The world we inhabit isn’t an undifferentiated bog. Everywhere there’s repetition and, importantly, we can even distinguish different types of repetition. We see one cat and then another cat. But we can also see that our cat is on top of the mat and subsequently notice that the cat from next door is on top of the fence. Repetition in the first sense requires only one thing and then another. By contrast, repetition in the second sense requires two (or more) things and then two (or more) other things. Properties are typically introduced to account for repetition in the first sense, whereas relations are typically introduced to account for repetition in the second sense. There’s cat repetition because there is more than one thing that has a certain property, the property of being a cat. There’s above-and-below repetition because there is more than one thing that bears a certain relation to something else, the relation that holds between two things when one is on top of the other.
It’s doubtful whether the distinction between properties and relations can be given in terms that do not ultimately presuppose it—the distinction is so basic. Nevertheless, there are elucidations on offer that may help us better appreciate the distinction. Properties merely hold of the things that have them, whereas relations aren’t relations of anything, but hold between things, or, alternatively, relations are borne by one thing to other things, or, another alternative paraphrase, relations have a subject of inherence whose relations they are and termini to which they relate the subject. More examples may help too. When we say that a thing A is black, or A is long, then we are asserting that there is some property A has. But when we say that A is (wholly) inside B then we are asserting that there is a relation in which A stands to B.
Be careful though not to be misled by these examples. A can only be (wholly) inside B if A is distinct from B. So the relation in which A stands to B if A is (wholly) inside B requires a distinct subject from terminus. By contrast, A can be black without prejudice to anything else. But we cannot infer straightaway that every relation holds between more than one thing because there may be some relations that a thing bears to itself, if, for example, identity is a relation. So we cannot distinguish properties from relations by appealing to the number of distinct things required for their exhibition, since these may be the same, viz. one. Hence the plausibility of thinking that a relation differs from a property because a relation, unlike a property, proceedsfrom a subject to a terminus, even if the subject and terminus are identical.
Theories of relations add to, subtract from or qualify this basic package of ideas in various ways. But with the distinction between properties and relations in hand we can already do some worthwhile conceptual geography, distinguishing four regions of logical space. (1) Rejection of both properties and relations. (2) Acceptance of properties but rejection of relations. (3) Acceptance of relations but rejection of properties. (4) Acceptance of both properties and relations. (1) amounts to a form of thoroughgoing nominalism and is typically motivated by a blanket aversion to the doctrine that general words like “black” or “before” refer to worldly items, whether properties or relations. By contrast, (2), (3) and (4) amount to different forms of realism, whether about properties and/or relations. Typically they are motivated either by an especial antipathy toward relations (2) or, alternatively, an appreciation of their distinctive utility ((3) and (4)). Here we will focus upon what it is about relations in particular that makes philosophers either love them or hate them.
[size=30]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
[ltr]n>1)n>1)[/ltr] 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. 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 is a contentious issue. Armstrong (1978b: 93–4, 1997: 85, 2010: 23–5) argues against them, whilst MacBride (2005: 569–93) argues for 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
xbears
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
xbears
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
xis 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
[ltr]n>2n>2[/ltr], 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: §§212–6, 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
bwhen
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
aand
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
adjacencyrelation 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
[ltr]ss[/ltr] (
[ltr]=⟨x,y⟩=⟨x,y⟩[/ltr]) is conceived as a “one” such that a relation
[ltr]RR[/ltr] is really only a monadic feature of it,
[ltr]R(s)R(s)[/ltr], then it’s mysterious what
[ltr]RR[/ltr]’s holding of
[ltr]ss[/ltr] has to do with
[ltr]xx[/ltr] and
[ltr]yy[/ltr]being arranged one way rather than another with respect to
[ltr]RR[/ltr]. If you don’t see the mystery straightaway, reflect that
[ltr]ss[/ltr] may have a monadic feature
[ltr]FF[/ltr], which is, so to speak, only to do with
[ltr]ss[/ltr]itself rather than its members. But if
[ltr]RR[/ltr] and
[ltr]FF[/ltr] are conceived alike, as monadic features of
[ltr]ss[/ltr], isn’t it mysterious that the possession by
[ltr]ss[/ltr] of
[ltr]RR[/ltr] has consequences for
[ltr]xx[/ltr] and
[ltr]yy[/ltr] whereas
[ltr]FF[/ltr] doesn’t? If, alternatively, a sequence is conceived as a “many”, just
[ltr]xx[/ltr] and
[ltr]yy[/ltr], such that
[ltr]RR[/ltr] 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.,
[ltr]⟨x,y⟩={{x},{x,y}}⟨x,y⟩={{x},{x,y}}[/ltr]). 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
[ltr]⟨x,y⟩⟨x,y⟩[/ltr]. 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 ownexperiences, i.e., the very notion that we set out to explain, viz. personal identity, is presupposed by our analysans. Similarly
[ltr]{{x},{x,y}}{{x},{x,y}}[/ltr] only serves as a model of
[ltr]⟨x,y⟩⟨x,y⟩[/ltr] relative to the assumption that the thing that belongs to a singleton member, namely
[ltr]{x}{x}[/ltr], of the first class
comes first in the pair
[ltr]⟨x,y⟩⟨x,y⟩[/ltr], so the notion of order is presupposed (Hochberg 1981).
The next idea that needs to be introduced is the idea of a
converse relation. For any given binary relation
[ltr]RR[/ltr], the converse of
[ltr]RR[/ltr] may be defined as the relation
[ltr]R[size=13]∗R∗[/ltr][/size] that
[ltr]xx[/ltr] bears to
[ltr]yy[/ltr] whenever
[ltr]yy[/ltr]bears
[ltr]RR[/ltr] to
[ltr]xx[/ltr] (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
[ltr]RR[/ltr]floating free from
[ltr]RR[/ltr] and holding between things independently of how
[ltr]RR[/ltr] arranges them. Rather it’s a more intimate relationship which, if it holds, holds of necessity. For
[ltr]xx[/ltr] to have one to
[ltr]yy[/ltr] is, so to speak, for
[ltr]yy[/ltr] to have the other to
[ltr]xx[/ltr]; 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
[ltr]RR[/ltr] is a non-symmetric relation which
[ltr]xx[/ltr] bears to
[ltr]y.y.[/ltr] Then the converse of
[ltr]RR[/ltr] is borne by
[ltr]yy[/ltr] to
[ltr]xx[/ltr]. But suppose that
[ltr]RR[/ltr], as it may, fails to be borne by
[ltr]yy[/ltr] to
[ltr]xx[/ltr]. Then something is true of the converse of
[ltr]RR[/ltr] that isn’t true of
[ltr]RR[/ltr] itself,
viz. that it is borne by
[ltr]yy[/ltr] to
[ltr]xx[/ltr]. Hence, the converse of
[ltr]RR[/ltr] must be distinct from
[ltr]RR[/ltr]. 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 final 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). 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
[ltr]RR[/ltr] is internal iff if
[ltr]xx[/ltr] bears
[ltr]RR[/ltr] to
[ltr]yy[/ltr] then
[ltr]xx[/ltr] does so necessarily. From the internality of
[ltr]RR[/ltr], in Moore’s sense, it follows that if
[ltr]xx[/ltr] exists at all then
[ltr]xx[/ltr] indeed bears
[ltr]RR[/ltr] to
[ltr]yy[/ltr]. But if it’s possible that
[ltr]xx[/ltr] exist whilst failing to bear
[ltr]RR[/ltr] to
[ltr]yy[/ltr], then
[ltr]RR[/ltr] 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
[ltr]RR[/ltr] is internal iff it’s holding between
[ltr]xx[/ltr] and
[ltr]yy[/ltr] is necessitated by the intrinsic natures, i.e., non-relational properties, of
[ltr]xx[/ltr] and
[ltr]yy[/ltr]; otherwise
[ltr]RR[/ltr] 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 of “external” is more involved:
[ltr]RR[/ltr] 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
[ltr]xx[/ltr] is a cube and
[ltr]yy[/ltr] is also a cube. It follows that
[ltr]xx[/ltr] is the same shape as
[ltr]y.y.[/ltr] So the relation that
[ltr]xx[/ltr] bears to
[ltr]yy[/ltr] when
[ltr]xx[/ltr] is the same shape as
[ltr]yy[/ltr] is internal in Armstrong’s sense. By contrast, spatio-temporal relations are external (in his sense) because the intrinsic characteristics of
[ltr]xx[/ltr] and
[ltr]yy[/ltr] don’t necessitate how close or far apart
[ltr]xx[/ltr] and
[ltr]yy[/ltr] are. Lewis intends his distinction to classify the same way. If
[ltr]xx[/ltr] is the same shape as
[ltr]yy[/ltr], but
[ltr]ww[/ltr] is not the same shape as z, then there must be some difference in intrinsic nature either between
[ltr]xx[/ltr] and
[ltr]ww[/ltr] or else between
[ltr]yy[/ltr] and
[ltr]zz[/ltr]. So the relation that
[ltr]xx[/ltr] bears to
[ltr]yy[/ltr] when
[ltr]xx[/ltr] is the same shape as
[ltr]yy[/ltr] is internal in Lewis’s sense. Lewis also claims that the distance between
[ltr]xx[/ltr] and
[ltr]yy[/ltr] is an external relation because the following conditions are met: (1)
[ltr]xx[/ltr] may be closer to
[ltr]yy[/ltr] than
[ltr]ww[/ltr] is to
[ltr]zz[/ltr] even though
[ltr]xx[/ltr] is an intrinsic duplicate of
[ltr]ww[/ltr], i.e., shares all and only the same intrinsic characteristics, and
[ltr]yy[/ltr] an intrinsic duplicate of
[ltr]zz[/ltr]; (2) the distance between
[ltr]xx[/ltr] and
[ltr]yy[/ltr] does supervene upon the nature of the composite
[ltr]x+yx+y[/ltr]. (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
[ltr]xx[/ltr] bears to
[ltr]yy[/ltr] when
[ltr]xx[/ltr] is the same shape as
[ltr]yy[/ltr] isn’t internal in Moore’s sense because, let us suppose,
[ltr]xx[/ltr] might have been spherical whilst
[ltr]yy[/ltr] remained a cube; so it doesn’t follow from the mere fact that
[ltr]xx[/ltr] exists that
[ltr]xx[/ltr] is the same shape as
[ltr]yy[/ltr].[/size]
Relations
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