Differential Application and the Standard Account

It is characteristic of many relations that they may both hold (in one way) and fail to hold (in another way) of the same relata. For example,

• (lovesAE) Abelard loves Eloise may be true, while
• (lovesEA) Eloise loves Abelard

is false. Evidently, (lovesAE) and (lovesEA) manage to say something different even though they claim that the same relation holds among the same relata. (lovesAE) and (lovesEA) express what Fine calls the ‘differential application’ of a single relation to fixed relata.⁴

Not every relation can apply in different ways to fixed relata. For example,

(next_toAE) Abelard is next to Eloise (next_toEA) Eloise is next to Abelard

cannot differ in truth value. Though their terms are arranged differently, (next_- toAE) and (next_toEA) seem to say the same thing about Abelard and Eloise.

For the discussion that follows, it will be helpful to develop these types of distinctions in more general terms.

Definition of Differential Application (Def_DA) Given any n-ary relation R,^[1] any n-place relational predicate ‘R’ standing for R, any terms ‘x₁’,..., ‘x_n’ referring to objects in the domain of R, and any permutation P of {1,..., n}, the relational claims

describe distinct ways for R to hold among x₁,..., x_n iff (*) and (*_P) are nonequivalent claims. In case (*) and (*P) are non-equivalent, we will say that they express the differential application of the relation R to x₁,..., x_n.

If we assume that any application of the n-ary relation R to relata may be described using any predicate standing for R in a sentence of the form (*_P), it follows that there are at most n! ways for R to hold among fixed relata.^[2] To see this, note that there are exactly n! permutations of {1,..., n}. Thus, there are at most n! non-equivalent sentences of form (*_P) for the predicate ‘R’ and fixed terms ‘x₁’,..., ‘x„’.^[3]

Fine (2000), p. 8.

The central question for this paper is—how are we to account for the differential application of relations with various symmetry properties? In particular, how are we to account for the difference in what (lovesAE) and (lovesEA) tell us about how the loves relation applies to Abelard and Eloise? And how are we to account for the difference in the number of ways loves, as opposed to next_to, may hold between two relata?

On what Fine calls the ‘standard view’ of relations, relations hold of their relata in a particular order, or direction.⁸ The standard account explains the distinction in the content of (lovesAE) and (lovesEA) by invoking a difference in the order in which loves may apply to Abelard and Eloise. (lovesAE) claims that loves applies in one order to Abelard and Eloise—let us suppose, from Abelard (first) to Eloise (second)—while (lovesEA) claims that loves applies to Abelard and Eloise in the opposite order. An early statement of the standard account is in Russell (1903), section 94:

... it is characteristic of a relation of two terms that it proceeds, so to speak, from one to the other. This is what may be called the sense of the relation, and is, as we shall find, the

source of order and series____We may distinguish the term from which the relation

proceeds as the referent, and the term to which it proceeds as the relatum. The sense of a relation is a fundamental notion which is not capable of definition.

One immediate problem with the standard account is that it is not clear what a direction of relational application (what I will call a ‘relational ordering’) is supposed to be and what, if anything, this kind of ordering has to do with the common-sense distinction between (lovesAE) and (lovesEA) in terms of differences in the lover/beloved roles of Abelard and Eloise. Russell tells us that this notion of the sense of a relation is not capable of definition. Given that relational application is not a process which unfolds over time or across space, a relational ordering clearly cannot be a familiar ordering of temporal precedence, say, or of spatial distribution along a vector. But without some content to the notion of relational ordering—some idea of what coming first as opposed to second in an application of loves amounts to—it is hard to see how the standard account can explain the difference between what is asserted in (lovesAE) and (lovesEA). At best, it can only assert that there is a difference in the implications of these claims with respect to the relational ordering of Abelard and Eloise.⁹ explains where these numbers come from. See Fine (2000), p. 8; MacBride (2014), p. 3. I take it that what they have in mind is something along these lines.

• 8 Fine (2000), p. 1.
• 9 Numerous further problems with the standard account are discussed at length in Williamson (1985); Fine (2000); Dorr (2004); Gaskin and Hill (2012); and MacBride (2014). I think that most of

A separate problem with the standard account is that it does not apply correctly (or at all) to relations with symmetries. Before explaining this second problem, it is worthwhile to pause briefly to introduce algebraic distinctions to be used throughout the paper. S_n, the symmetric group of degree n, is the group of all permutations of {1,..., n}. A group is a set which is closed under a binary associative operation and which includes an identity element and inverses for that operation. In the case of Sn, the group operation is function composition, where the composite PQ of permutations P and Q is the permutation mapping each i e {1,..., n} to P(Q(i)). The group identity element is the identity permutation IDn and inverses are inverse permutations.

For any n-place predicate ‘R’ standing for n-ary relation R, let SYM_R (the symmetry group for ‘R’) be the set of permutations such that for any terms ‘x₁’,..., ‘x_n’ referring to objects in the domain of R,

is equivalent to

It is straightforward to verify that for any relational predicate ‘R’, SYMR must be not just a subset of S_n, but a subgroup.¹⁰ In other words, SYM_R must itself be a group of permutations included within the larger group Sn. This result is important because it narrows down the possibilities for equivalences among claims of form (*_P). Since, according to definition (Def_DA), equivalence classes of such claims represent distinctions in ways for R to hold among x₁,..., x_n, our algebraic result tells us that general distinctions in ways for R to hold have the structure of a subgroup of S_n.¹¹ these more specific criticisms are consequences of the basic problem that the standard account’s notion of relational ordering is vacuous. For example, Fine (2000) criticizes the standard account for requiring that there are redundant relations and relational states. Fine’s focus is on converse relations—relations like above and below—that, according to the standard account, differ only in the order in which they apply to relata. Note that if the different orders of relational application amounted to a real difference in the way the world is structured, converse relations and converse relational states (e.g., the cat's being above the mat vs the mat's being below the cat) would not seem to be redundant.

• 10 Since S_n is finite, to show that SYM_R is a subgroup of S_n, it suffices to show that if P, Q e SYM_R, the composite permutation PQ is also in SYM_R. That SYM_R is closed under function composition follows immediately from the fact that the equivalence relation among claims of the form (*_P) is transitive.
• 11 I assume here that if ‘R’ and ‘R*’ are distinct n-place predicates standing for the n-ary relation R, then there is a permutation P e S_n such that for any terms ‘xf,..., ‘x_n’ referring to objects in the domain of R, Rxj... x_n is equivalent to R*x_P(j)... x_P(_n). Under this assumption, the symmetry groups of any two predicates standing for R are isomorphic and so represent the same structure. Note,

Let us say that the n-ary relation R is completely symmetric iff for some n-place predicate ‘R’ standing for R, SYM_R = S_n. In this case, any permutation of the relata-terms in (*) returns an equivalent claim and there is only one way for R to hold among x₁,..., x_n. For example, next_to is completely symmetric. The symmetry group for the predicate ‘ ...is next to... ’ is S₂. Any way of permuting the two relata-terms of an ‘ ...is next to... ’ claim results in an equivalent claim.

The n-ary relation R is non-symmetric iff for some n-place predicate ‘R’ standing for R, SYM_R = {ID_n}, the one-member subgroup of S_n consisting of the identity permutation. In this case, any way of permuting the relata-terms in a claim of form (*) generally results in a non-equivalent claim—in general, there are n! distinct ways for R to hold among x₁,..., x_n. (But see the comment at the end of n. 11 in this chapter for a minor caveat.)

Between the extremes of complete symmetry and non-symmetry, there are many other possibilities for relations of arity greater than 2. S₃ has two nonisomorphic subgroups other than itself and {ID₃}. S₄ has nine non-isomorphic subgroups other than itself and {ID4}. As n increases, so do possibilities for the symmetry structures of n-ary relations. I will say that the n-ary relation R is partially symmetric iff some n-place predicate standing for R has a non-trivial subgroup of S_n (i.e., a subgroup other than S_n or {ID_n}) as its symmetry group. An example of a partially symmetric relation is the ternary between relation (holding among three things just in case one of them is between the other two). The symmetry group for the predicate ‘ ...is between... and... ’ is the two-member subgroup of S₃ consisting of ID₃ and the transposition¹² (2 3) which maps 2 to 3, 3 to 2, and 1 to itself.

The second major problem for the standard account is that relational orderings cannot explain the differential application (or lack thereof) of completely or partially symmetric relations. In fact, if the standard account implies that all relations apply to their relata in a strict linear order, then it should follow that all relations are non-symmetric. If every non-identity permutation of the relata-terms of (*) results in a claim with distinct implications about the order in which R applies to x₁,..., x„, then every such claim should describe a distinct way for R to hold among x₁..., x_n.¹³ however, that there may be fewer distinct ways for R to hold among specific x₁,..., x_n than is represented by the symmetry structure of R-predicates. For example, there is only one way for the loves relation to hold among Abelard and Abelard.

• 12 A permutation Q of the set S is a transposition iff Q maps two members of S to each other and maps every other member of S to itself. For i j, the standard algebraic notation for the transposition that swaps i and j is ‘(i j)’.
• 13 I am assuming that the implications of a claim of the form (*_P) concerning the order of relational application are completely determined in some fixed way by the order of the terms denoting the relata. Note that this does not require that (*_P) implies that R applies to x₁ first, to x₂

The proponent of the standard account could claim that differences in relational ordering do not always result in different ways for a relation to hold among fixed relata. But if this were so, it would be unclear why differences in relational ordering would ever explain differential application even in cases like that of the loves relation where the number of orderings of two relata does happen to match the number of ways the relation might apply to the relata. Alternatively, the proponent of the standard account might claim that only non-symmetric relations apply to their relata in an order or that some relations apply to their relata in orderings which are not strictly linear.

I think there are problems with such complications of the standard account. I doubt that they can provide a satisfactory account of the differential application of partially symmetric relations, at least those with a non-cyclic symmetry structure (for example, the relation that holds between four objects when two of them are exactly as far apart as the other two). More importantly, given that we have no account of what it is for any relation to apply to its relata in an order, these sorts of case-by-case distinctions among the orderings of different relations seem ad hoc. Why should some relations apply to their relata in a strict linear order, others in a cyclic order, and others in no order at all? We cannot answer these questions if we don’t know what relational orderings are supposed to be.

On Fine’s own theory of relations (which he calls ‘antipositional’), there is no structure in a relation—or in its application to relata—which can account for its differential application to fixed relata. Fine claims that there is no explanation of why some relations may apply in multiple ways to fixed relata, while other relations can apply in only one way to fixed relata.¹⁴ Fine also claims that there is no explanation of the distinction between the relational states described in (lovesAE) and (lovesEA) in terms of the intrinsic features of these states. At best we can distinguish the different ways loves applies to Abelard and Eloise externally by comparing the states described in (lovesAE) and (lovesEA) to the results of substituting Abelard and Eloise into exemplar loves states (i.e., particular instances of one person loving another).¹⁵

The problem with Fine’s account (and with the similar positions of MacBride and Gaskin and Hill¹⁶) is that it seems that there is in fact a useful and coherent second,..., and so on, just that there is some regular correlation between the place of ‘x_i’ in the claim and the order of x_i in the application of R. If this were not so, it is unclear how (*_P) could distinguish a particular order for the application of R to x₁,..., x_n.

• 14 Fine (2000), p. 19. ¹⁵ Fine (2000), pp. 20-7.
• 16 MacBride and Gaskin and Hill all ultimately endorse Fine’s claim that nothing about a relation itself explains how and why it may apply differently to fixed relata. But they reject Fine’s strategy of explaining differential application through alignments among the relata of different relational states. See Gaskin and Hill (2012) and MacBride (2014) for further criticisms of Fine’s antipositionalism.

explanation of the distinction between the claims made in (lovesAE) and (lovesEA) in terms of intrinsic features of the loves relation. This is just the intuitive explanation that there are two distinct roles, or positions, associated with the loves relation—lover and beloved—and (lovesAE) and (lovesEA) make different assignments of Abelard and Eloise to these roles. Moreover, it seems that this type of explanation can be generalized to other relations, explaining why different relations may hold in different numbers of ways among fixed relata. Unlike loves, next_to holds in only one way because it has only one type of role for its relata to play, that of standing next to the other. In the remainder of this paper, I develop an account of differential application in terms of distinctions in positions relata occupy relative to one another.

• [1] In this paper, I consider only relations of finite fixed arity. However, I see no reason why thegeneral approach proposed here cannot be modified to apply also to relations of variable arity. To doso, we need only suppose that some of the relative properties introduced in Relative Positionalism’saccount of differential application may hold relative to sequences of varying lengths.
• [2] I am assuming that any way for R to hold among xj,...,xn can be described using any predicatestanding for R.
• [3] Fine and MacBride both claim that there are at most 2 ways for a binary relation to hold amongfixed relata, at most 6 ways for a ternary relation to hold among fixed relata, and so on, but neither