Explaining the differential application of non-symmetric relations

Abstract

Non-symmetric relations like loves or between can apply to the same relata in non-equivalent ways. For example, loves may apply to Abelard and Eloise either by Abelard’s loving Eloise or by Eloise’s loving Abelard. On the standard account of relations (Directionalism), different applications of a relation to fixed relata are distinguished by the direction in which the relation applies to the relata (e.g., from Abelard to Eloise rather than from Eloise to Abelard). But neither Directionalism nor its most popular rival, Positionalism, offer accounts of differential application that generalize to relations of arbitrary symmetry structure. Here, I develop an alternative account, Relative Positionalism, which distinguishes different applications of a relation to fixed relata in terms of the ways in which the relata are characterized relative to one another. In presenting and defending Relative Positionalism, this paper covers some of the same ground as my [2016], but avoids the latter’s algebraic approach and focuses on interpretative issues—in particular, how to make sense of relative property instantiation—that were not addressed in the earlier paper.

1 Introduction

In at least some cases, the order of relata-designating terms in relational claims is semantically significant. For example,

• (lovesAE) Abelard loves Eloise

• may be true, while

  (lovesEA) Eloise loves Abelard

is false. More generally, let R be any n-ary relation that is not completely symmetric¹ and let ‘R’ be a n-place predicate standing for R. Then there are non-equivalent claims of the form

• (*) Rx₁…x_n

• (*_P) Rx_P(1)…x_P(n)

(where P is a permutation of the indices 1, …, n).

For relation R designated by predicate ‘R’, call claims of the form (*) atomic R-claims. Call pairs of the form (*) and (*_P) permuted R-claims. The primary question addressed in this paper is how in general to understand the difference in the content of non-equivalent permuted R-claims. In other words, given non-symmetric R and fixed relata x₁, …x_n, what more is there to the content of (*) besides the claim—common to both (*) and (*_P)—that R holds, somehow or another, among x₁, …, x_n? An adequate answer to this general question would tell us, in particular, what more (lovesAE) says than that the loves relation holds, somehow or another, among Abelard and Eloise.²

Note that for a fixed predicate ‘R’ designating the n-ary relation R, the n! (= 1 × ··· × n) permutations of 1, …, n can be partitioned³ by grouping permutations P and Q in the same partition-class if and only if, necessarily, (*_P) and (*_Q) are equivalent for all x₁, …, x_n in the domain of R. I will assume that any such partition of the permutations of 1, …, n represents the symmetry structure of the n-ary relation R.⁴ A secondary question addressed in this paper is—what, in general, determines differences in the symmetry structures of n-ary relations. An adequate answer to this question would tell us, in particular, what is different about the binary loves relation as compared to the binary next to relation which allows the former to hold in two distinct ways among two relata while the latter can hold in only one way among two relata.

Borrowing vocabulary from Fine (2000, p. 8), we may say that nonequivalent permuted R-claims express the differential application of the relation R to fixed relata. Using this terminology, the two central questions for this paper are:

(DiffApp1)

What, in general, does the difference between different applications of a fixed relation to fixed relata consist in?

(DiffApp2)

What, in general, determines the different capacities for differential application in relations of the same arity?⁵

The standard answer to (DiffApp1) has its roots in Russell’s account of relations from his 1903 Principles of Mathematics. Russell claims here that, at least in the binary case, relations apply to their relata in a sense, or direction,—i.e., binary relations apply to relata by proceeding from one relata to the next.⁶ Following Gaskin and Hill (2012) and Ostertag (2019), I call Russell’s account of relations Directionalism.⁷ Directionalism explains the difference in the content of (lovesAE) and (lovesEA) by appealing to two distinct directions in which the loves relation might apply to Abelard and Eloise. (lovesAE) claims that loves holds from Abelard to Eloise, while (lovesEA) claims that loves holds from Eloise to Abelard.

Directionalism has been the target of criticism dating back at least to Williamson (1985) and, more recently, Fine (2000). Briefly, the primary complaint about Directionalism has been that, because it is committed to the existence of distinct converses, Directionalism leads to ontological excess or semantic indeterminacy (or both). To see how, note first that given Directionalism, any non-symmetric binary relation R would seem to have a distinct converse R⁻¹ which holds in the opposite direction—i.e., R⁻¹ holds from y to x just in case R holds from x to y. The purported ontological problem is that claims like (lovesAE) and

• (lovedbyEA) Eloise is loved by Abelard

seem to describe the same relational state or fact. But if Directionalism does indeed entail that all binary relations have converses, then either (i) (lovesAE) and (lovedbyEA) describe a single relational state/fact structured by the distinct relations loves and is loved by or (ii) (lovesAE) and (lovedbyEA) describe distinct but redundant relational states/facts.⁸

The purported semantic problem is that, given Directionalism, it must be indeterminate whether any relational predicate denotes a given relation or, instead, one of its converses. Very briefly, the complaint is that conventions for relational ordering and for fixing the denotations of relational predicates are interdependent. For example, claims of the form

• (lovesxy) x loves y

may be true in exactly the same circumstances as are claims of the form

• (lovedbyxy) x is loved by y

if the two sorts of claims are associated with opposite conventions for specifying the direction of the application of the relation to the relata through the order of the relata-designating terms in the claim. If there is no way of specifying the direction of relational application independently of assumptions about the denotations of relational predicates and no way of specifying the denotation of relational predicates independently of conventions about the direction of relational application, then the denotations of relational predicates must be indeterminate.⁹

I am concerned here with neither the ontological excess objection to Directionalism nor the semantic indeterminacy objection to Directionalism. Instead, I focus in the next section on a more fundamental problem afflicting Directionalism, as well as proposed alternatives to Directionalism—that these accounts of relations do not deliver satisfactory general answers to questions (DiffApp1) and (DiffApp2) concerning differential application. I take its failure to explain differential application to be the most fundamental problem for Directionalism because Directionalism is introduced for the sole purpose of explaining differential application and has no other apparent motivation. Moreover, it seems incredible that there should be no answer to at least question (DiffApp1). Distinctions between the different applications of a relation to fixed relata are so intuitive and easy to grasp, even in cases involving unfamiliar relations or relata, it would be surprising if there were no general account of distinctions in the content of claims like (lovesAE) and (lovesEA). (By contrast, the assumptions behind the ontological excess and the semantic indeterminacy objections to Directionalism are much less intuitive. I doubt, for example, that many people have strong intuitions about whether or not a single relational state can be structured by distinct converse relations.)

An important further reason for focusing on (DiffApp1) and (DiffApp2) is that while most alternatives to Directionalism deny that relations have distinct converses and thus avoid the ontological excess and semantic indeterminacy objections to Directionalism, none have provided answers to (DiffApp1) and (DiffApp2) that generalize to relations of arbitrary symmetry structure. Since Directionalism itself fairs no better with (DiffApp1) and (DiffApp2), we have reason to direct attention to the common problem of finding satisfactory general answers to (DiffApp1) and (DiffApp2).

The remainder of this paper proceeds as follows. After showing that Directionalism and its primary rivals fail to meet two minimal criteria for satisfactory explanations of differential application (Sect. 2), I introduce Relative Positionalism and explain its answers to (DiffApp1) and (DiffApp2). I focus initially on Relative Positionalism’s account of binary relations (Sect. 3), before indicating how this account can be expanded to cover also higher arity relations (Sect. 4). Admittedly, Relative Positionalism carries very strong metaphysical commitments of its own. It requires that individuals may instantiate certain properties relative to other individuals or to sequences of individuals. I address criticisms of Relative Positionalism in Sect. 5. My conclusion is that despite its strong commitments, Relative Positionalism deserves serious consideration because it is the only account of differential application proposed so far that extends to fixed arity relations with any symmetry structure.

In presenting and defending Relative Positionalism, this paper covers some of the same ground as my earlier [2016]. However, the main focus of the earlier paper is an algebraic proof that Relative Positionalism has adequate structural resources for a general account differential application. Here, I present Relative Positionalism in a way that does not rely on abstract algebra (which, I think, some readers find off-putting) and focus on interpretive issues—in particular, how to make sense of relative property instantiation—that were not addressed in the earlier paper.

2 Two minimal criteria for accounts of differential application

I take the following to be minimal constraints on satisfactory answers to (DiffApp1) and (DiffApp2). First, I assume that any satisfactory answer to (DiffApp1) must explain our obvious ability to distinguish cases of differential application without abstruse background assumptions concerning the structure of relations. Even small children can understand and correctly explain the difference between what is claimed in (lovesAE) and what is claimed in (lovesEA). A satisfactory general answer to (DiffApp1) must somehow underwrite the terms in which correct ordinary explications of specific cases of differential application are framed.

Second, I assume that satisfactory answers to (DiffApp1) and (DiffApp2) must be able to explain the differential application of any non-symmetric relation R of finite fixed arity. Recall that non-symmetric relations may be either completely non-symmetric (as is the loves relation) or partly symmetric. (See footnote 1 for definitions of terms distinguishing levels of symmetry.) For example, the ternary relation between (holding among three things just in case one is between the other two) is partly symmetric. To see that it is not completely symmetric, note that the following two claims may differ in truth-value.

• (betweenMLC) Moe is between Larry and Curly.

• (betweenLMC) Larry is between Moe and Curly.

To see that it is also not completely non-symmetric, note that (betweenMLC) is necessarily equivalent to:

• (betweenMCL) Moe is between Curly and Larry.

Whereas binary relations must be either completely symmetric or completely non-symmetric, relations of arity greater than two may be partly symmetric (as is between). To further complicate the situation for higher arity relations, partly symmetric relations of the same arity may have different symmetry structures. Whereas the ternary relation between may apply to fixed relata in three possible ways, the ternary stand clockwise in a circle relation applies to fixed relata in only two possible ways. The variety of possible symmetry structures for n-ary relations increases rapidly as n increases. There are four possible symmetry structures for ternary relations, eleven possible symmetry structures for quaternary relations, nineteen possible symmetry structures for quinary relations, and increasingly many more possibilities for n-ary relations with n greater than 5.¹⁰ I assume that satisfactory general answers to (DiffApp1) and (DiffApp2) must be applicable to finite fixed arity relations with any possible symmetry structure. Even if we are not committed to the strong assumption that there are relations of any possible symmetry structure, there seems to be no non-question-begging reason for ruling out relations with particular symmetry structures.

One reason why Directionalism cannot offer a satisfactory answer to (DiffApp1) is that its central assumption—that relations apply to their relata in an order—is obscure and fails to connect with ordinary thinking about relational claims like (lovesAE) and (lovesEA).¹¹ Non-philosophers do not explain the difference between (lovesAE) and (lovesEA) in terms an order in which Abelard and Eloise are supposed to stand in the loves relation. Not only can we apparently get by just fine without invoking orders of relational application in intuitive understandings of differential application, it 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.

An additional shortcoming of Directionalism is that it does not have the right structure to explain the differential application of partly symmetric relations like between or stand clockwise in a circle.¹² If the different ways R can hold among x₁,…, x_n amount to just different orders of application of R to x₁…x_n, then any difference in the order of x₁,…,x_n should correspond to a different way for R to hold among x₁,…,x_n. Otherwise, the order in which R applies to x₁…x_n would not, on its own, determine a distinctive application of R to x₁…x_n—some further ingredient would be required. But not every ordering of the stooges amounts to a different way for between or stand clockwise in a circle to hold among them. There are six ways to linearly order the stooges (corresponding to the six permutations of any three-membered set) but only three ways for between to hold among them and only two ways for stand clockwise in a circle to hold among them. Thus, Directionalism does not meet our second minimal criterion—it cannot explain the differential application of fixed arity relations with any symmetry structure.¹³

What about alternatives to Directionalism? Might they fare better in providing answers to (DiffApp1) and (DiffApp2) that satisfy our two minimal criteria? One alternative to Directionalism, Macbride’s Ostrich Realism (2014), denies that there are any informative answers to (DiffApp1). After examining and rejecting various accounts of differential application, Macbride concludes that “we should just take the difference between aRb and bRa as primitive. This means that our understanding of what makes the difference between aRb and bRa is schematic—it depends in particular cases upon the character of the R in question.”¹⁴

Another alternative to Directionalism, Fine’s Antipositionalism, explains differential application through substitution relations among relata in relational states (2000, pp. 25–32).¹⁵,¹⁶ Fine’s proposal is that the difference between, e.g., the state described in (lovesAE) and that described in (lovesEA) amounts to a difference in how these two states result from substitution into an exemplar state. For example, the state described in (lovesAE) is the result of substituting Abelard for Emily and Eloise for Laney Lou in the state of Emily’s loving Laney Lou, while the state described in (lovesEA) is the result of substituting Eloise for Emily and Abelard for Laney Lou in the state of Emily’s loving Laney Lou. In general, on Fine’s account two relational states involving the same relation and same relata differ insofar as their substitution relations with the relata in an exemplar state for that relation differ.

I don’t see how Fine’s Antipositionalism can give us an account of the difference in the content of (lovesAE) and (lovesEA). Not only is there no implicit or explicit reference to other loves states in either (lovesAE) or (lovesEA), it is also unclear how substituting into an exemplar state could help clarify the distinction between Abelard’s loving Eloise and Eloise’s loving Abelard if we don’t have an independent account of how the exemplar state itself differs from its differential opposite (viz., how Emily’s loving Laney Lou differs from Laney Lou’s loving Emily). Fine denies that there is any way of distinguishing different applications of a relation to fixed relata except through substitution relations among states involving that relation (2000, p. 30). In particular, Fine denies that relational states have any internal structure—e.g., a lover role and a beloved role—through which we might distinguish a particular application of a relation in a way that does not rely on connections with other relational states (Fine 2007, p. 57). Thus, according to Antipositionalism, any exemplar loves state is itself distinguished from its differential opposite only through the same substitution relations which are supposed to distinguish other loves states from their differential opposites through reference to the exemplar state. But this account never seems to cash out the distinctions among different applications of the loves relation that are tracked by its substitution relations.¹⁷

Antipositionalism’s account of differential application is also quite far from the ordinary way of distinguishing between the content of relational claims like (lovesAE) and (lovesEA). We do not normally invoke exemplar loving states or consider the results of substituting one person for another within loving states to distinguish the claims made in (lovesAE) and (lovesEA). Instead, we focus on what each of the claims says about Abelard and Eloise (in particular, what each claim says about who is loving whom). Thus, insofar as it accounts at all for the differential application of a relation to fixed relata, Antipositionalism fails to satisfy our first criterion of providing a framework to support correct ordinary explanations of specific cases of differential application.

A final alternative to Directionalism, Positionalism,¹⁸ does seem to fit ordinary ways of distinguishing between the content of claims like (lovesAE) and (lovesEA). According to Positionalism, each n-ary relation R has up to n associated unary properties called positions (or roles). The different ways for x₁,…,x_n to stand in R correspond the different ways of assigning x₁,…,x_n to R’s positions. For example, (lovesAE) describes one way for the loves relation to hold among Abelard and Eloise—that in which Abelard occupies the lover position and Eloise the beloved position. (lovesEA) describes a distinct way for the loves relation to hold among Abelard and Eloise—that in which Eloise occupies the lover position and Abelard the beloved position.

However, as has been pointed elsewhere,¹⁹ Positonalism does not offer answers to (DiffApp1) and (DiffApp2) that generalize to relations of arbitrary symmetry structure. For example, Positionalism does not account for the differential application of relations with a cyclical symmetry structure like the ternary stand clockwise in a circle relation. Roughly, this is because the two different ways for Moe, Larry, and Curly to stand in this relation are not differentiated by distinct absolute positions occupied by the stooges—no one of the stooges plays a distinctive role in any application of stand clockwise in a circle to the stooges. Instead, the two different ways the stooges might stand in the stand clockwise in a circle relation are distinguished by the ways the stooges are positioned relative to one another—namely, with either Moe, followed by Larry, followed by Curly in the clockwise direction or with Moe, followed by Curly, followed by Larry in the clockwise direction. Thus, though Positionalism, unlike Directionalism and Antipositionalism, does seem to offer an answer to (DiffApp1) that meets our first minimal criterion, its answer does not satisfy our second minimal criterion.

Unlike question (DiffApp1), question (DiffApp2) is rarely addressed in accounts of relations. I assume that both MacBride’s Ostrich Realism and Fine’s Antipositionalism would deny that there is any general account of differences in the symmetry structures of relations of the same arity. (Fine says, for example, that it “is a fundamental fact for …[the antipositionalist] that relations are capable of giving rise to a diversity of completions in application to any given relata and there is no explanation of this diversity in terms of a difference in the way the completions are formed from the relation by assigning the relata to different argument-places” (2000, p. 19).) And it is clear that because of the limits on the kinds of symmetry structures it can handle, Positonalism cannot provide an answer to (DiffApp2) that extends to relations of arbitrary symmetry.

3 Relative positionalism—binary relations

Since neither Directionalism nor the primary alternatives to Directionalism can provide satisfactory general explanations of differential application, we should look for an alternative account of relations that can. The account developed in this paper, Relative Positionalism, is a variation on Positionalism which, at least in the case of binary relations, shares Positionalism’s intuitiveness. However, unlike Positionalism, Relative Positionalism offers an explanation of differential application which extends to fixed arity relations with any symmetry structure.

I defer the discussion of Relative Positionalism’s account of higher arity relations to Section IV. In this section, I focus only on Relative Positionalism’s account of binary relations.

Consider the distinction between what is claimed in (lovesAE) and what is claimed in (lovesEA). Both statements claim that the loves relation holds among Abelard and Eloise. The obvious difference in the content of the two claims is that whereas (lovesAE) says that Abelard loves and Eloise is beloved, (lovesEA) says that Eloise loves and Abelard is beloved. This, I take it, amounts to a difference in the way (lovesAE) and (lovesEA) characterize each of the two relata, Abelard and Eloise, individually—either as lover or as beloved. But (lovesAE) does not merely claim that Abelard is a lover of someone or other and Eloise is beloved by someone or another. If that were all there is to the content of (lovesAE) besides the assertion that the loves relation holds (in some way or another) among Abelard and Eloise, then (lovesAE) would be true if, say, Eloise loves Abelard, Abelard loves Moe (but not Eloise), and Eloise is loved by Larry (but not by Abelard). Analogous comments apply to (lovesEA). Importantly, the crucial characterizations of the relata which distinguish the content of (lovesAE) from that of (lovesEA) are not absolute but relative. (lovesAE) characterizes Abelard as lover, not absolutely (or, so to speak, from the standpoint of the world at large), but relative to Eloise. In so doing, (lovesAE) also characterizes Eloise, relative to Abelard, as beloved—since what it is for Abelard to be a lover relative to Eloise is for Eloise to be beloved relative to Abelard. Thus, the difference in the content of (lovesAE) and (lovesEA) amounts to a distinction in the way the relata are characterized relative to one another.

Relative Positionalism proposes that any cases of the differential application of a binary relation to fixed relata are distinguished by the ways in which those relata are characterized relative to one another.²⁰ How might we fill out Relative Positionalism’s underlying assumption that one individual may be characterized, not absolutely, but relative to another individual? I propose that any individual may function not only as the target of characterizations (i.e., as itself an instance of properties), but also as a standpoint or parameter from which individuals are characterized.²¹ Note that certain familiar kinds of properties only make sense when assigned relative to a particular individual. From my own standpoint, there is a clear distinction between people who are beloved and people who are not beloved. But this cannot be an absolute distinction since people who are beloved for me are not beloved for everyone and people who are beloved for other people need not be beloved for me. Similarly, from my own (current) standpoint, there is a distinction between locations (cities, buildings, etc.) that are north and those that are south. But different locations are north or south relative to different people (or cities, buildings, etc., …), so this cannot be an absolute distinction. I take beloved, lover, north, and south to be unary properties—they characterize individuals like Eloise or Toronto, not pairs of individuals. But they are unary properties which are instantiated, not absolutely, but relative to particular individuals.

Note that the Empire State Building may be north relative to me even if I do not believe that it is, either because I do not know where the Empire State Building is or because I have an imperfect sense of direction. And individuals (other buildings, cities, people, etc…) are north or south relative to the Empire State Building even though the Empire State Building itself has no beliefs or perceptions at all concerning any individual’s location. Thus, relative properties like lover, beloved, north, or south, are not subjective characterizations. Whether or not Y is north relative to X does not depend on X’s perceptions and beliefs (or whether X has any perceptions or beliefs). However, there should at least be this connection between relative properties and subjective phenomena—if X is the kind of thing that has perceptions and beliefs, then X’s perceptions and beliefs are accurate only insofar as they correctly reflect the ways in which individuals are characterized relative to X. My belief that the Empire State Building is north is accurate only if the Empire State Building is, in fact, north relative to me.

Relative Positionalism makes the strong claim that the holding of a binary relation among two individuals consists in each of the relata being characterized in particular ways relative to one another. For relations like loves or north of, there are two distinct associated relative properties, R₁ $\ne$ R₂, such that for any x, y,

• relative to y (x is R₁) iff relative to x (y is R₂).       (CONJ)

Call relative properties satisfying (CONJ) conjugates. For example, north and south are conjugate relative properties, as are also lover and beloved.

Unlike non-symmetric binary relations, symmetric binary relations like next to have only one associated relative property. Let next be the property had by x relative to y when x is next to y. I take it that x qualifies as next from y’s standpoint just in case x is sufficiently nearby, relative to y, on some contextually determined standard of closeness. If x is in this sense next, relative to y, then y must be next, relative to x. Thus, next is self-conjugating, where relative property R₁ is self-conjugating just in case:

• relative to y (x is R₁) iff relative to x (y is R₁). (S-CONJ)

The general formulation of Relative Positionalism for binary relations is as follows.

Let R be any binary relation. According to Relative Positionalism, there are relative properties R₁ and R₂ (not necessarily distinct) such that

• (i) R₁ and R₂ are (self-)conjugates (i.e., if R₁ $\ne$ R₂, R₁ and R₂ are conjugates and if R₁ = R₂, R₁ is self-conjugating) and

• (ii) for any individuals x, y, R holds among x and y iff relative to y (x is R₁) or relative to y (x is R₂).

Call R₁ and R₂ the relative properties of the relation R. Note that given requirement (i), it follows from (ii) that R holds among x and y iff either:

• (*) relative to y (x is R₁) and relative to x (y is R₂), or

• (**) relative to y (x is R₂) and relative to x (y is R₁).

For binary relations, Relative Positionalism’s answer to question (DiffApp1) is that different applications of R to fixed relata x, y are distinguished by differences in the distributions of R’s relative properties R₁ and R₂ among x, y. It follows from conditions (i) and (ii) above that there are at most two ways in which R can hold among x and y—namely, the ways corresponding to the relative property distributions (*) and (**) above.

Note that (*) and (**) amount to the same way for R to apply to x and y in case R₁ = R₂. Relative Positionalism’s answer to (DiffApp2) for binary relations is that R has the capacity to hold in two ways among fixed relata if and only if R has two distinct relative properties, R₁ and R₂. Otherwise, R has only one relative property (i.e., R₁ = R₂) and can hold in only one way among fixed relata.

Note, crucially, that Relative Positionalism’s account of differential application must deny that relative property ascriptions are merely disguised relational claims. For example, if

• relative to Eloise(Abelard is a lover) (@E, loverA)

• relative to Abelard(Eloise is beloved) (@A, belovedE)

were merely alternative phrasings of the relational claims (lovesAE) and (lovedbyEA), then obviously such relative property ascriptions would not help distinguish the content of relational claims in the way proposed above. (@E, loverA) and (@A, belovedE) are instead what Fine calls external relativizations (Fine 2005, p. 279).²² Abelard functions in (@A, belovedE) not as a relatum of a relation but rather as the parameter (or, what I call the “standpoint”) at which the property ascription—Eloise is beloved—is assessed. In other words, Abelard’s function in (@A, belovedE) is analogous to that of the worlds at which propositions are evaluated in possible worlds semantics or that of the times at which propositions are evaluated in temporal logics.

So understood, relativized property ascriptions such as (@E, loverA) and (@A, belovedE) introduce a kind of orientation not present in relational claims like (lovesAE) or (lovedbyEA). Whereas (lovesAE) characterizes both Abelard and Eloise by predicating the loves relation of them, (@E, loverA) characterizes only Abelard by predicating the unary property lover of him (from the standpoint of Eloise) and (@A, belovedE) characterizes only Eloise by predicating the unary property beloved of her (from the standpoint of Abelard). In this way, relative property conjugates, like lover and beloved, differ from Directionalism’s relation converses. Relation converse pairs like, loves and islovedby, apply to the exact same relata, differing only in the supposed direction of their application to these relata, but not in which individual is the target and which is the parameter of a particular characterization.

4 Relative positionalism—higher arity relations

The second minimal criterion for satisfactory answers to (DiffApp1) and (DiffApp2)—that they generalize to relations having arbitrarily complex symmetry structures—can be evaluated only by considering n-ary relations for n > 2. As noted in Sect. 2, there are only two possible symmetry structures for binary relations. Any binary relation is either completely symmetric or completely non-symmetric. But there are four possible symmetry structures for ternary relations, eleven possible symmetry structures for quaternary relations, nineteen possible symmetry structures for quinary relations, and increasingly many more possibilities for n-ary relations as n increases. A minimally adequate account of differential application must be able to accommodate increasingly complex patterns of differential application in higher arity relations.

Unlike Directionalism, Positionalism, Antipositionalism and Ostrich Realism, Relative Positionalim offers an account of the differential application of any finite fixed arity relation in terms of the structure of relations. But I warn the reader in advance that Relative Positionalism’s treatment of higher arity relations posits an increasing complexity in relative property instantiation to match the increasing complexity of symmetry structures for higher arity relations. Relative Positionalism’s treatment of higher arity relations is thus less intuitive than its treatment of binary relations. Some may see this as a mark against Relative Positionalism. However, in Relative Positionalism’s defense, it is hard to imagine how any theory of relations could offer a simple account of differential application in quaternary or quinary relations—the eleven distinctions among the possible symmetry structures for quaternary relations and the nineteen distinctions among the possible symmetry structures for quinary relations are not themselves easy to grasp.

A simple example of a partly symmetric relation is the ternary relation between discussed in Sect. 2. Recall that between can apply in three different ways to fixed relata. These are illustrated below for the relata Moe, Larry, and Curly (Figs. 1, 2, 3).

What do the distinctions among these three applications of between to the stooges amount to? Intuitively, the different applications of between to the stooges are distinguished only by which one of the stooges occupies the middle position relative to the other two.

Can this intuitive explanation of the distinctions among different applications of between to the stooges be expressed in terms of the ways the stooges are characterized relative to one another? I think so. Here, though, we must be careful. As I have framed it, the intuitive explanation characterizes one stooge as middle relative to the other two. But it is not clear what sense can be made of a standpoint (i.e., a parameter of property attribution) that is supposed to be a combination of two individuals. There is a subjective perspective (that of Larry) which, insofar as it is accurate, reflects the way things in the world are characterized relative to Larry and another (that of Curly) which, insofar as it is accurate, reflects the way things in the world are characterized relative to Curly. But there is no subjective perspective which, insofar as it is accurate, reflects the ways things in the world are characterized relative to Larry and Curly. Moreover, such familiar parameters as possible worlds and times are not combined for joint assessments—there is no complete and consistent way things are at a combination of two times or at a combination of two worlds.

Ultimately, I propose that individuals may be characterized from embedded standpoints—i.e., from the standpoint of one individual as it is structured by that of another individual. To motivate this added complexity with intuitive observations, note first that two (or more) individuals may stand in certain relations, not absolutely, but only relative to another individual. For Moe to occupy the middle position relative to Larry and Curly (as in Fig. 1) is for Curly to stand opposite of Larry relative to Moe. Note that the holding of the opposite of relation between Larry and Curly is standpoint-dependent. If I were positioned to the right of Curly in Fig. 1, then Curly would not be opposite of Larry, from my standpoint (since both Larry and Curly would be on the same side of me).

Fig. 1

(betweenMCL) Moe is between Curly and Larry

(betweenMLC) Moe is between Larry and Curly.

Fig. 2

(betweenLCM) Larry is between Curly and Moe

(betweenLMC) Larry is between Moe and Curly.

Fig. 3

(betweenCML) Curly is between Moe and Larry

(betweenCLM) Curly is between Larry and Moe.

Other examples of standpoint-dependent spatial relations are closer than, farther than, in front of, and in back of where these relations are taken to depend on a framework imposed by an outside reference object. Note that the standpoint-dependent in front of and in back of relations match the uses of, respectively, “in front of” and “in back of” in which any object Y on a straight path between X and reference object RO counts as in front of X, relative to RO, and any object X such that a straight path between X and RO includes Y counts as in back of Y, relative to RO.²³ For example, if I am standing at the Capital Building in Washington D.C., then the Washington Monument is in front of the Lincoln Memorial (and the Lincoln Memorial is in back of the Washington Monument), relative to me. See Fig. 4. By contrast, the Washington Monument is in back of the Lincoln Memorial (and the Lincoln Memorial in front of the Washington Monument) from the standpoint of a person in Arlington Cemetery. Relative to the Washington Monument itself, the Lincoln Memorial is opposite of the Capital Building (and me, or anyone else, positioned at the Capital Building), since the Lincoln Memorial and the Capital Building lie on straight paths proceeding in opposite directions from the Washington Monument.

Fig. 4

Relative to the Capital Building, the Washington Monument is in front of the Lincoln Memorial. Relative to Arlington Cemetery, the Lincoln Memorial is in front of the Washington Monument

Ultimately, Relative Positionalism distinguishes applications of relative relations like opposite of, in front of, and in back of through doubly-relative unary properties. But before complicating our account by introducing doubly-relative properties, it is worth appreciating how the relative relations in front of, in back of, and opposite of distinguish the applications of between depicted in Figs. 1 and 2. In the arrangement depicted in Fig. 1, the stooges stand in the following relations relative to one another:

• (@M, oppofLC) relative to Moe (Larry is opposite of Curly);

• (@L, infrontMC) relative to Larry (Moe is in front of Curly);

• (@C, inbackLM) relative to Curly (Larry is in back of Moe).

In Fig. 2, by contrast, the stooges are arranged as follows relative to one another:

• (@L, oppofMC) relative to Larry (Moe is opposite of Curly);

• (@M, infrontLC) relative to Moe (Larry is in front of Curly);

• (@C, inbackML) relative to Curly (Moe is in back of Larry).

But note, crucially, that in front of and in back of apply in two different ways to fixed relata relative to a fixed standpoint. For example, (@C, inbackLM) and (@C, inbackML) describe different arrangements of Moe and Larry relative to Curly—the former holds in the arrangement of Fig. 1, but not in that of Fig. 2, while the latter holds in the arrangement of Fig. 2, but not in that of Fig. 1. To distinguish among different applications of between, we must ultimately distinguish among different applications of in back of. Merely noting that in back of holds, somehow or another, among Moe and Larry relative to Curly does not distinguish the Fig. 1 application of between from the Fig. 2 application of between.

Applying the method of the previous section to the present case, Relative Positionalism distinguishes (@C, inback LM) and (@C, inback ML) by positing unary properties, front and behind, had by the relata, Moe and Larry, relative to one another. Here, however, the unary properties characterize Moe and Larry relative to each other relative to Curly, since it is from Curly’s standpoint that in back of holds among Moe and Larry. In this way, the relative positionalist ultimately appeals to embedded standpoints in her account of n-ary relations for n > 2. The general idea is that from Moe’s standpoint on its own, objects are characterized as near or far, beloved or hated, north or south, and so on, as proposed in Sect. 3. But additional structure may be imposed on Moe’s standpoint when other individuals are characterized relative to Moe from an outside standpoint, such as that of Curly. Relative to Moe alone, no individual counts as front or behind (in the sense intended here—see footnote 23 for an alternative use of spatial prepositions). But to embed Moe’s standpoint within Curly’s standpoint is to supply external structure in terms of which other objects may be, e.g., front or behind, closer or farther, more beloved or less beloved, as characterized relative to Moe from Curly’s standpoint. These doubly-relative characterizations depend not just on Moe but also on the individual functioning as the outside reference point. If I stand to the left of Larry in Fig. 1, then relative to me, relative to Moe, Larry is front and not behind as he is relative to Moe, relative to Curly. As another example, Eloise may be more beloved, relative to Moe, relative to Abelard (i.e., more beloved than Moe from Abelard’s standpoint), but less beloved, relative to Moe, relative to Larry (i.e., less beloved than Moe from Larry’s standpoint).

Note the significance of ordering in doubly-relativized characterizations. X’s characterization relative to Y, relative to Z is, roughly, how X figures in comparison to Y within a framework imposed from Z’s standpoint (where Z is the outside parameter of the double-relativization). Thus, whereas Larry’s spatial characterization, relative to Moe, relative to Curly depends on a spatial framework centered at Curly, Larry’s spatial characterization relative to Curly, relative to Moe depends on a spatial framework centered at Moe. As another example, Eloise is more beloved, relative to Moe, relative to Abelard if given evaluations determined from Abelard’s standpoint, Eloise ranks as more beloved in comparison to Moe. Whether Eloise is characterized as more beloved, less beloved, or neither, relative to Abelard, relative to Moe depends on rankings determined from Moe’s (not Abelard’s) standpoint.

In terms of the doubly-relative unary properties front and behind, (@C, inbackLM) is the application of in back of to Larry and Moe relative to Curly in which:

• (@C, @M, behindL) relative to Curly (relative to Moe (Larry is behind));

• (@C, @L, frontM) relative to Curly (relative to Larry (Moe is front)).

By contrast, (@C, inbackML) is the application of in back of to Larry and Moe relative to Curly in which:

• (@C, @L, behindM) relative to Curly (relative to Larry (Moe is behind));

• (@C, @M, frontL) relative to Curly (relative to Moe (Larry is front)).

Unlike in back of and in front of, the relative relation opposite of applies in only one way to fixed relata, relative to a fixed standpoint. There is only one way for opposite of to hold among Larry and Curly relative to Moe—the way expressed in (@M, oppofLC) and depicted in Fig. 1. This is because when opposite of holds among x and y, relative to a third individual z, then each of x and y plays the same role relative to the other in standing on an opposing side of z. Thus, the binary relative relation opposite of has only one associated doubly-relative property, opposite.

In terms of the three doubly-relative properties front, behind, and opposite, Fig. 1 depicts the application of between to the stooges in which

• (@M, @L, oppositeC) relative to Moe (relative to Larry (Curly is opposite));

• (@M, @C, oppositeL) relative to Moe (relative to Curly (Larry is opposite));

• (@L, @M, behindC) relative to Larry (relative to Moe (Curly is behind));

• (@L, @C, frontM) relative to Larry (relative to Curly(Moe is front));

• (@C, @M, behindL) relative to Curly (relative to Moe (Larry is behind));

• (@C, @L, frontM) relative to Curly (relative to Larry (Moe is front)).

By contrast, Fig. 2 depicts the application of between to the stooges in which:

• (@L, @M, oppositeC) relative to Larry (relative to Moe (Curly is opposite));

• (@L, @C, oppositeM) relative to Larry (relative to Curly (Moe is opposite));

• (@M, @L, behindC) relative to Moe (relative to Larry (Curly is behind));

• (@M, @C, frontL) relative to Moe (relative to Curly (Larry is front));

• (@C, @L, behindM) relative to Curly (relative to Larry (Moe is behind));

• (@C, @M, frontL) relative to Curly (relative to Moe (Larry is front)).

Note that implication relations hold among instantiations of the three doubly-relative properties for the between relation. For any individuals x, y, z,

• relative to x (relative to y (z is front))

•                   iff relative to x (relative to z (y is behind))

•                   iff relative to z (relative to x (y is opposite))

•                   iff relative to z (relative to y (x is opposite))

•                   iff relative to y (relative to z (x is behind))

•                   iff relative to y (relative to x (z is front)).

Thus, any instantiation of any one of front, behind, opposite by x relative to y, relative to z determines the remaining five instantiations of front, behind, opposite among x, y, z and thus also determines a particular way for between to hold among x, y, z. It follows that the number of distinct ways the ternary between relation can hold among fixed relata must match its number of doubly-relative properties—three.

Recall that, unlike the ternary between relation, the ternary relation stand clockwise in a circle applies in only two different ways to fixed relata. Relative Positionalism explains why these two ternary relations differ in the number of ways they may hold among fixed relata by pointing out that, unlike between, stand clockwise in a circle has only two doubly-relative properties. To see this, consider the arrangements of the stooges represented in Figures 5 and 6.

Insofar as stand clockwise in a circle applies as depicted in Fig. 5, there are only two different ways in which one of the stooges may be characterized from the embedded standpoints of the other two. Either stooge x is ahead of stooge y along a path in the clockwise direction originating at stooge z (as, e.g., Larry is ahead of Curly in the clockwise direction from Moe in Fig. 5) or stooge x is behind stooge y along a path in the clockwise direction originating at stooge z (as, e.g., Curly is behind Larry in the clockwise direction from Moe in Fig. 5). The only other way for stand clockwise in a circle to hold among the stooges is that in which all of the stooges’ doubly-relative property instantiations are switched, as depicted in Fig. 6.

Fig. 5

(circleMLC) Moe, Larry and Curly stand clockwise in a circle. (circleLCM) Larry, Curly and Moe stand clockwise in a circle. (circleCML) Curly, Moe and Larry stand clockwise in a circle

Fig. 6

(circleMCL) Moe, Curly and Larry stand clockwise in a circle. (circleLMC) Larry, Moe and Curly stand clockwise in a circle. (circleCLM) Curly, Larry and Moe stand clockwise in a circle

For n-ary relations with n > 3, the relative positionalist claims that there are not only doubly-relative, but also triply-relative properties, quadruply-relative properties, and so on. Here we must allow that just as one individual may be characterized as, e.g., front or behind, relative to a second individual from the standpoint of a third individual, so also tuples of four or more individuals may be characterized relative to each other. As one brief example, consider the four stooges as they are arranged in Fig. 7. (Note that Fig. 7 depicts a situation in which both of (LineSLMC) and (LineCMLS) are true, since the linear arrangement of the stooges depends only on their positions relative to one another. Analogous comments apply also to Fig. 8.)

Fig. 7

(LineCMLS) Curly, Moe, Larry, and Shemp stand (in that order) in a line

(LineSLMC) Shemp, Larry, Moe and Curly stand (in that order) in a line.

Fig. 8

(LineCSLM) Curly, Shemp, Larry, and Moe stand (in that order) in a line

(LineMLSC) Moe, Larry, Shemp and Curly stand (in that order) in a line.

Here, Shemp is behind, relative to Larry, relative to Moe (i.e., Shemp is located after Larry along a straight path originating at Moe). But Shemp is not behind Larry relative to Moe in just any direction. He is located after Larry in the direction that counts as behind relative to Moe in the orientation determined by Curly’s standpoint (i.e., in the direction from Moe in which both Larry and Shemp count as behind Moe relative to Curly.) By contrast, in the arrangement depicted in Fig. 8, Shemp is still behind, relative to Larry, relative to Moe, but now both Larry and Shemp are in front of Moe relative to Curly.

In general, Moe’s standpoint alone can orientate individuals relative to Larry by determining what counts as behind, front, etc. relative to Larry, relative to Moe. When further structure is imposed on Moe from Curly’s standpoint, then we can further characterize individuals relative to Larry as: (i) behind Larry, relative to Moe in the direction that counts as behind Moe relative to Curly or (ii) behind Larry, relative to Moe in the direction that counts as in front of Moe relative to Curly. Admittedly, there are no English names for triply-relative properties like these. But we could introduce terms like “behind–behind” and “behind–front” where for all x, y, z, w

• relative to x (relative to y (relative to z (w is behind–behind))) = _def

• relative to y (relative to z (w is behind)) & relative to x (relative to y (w is behind));

• relative to x (relative to y (relative to z (w is behind-front))) = _def

• relative to y (relative to z (w is behind)) & relative to x (relative to y (w is front)).

Note that Shemp’s being behind–behind relative to Larry, relative to Moe, relative to Curly is one way for the quaternary stand (in that order) in a line relation to apply to the stooges. Another way for the stand (in that order) in a line relation to apply to the stooges is the arrangement illustrated in Fig. 8, in which Shemp is behind-front Larry, relative to Moe, relative to Curly.

In general, Relative Positionalism claims that n-ary relation R holds among x₁, x₂, …, x_n just in case each of x₁, x₂, …, x_n is characterized in certain ways from the (multiply embedded) standpoints of the other relata. More precisely:

Let R be any n-ary relation. According to Relative Positionalism, there are unary properties R₁, R₂, …, R_n! (not necessarily distinct)²⁴ such that for any individuals x₁, x₂, …, x_n, any permutation P of 1, …, n, and any natural number j such that 1 ≤ j ≤ n!, the following are equivalent:

• (i) R holds among x₁, x₂, …, x_n;

• (ii) for  some  1 ≤ i ≤ n!,  relative  to  x_P(1)  (relative  to  x_P(2)  (…  (relative  to x_P(n−1)(R_i x_P(n)))…));

• (iii) for some permutation Q of 1, …, n, relative to x_Q(1) (relative to x_Q(2) (… (relative to x_Q(n−1)(R_j x_Q(n)))…)).

Call R₁, R₂, …, R_n!, the relative properties of the relation R. For example, the relative properties of the ternary between relation are the three doubly-relative properties opposite, front, and behind. Note that the equivalence of (i)–(iii) entails that if R holds among x₁, x₂, …, x_n, then each of R’s relative properties characterizes at least one of x₁, x₂, …, x_n from some embedded standpoint of the remaining relata. In fact, Relative Positionalism holds that R’s relative properties are conjugates in the following sense:

• For any 1 ≤ i, j ≤ n!, there is a permutation P of 1, …, n such that for all x₁, x₂, …, x_n,

•                   relative to x₁ (relative to x₂ (… (relative to x_n−1(R_i x_n)))…)) iff                  (CONJ)

•                   relative to x_P(1) (relative to x_P(2) (… (relative to x_P(n−1)(R_j x_P(n)))…)).

Relative Positionalism’s answer to (DiffApp1) is that different applications of an n-ary relation R to fixed relata x₁, x₂, …, x_n are distinguished by the ways x₁, x₂, …, x_n are characterized relative to one another through R’s relative properties. More precisely, given distinct relative properties for R, R_i $\ne$ R_j, one way for R to hold among x₁, x₂, …, x_n is for x_n to be characterized relative to the other relata as:

• relative to x₁ (relative to x₂ (… (relative to x_n−1(R_i x_n)))…)).

Another way for R to hold among x₁, x₂, …, x_n is for x_n to be characterized relative to the other relata as:

• relative to x₁ (relative to x₂ (… (relative to x_n−1(R_j x_n)))…)).

Given that R’s relative properties are pairwise conjugates, it follows that the number of ways for R to hold among fixed relata equals the number of R’s distinct relative properties. Thus, Relative Positionalism partly answers (DiffApp2) by claiming that n-ary relations R and R* differ in the number of ways they can hold among fixed relata if and only if R and R* differ in their numbers of relative properties. At one extreme, if R has only one relative property, then R can apply to fixed relata in only one way and is completely symmetric. At the other extreme, an n-ary relation with n! distinct relative properties can apply in n! different ways to fixed relata and is completely non-symmetric. The ternary relations between and stand clockwise in a circle are intermediate cases of partly symmetric ternary relations with, respectively, three and two distinct relative properties.

Elsewhere (in Donnelly (2016)), I have used abstract algebra to prove that Relative Positionalism posits a relational structure complex enough to answer questions (DiffApp1) and (DiffApp2) for finite fixed-arity relations with any symmetry structure. More precisely, I show that given an arbitrary n-ary relation R, it is possible to assign relative properties to sequences of relata in such a way that non-equivalent R claims differ by permutations assigning different relative properties to some sequences of relata, while equivalent R claims differ by permutations assigning the same relative properties to all sequences of relata.

This shows that Relative Positionalism has a significant advantage over Directionalism, which can only explain the differential application of completely symmetric or completely non-symmetric relations, as well as Positionalism, which can only explain the differential application of relations whose symmetry structure is generated by two-cycles (i.e., by permutations which switch the places of two relata).²⁵ As far as I know, no proposed account of relations besides Relative Positionalim posits a relational structure complex enough to support the increasingly many possible patterns of differential application for n-ary relations as n increases.

Thus, Relative Positionalism meets the two minimum requirements for satisfactory explanations of differential application laid out in Sect. 2. It supports intuitive explanations of distinctions between different applications of a relation to fixed relata by invoking differences in the ways the relata are characterized relative to one another. And its account of differential application generalizes to finite fixed-arity relations with any symmetry structure.

5 Criticisms and responses

5.1 Criticism

Relative Positionalism has at least as much (if not more) of a problem with ontological excess as does Directionalism. Just as Directionalism commits us either to relational states structured by both a relation and its converse (e.g., by both loves and islovedby) or to duplicate relational states (e.g., both Abelard’s loving Eloise and Eloise’s being loved by Abelard), so also Relative Positionalism commits us either to relational states structured by a relation and its conjugate relative properties (e.g., by all three of loves, lover, and beloved) or to duplicate states (e.g., all three of: Abelard’s loving Eloise, Abelard’s being a lover relative to Eloise, and Eloise’s being beloved relative to Abelard).

5.2 Response

Directionalism commits its proponents to redundant relations or redundant relational states, because distinctions between converse relations or converse relational states depend on the vacuous notion of an order (or direction) in the application of a relation to its relata. If there were real differences among orders of relational application, then converse relations and converse relational states would not be redundant. Instead, there would be a real difference between Abelard’s loving Eloise and Eloise’s being loved by Abelard, since the relations structuring these states would apply to Abelard and Eloise in opposite orders. But given the difficulty of making sense of what it is for a relation to apply to one relata first and the other second, it seems, rather, that there is actually no difference at all between Abelard’s loving Eloise and Eloise’s being loved by Abelard. Directionalism has a problem with ontological excess not because its proponents are committed to distinct (but necessarily co-instantiated or co-occurring) relations and relational states, but because its proponents are committed to distinct relations and relational states with no clear account of the purported difference between them.

By contrast, Relative Positionalism can account for the distinctions between a relation and the conjugate relative properties for that relation. The relative positionalist can say that one relational state appears in different ways from the different standpoints of the relata. The state of Abelard’s loving Eloise is from Abelard’s standpoint, Eloise’s being beloved and from Eloise’s standpoint, Abelard’s being a lover. Unlike Directionalism’s converse relations, loves and islovedby, it is clear what the difference between being a lover and being beloved amounts to. The former consists in being the source of certain kinds of strong positive feelings, while the latter consists in being the target of certain kinds of strong positive feelings. It is also clear that Eloise’s being beloved and Abelard’s being a lover are characterizations of the state of Abelard’s loving Eloise from distinct standpoints—respectively, that of Abelard and that of Eloise. Thus, while Relative Positionalism does posit multiple aspects of a single relational state, these aspects differ both in content and in the standpoint functioning as the parameter of characterization.²⁶

5.3 Criticism

Relative Positionalism posits complicated nested standpoints to account for the differential application of partly symmetric relations. Wouldn’t it be simpler to deny that there are such relations? After all, most discussions of relations focus only on binary relations, for whose differential application either Directionalism or Positionalism might offer some account.

5.4 Response

We could deny that there are any higher-arity partly symmetric relations, but there is no apparent reason to do so. Granted, if we could establish that there is no way of accounting for differential application in partly symmetric relations, this could perhaps be a reason to deny that there are such relations. But Relative Positionalism does offer an account of differential application in partly symmetric relations (of arbitrary arity and with arbitrary symmetry structure). This shows that there are, after all, accounts of the differential application of partly symmetric relations.

5.5 Criticism

Granted, Directionalism and Positionalism cannot explain differential application in partly symmetric relations like forms a circle in the clockwise direction. Neither account can explain why, e.g., the relational state described in (circleMLC) differs from that described in (circleLMC) or why there are three ways for between to apply to the stooges but only two ways for forms a circle in the clockwise direction to apply to the stooges. But neither can Relative Positionalism explain why the relational states described in (circleMLC) and (circleLMC) differ or why different ternary relations can apply to fixed relata in different numbers of ways.

5.6 Response

The problem with other accounts of relations is not that they cannot explain why particular applications a relation to fixed relata differ or why some relations can apply to fixed relata in more ways than do other relations of the same arity. It is, rather, that they cannot explain what the distinction among different applications of a relation to fixed relata consists in or what determines different capacities for differential application in relations of the same arity. Unlike Directionalism and Positionalism, Relative Positionalism can explain what the difference in the content of (circleMLC) and (circleLMC) amounts to—a difference in the ways the stooges are characterized relative to one another—and what determines the different capacities for differential application of between and forms a circle in the clockwise direction—a difference in their numbers of relative properties.

5.7 Criticism

Relative Positionalism requires that any relatum of any relation—buildings, stones, refrigerators, and so on—has its own standpoint. That’s crazy.

5.8 Response

Relative Positionalism does not require that relata have standpoints. It requires that relata are standpoints, where (as explained in Sect. 3) a standpoint is a parameter at which property attributions are assessed. As emphasized in Sect. 3, being a standpoint in this sense does not presuppose an experiential point of view. For example, times or possible worlds are commonly taken to be parameters at which property attributions are assessed. But no one worries that times can function in this sense as parameters only if times have their own experiential point of view. Granted, allowing that not only times and possible worlds, but also buildings, stones, refrigerators, and all other relata are parameters of property attribution incurs the cost of a complicated account of property instantiation. The primary claim of this paper is that this is a cost that may be worth paying for a satisfactory account of differential application.

6 Conclusion

I have proposed here an account of differential application. Relative Positionalism can explain both what distinctions among different applications of a relation to fixed relata consist in (DiffApp1) and what creates different capacities for differential application in relations of the same arity (DiffApp2). Admittedly, Relative Positionalism’s account of relations makes the strong claim that individuals may have certain kinds of properties relative to other individuals, or to sequences of other individuals. In effect, Relative Positionalism requires that individuals may function as parameters (or standpoints) of truth assessment and that sequences of individuals may function as embedded parameters of truth assessment.

Such strong claims might be too high a price to pay for an account of differential application. But given that no other theory of relations proposed so far offers answers to (DiffApp1) and (DiffApp2) that generalize to relations of arbitrary symmetry structure, Relative Positionalism deserves serious consideration. It would be highly surprising if there were no general account of the distinctions among different applications of a relation to fixed relata, given that these kinds of distinctions are so easy to grasp, even in cases of unfamiliar relations. Though Relative Positionalism’s assumptions concerning relative property instantiation are complicated, they are based in the commonsense intuition that different applications of a relation to fixed relata are distinguished by the ways in which the relata are characterized relative to one another.