2 1 4 3]},

and that each has six relative properties, and thus that each can apply to four objects in six ways. When ^{2-2} applies to four objects, there are pairs of these objects which can be transposed without resulting in a new completion. But there are no such pairs in the case of any application of ^{4} to four objects. Any such transposition will result in a new completion. For example,

Alice and Bob’s being closer together than Carol and Diane

is the same state of affairs as

Bob and Alice’s being closer together than Carol and Diane

But

Alice, Bob, Carol, and Diane’s being arranged clockwise in that order

is a distinct state of affairs from any of the following (which represent all of the possible ways to transpose exactly two of Alice, Bob, Carol, and Diane, as so ordered, with one another).

Bob, Alice, Carol, and Diane’s being arranged clockwise in that order

Carol, Bob, Alice, and Diane’s being arranged clockwise in that order

Diane, Bob, Carol, and Alice’s being arranged clockwise in that order

Alice, Carol, Bob, and Diane’s being arranged clockwise in that order

Alice, Diane, Carol, and Bob’s being arranged clockwise in that order

Alice, Bob, Diane, and Carol’s being arranged clockwise in that order

This suggests that the ways these two relations can apply to four objects are different, despite the fact that they are the same in number. Another way to put the point is that there is a substantive difference in the symmetry structures of these two relations that has nothing to do with the number of ways in which they can apply to four objects.

Now that I have explicated (Q2), I turn to the question of why a theory of relations should supply an answer to it. (Q2) amounts to the question of why some relations have the same symmetry and others have different symmetries. Of course, relations with different (fixed) arities can’t have the same symmetry (nor can they apply in the same ways). But differences in symmetry (and manners of applicability) in these cases can be explained by appealing to the fact that the relations differ in arity, whether one is a relative positionalist or not. As the preceding discussion shows, however, even among relations of the same arity, there is more to the symmetry structure of any given one than just the number of ways in which it can apply. That is, there is more to the character of the way(s) such a relation can apply than just the

Having explicated and motivated (Q1) and (Q2), what remains is to show that relative positionalism provides answers to both of these questions while neither antipositionalism nor ostrich realism does so. I begin by showing the former. As far as (Q1) goes, the relative positionalist can explain why each relation can apply in the way(s) it can in terms of the specific number of relative properties it has, and the way(s) those relative properties can be instantiated by its relata relative to one another. ^{2}, for example, can apply to two things in only one way, since it has only a single relative property, which must be instantiated by each of its relata relative to the other whenever it applies. ^{2}, on the other hand, can apply to two things in two ways, since it has two relative properties, one of which must be instantiated by one of its relata relative to the other while the other must be instantiated by the other relatum relative to the one whenever the relation applies.

Concerning (Q2), the relative positionalist can explain identities and differences between the way(s) in which distinct relations can apply by appealing to identities and differences in the number of relative properties each relation has and identities and differences in the ways those relative properties can be instantiated by the relata of each relation. Differences of the former sort suffice to explain why certain ^{2} can apply to two objects on the one hand and the one way in which ^{2} can do so on the other by appealing to the fact that they have different numbers of relative properties. But to explain the difference between the six ways in which each of ^{4} and ^{2-2} can apply, she must appeal to the fact that the six ways in which the six relative properties of the former relation can be instantiated by four objects relative to one another are different from the six ways in which the six relative properties of the latter relation can be instantiated by four objects relative to one another. This can be established using the same reasoning I employed above to show that the ways in which these two relations can apply to four objects are different, despite their being the same in number. (Remember that the relative positionalist identifies the ways in which the relative properties of an

As mentioned, however, neither antipositionalism nor ostrich nominalism supplies answers to both (Q1) and (Q2). Neither supplies an answer to (Q2). Neither explains, for example, the fact that the two ways in which ^{2} can apply to two things are the same as the two in which ^{2} can do so. Antipositionalism supplies no way to compare the way(s) in which distinct relations can apply (cf. ^{2} can apply to two things in only one way, or for the fact that ^{2} can apply to two things in two ways. As just stated, she takes these facts as primitive.

Were it not for the fact that the relative positionalism can supply an explanation of the satisfaction of

Relative properties can be invoked, for example, by endurantists as a way to avoid the problem of temporary intrinsics (see, e.g.,

Assume that an _{1}, …, _{n}_{P(1)} relative to _{P(2)}, …, relative to _{P(n)} for some P ∈ S_{n}. The Base Thesis, along with SYM_{R}_{1}, …, _{n}_{P(1)} relative to _{P(2)}, …, relative to _{P(n)} singles out a distinct way for _{1}, …, _{n}

Recall that ◦ is function composition. The composite P ◦ Q of permutations P and Q is the permutation mapping each

([2 1] ◦ [1 2])(1) = [2 1]([1 2](1)) = [2 1](1) = 2 and

([2 1] ◦ [1 2])(2) = [2 1]([1 2](2)) = [2 1](2) = 1.

The permutation identities that follow can be computed using the same general method.

A completion is any object which results from a relation applying to some things in a certain way. See

The reader can check that the appropriate relative property assignments hold, when interpreted as Donnelly suggests, when Larry, Curly, and Moe are arranged clockwise in the two orders possible. The reader should think of _{1}’s being clockwise in front of _{2} relative to _{3} as _{1}’s being in front of _{2} when she (the reader) imagines herself looking around the spatial arrangement of them clockwise from the perspective of _{3}, and of _{1}’s being clockwise behind _{2} relative to _{3} as _{1}’s being behind _{2} when she so imagines herself.

It can also be proved that it does so for

For further discussion of such relations, see

I have used a binary manifestation of the putative variable arity relation to make my point — one which admittedly is at best a degenerate case of clockwise arrangement. But its

See

MacBride (

The relationless relative positionalist could employ an analogous strategy, and say that each relational claim, which one might have thought involved commitment to a variable arity relation, actually involves commitment to a number of relative properties appropriate to the arity of the claim’s relational predicate. Thanks to an anonymous referee for helping me to recognize the viability of this alternative.

Moreover, it may make the relative positionalist guilty of a ‘Russellian bias,’ assuming that all relations have fixed arities without argument (see

For more on variable arity predicates, see

One might worry that it is unclear how the account I have proposed would handle plurally quantified claims like ‘Larry, Curly, and some other things are arranged clockwise in that order.’ This concern is due to an anonymous referee. It is worth pointing out that it is not a special problem for the account I have proposed. It is an example of a general sort of limitation of relative positionalism that is a result of the fact that it is formulated in a singular first-order language. As such, it simply does not have the resources to deal with every plurally quantified claim. For this reason, I am tempted to leave this concern aside, writing it off as, for present purposes, being unrelated to a discussion about what the relative positionalist ought to say about variable arity relations. Still, the referee’s concern seems to loom larger in the context of such a discussion, since claims like these seem to call more loudly for a treatment in terms of variable arity relations than relational claims involving only singular terms. So I’ll say some things in response to it.

There are at least two things one might mean when one asks how an account handles a claim like this. First, one might be wondering how the account explains the truth of claims like these. So construed, I think the relative positionalist has a plausible answer. It is plausible and common in the truthmaker and grounding literatures to say that existential truths are made true by, or grounded in, facts concerning only particulars instantiating fixed arity relations. Armstrong (_{1} _{n} Rx_{1} _{n}_{1}_{n}_{1}_{n}_{1} _{n}_{1} _{n} Rx_{1} _{n}

_{1}… _{n}

_{1},

⋮

^{n}

for each _{1}_{n}_{1} _{n}

Second, however, one might be wondering what the semantic values of the relational predicates are in sentences like ‘Larry, Curly, and some other things are arranged clockwise in that order’ according to the account — what, for example, the semantic value of ‘

there is an

there is an

⋮

I will endorse none of these strategies today. It may be that this problem is insurmountable for the relative positionalist if these strategies turn out to be unfruitful and no alternative strategies can be found. But, at least for the time being, there appear to be some avenues of reply that the relative positionalist can explore.

One might be concerned that identifying the two ways in which ^{2} can apply to two things with the two ways ^{2} can do so commits one to the claim that each of the ways in which each one of these relations can apply to two things is identical to one of the ways in which the other can do so. This could be seen as problematic, since either possible assignment would seem unmotivated. One might think that there is reason to be found to make one assignment rather than another. Each of ^{2} and ^{2} has an agent role and a patient role, and so one might reasonably identify the way that has _{1} in ^{2}’s patient role and _{2} in its patient role with the way that has _{1} in ^{2}’s agent role and _{2} in its patient role. But one could easily have chosen a non-symmetric binary relation instead of ^{2} that does not obviously possess such roles, such as ^{2} or ^{2}. In such cases, there might be no basis for identifying either of the ways in which ^{2} can apply to two things with either of the ways in which the other relation can do so. Thanks to an anonymous referee for raising this concern. I am not convinced, however, that, in identifying ways

(Q2′)
Why are the ways in which some relations can apply to their relata more similar to one another than to the ways in which other relations can apply to their relata?

What would otherwise be understood as identical manners of applicability can be understood instead as maximally similar ones, where

The ways in which _{df}

It seems that one can safely say that the two ways in which ^{2} and another non-symmetric binary relation can apply are comparatively more similar to one another than they are to the way(s) in which a relation with a different sort of symmetry can apply without having to say anything about which one of the two ways that ^{2} can apply is more similar to which of the two ways the other non-symmetric binary relation can apply. And relative positionalism will be able to provide an answer to (Q2′), while neither antipositionalism nor ostrich realism will be able to, for reasons very similar to those I articulate below which establish that this is true of (Q2).

The same point can be made formally by noting that the symmetry groups of these two relations, while sharing the same index in S_{4} (i.e., as having the same number of left cosets in S_{4}, each left coset representing a way the relation can apply to four objects), are not isomorphic, in the sense that there is no one-to-one correspondence between their elements which respects their respective group operations, which is function composition in both cases (see ^{4} are of order four, while none of the elements of the symmetry group of ^{2-2} is of order four. It is a theorem that the group-isomorphic image of an element of a group and that element have the same order (see

This is for the simple fact that Fine defines sameness of manners of completion only for single relations. He does so in terms of simultaneous substitution (see n. 9 above). Specifically, he says,

to say that _{1}, _{2} _{m}, in the same manner as _{1}, _{2}, _{m} is simply to say that _{1}, _{2} _{m} that results from simultaneously substituting _{1}_{2} _{m} for _{1}, _{2}, _{m} in

But a modification of this definition to accommodate distinct relations is not forthcoming. It seems arbitrary whether the manner in which, for example, Goethe and Buff complete ^{2} in Goethe’s loving Buff is the same as or different from that in which they complete ^{2} in Goethe’s being to the left of Buff. (I am assuming here that each of Goethe and Buff is being substituted with itself.) Note that here we are attempting to identifying a single way one relation can apply with a single way a distinct relation can apply, and thus it would be of no help to the antipositionalist to invoke the argument I gave in n. 15 for the view that multiple manners of applicability of distinct relations can be identified without presupposing the identity of each way in which each of the relations can apply with some way in which the other can apply.

The antipositionalist can explain why a given relation can apply in the way(s) it can by appealing to the number of classes of co-mannered completions associated with it.

I am extremely grateful to Maureen Donnelly for invaluable discussions in helping me develop the ideas presented here, as well as for providing detailed comments on an earlier draft. Thanks also to Mike Raven and Cody Gilmore for reading and providing comments on earlier drafts. I am also grateful to Udayan Darji, Martin Glazier, Paul Hovda, Nicholas Jones, Jon Litland, and Tulsi Srinivasan for helpful discussions concerning aspects of this paper, and to audiences at the 2018 meetings of the Canadian Metaphysics Collaborative, the Central States Philosophical Association, and the 2019 meeting of the American Philosophical Association Central Division — in particular to Eileen Nutting — for helpful input on an ancestor of this paper. I would also like to thank two anonymous referees, whose comments significantly improved the paper. And finally, I would like to thank the Marc Sanders Foundation, the Alexander von Humboldt Foundation, Kit Fine, and Benjamin Schnieder, who helped to provide funding necessary to carry out this research.

The author has no competing interests to declare.