## 1. Preliminary Distinctions



To understand the contemporary debate about relations we will need to
have some logical and philosophical distinctions in place. These
distinctions aren't to be taken for granted. Part of the
development of the debate has consisted in the refinement of precisely
these distinctions. But you need to understand how, relatively
speaking, things got started.



To begin let's distinguish between the "degree" or
"adicity" or "arity" of relations (see, e.g.,
Armstrong 1978b: 75). Properties are "one-place" or
"monadic" or "unary" because properties are
only exhibited by particulars or other items, e.g., properties,
individually or one by one. Relations are "many-place" or
"\(n\)-adic" or "\(n\)-ary" (where \(n\gt 1)\)
because they are exhibited by particulars only in relation to other
particulars. A "2-place" or "dyadic" or
"binary" relation is exhibited by one particular only in
relation to another. A "3-place" or "triadic"
or "ternary" relation is exhibited by one particular only
in relation to exactly two others. And so on. Some examples. The
relation that \(x\) stands to \(y\) whenever \(x\) is
*adjacent* to \(y\) is binary. This is because it takes two
things to be adjacent to one another. The relation that \(x\) stands
to \(y\) and \(z\) whenever \(x\) is *between* \(y\) and \(z\)
is ternary because it takes three things for one to fall between the
other two.



In these terms, a relation can be distinguished from a
"relational property". Whereas a relation is borne by one
thing to another thing (possibly, in certain cases, to the same
thing), a relational property is the property of bearing a relation to
something. Suppose \(a\) bears a relation \(R\) to \(b\). Then \(a\)
has the relational property of bearing \(R\) to \(b\) whilst \(b\) has
the property of \(a\)'s bearing \(R\) to it. Here \(R\) is
binary whilst the relational properties are unary because they are had
by single things, \(a\) and \(b\) respectively. For example, marriage
is a relation between two people but being married to Cleopatra is a
relational property had by Antony whilst being married to Antony is a
relational property had by Cleopatra.



A "unigrade" relation \(R\) is a relation that has a
definite degree or adicity: \(R\) is either binary or ternary...
or \(n\)-ary (for some unique \(n\)). By contrast a relation is
"multigrade" if it fails to be unigrade (the expression is
owed to Leonard & Goodman 1940: 50). Putative examples include
causal, biological, physical, geometrical, intentional and logical
relations. Causation appears to be multigrade because a certain number
of events may be required to bring about an effect on one occasion,
and a different number of events may be required to bring about an
effect on another. Similarly, entailment appears to be a multigrade
relation because a certain number of premises may be required to
entail one conclusion but a different number of premises to entail
another. Whether there are any multigrade relations and exactly what
it means to be multigrade is a fraught matter (see Armstrong 1978b:
93-4, 1997: 85, 2010: 23-5 and Dixon 2019: 64-7 for
arguments against multigrade relations and Morton 1975, Fine 2000: 22,
MacBride 2005: 569-93 and Hossack 2007: 45-55 for
arguments in favour of them).



Next, let's draw a preliminary three-fold distinction between
binary relations in terms of their behaviour with respect to the
things they relate. A binary relation \(R\) is *symmetric* iff
whenever \(x\) bears \(R\) to \(y\), \(y\) bears \(R\) to \(x\). By
contrast, \(R\) is *non-symmetric* iff \(R\) fails to be
symmetric. *Asymmetric* relations are a special case of
non-symmetric ones: R is asymmetric iff whenever \(x\) bears \(R\) to
\(y\) then \(y\) does not bear \(R\) to \(x\). So whilst all
asymmetric relations are non-symmetric, not all non-symmetric
relations are asymmetric. Some examples. Whenever \(x\) is married to
\(y\), \(y\) is married to \(x\). So the relation in which \(x\)
stands to \(y\) when \(x\) is married to \(y\) is a symmetric
relation. But if \(x\) loves \(y\), it isn't guaranteed that
\(y\) loves \(x\). Because, alas, sometimes love is unrequited. So the
relation in which \(x\) stands to \(y\) when \(x\) loves \(y\) is a
non-symmetric relation. But love doesn't have to go unrequited
otherwise far fewer people would get married, only cynics and gold
diggers. So this relation isn't asymmetric. By contrast, the
relation in which \(x\) stands to \(y\) when \(x\) is taller than
\(y\) is asymmetric because if \(x\) is taller than \(y\) then \(y\)
isn't taller than \(x\). This doesn't exhaust the logical
classification of relations. It's easy to see that further
distinctions will need be drawn. Consider, for example, the
*between* relation. It's a ternary relation that is
symmetric in the sense that if \(x\) is between \(y\) and \(z\) then
\(x\) is between \(z\) and \(y\). But the *between* relation
isn't *completely* or *strictly* symmetric because
it's not the case that if \(x\) is between \(y\) and \(z\) then
\(y\) is between \(x\) and \(z\). So the distinction between symmetric
and non-symmetric relations for \(n\)-ary relations, where \(n \gt
2\), will need to be qualified. (In what follows we restrict ourselves
to the more straightforward cases discussed in the philosophical
literature on relations. But see Russell 1919: 41-52 for a more
wide-ranging discussion of the logical variety of relations.)



Nevertheless, equipped with this three-fold distinction for binary
distinction we can now appreciate the significance of one contemporary
debate about relations. Our ordinary and scientific views of the world
are replete with descriptions of things that are non-symmetrically or
asymmetrically related. For this reason, Russell advocating admitting
both non-symmetric and asymmetric relations alongside symmetric ones
(Russell 1903: SSSS212-6, 1914: 58-9, 1924: 339).
But, more recently, it has been questioned whether it is necessary to
admit non-symmetric and asymmetric relations at all, i.e., it has been
argued that the world contains only symmetric relations by Armstrong
1997: 143-4; Dorr 2004: 180-7; Simons 2010: 207-1,
but for counter-arguments see MacBride 2015: 178-94.



Now non-symmetric relations, including asymmetric ones, are typically
described as imposing an order upon the things they relate; they admit
of what is also called *differential application* (Fine 2000:
8). This means that for any such non-symmetric \(R\) there are
multiple different ways in which \(R\) potentially applies to the
things it relates. Why? If the way that *loves* applies to
\(a\) and \(b\) when \(a\) loves \(b\) were no different from the way
that *loves* applies to them when \(b\) loves \(a\), then the
*loves* relation couldn't be non-symmetric, because
otherwise \(a\) couldn't love \(b\) without \(b\)'s loving
\(a\). So we have to distinguish between the two different ways that
*loves* is capable of applying to \(a\) and \(b\). Similarly we
can distinguish the six different ways that the ternary
*between* relation potentially applies to three things. By
contrast, we're not required by the same reasoning to
distinguish between the different ways in which a (strictly) symmetric
relation applies to the things it relates. There's no
corresponding necessity to distinguish the way that the
*adjacency* relation applies to \(a\) and \(b\) when \(a\) is
adjacent to \(b\) from the way that the adjacency relation applies to
them when \(b\) is adjacent to \(a\) because \(a\) cannot be adjacent
to \(b\) without \(b\) being adjacent to \(a\).



Putting the ideas of the last three paragraphs together, philosophers
often seek to capture what is distinctive about relations by
describing them in the following manner. Unlike properties, binary
relations aren't exhibited by particulars but by *pairs*
of particulars. Similarly ternary universals aren't exhibited by
particulars but by *triples* of particulars, etc. But, we have
seen, non-symmetric relations, including asymmetrical ones, are
order-sensitive. So the pairs, triples, etc., of objects that exhibit
these relations must be themselves ordered. It is only ordered pairs,
ordered triples, more generally, \(n\)-tuples or sequences that
exhibit relations (Kim 1973: 222; Chisholm 1996: 53; Loux 1998:
23).



There are a number of reasons we should be wary of characterising
relations in such quasi-mathematical terms. First, sequences
presuppose order, the difference between a thing's coming first
and thing's coming second, so cannot be used to explain order
without our going around in a tight circle. Second, it's
problematic to try to explain a relation between two things in terms
of another thing, a sequence, which is those two things put together
in an order (MacBride 2005: 590-2). If a sequence \(s\)
(\(=\langle x, y \rangle\)) is conceived as a "one" such
that a relation \(R\) is really only a monadic feature of it,
\(R(s)\), then it's mysterious what \(R\)'s holding of
\(s\) has to do with \(x\) and \(y\) being arranged one way rather
than another with respect to \(R\). If you don't see the mystery
straightaway, reflect that \(s\) may have a monadic feature \(F\),
which is, so to speak, only to do with \(s\) itself rather than its
members. But if \(R\) and \(F\) are conceived alike, as monadic
features of \(s\), isn't it mysterious that the possession by
\(s\) of \(R\) has consequences for \(x\) and \(y\) whereas \(F\)
doesn't? If, alternatively, a sequence is conceived as a
"many", just \(x\) and \(y\), such that \(R\) applies to
them directly, then it's unclear what the appeal to sequences
has achieved.



We cannot avoid the difficulties associated with sequences by
appealing to the Kuratowski definition of sequences in terms of
unordered sets (where, e.g., \(\langle x, y\rangle = \{\{x\}, \{x,
y\}\}\)) (or Bergmann's Kuratowski inspired account of order in
terms of 'diads', see his 1981: 147). There are
indefinitely many other set-theoretic constructions upon which we may
rely to model sequences, e.g., Wiener's, hence the familiar
objection that we cannot legitimately fix upon any one such
construction as revelatory of the nature of sequences (see Kitcher
(1978: 125-6) generalising upon a point originally made by
Benacerraf (1965: 54-62)). But there is also the less familiar
objection that the Kuratowski definition cannot be used to analyse the
order inherent in the sequence \(\langle x, y\rangle\). Compare: we
cannot analyse personal identity in terms of memory because it's
built into the what we mean by memory that you can only remember
*your own* experiences, i.e., the very notion that we set out
to explain, viz. personal identity, is presupposed by our analysans.
Similarly \(\{\{x\}, \{x, y\}\}\) only serves as a model of \(\langle
x, y\rangle\) relative to the assumption that the thing that belongs
to a singleton member, namely \(\{x\}\), of the first class *comes
first* in the pair \(\langle x, y\rangle\), so the notion of order
is presupposed (Hochberg 1981, 2001: 176-7).



The next idea that needs to be introduced is the idea of a
*converse* relation. For any given binary relation \(R\), the
converse of \(R\) may be defined as the relation \(R^\*\) that \(x\)
bears to \(y\) whenever \(y\) bears \(R\) to \(x\) (Fine 2000: 3; van
Inwagen 2006: 459). Note that the relationship between a relation and
its converse isn't a matter of happenstance; there's no
possibility of a converse of \(R\) floating free from \(R\) and
holding between things independently of how \(R\) arranges them.
Rather it's a more intimate relationship which, if it holds,
holds of necessity. For \(x\) to have one to \(y\) is, so to speak,
for \(y\) to have the other to \(x\); they neither exist nor can be
observed apart from one another but can only be distinguished in
thought (Geach 1957: 33). *Before* and *after*,
*above* and *below* are *prima facie* examples of
mutually converse relations. Non-symmetric relations, including
asymmetric ones, are distinct from their converses (if they have
them). Suppose \(R\) is a non-symmetric relation which \(x\) bears to
\(y.\) Then the converse of \(R\) is borne by \(y\) to \(x\). But
suppose that \(R\), as it may, fails to be borne by \(y\) to \(x\).
Then something is true of the converse of \(R\) that isn't true
of \(R\) itself, *viz.* that it is borne by \(y\) to \(x\).
Hence, the converse of \(R\) must be distinct from \(R\). More
generally, whilst a binary non-symmetric relation has only one
converse, a ternary one has five mutual, distinct converses, a
quadratic relation has 23 converses, etc. The issue of whether
relations have converses is another issue to which we will return
later.



The next distinction we will need, or more accurately, family of
distinctions, is between *internal and external relations*.
What makes them members of a single family is that a relation is
internal if its holding between things is somehow fixed by the things
themselves or how those things are; external relations are relations
whose holding between things isn't fixed this way. Different
versions of the internal-external distinction correspond to different
explanations of how internal relations are fixed (Ewing 1934:
118-36; Dunn 1990: 188-192; Hakkarainen, Keinanen
& Keskinen, 2019). We don't need to describe every version
of the distinction but here are three that are essential to
understanding the contemporary debate.



The first version of the distinction is owed to Moore. According to
Moore's (1919: 47), a binary relation \(R\) is internal iff if
\(x\) bears \(R\) to \(y\) then \(x\) does so necessarily. From the
internality of \(R\), in Moore's sense, it follows that if \(x\)
exists at all then \(x\) indeed bears \(R\) to \(y\). But if
it's possible that \(x\) exist whilst failing to bear \(R\) to
\(y\), then \(R\) is external. The doctrine of the necessity of origin
provides one putative example of a relation that's internal in
Moore's sense. If you essentially come from your biological
parents then you could not have existed whilst failing to be their
offspring, i.e., the relation of *biologically originating
from* must be internal in Moore's sense.



The second version of the internal-external distinction is favoured by
Armstrong (1978b: 84-5, 1989: 43, 1997: 87-9, van Inwagen
1993: 33-4). According to Armstrong, a relation \(R\) is
internal iff it's holding between \(x\) and \(y\) is
necessitated by the intrinsic natures, i.e., non-relational
properties, of \(x\) and \(y\); otherwise \(R\) is external. The third
version is owed to Lewis. Lewis (1986: 62) advanced the view that an
internal relation is one that supervenes upon the intrinsic natures of
its relata. But Lewis's definition of "external" is
more involved: \(R\) is external iff (1) it fails to supervene upon
the nature of the relata taken separately, but (2) it does supervene
on the nature of the composite of the relata taken together.



Suppose that \(x\) is a cube and \(y\) is also a cube. It follows that
\(x\) is the same shape as \(y.\) So the relation that \(x\) bears to
\(y\) when \(x\) is the same shape as \(y\) is internal in
Armstrong's sense. By contrast, spatio-temporal relations are
external (in his sense) because the intrinsic characteristics of \(x\)
and \(y\) don't necessitate how close or far apart \(x\) and
\(y\) are. Lewis intends his distinction to classify the same way. If
\(x\) is the same shape as \(y\), but \(w\) is not the same shape as
z, then there must be some difference in intrinsic nature either
between \(x\) and \(w\) or else between \(y\) and \(z\). So the
relation that \(x\) bears to \(y\) when \(x\) is the same shape as
\(y\) is internal in Lewis's sense. Lewis also claims that the
distance between \(x\) and \(y\) is an external relation because the
following conditions are met: (1) \(x\) may be closer to \(y\) than
\(w\) is to \(z\) even though \(x\) is an intrinsic duplicate of
\(w\), i.e., shares all and only the same intrinsic characteristics,
and \(y\) an intrinsic duplicate of \(z\); (2) the distance between
\(x\) and \(y\) does supervene upon the nature of the composite
\(x+y\). (We'll return in
 section 3
 to the question whether, as Lewis claims, distance relation supervene
upon the nature of composites.) But the relation that \(x\) bears to
\(y\) when \(x\) is the same shape as \(y\) isn't internal in
Moore's sense because, let us suppose, \(x\) might have been
spherical whilst \(y\) remained a cube; so it doesn't follow
from the mere fact that \(x\) exists that \(x\) is the same shape as
\(y\).



The final distinction, or rather, collection of distinctions, which
we'll introduce here and which needs to be borne in mind when
considering the recent debate on relations, concerns the existence and
identity conditions for relations. Since Lewis it has become
commonplace to conceive relations as either "abundant" or
"sparse": according to an abundant conception there is a
relation for every arbitrary pair (or triple, or ...) of things
whereas according to a sparse conception there are only as many
relations as are required to characterise the relational aspects of
likeness and difference (Lewis 1986: 61). But this binary distinction
doesn't do justice to the variety of different conceptions of
the existence and identity of relations that have been put forward.
Distinguish, on the one hand, an *a posterori* view according
to which relations exist only if they are instantiated and whether
they are instantiated is an empirical matter (Armstrong 1978a:
76-77) from, on the other hand, an *a priori* view
according to which their existence is settled by purely logical
principles, for example strong comprehension principles whereby every
open formula determines a corresponding property or relation (Zalta
1988: 46-7; Williamson 2013: 226-7). Whereas the former
view might admit space-time separation as a relation (or family of
relations) whose instantiation is confirmed by physics, the latter
would affirm the existence of a relation corresponding to the open
formula "*x* is a much funnier anarchist than
*y*". Between these two extremes are views that allow
some relations, for example, relative velocities conceived as
relations between different things when one has a certain velocity
*relative* to another, to be uninstantiated for the purposes of
rounding out scientific theory (Bigelow and Pargetter 1990: 77) and
views that allow logically complex relations (RvS, R&S
*etc*.) insofar as they are built up by logical operations from
simple relations (R, S *etc*.) found in nature (Bealer 1982:
9-10).



There is a considerable variety of views concerning the identity
conditions for relations. Distinguish, for example, a
"coarse-grained" conception, according to which relations
are identical just in case they are necessarily equivalent,
*i.e.* identical just in case, as a matter of necessity, they
are exhibited by the same pairs, triples... of things, from a
"fine-grained" conception, according to which relations
are identical just in case they have the same analysis in terms of
simpler relations (Bealer 1982: 2-3, Menzel 1993: 62-4).
Another approach conceives relations as the same just in case they
confer the same causal powers on the pairs, triples... of things
related by them. A further approach to relation identity says that,
for example, the dyadic relations *R* and *S* are
identical if their corresponding relational properties (bearing
*R* to *a*, bearing *S* to *a*
*etc.*) are identical, where properties in general (which,
recall, are unary) are identical just in case, according to this view,
they are "encodings" of the same "abstract
object". The sense of abstract object here is a specialised one:
roughly speaking, an abstract object encodes a property by
encapsulating what it is to exhibit the property in question (Zalta
1988: 51-52). Even though these different conceptions of the
existence and identity conditions for relations yield different
results in different cases it is important to also bear in mind that
they need not invariably be seen as competing. It may be that they
provide different notions of relation relevant to different purposes,
for example, whether we are interested in relations because of their
role in causation, semantics, intentionality and so on (Lewis 1986:
55-59).


## 2. Eliminativism, External Relations and Bradley's Regress



Some philosophers are wary of admitting relations because they are
difficult to locate. Glasgow is west of Edinburgh. This tells us
something about the locations of these two cities. But where is the
relation that holds between them in virtue of which Glasgow is west of
Edinburgh? The relation can't be in one city at the expense of
the other, nor in each of them taken separately, since then we lose
sight of the fact that the relation holds between *them*
(McTaggart 1920: SS80). Rather the relation must somehow share the
divided locations of Glasgow and Edinburgh without itself being
divided. This may sound peculiar if we assume that middle-sized things
like cities, which have locations in a relatively straightforward
sense, set a standard for how entities in general ought to behave. But
why shouldn't we admit other kinds of entity that have a
different kind of relationship to space and time from such readily
located things. Or even go down the route of conceiving relations as
abstract, i.e., as "Nowhere and nowhen" (Russell 1912:
55-6). As Lewis reflected, "if the price is right, we
could learn to tolerate it" (Lewis 1986: 68). But is the price
right? Do the theoretical benefits of admitting relations outweigh the
cost of offending pre-theoretic intuition?



Other philosophers have been wary of admitting relations because, they
complain, relations are unsatisfactorily characterised, "neither
fish nor fowl", i.e., neither substances nor attributes (Heil
2012: 141, 2016: 130) or "ontologically weird" (Lowe 2016:
111). Now it's certainly true that \(n\)-adic relations, where
\(n\gt 1\), cannot be characterised as either substances or (monadic)
attributes. But it only follows that relations are unsatisfactorily
characterised if we assume that substances and attributes provide the
benchmark for satisfactory characterisation. But we shouldn't
assume this ahead of investigating whether the theoretical benefits of
admitting relations outweigh the costs. It's also been
complained that relations are suspect because they *depend*
upon the existence of substances that bear them (Campbell 1990:
108-9) or, as Lowe makes the point, "A 'relational
accident', if there could be such a thing would not be
'in', to at least not be wholly 'in', any of
its two or more 'subjects', nor even wholly
'in' the totality of them. I consequently find it hard to
conceive what such an entity could really be" (Lowe 2016:
111-2). Of course it's true that in order for a relation
to be borne by one thing to another thing, then those things must
exist. But it doesn't follow that relations don't exist if
nothing bears them. Moreover, it doesn't follow straightaway
from the reflection that to be borne by something a relation needs
things to bear it, that relations are suspect. It doesn't follow
any more than it follows from the reflection that to be had by
something an attribute needs things to have it, that attributes are
suspect. If the price is right we should open our minds to the
possibility of things which are neither fish nor fowl but vegetables,
i.e., neither substances nor attributes but relations.



In light of the internal/external distinction introduced above, we can
distinguish two distinct questions here: (a) should we acknowledge
external relations? (b) should we acknowledge internal relations? In
the present section our focus will be upon whether to accept or reject
external relations, before turning to internal relations in the
next.



F.H. Bradley regarded himself the nemesis of external relations, but
not only them. Famously, Bradley brought a vicious regress argument
against external relations. In his original version (1893:
32-3), Bradley presented a dilemma to show that external
relations are unintelligible. Here's the dilemma. Either a
relation \(R\) is *nothing* to the things \(a\) and \(b\) it
relates, in which case it cannot relate them. Or, it is
*something* to them, in which case \(R\) must be related to
them. But for \(R\) to be related to \(a\) and \(b\) there must be not
only \(R\) and the things it relates, but also a subsidiary relation
\(R'\) to relate \(R\) to them. But now the same problem arises with
regard to \(R'\). It must be something to \(R\) and the things it
related in order for \(R'\) to relate \(R\) to them and this requires
a further subsidiary relation \(R''\) between \(R'\), \(R\), \(a\) and
\(b\). But positing more relations to fix up the problem is only
throwing good money after bad. We fall into an infinite regress
because the same reasoning applies equally to \(R'\) and however many
other subsidiary relations we subsequently introduce--this is
commonly called "Bradley"s Regress'. As Bradley
later summed up,



> 
> 
> "while we keep to our terms and relation as external, no
> introduction of a third factor could help us to anything better than
> an endless renewal of our failure" (Bradley 1935: 643; van
> Inwagen 1993: 35-6).
> 
> 
> 



So whichever horn of Bradley's dilemma we choose, whether a
relation \(R\) is nothing or something to the things it relates, we
cannot make sense of \(R\) relating them. Bradley concluded that we
should eliminate external relations from our ontology.



It is often claimed that Bradley's eliminativism may be
addressed by positing a special category of entities, either (1) facts
(Hochberg 1978: 338-40; Armstrong 1989: 109-110, 1997:
118-9; Hossack 2007: 41-5) or (2) tropes (Maurin 2010:
321-3; Simons 2010: 201-3). The strategic idea they share
is that we can embrace the second horn of Bradley's dilemma
without being stymied by his regress argument. Bradley assumed
external relations to be *general* entities, i.e., universals,
capable of relating different things from whatever things they happen
to actually relate. It follows from this assumption that the mere
existence of \(R\) and the things it relates cannot suffice for \(R\)
to relate them, because \(R\) might have been engaged elsewhere
relating other things, and this problem will persist however many
subsidiary (universal) \(R\)s we introduce. But, it is argued, this
problem can be avoided by either positing facts or tropes, which are
particular rather than general entities. Both of these purported
solutions are open to question. It seems that they presuppose what
they set out to show: the capacity of relations to relate (MacBride
2011: 168-72). Why is this?



First, take (1): Hochberg, Armstrong and Hossack argue that the fact
that \(a\Rel b\) does indeed suffice for \(R\) to hold between \(a\)
and \(b\), because the fact that \(a\Rel b\) cannot exist unless \(R\)
does hold between \(a\) and \(b\). So once the fact that \(a\Rel b\)
has been admitted into our ontology, the later stages of
Bradley's regress either don't arise or they harmlessly
supervene upon the admission of this fact. But, MacBride counters,
this presupposes that Bradley's argument has already been
disarmed. Ask yourself: what is a fact? What distinguishes the fact
that \(a\Rel b\) from the mere plurality of its constituents: \(R\),
\(a\), \(b\)? Just this: that the constituents of the fact are
*related.* But Bradley's argument is intended to
establish that we cannot understand how it is possible for things to
be *related*. So unless we have already established where
Bradley's argument goes wrong, we cannot appeal to the existence
of facts because facts are cast into doubt by his argument too.



Next, take (2). Maurin and Simons argue that if we reject
Bradley's assumption that relations are general entities, i.e.
universals, and hold instead that relations are a special class of
particular entities, i.e., non-transferable tropes, then we can avoid
Bradley's Regress. What is a relation conceived as a
"non-transferable trope"? It's a relation \(r\_1\)
that's borne essentially by the things it relates and which
could not have been borne by anything else (Simons 2002/3: 6). Suppose
\(r\_1\) is borne by \(a\) to \(b\). Then \(r\_1\) is a trope in the
sense that it is non-repeatable and so is not a "one-over
many", not a universal. And \(r\_1\) is non-transferable in the
sense that it could not be transferred from one pair of particulars to
another, i.e., \(r\_1\) could not be borne by \(c\) to \(d\) where \(c
\ne a\) and \(d \ne b\). So, necessarily, if \(r\_1\) exists then (i)
\(a\) and \(b\) exist and (ii) \(a\mathbin{r\_1} b\). So in virtue of
the manner in which \(r\_1\) depends upon the things it relates it
follows that the mere existence of \(r\_1\) accounts for \(a\) and
\(b\)'s being related without having to go down the route of
appealing to subsidiary relations to glue them together. But again,
this proposal assumes that relations really do relate, and therefore
that Bradley's regress has already been disarmed. By way of
analogy, consider the Problem of Evil: the difficulty of theoretically
squaring God's omnipotence, omniscience, omnibenevolence, etc.,
with the (apparent) fact of unnecessary evil in the world. Of course
if God exists then there cannot be any unnecessary evil, i.e., there
must be a solution to the Problem of Evil. But we don't provide
a theoretical solution to the Problem by simply positing God. That
doesn't make any easier the theoretical task of squaring what
appears to be unnecessary evil with the nature of an omnipotent,
omniscient, omnibenevolent deity. Similarly we don't provide an
effective theoretical response to Bradley's argument by simply
positing relational tropes. Positing them doesn't provide an
explanation of how a relation can succeed in relating some things
without requiring to be related by a further relation to them.



Bradley's problem with relations would be resolved if we could
achieve a perspective whereby we appreciated that the bearing of a
relation isn't itself a relation or require a further relation,
whatever grammar may suggest (Armstrong 1978a: 106-111; Grossman
1983: 167-8; Lewis 2002: 6-7). But we won't achieve
that perspective until we can provide a diagnosis of Bradley's
reasoning that relieves us of the temptation to think that the bearing
of a relation is itself a relation.



Bertrand Russell dismissed Bradley's argument on the grounds
that philosophers who disbelieve in the reality of external relations
cannot possibly interpret those numerous parts of sciences which
appear to be committed to external relations (Russell 1924: 339).
Russell's argument bears comparison to Quine's famous
indispensability argument for mathematics: just as we cannot make
sense of scientific discourse without taking seriously those portions
irredeemably committed to, e.g., numbers, we cannot make sense of
scientific discourse without taking seriously those portions committed
to, e.g., spatio-temporal relations (Quine 1981: 149-50). From a
methodological point of view, Russell deemed it more likely that an
error is concealed somewhere in Bradley's argument than that
modern science should have incorporated so fundamental a falsehood,
i.e., portrayed a world of spatio-temporal relations when really there
are none. Of course even if we have good or even overwhelming reason
to think that Bradley's argument must be wrong, this
doesn't relieve us of the philosophical responsibility of
determining where exactly Bradley slipped up.


## 3. Reductionism about Internal Relations



What about internal relations? Do we need to acknowledge their
existence? We distinguished 3 senses of "internal
relation" in our preliminary discussion: internal relations are
determined by the mere existence of the things they relate, or
internal relations are determined by the intrinsic characters of the
things they relate, or internal relations supervene upon the intrinsic
characters of the things they relate. So, in one sense or another, it
suffices for the obtaining of an internal relation that either the
things it relates exist or that the things it relates have
such-and-such intrinsic characters. On this basis, some philosophers
have concluded that there is no need to admit internal relations as
additional pieces of furniture in the Universe (Fisk 1972:
146-9; Armstrong 1978b: 86; Campbell 1990: 99-101; Simons
2010: 204-5; 2014: 314-5; Lowe 2013: 242, 2016:
105-6; Heil 2009: 316-7, 2012: 144-6, 2016: 130,
Lowe 2016:102-11, Keinanen 2020). Why do they think this?
Truths about internal relations aren't necessarily equivalent to
truths about the things they relate or the intrinsic characters of the
things they relate. Nevertheless, all the truths about internal
relations are determined by the existence or the intrinsic characters
of the things they relate, or supervene upon them. So internal
relations are variously declared to be "no addition of
being" (Armstrong) or "absent from the fundamental
ontology" (Heil) or, more clearly, "there are no such
things" (Simons).



Consider an example:


1. Ben Vorlich, a Scottish mountain, is taller than Ben Vane, its
neighbour.
2. Ben Vorlich is 3094 ft high whilst Ben Vane is 3002 ft.



(1) and (2) aren't necessarily equivalent because Ben Vorlich
might have been taller than Ben Vane even if Ben Vorlich and Ben Vane
had had different heights from the ones they actually have, i.e., (2)
doesn't follow from (1). Nevertheless, Ben Vorlich's being
taller than Ben Vane is determined by its having the height it does
(3094 ft) and Ben Vane's having the height it does (3002 ft); it
isn't possible for these mountains to have the heights they do
whilst Ben Vorlich fail to be taller than Ben Vane. So the *being
taller than* relation is internal in Armstrong's sense. This
relation is also internal in Lewis's sense which, recall, relies
upon comparison between pairs of a relation's relata (Lewis
1986: 62). Ben Lomond doesn't stand to Ben More as Ben Vorlich
does to Ben Vane, because Ben Lomond is 3196 ft, Ben More, 3852 ft.
Hence Ben Vorlich's being taller than Ben Vane supervenes, in
Lewis's sense, upon the heights of Ben Vorlich and Ben Vane. So
once Ben Vorlich and Ben Vane have been bestowed with their respective
heights there is no need to admit a further (internal) relation to
account for Ben Vorlich's being taller than Ben Vane (Campbell
1990: 100, 103; Lewis 1994: 294).



This, broadly speaking, reductionist argument is open to question in a
variety of respects. Some philosophers have maintained that it cannot
be effective because we literally perceive proportions and other
internal relations (Mulligan 1991; Hochberg 2013: 232). These
relations must exist, otherwise we couldn't perceive them. So,
they conclude, something must have gone wrong with this argument
because it leads to the mistaken conclusion that there aren't
any internal relations out there to be perceived.



More generally, this reductionist argument assumes that if an internal
relation \(R\) isn't required to perform the role of determining
that \(R\) holds between the things it relates, because that role has
already been discharged by the intrinsic characteristics of the things
it relates, then \(R\) doesn't exist at all or doesn't
constitute "an addition to the world's furniture"
(Armstrong 1997: 87). But from the fact that \(R\) doesn't
fulfill one role it doesn't follow that it isn't required
to fulfill another. It doesn't follow because there may still be
other good reasons for us to believe that internal relations exist, or
constitute "an addition to the world's
furniture".



To bring the issues here into focus we will need to take on board two
further distinctions. First, the distinction between
*truth-making* and *ontological commitment*. A
truth-maker for a statement \(S\) is (at least) something the
existence of which determines (necessitates) that \(S\) is true
(Armstrong 2004: 5-7). By contrast, something \(x\) is an
ontological commitment of \(S\) if, roughly speaking, \(S\) could not
be true unless \(x\) existed (Quine 169: 95). Typically, a statement
\(S\) incurs ontological commitment to an entity or some entities
because we refer to it or quantify over them when we make the
statement. It's easy to run *truth-making* and
*ontological commitment* together but it's important to
keep them separate. A statement \(S\) cannot be true unless the
entities to which \(S\) is ontologically committed exist. So
it's a *necessary* condition of the truth of \(S\) that
the entities to which \(S\) is ontologically committed exist. By
contrast, it's a *sufficient* condition for the truth of
\(S\) that a truth-maker for \(S\) exist because a truth-maker for
\(S\) determines that \(S\) is true.



Armed with the distinction between truth-making and ontological
commitment, we are now able to state a very general shortcoming of the
argument. Let it be agreed that Ben Vorlich's being the height
it is and Ben Vane's being the height it is together make true
the claim that Ben Vorlich is taller than Ben Vane; so no further
truth-makers are required for this claim to be true. But it
doesn't follow that we aren't ontologically committed to
the existence of an internal relation between Ben Vorlich and Ben Vane
when we make this claim. We surely are ontologically committed to Ben
Vorlich and Ben Vane when we make the claim that one is taller than
the other even though they're not themselves truth-makers for
this claim. They're not truth-makers for this claim because they
might have existed even though Ben Vane was taller than Ben Vorlich or
the same height. But recognising that the class of ontological
commitments and the class of truth-makers for a given claim may be
different classes opens up the possibility that we may be
ontologically committed to an internal relation between Ben Vorlich
and Ben Vane even though this relation isn't a truthmaker for
this claim either (MacBride 2011: 162-6).



The second distinction we need is between *thick* and
*thin* relations (Mulligan 1998: 342-7). Some examples.
Thick relations: *loves*, *kills*, *gives*. Thin
relations: *identity*, *resemblance*, *greater
than*. Thick relations have more of a "material"
content and are less "formal" than thin relations. It
correspondingly makes more sense to think of thick relations as
"topic-neutral". Thin relations, by contrast to thick
relations, are typically internal (see Clementz 2014 and Johansson
2014 for further discussion of the thick/thin distinction).



We are now equipped state another, more specific gap in the
reductionist argument. Whilst Ben Vorlich and Ben Vane's having
the heights they do avoids the need to admit the existence of a thick
(internal) relation that Ben Vorlich bears to Ben Vane, this
doesn't avoid the need to admit the existence of a thin
(internal) relation. This is because we need to explain why Ben
Vorlich's having the height it does (3094 ft) and Ben
Vane's having the height it does (3004 ft) entails that Ben
Vorlich is taller than Ben Vane. The answer is that the thin relation
*being greater than* holds between the number of feet in height
that Ben Vorlich has and the number of feet in height that Ben Vane
has (Russell 1903: SS214; Bigelow & Pargetter 1990:
55-6. So internal relations may not (in general) be required to
perform the role of truth-makers for comparative claims. Nonetheless
thin internal relations may perform an indispensable role in
explaining why the truth-makers for comparative claims are equipped to
make such claims true, rather than leaving this a matter of brute
inexplicable necessity--because, in this case, the heights of the
mountains in question lie in a certain relation of proportion to one
another. Similarly Ben Vorlich's being taller than Ben
Vane's supervenes upon their heights but this is only
intelligible because the height of Ben Vorlich is larger than the
height of Ben Vane.



This leaves us a choice. Either we can rewind the argument and admit
internal relations, whether thick or thin, as additions to the
world's furniture (Bigelow & Pargetter 1990: 56-60).
Or we can adopt the more austere view that only admits thin (internal)
relations to hold amongst intrinsic characteristics (Mulligan 1998:
347).



The situation may be summed up in schematic terms. There are true
comparative claims of the form \(a\Rel b\) which are either determined
to be true by, or supervene upon, the intrinsic natures of the things
they relate, \(Fa \amp Gb\). Philosophers of a reductionist persuasion
have argued that this shows there is no need to admit an internal
relation \((R)\) over and above the intrinsic characteristics or
monadic foundations of the things related \((F, G)\). But the
intrinsic characteristics of the things related only determine the
truth of the claim that \(a\Rel b\), or the truth of the claim that
\(a\Rel b\) only supervenes upon \(Fa \amp Gb\), because these
characteristics stand in an internal relation themselves, \(F \fR G\).
Because the monadic foundations, \(Fa\) and \(Gb\), are only empowered
to determine that \(a\Rel b\) because \(F\) and \(G\) lie in this
internal relation, \(\fR\), it follows that internal relations still
perform an indispensable role in our theorising about the world. One
line of response to this argument is to deny that the ontological
commitments and the truth-makers for a claim might come apart by
defining ontological commitment in terms of truth making, by roughly
speaking, maintaining that to be is to be a truth-maker, but then an
alternative theory of truth will need to be provided for ordinary and
formal languages if quantificational structure is never ontologically
committing and it unclear whether this can be done (Heil 2016: 129;
MacBride 2018: 157).



The case against reductionism about internal relations has proceeded
so far from a relatively *a priori* basis. But there's
also a case to be made against it on *a posteriori* grounds.
Grant for the sake of argument the claim that if a relation is
internal then it doesn't count as an ontological addition. The
significance of this claim, when it comes to providing an inventory of
the world's furniture, depends upon the extent to which
relational truths have monadic foundations. But relativity theory,
quantum theory and even classical mechanics provide an *a
posteriori* basis for doubting that there are very many, if any,
\(F\)s and \(G\)s capable of serving as monadic foundations for target
\(R\)s, i.e., intrinsic properties to which a significant range of
relations may be reduced.



It is often taken for granted by metaphysicians that the fundamental
quantities of physics provide a ready source of monadic foundations
because physical quantities are intrinsic characteristics. The vision
of reality metaphysicians typically ascribe to physics is akin to
Lewis's doctrine of "Humean Supervenience",
according to which there are intrinsic properties which need nothing
bigger than a point to be instantiated and there's no difference
without a difference in the arrangement of intrinsic properties over
points (Lewis 1986: ix-xvi). And certainly some metaphysicians
have endeavoured to make good upon this vision, arguing, for example,
that velocity is an intrinsic property (Tooley 1988; Bigelow &
Pargetter 1990: 62-82). But not only is this view contentious,
there are many other notions in classical mechanics, such as force,
stress, strain and elasticity, represented using vectors and tensors,
that belie the conception of fundamental physical quantities as
intrinsic (Butterfield 2006, 2011).



When we turn from classical mechanics to relativity theory and quantum
theory, the *a posteriori* case against conceiving physical
quantities as intrinsic is even stronger. Consider mass and charge,
most metaphysicians' favourite and commonly relied upon examples
of intrinsic properties of a piece of matter (in particular a point
particle). In relativity theory, mass is identified with energy
("mass-energy"). In a relativistic field theory, such as
Maxwell's theory of electromagnetism, the all-pervading
electromagnetic field has mass-energy. It also has momentum, stress
and other traditionally mechanical properties. So mass is no longer
conceived as an intrinsic property of localized lumps of stuff, let
alone of point particles. Furthermore, the attribution of mass,
momentum, stress, etc., to the electromagnetic field, or to a matter
field such as a continuous fluid, depends upon the metrical structure
of spacetime. So these quantities are not intrinsic, but, rather,
relational properties. They are properties that the radiation or
matter field in question has in virtue of its relation to spacetime
structure (Lehmkuhl 2011).



In quantum theory there are even fewer candidates for intrinsic
properties. Even in elementary quantum mechanics, the closest analogue
of a point particle as classically understood ("a quantum
particle") is an all-pervading field, *viz*. an
assignment, for each instant of time, of a (complex) number to every
point of space. Here, the mass of the quantum particle cannot be
attributed to any single point of space, but must be associated with
the whole field, although not in the simple manner of "spreading
the mass out" as a classical mass density distribution. (The
same remark applies to the quantum particle's charge.) Turning
to quantum field theory: here again, the notion of particle is
radically changed from the idea of a classical point particle
*and* from the idea of a quantum particle. There are not
various quantum particles, each of them represented by a
complex-valued field as above, and each of them, say, electrons.
Rather there is a single all-pervading *electron field* and
each electron, as treated in elementary quantum mechanics, is replaced
by a unit of energetic excitation of the electron field. The same goes
for other types of matter, such as quarks. According to quantum field
theory, quarks are really excitations of an all-pervading quark field.
Besides, quantum field theory exhibits a second, and fundamentally
different, way in which mass and charge fail to be intrinsic
properties. This goes under the label of
"renormalization". In short, the mass and charge of a
particle *a la* quantum field theory, i.e., a unit of
excitation of an all-pervading field, depends upon the length-scale at
which you choose to describe, or to experimentally probe, the field
(Butterfield 2014, 2015).



So, to sum up this case against reductionism about relations: there
seem to be far fewer (if any) monadic physical properties than the
reductionist view needs. But this it is not the only case that has
been made. It has also been argued on the basis of quantum
entanglement, the fact that the quantum states of entangled particles
cannot be described independently of one another, that physics is
committed to relations that don't supervene upon intrinsic
properties (Teller 1986). More radically, it has been argued that a
consideration of quantum entanglement pushes us towards the
metaphysical outlook that there are no intrinsic properties (Esfeld
2004, 2016) or even that there are no objects but only relations
(French & Ladyman 2003; Ladyman & Ross 2007: 130-89,
Ladyman 2016; for objections see Psillos 2001, 2006, Brinceno
& Mumford 2016).



Consider next the related suggestion that spatio-temporal relations
are internal relations between, variously, space-time points, regions
or events (Simons 2010: 207-8, 2014: 312-4; Simons 2016:
2010: 207-8; Heil 2012: 147; Lowe 2016: 109-111). General
relativity makes serious trouble for this suggestion. The thinking
behind the suggestion is that it belongs to the essential natures of
space-time points, etc., that they bear the catalogue of
spatio-temporal relations they do. In philosophy of physics, the view
is called "metrical essentialism", the doctrine that
points have their metrical properties and relations essentially
(Maudlin 1988). From metrical essentialism it follows that the mere
existence of a point, etc., already suffices to determine that it
inhabits this network of spatio-temporal relations. Hence, it is
argued, there is no need to admit spatio-temporal relations, or at
least consider spatio-temporal relations as ontological additions to
the furniture of the Universe. But, as well as the *a priori*
concerns already aired, this argument for refusing to admit
spatio-temporal relations as ontological additions is encumbered by
the *a posteriori* difficulties of the metrical essentialism it
presupposes.



These difficulties for metrical essentialism arise from the fact that
in general relativity space-time structure varies from one model,
i.e., possible world, to another (Butterfield 1989: 16-27). We
cannot leave points or regions to themselves to settle their internal
relations, because general relativity denies the idea of a
spatial-temporal framework whose make-up is settled independently of
the matter or radiation. So there isn't a stable
once-and-for-all class of points or regions which once posited suffice
for the purposes of modern physics, hence no stable class of
truth-makers that suffice for making all the distance-duration claims
true.



It may help to understand this debate about space-time points if we
consider the following line of response on behalf of metrical
essentialism. According to metrical essentialism, points have their
metrical properties and relations, i.e., space-time framework,
essentially. So if the space-time framework differs between models
then the points must differ between models too. But this view
isn't compromised by the fact, as Butterfield (1989) points out,
that in general relativity the space-time framework varies from one
model to another; it only follows that the points must vary between
these models too. What, however, this line of response fails to take
into account is how extremely fragile spatio-temporal relations must
be conceived to be when metrical essentialism is combined with general
relativity.



Take a possible world which is truly described by a dynamical theory
of space-time (i.e., described by general relativity and any likely
successor to it). Now take a second possible world which differs from
the first only with respect to a tiny change, on a millimeter scale as
regards length, and on a milligram scale as regards mass. This
involves altering spatio-temporal relations throughout the entire
universe. Hence it's a consequence of metrical essentialism that
in the second possible world, all space-time points are non-identical
to those in the first. But surely it's implausible that such a
small adjustment should add up to such a world-altering difference
(thanks here to Jeremy Butterfield).



We are now in a position to evaluate David Lewis's claim that
whilst distance relations fail to supervene on the intrinsic
characters of the things they relate, there is nevertheless a
different way in which relations of distance do supervene on intrinsic
character:



> 
> 
> If, instead of taking a duplicate of the electron and a duplicate of
> the proton, we take a duplicate of the whole atom, then it will
> exhibit the same electron-proton distance as the original atom.
> Although distance fails to supervene on the intrinsic nature of the
> relata taken separately, it does supervene on the intrinsic nature of
> the composite of the relata taken together - in this case the
> composite hydrogen atom. (1986: 62)
> 
> 
> 



This remark has suggested to some philosophers that we can do away
with relations generally because we need only posit the composites
upon whose intrinsic nature the relation supervenes (Parsons 2009:
223-5). But Lewis's view and the more general form of
reductionism inspired by it encounter a dilemma. Either (1) the
composite is just the proton and electron taken together. But then the
distance between them fails to supervene upon the intrinsic character
of the composite, because there are duplicates of the proton and
duplicates of the electron that vary in distance. Or (2) the composite
is more than just the proton and electron, i.e., the proton and
electron related together in some (external) way. But then we
haven't gotten rid of external relations in favour of internal
ones but only presupposed them.



Can this dilemma for Lewis be sidestepped by the following manoeuvre?
Suppose the composite is the mereological fusion of the proton and the
electron. Then can't we say that the fusion has its intrinsic
character, and the distance relation between the parts of the fusion
supervenes on this character?



But this can't be enough to say. We don't really have any
idea of what the intrinsic properties of fusions are because the
axioms of mereology are silent about the intrinsic characters of
fusions. So we don't really have any idea of whether the
distance relations of their parts supervene upon their properties.
Perhaps fusions don't have much, if any, intrinsic character!
Since we don't know much about these things, we're not in
a position to rule out the possibility of a duplicate of Lewis's
proton-electron fusion being a fusion that has parts which are a
different distance apart than the original proton and electron Lewis
described.



Of course, there's no logical bar to putting forward a theory
that posits a rich supply of fusions whose intrinsic natures are such
that the distance relations that hold between the parts of these
fusions supervenes upon their intrinsic characters. But why believe
such a theory? It's difficult to see how the supervenience of
distance relations upon intrinsic characters of fusions could ever be
explained ('superdupervenience'), because it's
difficult to see how an intrinsic property \(F\) of one thing \(X\)
can intelligibly give rise to an external relation \(R\) between two
other things \(x\) and \(y\), even if \(x\) and \(y\) are parts of
\(X\).



Finally, in the light of the foregoing discussion, Lewis's view
is open to an objection from physics: there is no absolute fact of the
matter about the distance between the electron and proton
independently of facts about the physical fields more generally, e.g.,
the electromagnetic field and the electron field, i.e., facts
concerning what is happening elsewhere. So the distance relation of
the electron and fusion cannot supervene upon the intrinsic character
of the proton-electron fusion. It follows that distance relations
cannot be external in Lewis's sense.


## 4. The Nature of Relations: Order and Direction



Supposing they exist, what is the nature of relations, whether
internal or external? It is the crucial feature of binary
non-symmetric relations, which distinguishes them from binary
symmetric relations, or, more generally, distinguishes relations which
fail to be strictly symmetric from strictly symmetric ones, that they
bestow order upon the things they relate. Non-symmetric relations do
so, because they admit of *differential application*, i.e., for
any such relation there are multiple ways in which it potentially
applies to the things it relates. So, in particular, for every binary
non-symmetric \(R\), \(a\Rel b \ne b\Rel a\). But can we say anything
more about how it is possible for non-symmetric relations to admit of
differential application? More generally, can we say anything
discursive about the origins of order?



According to what is sometimes called the "standard"
account of non-symmetric relations, we can (Russell 1903: SS94;
Grossman 1983: 164; Paul 2012: 251-2). A non-symmetrical
relation \(R\) has a *direction* whereby it travels
*from* one thing *to* another. What distinguishes
\(a\Rel b\) from \(b\Rel a\) is that \(R\) runs from \(a\) to \(b\) in
the former case and \(b\) to \(a\) in the latter. So differential
application is possible because non-symmetric relations have
direction, i.e., because non-symmetric relations run one way, rather
than another, amongst the things they relate. (See Johansson 2011 for
an argument that 'direction' is really a cluster term
covering different logical and intentional notions.)



The standard account faces the serious objection that it is liable to
*over-generate* relations, specifically converse ones, giving
rise to an *over-abundance* of states that results from the
holding of converse relations (Fine 2000: 2-5; cf. Russell 1913:
85-7; Castaneda 1975: 239-40; Armstrong 1978b: 42,
94; for related puzzles arising out of the commitments of Armstrong
1997, see Hochberg 1999: 157-60, Cross 2000 and Newman 2002).
Employing the notion of *direction* that the standard account
uses to explain order, we can define the converse of a non-symmetric
\(R\) as the relation \(R^\*\) that runs from \(b\) to \(a\) whenever
\(R\) runs from \(a\) to \(b\). But once given this notion of a
(distinct) converse, it is difficult to deny that \(R\) has a
converse. This is because it would be arbitrary to admit the existence
of \(R\) whilst denying its converse (or *vice versa*). Hence,
the standard view is committed to the general principle,



(*Converses*) Every non-symmetric relation has a distinct
converse.



But the admission of converse relations threatens to conflict with two
plausible sounding principles about the number of states that arise
from the holding of non-symmetric relations and their converses.



(*Identity*) Any state that arises from the holding of a
relation \(R\) is identical to a state that arises from the holding of
its converse \(R^\*\).



(*Uniqueness*) No one state arises from the holding of more
than one relation.



Support for *Identity* is garnered from intuitions: that the
cat's being *on top of* the mat is the very same state as
the mat's being *underneath* the cat, that Obama's
being *taller* than Putin is the very same state as
Putin's being *shorter* than Obama, and so on.
*Uniqueness* is based upon more overtly theoretical grounds.
States are often conceived as complexes of things, properties and
relations. They are, so to speak, metaphysical molecules built up from
their constituents, so states built up from different things or
properties or relations cannot be identical. Hence it cannot be the
case that the holding of two distinct relations give rise to the same
state.



Now *one* problem with the standard view is that
*Converses*, *Identity* and *Uniqueness* form an
inconsistent triad, assuming there are non-symmetric relations. Take
the cat's being on top of the mat. *Identity* dictates
that this state is the very same state as the mat's being under
the cat. But if there is only one state in play here,
*Uniqueness* tells us that this state can only arise from the
holding of one relation. So the relation that holds between the cat
and the mat in virtue of which the cat is on top of the mat must be
identical to the relation that holds between the cat and the mat in
virtue of which the mat is underneath the cat. But this conflicts with
*Converses* which tells us these relations are distinct.



In the face of this inconsistency a number of theoretical options are
open to us. If we wish to maintain the standard view that
non-symmetric relations have direction then something has to give. (1)
Either we can give up *Identity*, *Uniqueness,* or both,
or we can ditch *Converses*. Indeed it has been argued that
*Converses* should be ditched anyway because it gives rise to a
special kind of semantic indeterminacy: we cannot establish by our use
of a sign for a binary non-symmetric relation \(R\) that we mean to
pick *it* out rather than its converse \(R^\*\). This is because
we can shift the referent of a predicate from a given relation to its
converse whilst leaving undisturbed the truth conditions of the
statement in which the predicate occurs by making compensating
adjustments to the manner in which the names flanking the predicate
are interpreted (Williamson 1985: 252-55, 1987; van Inwagen
2006: 458-68 but see Liebesman 2013 for a reply to Williamson).
But whether *Converses* is ditched to avoid semantic
indeterminacy, or because *Converses*, *Identity* and
*Uniqueness* form an inconsistent triad, or for some other
reason, the link between *Converses* and the standard view will
have to be broken. (2) Alternatively we can jettison
*Converses* but throw out the standard view with it. But if
that's the option we choose, then we will have to go back to the
drawing board to provide a fresh account of differential application
that doesn't appeal to direction.



There is certainly room for development if we go down route (1). The
intuitions that provide evidence for *Identity* are, like
intuitions generally, considered by many philosophers to be relatively
weak sources of evidence, because there may be all variety of
psychological, historical and sociological forces responsible for our
holding this or that view to be intuitive which have little to do with
whether the view is true or credible. There's also a tradition
of conceiving states or facts as capable of being analysed into more
than one collection of constituents, i.e., a tradition that denies
*Uniqueness* (Frege 1884: SS64; Hodes 1982; see Macbride
2007: 54-55 and Liebesman 2014 for counterarguments to
Fine's assumption; but see Leo 2013 for a view which denies
*uniqueness* for states).



Alternatively we may leave *Identity* and *Uniqueness*
in place but question whether the standard view really does give rise
to *Converses*. According to one line of thought (Russell 1913:
86-7), it would be arbitrary for us to choose from the inside,
so to speak, one binary non-symmetric relation at the expense of its
converse. But, it may be argued, this doesn't preclude nature
choosing for us from the outside, providing an externalist rather than
an internalist foundation for admitting a non-symmetric relation but
not its converse. Nevertheless, even if *Converses* is ditched
there is another obstacle to the standard view, namely that it appears
to give rise to questions about direction for which there are no
sensible answers, even if doesn't over-generate converse
relations and states to house them (MacBride 2014: 5-7,
9-10). Consider the states, (a) Antony is married to Fulvia and
(b) Antony is to the left of Fulvia. If these states arise from their
respective relations travelling from one thing they relate to another,
then there must be a fact of the matter about which things proceed
from and which to. So, we may ask, do they both proceed from the same
thing or do they proceed from different things? It doesn't seem
that there could be a sensible answer to this question and so we
should avoid an account of differential application that commits us to
there being such recherche and undetectable facts of the
matter. (But note that whilst many philosophers have felt the pull of
these kind of intuitive considerations, others have argued that is is
necessary to introduce ordinal properties or relations to impose
structure upon relational facts. For example, Hochberg posits
"ordering relations" (Hochberg 1987, 1999: 180-81)
whilst Tegtmeier posits "ordinators", such as
"coming first" and "coming second" in a fact
(Tegtmeier 1990, 2004, 2016).)



What are the options if we go down route (2)? Some philosophers have
suggested that if we ditch *Converses* then we will have to
give up non-symmetric relations altogether (van Inwagen 2006: 453).
But this presupposes that the only way to account for the distinctive
feature of non-symmetric relations, *viz.* differential
application, is to appeal to the standard view. However, philosophers
that go down this second route endeavour to account for differential
application by different means than *direction* and thereby
earn the right to hold onto non-symmetric relations. We can
distinguish three broad strategies for explaining differential
application without direction, namely (1) *positionalism*, (2)
*anti-positionalism*, and (3) *primitivism*, although
this does not exhaust the range of possibilities considered in the
literature (consider, for example, the view that relations are
properties of sequences discussed above (sec. 1).



First, *positionalism* may be roughly sketched as the view that
an \(n\)-ary relation has \(n\) argument positions, where argument
positions are conceived as themselves entities, and, crucially, there
is no intrinsic order to the argument places of a relation; neither
argument position of a binary relation is the first or the second,
etc. (Castaneda 1972, 1975; Williamson 1985: 257; Grossman
1992: 57; Armstrong 1997: 90-1; Cross 2002: 219-223, Fine
2000: 10-16; Orilia 2011; Gilmore 2013). In the case of a binary
non-symmetric relation \(R\), this amounts to the idea that \(R\) is
associated with two further entities, % and #, its argument positions,
and \(R\) holds between two given relata \(a\) and \(b\) relative to
an assignment of objects to % and #. The difference between \(a\Rel
b\) and \(b\Rel a\) (differential application) is then explained in
terms of the different relative assignments of \(a\) and \(b\) to %
and #: \(a\) to % and \(b\) to # in one case, and \(b\) to % and \(a\)
to # in the other. But since the argument positions do not themselves
exhibit any significant order or direction, there is no basis here for
introducing converses, i.e., relations that differ only with respect
to their direction of travel.



Different versions of *positionalism* vary with regard to
\((i)\) the nature of argument positions, crucially upon whether they
can be shared by different relations, and \((ii)\) the variety of
argument positions, whether there are different kinds of position
specifically suited to different kinds of relata, and \((iii)\) the
complexity of the relation whereby things are assigned to positions.
According to Orilia's *onto-thematic role* view,
positions are capable of recurring in different relations and suited
by their nature to different sorts of filling (Orilia 2011, 2014,
Paolini Paoletti 2016, 2018b sec. 2, and for an earlier version
inspired by Leibniz, Orilia 2008). For example, the *loves*
relation has agent and patient roles, roles which also recur in the
*hates* relation. Other more complex relations involve more and
different roles, for example, theme, location, instrument, goal
*etc*. Orilia's idea is that the schedule of ontological
roles belonging to relations should be informed by the linguistic
programme of identifying thematic roles in natural language because
this helps us to provide informative generalisations about what
different relations have in common (see Davis 2011 for a helpful
introduction to the thematic role literature in linguistics).



By contrast, other versions of *positionalism* conceive of
positions as unique to the relations in which they figure. Some
positionalists of this stripe conceives of each position as being
assigned a unique occupant (Gilmore 2013). But others whilst sharing
the assumption that positions are unique to the relations in which
they figure, also maintain that it is a mistake to assume that there
is a 1-1 correspondence between positions and their occupants. So this
version of positonalism conceives relations more liberally as having
positions capable of being assigned several distinct occupants at once
(see, for example, Armstrong 1997: 91, Yi 1999: 169; Fine 2000: 17,
MacBride 2005: 588). Further sophistication is added by the idea that
positions can be occupied by a number of things varying between fixed
limits (Hazen&Taylor 1992: 376, 390, Dixon 2018: 207) and by the
suggestion that the assignment of things to positions should also be
relative to a number to allow for one and the same thing to
simultaneously occur more than once in the same position (Dixon 2018:
213-6).



A different line is taken by Donnelly's *relative
positionalism*, according to which positions are a kind of
"relative property" (Donnelly 2016). Donnelly takes
*being north* as an example of a relative property: a property
because it is had by one thing but only relative to something else, in
this case a location. Her basic idea is that corresponding to each
*n*-ary relation *R* is a package of relative properties
(as many as *n* factorial properties) such that when *n*
things instantiate R they instantiate these properties relative to one
another (Donnelly 2016: 91). It is a distinctive feature of the view
that by positing for each relation a bespoke package of relative
properties which models the logical character of the relation in
question, *relative positonalism* is able to handle relations
of arbitrary complexity, including symmetric and partially symmetric
relations (Donnelly 2016: 88-96, Dixon 2019: 56-67). By
contrast, the *onto-thematic role* approach doesn't
provide a once and for all recipe for positing roles but takes this to
be a matter to be settled by systematic "ontological
investigation" into the character of relations as we find them
(Orilia 2014: 296-301 but see objections from MacBride 2014:
11-12, Ostertag 2019: 1492-5).



There are two immediate challenges faced by the different versions of
*positionalism*. Can the account be generalized to cover all
relations? If, for example, the binary symmetric relation that holds
between two things \(a\) and \(b\) whenever \(a\) is next to \(b\) has
two argument positions, PS and $, then there should be two
possible ways of assigning \(a\) and \(b\) to PS and $. But this
means (absurdly) that there are two different ways for \(a\) and \(b\)
to be next to one another, *i.e.*, symmetric relations admit of
differential application too! Orilia's *onto-thematic
role* approach handles this kind of case by conceiving of thematic
roles as "intra-repeatable" (Orilia 2014: 294), so there
is only one role or position in play in this relation (PS = $)
and so the possibility of different assignments to different roles or
positions doesn't arise. Similarly, Dixon's
"pocket" theory handles this case by conceiving the
*next to* relation as having only one pocket. But, as Fine
(2000: 17-18, 2007: 58-9) and MacBride (2007: 36-44)
argue, and as Dixon (2018: 31, n.21) acknowledges, this kind of
strategy doesn't straightforwardly extend to more complex
partially symmetric relations such as *being arranged clockwise in
a circle in that very order*, or, *playing-tug-of-war*.
Accordingly Orilia (2011: 6-9, 2014: 296-301) proposes a
programme of analysis whereby more complex relations are reduced to
simpler ones that the onto-thematic role approach can accommodate. By
contrast, *relative positionalism* can handle these kinds of
partially symmetric relations without any analysis (Donnelly 2016:
88-96, Dixon 2019: 56-67), but not multigrade relations
which lack a fixed arity *n*, hence cannot be modelled by a
*n*! package of relative properties (Donnelly 2016: 94, Dixon
2019: 59) (For further criticisms of *positionalism* see
Hochberg 1999: 153, 2000: 40-1).



The more general challenge that faces the different versions of
positionalism is whether we have a better understanding of positions
and the assignment relation that the different versions of
positionalism propose, than we do of the relations that positions and
assignment are used to explain (Fine 2000: 16; MacBride 2014:
11-12; Orilia 2014: 302-3). Consider, for example,
Donnelly's claim that the application of *n*-ary
relations needs to be explained in terms of packages of *n*!
relative properties. One problem, acknowledged by Donnelly (2016: 98),
is the ontological cost of having to trade in each relation for so
many properties. Another problem concerns why relative properties of a
given package cannot be instantiated independently of one another but
only relative to other properties belonging to that package, even
though each relative property is meant to be unary. The obvious
explanation is that such properties aren't unary at all but to
be explained in terms of an underlying relation (Bigelow &
Pargetter 1990: 55, MacBride: 2000: 83). So saying that one thing
*x* has a property relative to something else *y* having
another relative property *etc* is more perspicuously rendered
by saying that *x* bears a relation to *y*.



Both the standard view and the different versions of
*positionalism* assume that the differential application of a
non-symmetric relation \(R\) with respect to \(a\) and \(b\) is to be
explained "locally" in terms of \(R\), whether its
direction or argument positions, and the things it relates, \(a\) and
\(b\). *Anti-positionalism* is the doctrine, due to Kit Fine,
which eschews direction and argument positions and explains
differential application by abandoning the assumption that
differential application is to be explained locally (Fine 2000:
20-32). According to anti-positionalism, the difference between
\(a\Rel b\) and \(b\Rel a\) arises from how these states are
inter-connected with other states that arise from \(R\) holding of
other relata. So order is not a feature of given state capable of
being isolated, but only emerges over a totality of states to which
\(R\) gives rise. What distinguishes \(a\Rel b\) from \(b\Rel a\) is
that inter-connectedness with other states such as \(c\Rel d\). Pay
special attention to the relative arrangement of the schematic names
in the following sentence. Whereas the state \(a\Rel b\) is a
completion of \(R\) by \(a, b\) in the same manner as \(c\Rel d\) is a
completion of \(R\) by \(c, d\), the state \(b\Rel a\) isn't.
The state \(b\Rel a\) is a completion of \(R\) by \(b\), \(a\) in the
same manner as \(c\Rel d\) is a completion of \(R\) by \(c, d\) (Fine
2000: 21). However, Fine doesn't take the notion of two states
being put together "in the same manner" as primitive, but
explains this in terms of a substitution relation. To say that a state
\(s\) is a completion of relation \(R\) by \(a\_1, a\_2 \ldots, a\_n\) in
the same manner as a state \(t\) is completion of \(R\) by \(b\_1, b\_2
\ldots, b\_n\) is to say that \(s\) is completion of \(R\) by \(a\_1,
a\_2 \ldots, a\_n\) that results from simultaneously substituting \(a\_1,
a\_2 \ldots, a\_n\) for \(b\_1\), b\(\_2 \ldots, b\_n\) in \(t\) (and
*vice versa*) (Fine 2000: 25-6). Fine goes on to suggest
that the notion of simultaneous substitution of many objects may be
defined, under certain conditions, in terms of the single substitution
of one object (Fine 2000: 26 n. 15). *Converses* is avoided
because the notion of direction, in terms of which the distinction
between converse relations is drawn, is eschewed. This is because
substitution is doing all the work.



Fine's version of *anti-positionalism* has certain
limitations, including the assumption that states arising from a
single relation have a unique composition and that substitution
operates on objects rather than occurrences of objects. Leo (2013)
provides a more fully developed version of *anti-positonalism*
that addresses these limitations from a more formal point of view.
According to Leo, relational complexes are structured perspectives on
states 'out there' in reality, relational complexes have
occurrences of objects, and different complexes of the same relation
may correspond to the same state. It is key to this approach that
substitution is a primitive operation taking a complex and an
assignment of objects to the occurrences of objects in the complex as
input and returning another complex as output. For example, for the
complex of Anthony's loving Cleopatra, substituting Abelard for
the occurrence of Anthony and Eloise for the occurrence of Cleopatra
results in the complex of Abelard's loving Eloise. Substitution
is identified in terms of basic principles governing substitutions, in
particular a composition principle, which forms the heart of an
antipositional theory of relations (Leo 2013: 363; for further
developments and defence of this version of
*anti-positionalism*, see Leo 2008a, 2008b, 2010, 2014,
2016).



By abandoning the assumption that differential application is to be
explained locally, *anti-positionalism* in effect tells us that
we're misconceiving the logical form of our ordinary relational
judgements. Contrary to their surface form, relational judgements
always have a suppressed argument position for other states, by which
means the state we are making a judgement about is implicitly compared
to another. The following two objections arise (MacBride 2007:
47-53). First of all, *anti-positionalism* presupposes a
non-symmetric relation of substitution to explain differential
application. But this suggests that the explanation anti-positonalism
provides involves a circularity because it uses a non-symmetric
relation, substitution, whose differential application is take for
granted. Take the state \(a\Rel b\). The result of substituting \(a\)
for \(b\) is the state \(a\Rel a\); where the result of substituting
\(b\) for \(a\) is the state \(b\Rel b\). The outcomes are different
so the substitution relation must be order sensitive. Secondly,
anti-positionalism entails that there cannot be lonely binary
relations, i.e., relations which only hold of exactly two things. But
*prima facie* we should be able to distinguish \(a\Rel b\) from
\(b\Rel a\) even if \(R\) holds between nothing except \(a\) and
\(b\), Antony's loving Cleopatra from Cleopatra's loving
Antony even if the world is otherwise loveless (see, for replies to
these objections, Fine 2007: 59-62, Wieland 2010: 487-93,
Leo 2014: 279-81).



The third strategy, *primitivism*, seeks to avoid the
difficulties that threaten *positionalism* and
*anti-positionalism* by taking the radical step of asking us to
reconfigure our explanatory settings. It's a familiar thought
that we cannot account for the fact that one thing bears a relation
\(R\) to another by appealing to a further relation \(R'\) relating
\(R\) to them--that way Bradley's regress beckons. To avoid
the regress we must recognise that a relation is not related to the
things it relates, however language may mislead us to think otherwise.
We simply have to accept as primitive, in the sense that it cannot be
further explained, the fact that one thing bears a relation to
another. But, according to primitivism, it is not only the fact
*that* one thing bears a (non-symmetric) relation \(R\) to
another that needs to be recognised as ultimate and irreducible.
*How* \(R\) applies, whether the \(a\Rel b\) way or the \(b\Rel
a\) way, needs to be taken as primitive too (MacBride 2014:
14-15). The difficulties encountered by attempts to provide an
analysis of differential application provide corroborative evidence
for primitivism. Does refraining from providing an analysis of
differential application constitute shirking an obligatory task for
systematic philosophy? Hardly. Remember what Lewis correctly said
about predication: "Not every *account* is an
*analysis*" (Lewis 1983: 352). What we need, according to
this third strategy, is a form of realism committed to the existence
of relations but which denies that the manner of their application,
their differential application, can be further explained.



One of the most intriguing features of the debate about relations and
differential application is that there is no single sense in which the
different views we have considered seek to "explain" or
"account" for differential application. They all run on
different explanatory settings. Some think there is a genuine problem
to do with the capacity of relations to differentially apply which
needs to be explained by any successful theory of relations. Hence,
because he thinks there is a genuine problem here, Hochberg claims
that the question which any adequate theory of relations must answer
"is about how the order is ontologically grounded in
facts" (Hochberg 2001: 178). But others think that there
isn't a problem of differential application at all, much less
one that requires an ontological analysis of relations or facts. This
means that the debate cannot be reduced to simply an exchange of
opinions about which view explains the most at the least cost.



For example, positionalists typically maintain that their positing
roles, slots, pockets and relative properties enables them to answer
what they take to be obligatory questions about why each relation
applies the way it can and why some relations are capable of applying
differently from the ways others can (Dixon 2019: 68). So the
different ways that the relations *between* and
*adjacency* are capable of applying is explained by
positionalists in terms of the different ontological structures
attributed to these relations, *i.e.* their number of roles or
positions and the nature of the relation that assigns things to roles
or positions. By contrast, anti-positionalists and primitivists deny
that such questions about why relations have the capacity to
differentially apply need answering at all. As Fine describes his own
version of *anti-positionalism*, "It is a fundamental
fact for [the anti-positionalist] 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 and its
relata" (Fine 2000: 19).



However, the explanatory differences between the parties to the debate
run deeper than merely whether they think the capacity of relations to
differentially apply is open to informative explanation. Whilst Fine
takes the capacity of relations to differentially apply as primitive,
Fine also holds that it is necessary to provide an account of wherein
the difference consists between two states which arise from the same
constituents: "Yet clearly the state of Anthony's loving
Cleopatra and the state of Cleopatra's loving Anthony are
distinguishable; they are not merely two indiscernible
"atoms" within the space of states. But if these states
are not to be distinguished by how they derive from the given relation
and its relata, then how are they to be distinguished?"(Fine
2000: 20). Fine's answer, as we have seen, is spelled out in
terms of substitution. But Leo's version of anti-positionalism
has different explanatory settings again. He doesn't engage with
metaphysical questions about wherein the difference between states
consists and instead provides principles of substitution intended to
explicate rather than explain the differential application of
relations (Leo 2013: 361-8). And primitivism, we have seen,
takes the fact that relations admit of differential application as
unanalysable too but also takes the difference between states arising
from the same constituents as primitive (MacBride 2014: 8,
*cf.* van Fraassen 1982). The dispute about relations and
differential application continues because it remains to be
established which questions about relations are genuine and which are
pseudo-problems.



In conclusion, it should be emphasised that nothing written here is
intended to imply that there is just one problem of order
(differential application). It's plausible that there are
several problems in the vicinity, whether logical, metaphysical or
epistemological and they will require different if co-ordinated
solutions.