## 1. Introduction




All theorizing about relations in the Middle Ages begins with
Aristotle's short treatise, the *Categories*. Due to
historical circumstances, this Aristotelian text was one of the very
few pieces of ancient Greek philosophy available in the Latin west
between the seventh and twelfth centuries, and the only one to contain
a systematic philosophical treatment of relations. In the
*Categories*, Aristotle identifies relations as one of the ten
highest kinds or categories (one of the so-called *summa
genera*), and he devotes an entire chapter--the seventh
chapter of the treatise--to distinguishing its members from those
of the other categories (most notably, substance, quantity, and
 quality).[1]
 On the standard medieval interpretation of this chapter, Aristotle is
attempting to characterize relations at least partly on the basis of
semantic or logical considerations--that is, on the basis of the
differences between statements containing relational (or
'relative') terms and those containing only non-relational
(or 'absolute') terms. As we shall see, this approach to
characterizing relations helps to shape the way medieval philosophers
understand them, and gives rise to the common medieval distinction
between relations merely according to speech (*relationes secundum
dici*) and relations according to nature or being (*relationes
secundum esse*).




In addition to defending a particular characterization of relations in
the *Categories*, Aristotle also suggests a general model or
paradigm for analyzing what we shall call 'relational
situations'--that is, the situations or states of affairs
that explain the truth of genuinely relational statements. According
to this *Categories* model, whenever two (or more) substances
are related, this is to be explained by certain monadic properties or
accidents inhering in the
 relata.[2]
 Thus, if Socrates is similar to Theaetetus (i.e., resembles him with
 respect to some
 quality),[3]
 this is not to be explained by an entity to which Socrates and
Theaetetus are somehow jointly attached (namely, the dyadic or
two-place property, *being-similar-to*). On the contrary, it is
to be explained by a pair of accidents, one of which inheres in
Socrates and relates him to Theaetetus, and the other of which inheres in
Theaetetus and relates him to
 Socrates.[4]




This Aristotelian model exercises enormous influence during the Middle
Ages, and until at least the fourteenth century, medieval philosophers
develop their own analysis of relational situations in terms of it. In
this context, the most important medieval debates concern the precise
nature of the properties or accidents that relate particular
substances. Some philosophers, such as Peter Abelard and William
Ockham, adopt a form of reductionism, according to which
the properties in question are accidents falling under categories
other than relation. Thus, according to Abelard and Ockham, when
Socrates is similar to Theaetetus, this is to be explained in terms of
particular qualities, say, their respective colors. Other
philosophers, however, such as Albert the Great and John Duns Scotus,
reject this form of reductionism, maintaining instead that relations
are accidents of a *sui generis* type. Thus, according to
Albert and Scotus, when Socrates is similar to Theaetetus, this is to
be explained by a pair of accidents whose members are distinct from,
and irreducible to, their respective colors.




Although the analysis of relational situations that Aristotle suggests
in the *Categories* is perfectly general, it is clear from his
later writings that he does not think that all relational situations
conform to it. Thus, in the *Metaphysics* he claims that there
are relational situations (such as Simmias's thinking about Socrates)
in which substances are related not in virtue of a *pair* of
accidents, but rather in virtue of a *single* accident
possessed by just one of the substances. "An object of thought
[e.g., Socrates]" he says at one point "is said to be
related because something else [e.g., Simmias] is related to
 it".[5]
 And his point just appears to be that some relational situations are
grounded in a single property or accident of a single
*relatum*.




Aristotle's *Metaphysics* did not become available to
philosophers in the Latin west until the mid-twelfth century, and it
did not circulate widely before the thirteenth. Nonetheless,
philosophers throughout the medieval period denied that all relational
situations conform to the *Categories*
paradigm. Unlike Aristotle, however, the basis of the medievals'
denial was largely theological in nature. Considerations associated
with the doctrine of creation, for example, led them to accept
relational situations in which two (or more) substances are related by
a single property or accident. And considerations associated with the
Christian doctrine of the Trinity forced them to admit that, in
certain relational situations, even substances themselves can qualify as
 relations.[6]
 Prior to the fourteenth century, however, these sorts of situations
were regarded as special cases, rare exceptions to a general
rule. Indeed, prior to this time, most philosophers worked hard to
show that even these sorts of situations conform, at least to some
extent, to the paradigm of the *Categories*. This effort, as we
shall see, helps to explain the pervasiveness in late medieval
philosophy of the distinction between so-called real relations
(*relationes reales*) and mere relations of reason
(*relationes rationis*).




As the above remarks suggest, the fourteenth century marks an
important turning point in the medieval discussion of relations. After
this point, there is a noticeable shift away from the
*Categories* paradigm, with the result that many philosophers
come to regard those situations in which substances are related by
virtue of their accidents as the exception rather than the rule. Part
of the explanation is due, perhaps, to a gradual waning of
Aristotle's influence, but part of it is also due to substantive
doctrinal changes. By the fourteenth century, it is common to deny the
existence of any real distinction between substances and most of their
accidents, and hence philosophers are increasingly willing to say that
substances themselves provide the ontological grounding for relational
situations. Again, due to important semantic innovations around the
same time, and the subsequent emergence of late-medieval nominalism,
philosophers also begin to depart from the traditional Aristotelian
characterization of relations. Thus, instead of thinking of relations
as the items responsible for relating two or more substances, they now
begin to think of them as items existing only in the mind--that
is, as mere beings of reason or concepts. With respect to both of
these developments, as we shall see, Ockham is an important
transitional figure.


## 2. Aristotelian Background




Although Aristotle discusses relations at various places in his work,
his discussion in the *Categories* is the most important for
understanding the development of medieval theories of relations. Part
of the reason, as we have already indicated, is historical: until the
twelfth century, the *Categories* is the medievals' sole
direct source for Aristotle's views about relations. But part of
the reason is more theoretical: in the *Categories* Aristotle
develops a basic account of relations that is presupposed by his other
discussions. Thus, even when some of these other discussions become
available--discussions such as those in the *Physics* and
*Metaphysics*--medieval philosophers continue to interpret
them in light of their understanding of the *Categories*.




Aristotle's discussion in the *Categories* influences the
development of medieval views about relations in several discernible
ways. First of all, it introduces the basic framework and terminology
that comes to dominate all medieval theorizing about
relations. Second, and perhaps most importantly, it articulates the
main claims in response to which medieval philosophers develop their
own views, including each of the following: (1) relations are the
items that relate substances, (2) the items that relate substances are
accidents, and (3) no substance is a relation.




Finally, the details of Aristotle's discussion in the
*Categories*, especially in chapter 7, give rise to a number of
important medieval debates, as well as the traditional distinction
between relations merely according to speech (*relationes secundum
dici*) vs. relations according to being (*relationes
secundum* *esse*). In order to understand the development
of medieval views about relations, therefore, we must begin with some
account of Aristotle's discussion in the
*Categories*.


### 2.1 Framework and Terminology of the *Categories*




Throughout the *Categories* Aristotle assumes that relations
comprise one of the accidental categories, and hence that they must be
understood as items *inhering* in particular substances. This
helps to explain why, unlike contemporary philosophers, who speak of
relations as holding *between* two or more things, Aristotle
prefers to speak of them as inhering in one thing and somehow pointing
toward (*pros*) another. Indeed, Aristotle's preferred
name for relations is just 'things toward something'
(*ta pros ti*).




During the Middle Ages, it is customary for philosophers to refer to
relations not only by Aristotle's term, 'toward
something' (or *ad aliquid*, the verbatim Latin
equivalent of *pros ti*), but also by two others as well:
'relative' (*relativum*) and 'relation'
(*relatio*). In the early Middle Ages, philosophers often move
freely among these terms without paying much attention to their
various
 senses.[7]
 Over time, however, they come to emphasize their differences,
sometimes giving elaborate explanations as to why one must be regarded
as more appropriate than the others. To give just one example,
consider the following passage from Albert the Great:



> 
> 
> Now the most general genus in the arrangement of this predicable is
> toward something (*ad aliquid*), or relative
> (*relativum*), or less properly speaking, relation
> (*relatio*), as some people say. But it must be recognized that
> the most general genus is signified most clearly by the name
> 'toward something', which is a preposition together with
> [the term] 'something' in [the accusative] case. For this
> name conveys the two things that are in a relative, namely: [i]
> diversity, which the preposition indicates through its taking an
> object (*transitionem*), and [ii] the direction of the
> comparison, which the accusative case indicates when something is
> called 'toward something'. (*Liber de praedicamentis*,
> 225b.)
> 
> 




Whatever their disagreements about the appropriateness of these
terms, it was generally agreed that the terms 'toward
something' and 'relative' are related to the term
'relation' in the way that a concrete term (such as
'white') is related to its abstract counterpart (such as
'whiteness').




Since 'relation' is the term most familiar to us, in
what follows we shall rely on it whenever possible. It is important to
recognize, however, that the medievals introduce a host of other
abstract terms more or less synonymous with 'relation'
(*relatio*) but whose connotations they often regard as more
informative than any of those discussed so far. A partial list of such
terms includes:


* 'Comparison' (*comparatio*), which has
psychological overtones, and so is often (though not exclusively) used
by conceptualists;
* 'Outward-looking-ness' (*respectus*), which
draws on a visual metaphor to suggest that relations are that in
virtue of which a substance 'looks out toward something'
(*respicit ad aliquid*);
* 'Disposition' (*habitudo*) or 'relative
disposition' (*habitudo relativa*), which suggests that
relations account for the way a thing 'holds itself toward
something' (*se habere ad aliquid*);
* 'Order' (*ordo*), 'ordering'
(*ordinatio*), and 'directionality' or
'toward-ness' (*aditas*), which are used to
indicate that relations account for the order or structure we find in
 the
 world.[8]




Although Aristotle makes scattered remarks about relations throughout
the *Categories*, it is only in chapter 7 that he singles them
out for detailed independent consideration. Thus, in order to complete our
survey of the relevant Aristotelian background, we need to consider
briefly how this chapter was commonly understood by medieval
philosophers.


### 2.2 Relations in *Categories*



*Categories* 7 opens rather abruptly with a definition of
relations that Aristotle ultimately regards as unacceptable. According
to this definition, which medieval philosophers attribute to Plato,
relations are things "spoken of" in a certain way. Whether
or not this definition is really Plato's, it certainly appears
to be well entrenched by Aristotle's time, as is indicated by
the fact that Aristotle says we are at least prepared to *call*
something a relation just in case it satisfies the conditions
specified by this definition:



>  
> 
> We call the following sorts of things toward something: all those
> things said to be just what they are *of* or *than*
> something, or *toward* something in some other way (any other
> way whatsoever). Thus, what is larger is said to be what it is
> *than* another (it is said to be larger than something); and a
> double is said to be just what it is *of* another (it is said
> to be double of something); similarly with all other such
> cases. (*Categories* 7, 6a36-6b.)
> 
> 




As the medievals interpret it, this definition identifies relations in
terms of the predicates by means of which they are signified.
Medievals refer to the predicates in question as 'relative
terms' (*ad aliquid* or 
 *relativa*),[9] 
 and understand them, roughly speaking, as those terms whose true
predication requires a comparison to something other than the subject
of which they are predicated. Thus, 'taller'
(*maius*) counts as a relative term because when we assert of
something that it is taller--that is, when we
predicate the term 'taller' of it--we necessarily do
so in comparison to something else. We don't say merely that
Simmias is taller; we say that he is taller *than* Socrates, or
Theaetetus, or the average man. Similar remarks apply to
'double', as well as to all other relative terms. Borrowing
on the medievals' behalf the notation of first-order logic, we
can make this characterization of relative terms precise by saying
that a term *F* is relative just in case a predication of the
form '*Fx*' is more perspicuously represented as a
predication of the form '*Rxy*'. (If a term is not
relative, then the medievals say that it is *absolute*.)




The first or Platonic definition, therefore, characterizes relations
as the items signified by relative terms--or what we would
nowadays call 'polyadic' or 'many-place
predicates'. The medievals regard this characterization as
initially promising, and often employ it in their writings. In the
end, however, they think it must be rejected--or at least
modified. For according to them, not everything we are prepared to
*call* a relation actually *is* a relation. In rejecting
the first or Platonic definition, the medievals take themselves to be
following Aristotle himself, who rejects this definition on the
grounds that it allows certain substances (namely, heads and hands) to
qualify as relations. As Aristotle says later in chapter 7:



> 
> 
> If the [first] definition of things toward something was adequately
> assigned, then it is exceedingly difficult, or impossible, to reach
> the conclusion that no substance is toward something. But if, on the
> other hand, it was not adequately assigned, and things toward
> something are rather [defined as] *those things for which this is
> their very being: to be toward another in a certain way*, then
> perhaps something may be said about the problem [of heads and
> hands]. (*Categories* 7, 8a29-35.)
> 
> 




Here Aristotle seems to be suggesting that some relative terms fail to
signify genuine relations. Although terms such as
'head' and 'hand' clearly count as
relative--since predications of the form '*x* is a
head' and '*x* is a hand' are more
perspicuously represented as of the form '*x* is a head
of *y*' and '*x* is a hand of
*y*'--these terms signify parts of substances rather
than relations. On the basis of these and other such examples, the
medievals conclude that relations must be identified not with the
items signified by relative terms, but rather with a proper subset of
them.




Of course, this still leaves us with the question of how in general we
are to distinguish genuine relations from the mere significata of
relative terms. Here medieval philosophers think that Aristotle's
second definition--which we italicized in the passage quoted
above--can be of some help. This definition, as it was commonly
interpreted, highlights an important metaphysical or functional role
that relations must play: in addition to being signified by relative
terms, they are what actually serve to relate two (or more)
things--or to put the point in slightly different terms, they are
that *in virtue of which* such things are related. On this
interpretation, Aristotle's definition provides a clear explanation
for why heads and hands fail to qualify as genuine relations. Although
they are signified by relative terms, they do not relate anything. On
the contrary, they are things standing in relations.




The distinction between what actually serves to relate things (i.e.,
genuine relations) and what merely stands in some relation (i.e., the
*relata* of some relation) is often used by medieval
philosophers to explain the difference between the two definitions in
*Categories* 7. Whereas the first or Platonic definition
includes both relations and *relata* (under relational
descriptions), the second or Aristotelian definition includes only
relations. This understanding of the difference between the two
definitions provides the basis for a common medieval distinction
between relations merely according to speech (*relationes secundum
dici*) and relations according to being or nature (*relations
secundum esse*). As Aquinas says, in a well-known passage of his
*Summa Theologiae*:



> 
> 
> Some relative terms--such as 'master' and
> 'slave', 'father' and
> 'son'--are introduced to signify relative
> dispositions themselves (*ipsas habitudines relativas*); these
> terms express things relative *secundum esse*. But other
> relative terms--such as 'mover' and
> 'moved', 'head' and 'headed', and
> terms of this sort--are introduced to signify things on which
> certain relations follow; these terms express things relative
> *secundum dici*. (*Summa Theologiae* I, q. 13, a. 7, ad
> 1.)
> 
> 




Although medieval philosophers generally accept the distinction
between relations *secundum dici* and relations *secundum
esse*, and place heads and hands (and other body parts) in the
former category, they sometimes disagree about the placement of
certain other examples--most notably, knowledge and
perception. Clearly, the names of these accidents are relative terms,
since predications of the form '*x* is knowledge'
or '*x* is a perception' are more perspicuously
represented as of the form '*x* is knowledge of
*y*' and '*x* is a perception of
*y*'. But should we characterize the accidents themselves
as genuine relations (i.e., relations *secundum esse*)? Or
should we rather say that they are qualities on which certain
relations follow (i.e., relations *secundum esse dici*)? Near the
end of *Categories* 7, Aristotle suggests an epistemic
criterion that promises to resolve these questions. But because the
criterion itself is difficult to understand, and there is no clear
alternative to it, debate about these sorts of examples continues
throughout the Middle Ages.



*Categories* 7 ends almost as abruptly as it began. Having
defended his own preferred definition of relations, Aristotle leaves
us with the following cautionary remarks:



> 
> 
> It is perhaps hard to make firm statements on such questions without
> having examined them many times. Still, to have gone through the
> various difficulties is not unprofitable. (*Categories* 7,
> 8b22-24.)
> 
> 




Boethius, whose Latin translation is responsible for transmitting
these words to the Middle Ages, takes Aristotle's remarks here
to be inviting further refinements and improvements on his own
account:



> 
> 
> Aristotle would never have said this if he were not prompting us to
> further reflection and to even greater exercise of subtlety. Because
> of his exhortation, we shall not hesitate in the least to raise
> [further] questions and offer [our own] solutions to them in other
> places. (*In Categorias Aristotelis*, 238.)
> 
> 




Like Boethius, medieval philosophers are happy to take up this
invitation--and, indeed, as we shall see, to develop
Aristotle's account of relations in some striking and original
ways of their own.


## 3. Relations in Medieval Philosophy




As we have seen, Aristotle begins his discussion of relations in the
*Categories* with the assumption that relations are a certain
type of accident, and then inquires after their proper
characterization, eventually arriving at the view that they are a
proper subset of the entities signified by relative
terms--namely, all and only those that actually serve to relate
two or more things. Medieval philosophers tend to reverse Aristotle's
approach, beginning where he ends. That is to say, they tend to start
their discussion with the view that relations are entities that are
both signified by relative terms and play a certain metaphysical
role, and then proceed to ask about the precise nature and
ontological status of these entities--in particular whether they must,
in every case, be identified with Aristotelian accidents. There are at
least two reasons for this reversal. First, the medievals see nothing
in Aristotle's preferred characterization of relations that requires
relations to be accidents (or indeed, entities of any ontological
type). Second, and perhaps more importantly, the medievals think that
there are certain situations that require relations to be something
other than accidents (more on this in Section 5 below).


### 3.1 The Rejection of Polyadic Properties




Although medieval philosophers generally regard Aristotle's
characterization of relations as ontologically neutral, there is one
type of entity they standardly assume that relations could never
be--namely, polyadic or many-place properties. For all their
differences, the medievals tend to agree on the following point: polyadic properties
cannot exist outside the mind.




Historians of philosophy sometimes speak as if the medievals could not even
have possessed the concept of a polyadic property--as if
conceiving of relations in this way only became possible in the
nineteenth and twentieth centuries, with the advent of a formal logic
of relations and multiple quantification (Cf. Weinberg 1965,
esp. 61-63). But this is surely mistaken. What recent advances in
logic have made possible is not the concept of a polyadic property,
but merely its representation within a formal system. Moreover, as we
have already seen, the notion of a polyadic predicate, from which
our own concept of a polyadic property is derived, corresponds exactly
to the medieval notion of a relative term. And if that by itself
weren't enough, there is the fact that medievals habitually speak of
relations in polyadic terms, explicitly comparing them to a road
that runs between two cities (*via*), or to a palisade running
between two watchtowers (*inter*-*vallum*) (Cf. Albert
the Great, *Liber de praedicamentis*, 241a-241b; Aquinas
*Summa Theologiae* I, q. 28, a. 4, obj. 5; and the texts of
Harclay). Nor do they take themselves to be original in this
regard. On the contrary, they take themselves merely to be following a
suggestion of Aristotle's in the *Physics*, which itself just
appears to be part-and-parcel of the common sense conception of
relations (see *Physics* III, 202b11-15). As Peter Aurieol says
at one point:



> 
> 
> In the third book of his commentary on the *Physics*, comment
> twenty, the Commentator [=Averroes] says that a relation is a
> disposition (*dispositio*) existing *between* two
> things. But even apart from him it is clear that fatherhood is
> conceived of as if it were a kind of medium connecting a father with
> his son. And the same is true of other relations. (*Scriptum super
> Primum Sententiarum*, fols. 318va-b. See Henninger
> 1989, 153-4, n. 12, for the Latin text.)
> 
> 




In light of passages such as these, it is hard to take seriously the
suggestion that the medievals--or indeed, any reflective
beings--lack the concept of a polyadic property.




In addition to possessing this concept, there is evidence that at
least some medieval philosophers also posit the existence of polyadic
properties. In an important paper, Heine Hansen has argued that this
is certainly true in the case of the thirteenth-century philosopher,
Nicholas of Paris. (See Hansen 2012.) And further research may well
turn up other medieval philosophers holding the same view. At present,
however, Nicholas appears to be something of an anomolous case.




Indeed, despite possessing the concept of a polyadic property, medieval
philosophers are often explicit in their denial of anything in extramental
reality exactly corresponding to such a concept. If we return to the passage just
quoted from Auriol, we can see that, immediately after describing
relations as a sort of 'interval' (*intervallum*),
he proceeds to deny the existence of any such intervals in extramental
reality.



> 
> 
> It appears that a single thing, which must be imagined as some sort of
> interval (*intervallum*) existing between two things, cannot
> exist in extramental reality, but only in the intellect. [This appears
> to be the case] not only because nature does not produce such
> intervals, but also because a medium or interval of this sort does not
> appear to be *in* either of the two things [it relates] as in a
> subject, but rather *between* them where it is clear that there
> is nothing which can serve as its subject. (*Scriptum super Primum
> Sententiarum*, fols. 318va-b.)
> 
> 




As this passage helps to make clear, the root of the standard medieval
objection to polyadic properties is ontological in nature, stemming
from a particular conception of subjects and attributes. Following
Aristotle, the medievals divide extramental beings into
substances and accidents, a division which they take to be both exclusive and
exhaustive (since it is given in terms of a contradictory pair of
properties, namely, *being in a subject* vs. *not being in a
subject*) (Cf. *Categories* 1, 1a20-1b6). Now in the case
of substances, it is perhaps clear that they are not polyadic in
nature. But medievals typically think the same is also true of their accidents.
Thus, as Aquinas says at one point in his *Sentences*
commentary, in response to the question whether it is possible for
"numerically one and the same relation" to belong to
belong to two subjects at a time: "No, this cannot be, for one
accident cannot belong to two subjects" (*In Sent.* I,
d. 27, q. 1, a. 1, ad 2).




Although the medievals' objection to polyadic properties stems in large
part from ontological considerations, they also offer
other reasons for rejecting them. Sometimes these involve
considerations of theoretical parsimony. As Ockham is so fond of
saying: "It is futile to do with more [entities] what can be
done with
 fewer".[10]
 Other times, however, these reasons involve phenomenological or
epistemological considerations. Albert the Great, for example, objects
to polyadic properties on the grounds that we are not presented in
experience with anything but individuals and their so-called absolute
attributes--that is, substances and their quantities and
 qualities.[11]


### 3.2 Anti-Realism about Relations




The rejection of polyadic properties might seem to commit one to a
form of anti-realism about relations--that is, to the view that
relations exist only in the mind. This was, in fact, the position of
Peter Auriol. If there are no polyadic properties or
'intervals', he assumes, then there is nothing in
extramental reality to correspond to our relational concepts. But in
that case, he says, "a relation cannot be posited except in
apprehension
 alone".[12]




Given the current state of medieval research, it is difficult to know
how widespread this (or any other) form of anti-realism was during the
Middle Ages. Shortly after Auriol's death, certain aspects of
his anti-realism appear to have met with considerable success. Ockham
and his followers, for example, accept Auriol's view that
relations exist only in the mind as concepts or beings of reason
(*entia rationis*), though unlike Auriol, as we shall see, they
think this is compatible with saying that things are related
independently of the activity of the mind. And we can find echoes of
the same sort of view in the writings of early modern
 philosophers.[13]




Prior to the fourteenth century, however, it is difficult to identify
any unambiguous representatives of anti-realism, at least in the Latin
West. Albert the Great claims to find something akin to Auriol's
argument in the writings of Alfarabi (d. 950) and Avicenna (d. 1037),
but he does not say whether either of these figures actually endorses
it or merely presents it for consideration (See *Liber de
praedicamentis*, 222b-223a).
 Again, Aquinas suggests that a similar view can be found in the
writings of Gilbert of Poitiers and his followers, the so-called
Porretani, but his suggestion is controversial, and in any case
Aquinas reports that Gilbert later retracted the view in the face of
theological controversy (Cf. *Summa Theologiae* I, q. 28,
a. 2, corpus; cf. also Veckley 1965, 28-9). Unless further research
alters the current picture, therefore, it would appear that, although
anti-realism may have had something of a foothold in the Arabic
speaking world, most notably among the members of a group of orthodox
Muslim theologians known as the Mutakallimun, in its more radical
forms it was always a minority position in the Latin
 West.[14]


### 3.3 Realism without Polyadic Properties




If we take anti-realism to be the view that (a) nothing in extramental
reality corresponds to our relational concepts and (b) nothing is
related independently of the activity of the mind, it is not hard to
see why most medieval philosophers would reject it. Like most of us,
they recognize the utter implausibility of saying that facts like
Simmias's being taller than Socrates are somehow dependent on
the activity of the
 mind.[15]
 Indeed, as they often point out, this sort of anti-realism is
inconsistent with even such basic facts as the real structure of the
universe, the existence of real composition, causality, spatial
proximity, or even the objectivity of mathematical knowledge. Speaking
for the majority, Ockham says in his *Ordinatio*:



> 
> 
> The intellect does nothing to bring it about that the universe is one,
> or that a whole is composed [of its parts], or that causes in spatial
> proximity [to their effects actually] cause [their effects], or that a
> triangle has three [angles], etc. ... any more than [the
> intellect] brings it about that Socrates is white or that fire is hot
> or water cold. (*Ordinatio* I, d. 30, q. 1 in *Opera
> Theologica* iv, 316-317. Cf. also Scotus, *Ordinatio* II,
> d. 1, q. 5, n. 224 in *Opera Omnia*; Aquinas, *De
> potentia*, q. 7, a. 9.)
> 
> 




In addition to such common sense considerations, the medievals also
identify various philosophical and theological grounds for opposing
anti-realism about relations. Here we shall mention just two. First, it
is hard to see how such anti-realism is to be squared with the
standard interpretation of the Aristotelian categories. "For
nothing is placed in a category" as Aquinas says "unless
it is something existing outside the soul" (*De
potentia*, q. 7, a. 9). Again, this sort of anti-realism seems
incompatible with the orthodox Christian view of the Trinity. Indeed,
it is precisely for this reason that the existence of real relations
in God is affirmed at the Council of Rheims in 1148, and that shortly
thereafter it becomes customary to say that the denial of this entails
a form of heresy known as Sabellianism (Cf., e.g., Aquinas, *Summa
Theologiae* I, q. 28, a. 1 and *De potentia* q. 8,
a.1).




But of course, all of this raises a question. How is it that
medieval philosophers maintain a form of realism about relations while
at the time rejecting the existence of polyadic properties? In
particular, what do they say about arguments such as Auriol's,
which in effect deny the coherence of realism without polyadic
properties? Here it is useful to consider the work of Albert
the Great. Although he writes a generation before Auriol, he explicitly
addresses his sort of argument, and the reply he offers appears to be
perfectly standard.




According to Albert, the problem with arguments of the sort Auriol
gives is that they rely on a questionable assumption, namely: *if
there are no real polyadic forms or properties, then there is nothing
in extramental reality to correspond to our relational concepts*.
This assumption would be true, he suggests, if our conceptual
framework displayed an exact isomorphism to the structure of the
world. But it does not, and hence there is no reason in principle why
a polyadic concept cannot have something non-polyadic corresponding to
 it.[16]
 In making this sort of reply, Albert aligns himself with a medieval
tradition of rejecting the view that *for every distinct type of
concept there is a distinct type of entity*. This view is often
associated in medieval philosophy with Plato and his followers, and by
the time of Ockham it is regarded, somewhat hyperbolically, as the
source or root (*radix*) of the greatest errors in
philosophy. "To multiply beings according to the multiplicity of
terms" Ockham says "is erroneous and leads far
away from the truth" (*Summa Logica* I, c. 51 in
*Opera Philosophica* i, 171; cf. Loux, 1974, 171).




For most medieval philosophers, then, the question is not
*whether* there are any things in extramental reality
corresponding to our relational concepts, but *what* these
things are like in themselves. Having ruled out the possibility of
their being polyadic, however, the medievals don't leave
themselves with many options. Indeed, because they accept a broadly
Aristotelian ontology, they have no choice but to identify relations
with either individual substances or their monadic properties.




As we noted earlier, philosophers throughout the medieval period are
attracted to the view suggested by Aristotle's
*Categories* that relations (or the items corresponding to our
relational concepts) are accidents. Of course, as we also noted, the
medievals cannot take this view to be perfectly general, since they
also admit cases, especially theological ones, which involve relations
but no accidents. But even so, until the fourteenth century, this view
is taken to apply so widely that it functions as a kind of paradigm
for thinking about relational situations. Thus, if a predication of
the form '*aRb*' is true, it is generally assumed
that what makes it true is a pair of individuals, *a* and
*b*, and a pair of accidents, *F* and *G*. As we
shall see in the next section, it is within the context of this
paradigmatic analysis of relational situations that one of the most
hotly disputed and intractable debates of all the Middle Ages
arises.


## 4. Paradigmatic Relational Situations




In his commentary on the *Categories*, Boethius highlights an
interesting feature of Aristotle's discussion of relations. Whenever
Aristotle refers to the category of relations, he always does so in
the plural ('things toward something'), thereby departing
from his usual practice of referring to categories in the singular
('substance', 'quantity',
'quality', etc.). This certainly reinforces the
paradigmatic analysis of relational situations, which requires
relations to come in pairs. But Boethius thinks there is a further
reason why Aristotle refers to the category in this way--namely,
because its members are unique in that they cannot be
*understood* to exist by themselves:



> 
> 
> Things toward something [or relations] cannot be grasped by the
> intellect by themselves or individually, so that we could say that
> things toward something exist individually. Whatever is known
> regarding the nature of a relation must be considered together with
> something else. For example, when I speak of a master, this by itself
> means nothing if there is no slave. The naming of one relative
> immediately brings with it another thing toward something. (*In
> Categorias Aristotelis*, 217.)
> 
> 




Following Boethius, medieval philosophers often take Aristotle's use
of the plural to indicate that the category of relations must be at
least *conceived of* as containing pairs of correlatives or
converse relational accidents (see, e.g., Abelard, *Logica
'ingredientibus'*, 80-95; and Albert the Great,
*Liber de praedicamentis*, 22). As we shall see, this point
becomes important for their understanding of those relational
situations that do not conform to the paradigm analysis of the
*Categories*.




For now, however, let us focus on the those relational situations
that do conform to the paradigm. As the medievals recognize, there are
a number of questions that can be raised about the relations they
involve--call them 'paradigmatic relations'. What sort
of monadic properties or accidents are they, and how are we to
understand them? As it turns out, there are only two main
positions one can take. For convenience, we shall refer to them,
respectively, as 'reductive' and 'non-reductive
realism'.




According to reductive realism, which is the simplest or most
ontologically parsimonious form of realism (without polyadic
properties), paradigmatic relations are to be identified with
ordinary, non-relational accidents--that is to say, with
accidents falling under Aristotelian categories other than
relation. Thus, if Simmias is taller than Socrates, this is to be
explained by their respective heights, which fall under the category
of quantity. Again, if Socrates is similar (say, in color) to Theaetetus,
this is to be explained by Socrates's and Theaetetus's
particular colors, which fall under the category of quality. And so on
for other paradigmatic relations.




According to non-reductive realism, by contrast, paradigmatic
relations are to be identified not with ordinary, non-relational
accidents, but rather with accidents of a *sui generis* type.
Thus, if Simmias is taller than Socrates, this is to be explained by
appealing to a pair of *sui generis* accidents that are distinct
from, but nonetheless necessitated by, Simmias's and
Socrates's heights. Again, Socrates's similarity to Theaetetus
is to be explained not by their respective colors, but by a pair of
*sui generis* accidents necessitated by them. And so on for the
relations involved in other paradigmatic relational situations.




In the medieval discussion of relations, it is this
difference--the difference between reductive and non-reductive
realism--that constitutes the greatest divide among philosophers,
and representatives of both positions can be found throughout the
medieval period. Peter Abelard (d. 1142) and Albert the Great
(d. 1280) are, perhaps, the clearest representatives of reductive and
non-reductive realism, respectively, in the early and high Middle
Ages, whereas William Ockham (d. 1347) and John Duns Scotus (d. 1308)
are well-known representatives of these positions in the later Middle
Ages. Again, some philosophers appear to have held different
positions at different stages of their career. Henry Harclay
(d. 1317), for example, began his career as a staunch defender of
Scotus's non-reductive realism, but by the end of it gravitated
towards a position that is much closer to, and in many ways
anticipates, Ockham's specific form of reductive
 realism.[17]




Beginning in the mid-thirteenth century, the debate between reductive
and non-reductive realists is often carried out in the context of the
discussion whether relations are identical to their foundations. (This
is the context, for example, in which Scotus and Ockham develop their
positions.) By the mid-thirteenth century, moreover, it is customary
to admit at least a conceptual distinction between relations and the
non-relational accidents in virtue of which these relations hold. The
relevant non-relational accidents come to be referred to as the
'grounds' or 'foundations' of relations
(*fundamenta*), since their possession is thought to
necessitate the holding of relations. Hence, in this context, the
important question is whether the distinction between relations and
their foundations is *merely* conceptual.




We might expect that reductive realists would always answer this
question in the affirmative, whereas non-reductive realists would
always answer it in the negative. It turns out, however, this is not
the case. For reasons to be explained below, there are some reductive
realists who, in spite of rejecting that relations comprise a *sui
generis* type of monadic property, nonetheless maintain that
relations are in an important sense really distinct from their
foundations. In light of this complication, we shall continue to cast
the medieval debate about the nature of paradigmatic relations in terms
of reductive and non-reductive realism rather than in terms of the
identity or real distinctness of relations and their foundations.


### 4.1 Reductive Realism without Polyadic Properties




As mentioned above, on the simplest or most ontologically
parsimonious form of realism without polyadic properties, what we are
calling 'reductive realism', paradigmatic relations are
identified with ordinary, non-relational monadic properties or
accidents. Now we might expect this form of realism to appeal to
anyone committed to realism without polyadic properties. After all,
failure to reduce such relations to such accidents threatens to make
them mysterious. If paradigmatic relations are monadic properties, but
not ordinary, non-relational accidents, then how are we to understand
them?




Considerations of intelligibility certainly play a role in the debate
between medieval reductive and non-reductive realists. As we shall
see, non-reductive realists typically recognize that the chief
difficulty for their view lies in specifying the precise nature of the
*sui generis* monadic properties to which they appeal. More often
than not, however, reductive realists object to non-reductivists on
grounds of theoretical parsimony. This is certainly true in the case
of Abelard and Ockham, whom we earlier identified as reductive
realists. In the case of Abelard, such considerations are not
explicitly formulated, though they are part-and-parcel of his general
approach to metaphysics and philosophy of
 language.[18]
 In the case of Ockham, by contrast, the appeal to parsimony is much
more explicit, often taking the form of what has come to be known as
Ockham's razor: "Plurality should not be assumed without
 necessity" (*Ordinatio* I, d. 30, q. 2 in *Opera
Theologica* iv, 322).




Another consideration that plays an important role in the debate
between reductive and non-reductive realists has to do with the proper
interpretation of authoritative texts. Since medieval philosophers
develop their theories of relations in the course of reflecting on
Aristotle's *Categories*, they often present their views as the
one actually suggested (or even required) by the text itself. Abelard,
for example, suggests that his reductive theory is a direct
consequence of Aristotle's second definition of relations. Thus, when
Aristotle characterizes relations in *Categories* 7 as
"those things for which this is their very being: being toward
something in a certain way," Abelard argues that this should be
interpreted to mean that relations are items that *make other
things to be relatives*. Just as certain attributes (say, being
red, juicy, and sweet) make their subjects good, so too, he suggests,
certain accidents (such as being-six-feet-tall and five-feet-ten)
relate their subjects (See *Logica
'ingredientibus'*, 216-17. For further discussion, see
Brower 1998).




Textual considerations and direct appeals to Aristotle's
authority play a much bigger role in early medieval debates than they
do in the high and later Middle Ages, when literal commentary was no
longer the dominant form of philosophical literature. Nonetheless,
these sorts of considerations continue to be important throughout the
Middle Ages. Even Ockham, in his *Summa logicae*, addresses the
issue of whether Aristotle should be interpreted as holding some form
of non-reductive realism:



> 
> 
> There are many theologians of this opinion, and at one time even I
> believed it to have been Aristotle's opinion. Now, however, it seems
> to me that the contrary opinion follows from his principles. (*Summa
> logicae* I, c. 49 in *Opera Philosophica* i, 154. Cf.
> Henninger 1989, 120.)
> 
> 




One final argument used by reductive realists to motivate their
position is worth mentioning here. This argument has to do with the
nature of relational change. Almost all medievals accept the
intuitively plausible view that things can acquire (and lose)
relations without undergoing any real (as opposed to merely Cambridge)
change. Thus, Socrates can come to be shorter than Simmias solely in
virtue of a change in Simmias's
 height.[19]
 As reductive realists such as Ockham often point out, this fact about
relational change admits of a ready explanation on their position,
since Socrates's relation is nothing over and above
Simmias's and Socrates's heights, the latter of which
never changes. By contrast, the same fact appears to pose a serious
challenge for the non-reductive realist. For if Socrates's
relation is something distinct from Simmias's and
Socrates's heights, then apparently Socrates must acquire
something new when Simmias's height increases. But then,
contrary to the original intuition, the non-reductive realist must say
that Socrates undergoes a real change after all.




These considerations are not the only ones that reductive realists
appeal to, but they are among the most important for understanding
their position, and we shall rely on them in what follows to motivate
and clarify the alternative position of the non-reductive
 realists.[20]




Before leaving the topic of reductive realism, we need to take account
of a slight complication. In our discussion to this point, we have been
speaking as if reductive realists always intend to identify individual
paradigmatic relations such as tallness or shortness with pairs of
monadic properties *taken jointly*. We have done this because,
at least initially, this seems to be the most natural way to
understand their position. After all, it is not Simmias's height
taken by itself that necessitates his being taller than anyone; it is
only his height taken together with that of another, say,
Socrates. And yet, despite the naturalness of this way of construing
reductive realism, it is not the way its proponents typically
understand
 it.[21]
 On the contrary, instead of identifying relations such as tallness or
shortness with pairs of monadic properties taken jointly, reductive
realists typically identify them with the individual members of such
pairs. Thus, if Simmias is taller than Socrates, it is only Simmias's
height that is identified with Simmias's tallness (and hence said to
correspond to the concept 'taller than'), whereas
Socrates's height is identified with Socrates's shortness (and hence
said to correspond to the concept 'shorter than'). But how
is this possible, especially if the properties in question are
supposed to be just ordinary heights?




Reductive realists recognize the worry here, as well as the initial
attraction of identifying relations with pairs of accidents taken
jointly. As evidence, consider the following passage from one of Henry
Harclay's later works. Here Henry not only entertains the suggestion
that relations could be identified with pairs of monadic properties,
but draws on a common medieval view about number to make it
plausible:



> 
> 
> A relation is clearly a reality distinct from one foundation, but
> not a reality distinct from both foundations. For when one white thing
> [such as Simmias] is posited and then another white thing [such as
> Socrates] is posited, there is a relation beginning at that point.
> Thus, the similarity is not a [single] whiteness, but two whitenesses
> existing at once. Indeed, the two whitenesses can be sufficient to
> constitute a single species in the genus of relation. Just as two
> discrete unities of whatever magnitude are sufficient for producing a
> single species of number, so too the same thing holds for the case at
> hand. Again, just as plurality or multitude is not a reality different
> from the constituents of the plurality or multitude, neither is the
> similarity [of the white things] different [from their
> whitenesses]. ("Utrum Dei," n.
>  46)[22]
> 




Despite Henry's attraction to this sort of view, he
feels compelled to reject it on the grounds that it leads
to certain absurdities. Drawing on Avicenna for support, he says:



> 
> 
> Avicenna argues against this on the basis of relations involving
> inferiority and superiority, for it is clearer in those cases than it
> is for other relations, such as those involving equals. For fatherhood
> is in the father alone and not in the son ... and the same thing
> holds of sonship. Therefore one must hold that there are two relations
> [here]. And in this case it is clear that the relation is not the same
> reality as [the two] foundations because the relation is not in
> everything in which the foundation of the *relatum* is. This is
> the case, too, for relations involving equals, even if it is not
>  as
>  clear. ("Utrum Dei," n. 46)
> 
> 




Henry's worry here can be expressed as follows. If a relation such as
fatherhood is identified with a pair of accidents, then it will follow
that fatherhood is partly in the son (since the pair is such that one
of its members is in the son). But this is absurd. Fatherhood is an
asymmetrical relation (or as Harclay prefers to put it, a relation
involving superiority and inferiority). That is to say, fatherhood is
such that if an individual *a* has it in respect of another
individual, *b*, then *b* cannot have fatherhood in
respect of *a*. But according to Harclay, this is tantamount to
saying that "fatherhood is in the father alone and not in the
son". And of course he doesn't think the point is restricted to
fatherhood. On the contrary, it also holds true of all relations,
including symmetrical relations or "those involving equals, even
if it is not as clear" ("Utrum Dei," n. 47).




But if relations are not to be identified with pairs of accidents, how
can reductive realists think of them in terms of ordinary,
non-relational accidents? After all, the only other option seems to be
identifying them with ordinary accidents taken
individually--which as we have seen is problematic. An ordinary
accident such as Simmias's height cannot necessitate Simmias's being
taller than anyone. Indeed, it was precisely this point that led us
to conclude that 'relative-making' is a description best
reserved for pairs of accidents taken jointly.




Note, however, that it is open to the reductive realist to respond by
saying that, although Simmias's height is not by itself
relative-making, it is nonetheless *potentially*
relative-making. In that case, however, the reductive realist can insist
that in the presence of another height, such as Socrates's, it
becomes *actually* relative-making--that is to say, it
comes actually to relate Simmias to Socrates. This suggests a third
option for reductive realists: to maintain that relations are to be
identified with ordinary, non-relational accidents *in certain
circumstances*.




In fact, it is this last sort of view that most reductive realists
during the medieval period actually hold. Indeed, as we are
interpreting Harclay, this is precisely the view he goes on to develop
in the discussion we have been
 following.[23] 
 Simmias's being taller than Socrates is to be identified, on his
view, not with Simmias's height *tout court*, but with
Simmias's height in certain circumstances--including the
circumstance that Socrates is
 five-feet-ten.[24]
 To put Harclay's view in a slightly different way, which will
help to bring out its semantic consequences, we might say that in a
world in which Simmias exists by himself a predication of the form
'Simmias is taller than Socrates' will be false, and hence
nothing will correspond to the relational term 'taller
than'. But in a world in which Simmias is six-feet-tall while
Socrates is five-feet-ten, a predication of the same form will be
true, and in this world there will be something corresponding to
'taller', namely, Simmias's height. In such a world,
medieval philosophers would say that 'taller' primarily
signifies Simmias's height, but indirectly signifies or connotes the
height of
 Socrates.[25]




As it turns out, it is not only reductive realists, but non-reductive
realists as well that speak of relations as accidents of single
subjects--that is, of Simmias's tallness as something belonging
only to him, of Socrates's shortness as something belonging only to
him, and so on for all other particular relations. In light of what
has just been said, therefore, we can clarify our understanding of the
paradigmatic analysis of relational situations. As indicated earlier,
this analysis requires that when a predication of the form
'*aRb*' is true, what makes it true is a pair of
individuals, *a* and *b*, and a pair of monadic
properties or accidents, *F* and *G*. Reductive
realists, as we have just seen, identify *F* and *G*
with ordinary categorial accidents, whereas non-reductive realists
identify them with accidents of *sui generis* type. But however
we construe the nature of these accidents, both reductive and
non-reductive realists deny that it is the pair of
accidents--that is, *F* and *G* taken
jointly--that corresponds to our relational concepts. On the
contrary, they say, *F* directly corresponds to one of our
relational concepts, whereas *G* directly corresponds to its
converse. Thus, when Simmias is taller than Socrates, reductivists and
non-reductivists are agreed that it is an accident of Simmias that
directly corresponds to (or as they would prefer to say, is primarily
signified by) the concept 'taller than', and an accident
of Socrates that is primarily signified by the concept 'shorter
than'. To the extent they disagree, therefore, their
disagreement concerns only whether the accidents primarily signified
by relational concepts can also be signified by ordinary
non-relational concepts (and if so, precisely how and under what
circumstances).




With these clarifications in mind, let us turn now to the
considerations that led some medieval philosophers to reject reductive
realism in favor of some form of non-reductivism.


### 4.2 Non-Reductive Realism without Polyadic Properties




Of all the considerations favoring non-reductive realism, perhaps none
is more compelling than the intuition that relations have a different
nature or "quiddity" from that of ordinary categorial
accidents. As we have seen, medieval philosophers recognize that
predications involving relative terms (such as '*x* is
taller') are incomplete in a way that monadic or absolute
predications (such as '*x* is white') are not. They
also recognize, moreover, that unlike absolute terms relative terms
come in pairs or sets. Every relative term has a correlative, and the
meaning of the one (say, 'taller') cannot be understood in
isolation from the meaning of the other ('shorter'). These
facts about relative terms and predication are often thought to show
that their significata have a distinct nature or quiddity from the
ordinary, absolute accidents that serve as their foundations. Here
again it is useful to quote Harclay:



> 
> 
> Avicenna (*Metaphysics* III, the chapter on relation) says that
> a relation has its own quiddity distinct from the quiddity of its
> foundation. Therefore it is a distinct reality. Moreover, in the
> *Categories* Aristotle says that the being associated with
> relatives is being-toward-something-else (*ad aliud se
> habere*). But it is not the case that the being of a foundation
> is being-toward-something-else. Therefore they are not the same.
> ("Utrum Dei," n. 43.)
> 
> 




Reductive realists can, of course, explain the uniqueness of relative
terms and predications without introducing relations over and above
their foundations. For they can always (a) assert that relative terms
are associated with concepts whose content is distinct from that of
any non-relational or absolute concepts, and yet (b) insist that this
is all that is required to explain the peculiar nature of relative
terms and predications. Even so, the question remains: If Simmias's
being taller than Socrates is nothing ontologically over and above
Simmias's and Socrates's heights, why do we represent it *as
if* it were? At this point, it would seem that reductive realists
have no choice but to appeal to our psychological make-up. We simply
do (or at least can) represent one and the same situation in two very
different ways. As Ockham says in one of his *Quodlibetal*
questions, using a slightly different example:



> 
> 
> Socrates is similar to Plato by the very fact that Socrates is white
> and Plato is white ... Yet, despite this, the intellect can
> express these many absolute things by means of concepts in diverse
> ways: in one way, by means of an absolute concept, as when one says
> simply 'Socrates is white' or 'Plato is
> white'; in a second way, by means of a relative concept, as when
> one says 'Socrates is similar to Plato with respect to
> whiteness'. (*Quodl*. VI, q. 25 in *Opera
> Theologica* ix, 679.)
> 
> 




Even if one does not find this sort of appeal to psychology
implausible, the non-reductive realists would seem to have a more
satisfying reply to our original question. For they can say that the
reason we represent relational situations as distinct from
non-relational situation is because they *are*
distinct. Indeed, non-reductive realists can add that the logical
incompleteness of predicates such as 'taller' (and its
correlative, 'shorter') calls our attention to precisely
what makes these situations distinct--namely, the *sui
generis* accidents that are possessed by Simmias and Socrates in
addition to their respective heights.




Now as most non-reductive realists recognize, there is a difficulty
posed by their position--namely, that of giving a perspicuous
account of the nature of paradigmatic relations or relational
accidents. But they often attribute this difficulty to the fact that
the nature in question is *sui generis*. As Albert the Great
says, when he turns to the discussion of relations in his commentary
on Aristotle's *Metaphysics*:



> 
> 
> It is difficult for us to speak about [the category of] toward
> something or relation because it has a nature and being altogether
> different from the genera of being which have been considered so far
> [namely, substance, quantity, and quality]. (*Metaphysica*,
> 266a.)
> 
> 




Given that non-reductivists construe the nature of paradigmatic
relations as *sui generis*, it is not surprising that they feel
the need to resort to metaphors to describe it. Albert himself appeals
most often to a visual metaphor of outward-looking-ness
(*respectus*), and describes individual relations as that in
virtue of which a subject "looks out toward another"
(*respicit ad aliud*). Other philosophers, rely on other
metaphors and variously describe the nature of relations as a kind of
directionality or toward-ness (*aditas*), as a type of
disposition or way of holding oneself (*habitudo* or
*relativa habitudo*), or again as the source or principle of
structure and order 
 (*ordinatio*).[26]




Of all these metaphors, the ones involving directionality--or
intentionality--are likely to be the most helpful to us. For
there are contemporary philosophers who characterize
intentionality, not in terms of a polyadic or many-place property, but
as a *sui generis* type of monadic
 property.[27]
 According to these philosophers, intentionality is a property whose
intrinsic nature is such that, when it is exemplified by a subject in
appropriate circumstances, which include the presence of an
appropriate object, it relates its subject to the object in
question--in particular, it relates the subject to it as thinker
to object thought. This analogy is useful because non-reductive
realists typically regard intentionality as a special case (or type)
of relation. According to them, all paradigmatic relations (or at
least all those that qualify as *sui generis* accidents) are
properties whose intrinsic nature is such that their exemplification
in the appropriate circumstances will relate their subjects to
something else. It is just that in certain cases, these properties
relate their subjects specifically as thinkers to objects thought.




In the end, therefore, it would appear that the non-reductive
realists do have something to say in response to the
reductivists' worries about the mysterious nature of relations
and their appeals to considerations of theoretical parsimony. But what
about the phenomenon of relational change?




As we said earlier, medieval philosophers share the intuition that
the acquisition (or loss) of relations can occur without
any real change on the part of their subject. To explain this intuition,
some non-reductivists think it is enough to say that a relation can be
acquired (or lost) without its subject undergoing any real change
*with respect to its absolute accidents*. Thus, the example of
Socrates's coming to be shorter than Simmias is sometimes glossed
by saying that, with respect to absolute accidents, it is true that
Socrates comes to be shorter than Simmias solely in virtue of a real
change in Simmias. Nonetheless, if we consider Socrates's
relative accidents as well, such non-reductivists will say that
Socrates has, in fact, undergone a real 'relative'
change--that is, a real change with respect to one of his
relations.
 [28]




But non-reductivists need not take this line. Some, more sympathetic
to the reductive realist position on relational change, take cases
like that involving Simmias and Socrates as an opportunity to further
clarify the nature of relations. Albert the Great, for example, argues
that relational accidents are so closely tied to their foundations
that they are acquired along with them (cf. *Metaphysica*,
266b-267a, ad 1 and Brower 2001). Thus, if Socrates comes into
existence in a world all by himself at a height of
five-feet-ten-inches-tall, not only will he possess a certain quantity
(namely, his height), but also, in virtue of it, he will possess a
*sui generis* relational accident. Obviously, in such a world,
the relational accident will not *actually* relate Socrates to
anyone or anything (since there's nothing for it to relate him
to)--though it will explain his being *potentially*
related to other things. Suppose, however, that Simmias now comes
into existence in this world at six-feet-tall. In that case,
Socrates's relational accident will come actually to relate him to
Simmias as shorter. Even so, Albert insists, Socrates himself will not
have undergone any real change, relative or absolute. Not surprisingly
these sorts of consideration lead Albert to distinguish sharply
between relational accidents and relations (which are just relational
accidents in certain circumstances), and to suggest that strictly
speaking we should think of the relevant Aristotelian category as
comprised, not of relations, but of relational accidents. These same
considerations also help to explain why, in referring to the members
of this category, Albert prefers the concrete terms 'toward
something' (*ad aliquid*) and 'relative'
(*relativum*) to the abstract term 'relation'
(*relatio*)--for relational accidents are 'toward
something' or outwardly directed even if they are not actually
relating their subject to anything. (*Metaphysica*, 267a,
ad. 3.)




Like Albert, Aquinas too suggests that there is a close connection
between relational accidents and their foundations. Following Albert's
terminology, he suggests that in virtue of possessing a specific
height or quantity, an individual such as Socrates is
*potentially* equal to all those who have the same height, and
*potentially* unequal to--that is, shorter or
taller than--all those with a different height (*In V
Phys*., lect. 3, n. 8). Like Albert, moreover, Aquinas says that
such an individual can come to be *actually* equal (or unequal)
to another solely in virtue of a change in that other, and concludes
from this that relations must be in their foundations as in a root
(*in radice*) (*In V Phys*., lect. 3, n. 8). On some
interpretations, Aquinas's metaphor can be explicated by saying that
although relations and their foundations differ formally--that
is, involve different accidental forms or properties--nonetheless
their act of being (*esse*) is the same. Thus, when Socrates
comes to be shorter than Simmias as a result of Simmias's growth,
Socrates does not undergo a change properly speaking, since he does
not acquire any new act of being. Rather, the "old" act of
being of his height (sometimes referred to as the
'*esse-in*' of the relation) merely acquires a new
determination (sometimes referred to as the
'*esse-ad*' of the relation), in this case
 to
 Simmias.[29]




We have seen enough to appreciate the main ontological
differences that divide reductive and non-reductive realists, and even
some of the differences that divide non-reductivists among themselves.
As we indicated earlier, in the later Middle Ages these differences tend
to emerge in the context of the debate over whether relations are
identical with their foundations. As we also indicated, however, we have
to be careful not to assume that how philosophers answer this
question provides an infallible guide to their positions. For although
reductivists typically affirm that relations are identical to their
foundations, and non-reductivists typically deny it, there are some
reductivists who side, at least verbally, with the non-reductivists on
this question. Thus, the later Harclay, as we are interpreting him, is a
reductive realist--he not only rejects the existence of
*sui generis* relational properties, but maintains that in
paradigmatic relational situations, substances are related by their
ordinary, non-relational accidents. Nonetheless, Harclay denies that
relations are identical with their foundations, since he thinks this
would amount to saying that relations can be straightforwardly
identified either with ordinary accidents taken by themselves or with
pairs of such accidents taken jointly.


 

Again, as we shall point out in the next section, a number of later
medieval reductivists, including Ockham, eventually come to reject the
traditional Aristotelian characterization of relations as items that
relate substances in favor of the view that relations are items
existing only in the mind (as concepts). As we shall see, this does
not mean that Ockham and others deny that there are extramental
grounds for our relational concepts or even that things can be related
by their foundations independently of the mind. On the contrary, it
means only that, unlike their predecessors, they refuse to call
anything a relation merely because it grounds a relational concept. The fact that
these reductivists regard relations as concepts, however, explains why
they too are unwilling to identity relations with their foundations:
for the relevant foundations, according to these philosophers, are
ordinary, extramental accidents, and obviously no concept (or act of
the mind) could be identical with them.




Finally, it must be noted that even the notions of identity and real
distinction come to be the subject of controversy during the high and
later Middle Ages, and this too has the result of complicating the
debate over whether relations are identical to their
 foundations.[30]
 For all these reasons, therefore, we must be careful not to identify
too closely the debate between reductive and non-reductive realists
with the debate over whether relations are identical to their
foundations.


## 5. Non-Paradigmatic Relational Situations




So far we have been focusing only on the medieval discussion of
paradigmatic relational situations--that is, relational
situations conforming to the analysis suggested in Aristotle's
*Categories*. But as we indicated earlier, it is clear from
some of Aristotle's later works, most notably the
*Metaphysics*, that he thinks not all relational situations can
be made to conform to it. Thus, in *Metaphysics* V, he suggests
that there are some relational situations in which substances are
related, not by a pair of accidents, but by a single accident
belonging to just one of them. Here he cites the example of
intentional relations: if Simmias is thinking about Socrates, this is
to be explained in terms of nothing but Simmias, Socrates, and an
accident of Simmias. For the sake of convenience, we shall hereafter
refer to relational situations that do not conform to the
*Categories* paradigm as 'non-paradigmatic relational
situations'.




Although medieval philosophers did not have direct access to
Aristotle's *Metaphysics* until the mid-twelfth century, they
did feel pressure from other, largely theological sources to admit
something like his non-paradigmatic relational situations. The
standard medieval conception of deity, for example, requires that God
is an absolutely perfect being, possessing no accidents
whatsoever. But this conception makes it difficult to explain how God
can enter into relational situations. Indeed, as Augustine points out
in book V of his *De Trinitate*, there is a special difficulty
for those want to combine this conception with the Christian doctrine
of creation. For this doctrine would seem to require that God first
lacks, and then acquires a contingent or accidental
relation--namely, that of being creator. But then does it not
also require that in creating the universe God underwent a real
change? And what about the claim that there are no accidents in
God?




Augustine's solution, which is adopted by medieval philosophers
generally, is to say that when God acquires a new relation, this is to
be explained in terms of properties or accidents of something other
than God. Thus, in the case of creation Augustine says:



> 
> 
> Even though [God's substance] begins to be [truly] spoken of
> [as related to a creature] at some time, still nothing is to be
> understood as having happened to the divine substance itself, but only
> to the creature in relation to whom it is spoken
>  of.[31]
> 




On the basis of theological considerations, therefore, Augustine is
led to something like Aristotle's non-paradigmatic relational
situations. What makes it true that God is related to his creatures is
nothing but God, the creatures, and a monadic property or accident of
the latter.




Interestingly, Augustine does not think that the case of God is
unique, but also suggests that there are non-theological cases in which
things are related solely in virtue of the properties or accidents of
other things. Thus, a coin, he says can increase or decrease in value
solely in virtue of the intentional states of human
 beings.[32]
 Boethius, in a treatise also known as *De Trinitate*,
discusses the same sorts of issues, and in the course of doing so adds
yet another non-theological example--a variation of which comes
to be the standard medieval example of non-paradigmatic relational
situations. Consider a man who walks up beside another man (or beside
stationary column, as the standard medieval variation has it). In this
sort of case, says Boethius, the latter comes to be to the right of
the man, but solely in virtue of a property of the
 man.[33]




Thus, even before medieval philosophers had direct access to
Aristotle's *Metaphysics* they were led to acknowledge the
existence of the same type of relational situations mentioned
there. Of course, once the relevant texts of Aristotle became
available, the medievals worked hard to connect their discussions with
these texts. Indeed, when the issue of God's relation to his
creatures is taken up in the thirteenth century--an issue that
becomes one of the foci of medieval discussions of relations
generally--medieval philosophers often made direct appeals to
Aristotle's discussion in *Metaphysics*
 V.[34]


### 5.1 Relations of Reason




So far, then, we have seen that medieval philosophers are committed to
recognizing at least one type of non-paradigmatic relational
situation--namely, those involving a pair of substances and a
single accident inhering in just one them. As we might expect,
medieval philosophers disagree about the precise nature of the
accidents involved in these situations--with reductive realists
identifying them with ordinary, non-relational accidents, and
non-reductive realists identifying them with accidents of a *sui
generis* type.




For our purposes, however, what is most interesting about this type of
relational situation has to do with complication it presents for
reductivists and non-reductivists alike. In paradigmatic relational
situations, there is always a distinct property or accident
corresponding to each member of the relevant pair of converse
relational concepts. Thus, when Simmias is taller than Socrates, there
is one accident corresponding to the concept 'taller than'
(namely, an accident of Simmias) and another accident corresponding to
the concept 'shorter than' (namely, an accident of
Socrates). But now consider a situation such as Simmias's thinking
about Socrates (or God's being related to a creature)--that is, a
situation where two substances are related by a property or accident
inhering in only one of the *relata*. What is to be said about
the relationship between the relevant properties and concepts in
situations of this sort?




Initially, we might expect medievals to respond to this by saying
that, in such situations, the single accident responsible for relating
both subjects corresponds to both members of the relevant pair of
relational concepts. Thus, if Simmias is thinking about Socrates, a
quality of Simmias corresponds to both the concepts 'thinking
of' and 'thought about'. The problem with this
suggestion, however, is that it leads to the same sort of difficulties
we encountered earlier for identifying relations with pairs of
accidents taken jointly. Socrates's being thought about appears to be
something inhering in Socrates and Socrates alone. But if we identify
this relation with an accident of Simmias, then we shall have to say
that Socrates's being thought about does not exist in Socrates after
all, but rather exists only in Simmias. Although medievals do allow
for cases of *extrinsic denomination*--that is, cases
where a property or accident of one thing is predicated of something
else--they typically do not allow for this in the case of
relations. That is to say, they typically regard relations as
intrinsic to the subjects of which their corresponding terms or
concepts are
 predicated.[35]
 Evidently, therefore, the relevant accident of Simmias can correspond
to the concept 'thinking of', but not to the concept
'thought about'. But, then, we still need to know what
corresponds to this latter concept.




Now perhaps it will be suggested that, in the absence of any property
or accident of Socrates to correspond to the concept 'thought
about', Socrates himself can serve as the correspondent. In that
case, however, we should have to admit that an individual substance is
a relation (since, as we have seen, whatever corresponds to a
relational concept and plays the relevant metaphysical role qualifies
as a relation). But such an admission goes against deep-seated
intuitions deriving from the *Categories*. As we have seen,
Aristotle not only assumes that relations are accidents in the
*Categories*, but intentionally characterizes them in such a
way as to exclude substances from this category. But then we still
lack a solution to our problem. If there is no property or accident of
Socrates to correspond to the concept 'thought about', and
neither Socrates nor a property of anything else can correspond to it,
what if anything can?




It is at this point that the medieval notion of a relation of reason
(*relatio rationis*) becomes relevant. Medieval philosophers
often say that, even if there are relational situations involving only
a single accident, nonetheless these situations must be conceived
*as if* they involved a pair, one belonging to each of the
related things (cf., e.g., Aquinas, *De potentia* q. 1, a. 1,
ad 10). Like Boethius, the medievals accept the view that relations
cannot be understood to exist by themselves or apart from their
correlatives. At least conceptually, they say, relations always come
in pairs. Thus, even if there is no real or extramental property in
Socrates that accounts for his being thought about by Simmias, we must
still *conceive* of this situation as if there were one and, as
it were, *project* this property onto Socrates. Since such
projections depend for their existence on the activity of the mind,
medieval philosophers refer to them as beings of reason (*entia
rationis*). And their suggestion is that we take these beings of
reason to be the items corresponding to or signified by concepts such
as 'thought about'.




The notion of a being of reason, or more specifically the notion of a
relation of reason, does not appear to have been invoked in the Latin
West before the thirteenth century, when Aristotle's
*Metaphysics* and certain Muslim philosophical commentaries and
treatises derived from it began to circulate widely. The notion of a
relation of reason is not to be found explicitly in the works of
Aristotle, though he does distinguish beings of reason in general from
real beings (cf. *Metaphysics* IV, 1, esp. 1003a32-b11 and V,
7, esp. 1017a31-35). Even so, this notion is explicitly invoked by
certain Muslim philosophers, most notably Avicenna (d. 1037), and it
may well be that it makes its way into the Latin west through
them.




However that may be, once the notion of a relation of reason is
introduced in the Latin West, it becomes pervasive--so pervasive,
in fact, that even philosophers, such as Ockham, who complain that
such a notion is "not to be found in the writings of
Aristotle" and that "'relation of reason' is
not a philosophical term," nevertheless feel compelled to give
some account of it in order to preserve common
 usage.[36]
 The pervasiveness of this notion is explained at least partly by the
fact that medieval philosophers think it can be used to clarify and
explain a number of troublesome non-paradigmatic relational
situations. For example, by the end of thirteenth century, most
philosophers use it to explain the doctrine of creation, saying that
creatures are related to God by a real relation, whereas God is
related to them by a mere relation of reason (cf. Henninger
1989). Again, many use the notion to clarify certain cases of
relational change. Thus, when a substance acquires a new relation
without undergoing any real change this is often explained by saying
that the substance acquired a mere relation of
 reason.[37]
 Finally, some medieval philosophers use relations of reason to
identify a sense in which God can have accidents after all. Since
relations of reason are mere projections (or properties a thing has by
virtue of the activity of some mind), we can conceive of
them as accidents in a broad sense--that is, as properties or
features that a thing can both acquire and lose. But since the
acquisition or loss of these properties does not require a subject to
undergo any real change, some medievals claim that there is no reason
in principle why even God should not have accidents of this
 sort.[38]


### 5.2 A Shifting Conception of Relations




The distinction between real relations and relations of reason has a
number of important consequences for the development of medieval
theories of relations. For one thing, it enables philosophers to
introduce a number of refinements and distinctions within the
category of relations. By the mid-thirteenth century, for example, it
becomes common to say that the category of relations is unique in
allowing mere beings of reason among the things signified by its terms.
To quote from Aquinas who is representative in this regard:



> 
> 
> The other genera introduce something in extramental reality just by
> virtue of what they are. Thus, quantity introduces something just by
> virtue of the fact that it is quantity. By contrast, relation alone
> is such that it does not introduce anything in extramental reality,
> just by virtue of what it is--for it does not predicate a
> *something* but a *toward something*. Hence we find
> certain relations that do not introduce anything in extramental
> reality, but only in reason. (*De veritate*, q. 1, a. 5, ad 15.)
> 
> 




Again, Aquinas and others appeal to this special feature of the
category of relations (namely, that the things signified by its
terms can either be real beings or mere beings of reason) to provide a
systematic division of relations or pairs of correlatives into three
different types, depending on whether their members are
both real, both conceptual, or mixed (one real, one conceptual). To
quote again from Aquinas:



> 
> 
> It must be known that, since a relation requires two
> *relata*, there are three ways in which it can be something real
> or conceptual:
> 
> 
> 
> [1] Sometimes it is a mere being of reason on the part of both
> *relata*, namely, when the order or [relative] disposition
> cannot exist between things except in virtue of the apprehension of
> reason alone. For example, when we say that something is identical to
> itself. For in virtue of the fact that reason apprehends the one thing
> twice, it regards it as two; and in this way it apprehends a certain
> [relative] disposition of a thing to itself. And the same thing is true
> of all relations between being and non-being, which reason forms
> insofar as it apprehends a non-being as a certain *relatum*.
> Again, the same is true of all relations that follow upon the activity
> of reason, such as genus and species, and the like.
> 
> 
> 
> 
> [2] Now there are other relations that are real as regards both
> *relata*, namely, whenever there is a [relative] disposition
> between two things in virtue of something really belonging to each of
> them--as is clear from all relations that follow on quantity, such
> as large and small, double and half, and things of this sort. For there
> is a quantity in both *relata*. And the same is true of
> relations that follow on action and passion, such as mover and movable,
> father and son, and the like.
> 
> 
> 
> 
> [3] Sometimes, however, a relation is something real in one of the
> *relata* and a mere being of reason in the other. And this
> happens whenever the two *relata* do not belong to a single
> order. For example, sense perception and knowledge are related to
> things that are sensible and intelligible. But insofar as the latter
> are things existing in extramental reality, they are outside the order
> of sensible and intelligible being. And so there is a real relation in
> the knowledge and sense perception in virtue of the fact that they are
> [really] ordered to things that can be known or sensed. However,
> things [that can be known or sensed] are outside this sort of order
> [when] considered in themselves, and hence there is not really a
> relation in them to knowledge or sense perception. On the contrary,
> there is only a relation in them according to reason, insofar as
> the intellect apprehends them as terms of the relations of knowledge
> and sense perception. This is why the Philosopher says in
> *Metaphysics* V that they are spoken of relatively, not because
> they are related to other things, but because other things are related
> to them. Similarly, being to the right is not said of a column unless
> it is placed to the right of some animal. Hence a relation of this
> sort is not really in the column but in the animal. (*Summa
> Theologiae* I, q. 13, a. 7, corpus.)
> 
> 
> 




In this passage, Aquinas contrasts the relations involved in
paradigmatic relational situations--namely, relations of the
second type--with the relations involved in two sorts
of non-paradigmatic relational situation. We are already familiar with
relations of the third or 'mixed' type from our discussion
of creation and intentional relations. Moreover, we can see that
Aquinas follows the common medieval practice of connecting relations
of this type with both Aristotle's discussion in the
*Metaphysics* and the Boethius-inspired example of the column.
We have yet, however, to encounter relations of the first type. These
are relations comprised by pairs of properties or accidents both of
whose members are beings of reason. As an example of this type of
relation, Aquinas gives self-identity. When we conceive of a situation
involving this sort of relation, he says, we conceive of it as if
it involved two things ("a relation requires two
*relata*" and hence "reason apprehends the one
thing twice"), and also as if the two things were ordered to each
other by a pair of properties (or "[relative]
dispositions"). Obviously, however, there are not distinct
things in extramental reality serving as the *relata* of the
relation of self-identity, much less two properties by which such
*relata* are related. Like many other medievals, therefore,
Aquinas concludes that in this case the relations (or relative
dispositions) are not real, but mere beings of
 reason.[39]




The claim that self-identity is a relation of reason might seem
worrisome. For insofar as relations of reason depend for their
existence on the activity of the mind, it would seem to follow that
something's being self-identical is mind dependent. But *that* is
absurd.




Aquinas is aware of this sort of worry. In fact, in one of his
disputed questions, he imagines a similar worry leading someone to
doubt the view that even God's relation to his creatures can be
considered a relation of reason:



> 
> 
> After all, if there were no created intellect in existence, God would
> still be Lord and Creator. But if there were no created intellect in
> existence, there would not be any beings of reason. Hence
> 'Creator', 'Lord', and terms of this sort, do
> not express relations of reason. (*De potentia*, q. 7, a. 11,
> obj. 4.)
> 
> 




This sort of worry about relations of reason helps to explain what is
perhaps their most significant effect on the medieval discussion of
relations--namely, a gradual shift away from the traditional
Aristotelian characterization of relations. On this characterization,
as we saw earlier, relations are identified at least partly in terms
of their metaphysical function--that is to say, they are
identified as items that actually serve to relate things. But in order
to maintain that things can be self-identical apart from the activity
of any mind, while at the same time maintaining that self-identity is
a relation of reason, medieval philosophers have little choice but to
move away from the traditional characterization. And of course the
same thing is true in the case of God's relation to his creatures.
Thus, as Aquinas says in reply to the abovementioned doubt:



> 
> 
> A man is really (and not merely conceptually) identical to himself,
> even though his relation [of self-identity] is a being of reason. And
> the explanation for this is that the cause of his relation is
> real--namely, the unity of his substance, which our intellect
> considers under the aspect of a relation. In the same way, the power
> to compel subjects is really in God, and our intellect considers this
> power as ordered to the subjects because of the subjects' order to
> God. It is for this reason that he is really said to be Lord, even
> though his relation is a mere being of reason. And for the same reason
> it is evident that he would be Lord [Creator, etc.] even if there were
> no created intellect in existence. (*De potentia* q. 7, a. 11,
> ad 3-5.)
> 
> 




In this passage, Aquinas makes it clear that in cases involving
relations of reason--such as self-identity or God's
relation to the world--the *relata* are related, not by
their relations (since these are mere beings of reason and hence
dependent on the activity of the mind), but by what he refers to here
as the *cause of* their
 relations.[40]
 Now, in the case of a man's being self-identical, Aquinas says
the cause is just "the unity of his substance," where by
this he seems to mean that what makes a man identical to himself is
just the man himself. Again, in the case of God's being Lord he
says that the cause is "the power to compel subjects". In
the case of God, however, Aquinas does not think the power to compel
subjects is distinct from its subject, namely, the divine
nature. Hence, he maintains that what makes it true that God is Lord
is nothing but God, his creatures, and some property or attribute of
the creatures.




In effect, therefore, reflection on relations of reason brings about a
shift away from the conception of relations as items that relate, and
thus forces medieval philosophers to fall back on what they might
otherwise have thought of as an equivalent characterization, namely,
the view that relations are items corresponding to or signified by our
genuinely relational concepts. Thus, even if self-identity or God's
relation to the world is a mere being of reason, and hence does not
actually relate its subject to anything, nonetheless it can still be
regarded as a relation on the grounds that it is signified by a
relational concept. Now obviously this shift away from the traditional
Aristotelian conception has the awkward consequence that things can be
related even if their relations do not exist. Thus, Socrates can be
identical to himself even if there is no self-identity, God can be
Lord of creation even if his relation of Lordship does not exist, and
more generally, a predication of the form '*aRb*'
can be true, even when no predication of form '*R-*ness
exists' is true. Of course, there is nothing ultimately
incoherent about this consequence, provided we keep in mind that the
relations in such cases are mere beings of reason. Nonetheless,
accepting this consequence does force medieval philosophers to deny
what at least initially appears to be a truth of reason, and at any
rate is part of common sense--namely, that things are related by
their
 relations.[41]




Some philosophers, such as Aquinas, assume that the departure from the
traditional Aristotelian characterization of relations is required
only in non-paradigmatic relational situations. By the end of the
Middle Ages, however, this sort of departure is so common and familiar
that philosophers no longer feel the need to regard it as exceptional.
Thus, Ockham eventually adopts a view according to which all relations
depend for their existence on the activity of the mind. Indeed, things
are so changed by the time of Ockham that he feels free not only to
reject the traditional Aristotelian characterization of relations, but
also to modify the standard medieval alternative to it. On his
preferred characterization, relations are the items corresponding, not
to all of our relational concepts, but to just one of
them--namely, the concept 'relation'. As Ockham sees
it, moreover, the concept 'relation' is a term of second
intention--that is to say, a term to which only concepts
correspond. Thus, even though Ockham insists that many predications of
the form '*aRb*' are true independently of the
mind, he nonetheless maintains that properly speaking relations exist
only in the mind as
 concepts.[42]
 This helps to explain why he often expresses his view using such
otherwise unintelligible formulas as: "This white thing really
is similar, even though similarity is not really in this white
thing" (*Quodl.* VI, q. 22 in *Opera Theologica*
ix, 669).


### 5.3 Relations as Substances




Although relations of reason force a shift away from the Aristotelian
characterization of relations, it is important to recall that they
were originally invoked to preserve the deep-seated Aristotelian
conviction that no substance is a relation. As we have seen, it was to
avoid saying that Socrates is signified by the concept 'thought
about', when he is being thought about by Simmias, that medieval
philosophers invoked the notion of a relation of reason: it is not
Socrates, but a relation of reason, they say, that corresponds to the
concept in question, and in this way avoid the consequence that a
substance such as Socrates is a relation. And of course it was the
same sort of invocation that enabled them to avoid saying that
Socrates corresponds to the concept of 'self-identical' or
that God corresponds to the concepts 'Lord' or
'Creator'. Given the lengths to which medieval
philosophers are willing to go, in situations such as these, to
preserve the Aristotelian thesis that no substance is a relation, it
is all the more surprising that there is at least one
case--namely, the Christian doctrine of the Trinity--in
which they are forced to admit that even it cannot be upheld.




According to Christian doctrine of the Trinity, God exists in three
persons: Father, Son, and Holy Spirit. As this doctrine was typically
understood during the Middle Ages, it implies not only that God
possesses certain relations--such as fatherhood and
sonship--but also that he possess them independently of the
activity of any mind. As Aquinas says in his *Summa
Theologiae*:



> 
> 
> Someone is said to be a father only by virtue of his fatherhood, and
> someone is said to be a son only by virtue of his sonship. Therefore,
> if [the relations of] fatherhood and sonship are not really in God, it
> follows that God is not a Father or Son really, but merely according to
> a concept of the mind--which is the Sabellian
>  heresy.[43]
> 




Now when the claim that there are real relations in God is combined
with another doctrine that was ubiquitous in the Middle Ages, namely
the doctrine of divine simplicity, the conclusion that God (and hence
at least one substance) is a relation seems to follow necessarily. For
as the doctrine of divine simplicity is typically understood, there is
no real distinction to be drawn between God and any of his intrinsic
or essential attributes. Thus, if God is good, he is identical to his
goodness; if he is wise, he is identical to his wisdom. By parity of
reasoning, therefore, if God is a father or son, he must be identical
to his fatherhood and sonship. Again, Aquinas is perfectly
representative in this regard:



> 
> 
> Whatever is in God *is* his nature ... It is thus clear
> that a relation really existing in God is identical to his nature
> according to reality, and does not differ from it except according to
> a concept of the mind. (*Summa Theologiae* I, q. 28, a. 2,
> corpus.)
> 
> 




With the doctrine of the Trinity, therefore, we arrive at what is perhaps
the medievals' greatest departure from Aristotle.




Some medievals, however, see the Trinity not as providing a
counterexample to Aristotle's thesis that no substance is a relation,
but rather as calling our attention to a restriction on its range of
applicability. Aquinas, for example, appears to think that the thesis
was specifically formulated to apply only to the case of creatures
(cf. *Summa Theologiae* I, q. 28, a. 2, corpus), and to some
extent this is plausible, since obviously Aristotle was not thinking
about theological examples such as the Trinity when he formulated
it. Interestingly, however, other philosophers think that the thesis
does not hold even in the case of all creatures. Thus, Gilbert of
Poitiers (d. 1154), in a discussion of Boethius's *De
hebdomadibus*, suggests that creaturely goodness is a relation,
indeed, just the relation of being created by God. On this
interpretation, moreover, Gilbert takes this relation to be nothing
over and above creatures themselves, so that in their case it is
individual substances that correspond to the concept 'created
by' and hence qualify as relations. Like Aquinas, however,
Gilbert appears to think of these sorts of cases, not as providing
counterexamples to Aristotle's thesis, but rather as telling us
something about the scope of its applicability. According to Gilbert,
there is a distinction to be drawn between *natural
philosophy*, which deals with natural things, and other areas of
intellectual inquiry, including *theology* and *ethics*,
which deal with a broader scope of
 things.[44]
 And Gilbert's suggestion is that if we restrict our attention
to natural philosophy, as no doubt Aristotle did, then like him we
will be led to the conclusion that no substance is a
 relation.[45]




Although Gilbert and Aquinas work hard to preserve something like the
Aristotelian thesis that no substance is a relation, at least in
non-theological contexts, not all medieval philosophers feel the need
to do so. In fact, as Middle Ages progress, there appears to be a
gradual shift, even in non-theological contexts, towards allowing
substances to be relations, or at least towards allowing them to be
the primary *significata* of our relational concepts. Here
again Ockham appears to be an important transitional figure. Thus, he
breaks with tradition in allowing that even self-identity is a real
relation, or at least "real ... in the same way that
similarity and equality are" (*Quodl*. V, q. 27 in
*Opera Theologica* ix, 685). Indeed, on Ockham's view, which
becomes influential in the generations following him, it turns out
that substances are signified by most of our relational
concepts. Apart from some species of quality, Ockham thinks there is
no real distinction to be drawn between substances and any of there
accidents. According to him, therefore, relational situations do not
typically involve anything more than individual
 substances.[46]


## 6. A Shifting of Paradigms




We are now in a position to appreciate the main types of views that
medievals developed concerning the nature and ontological status of
relations, as well as the main historical and dialectical
considerations that helped to shape them. As we have seen, with the
exception of thinkers such as Peter Auriol, medievals appear to have
been drawn (almost to a person) to a form of realism about relations,
one according to which at least some predications of the form
'*aRb*' are true independently of the mind. There
is some disagreement as to the precise analysis of the situations that
makes these sorts of predications true, but even here the medievals typically 
develop their views from within a common framework provided by
Aristotle's *Categories*: relational situations do not include
anything corresponding to the notion of a polyadic property, but
instead include only substances and their monadic properties or
accidents. The main medieval disagreements, therefore, are best
characterized as disagreements about the extent to which the proper
analysis ought to conform to the paradigm suggested by the
*Categories.* Prior to the fourteenth century, as we have seen,
medievals tended to follow Aristotle in claiming that things are
related by their accidents, and hence that relational situations
typically involve not only pairs of substances, but pairs of monadic
properties or accidents as well. As a result, one of the most pressing
questions during this period concerns the precise nature of the
accidents involved in relational situations. Reductive realists, as we
have seen, identify them with ordinary, non-relational accidents such
as quantities or qualities, whereas non-reductive realists identify
them with monadic properties of a *sui generis* type.




With the advent of the fourteenth century, however, important changes
begin to take place in the medieval discussion of relations.
Philosophers and theologians continue, of course, to allow for
situations in which substances are related by their accidents, and
hence to worry about the precise nature of the accidents involved in
these situations. But at this point there is a decided shift toward
regarding such situations as exceptional. As we have seen, there was
always strong theological pressure to allow for at least some
departures from the Aristotelian paradigm. But what appears to have
happened over time is that these departures come to seem less and less
peculiar, and eventually to provide the basis for a new analysis of
relational situations--one according to which substances
themselves are the items responsible for relating. Around this same
time, moreover, Ockham and his followers, most notably John Buridan,
institute another sort of change, namely, a shift away from the traditional
Aristotelian characterization of relations. Prior to the
fourteenth century, philosophers and theologians typically assume that
the term 'relation' signifies whatever it is in a
relational situation that does the relating--though here again
theological considerations force them to allow for certain exceptions.
By the time of Ockham, however, we get a complete break with this
standard characterization. Thus, whereas earlier philosophers and
theologians would allow the term 'relation' to signify a
being of reason only in certain cases, such as creation, Ockham and
his followers maintain that 'relation' is a term of second
intention, and so strictly speaking always signifies beings of reason.
With Ockham, therefore, we have not only a complete severing of the
connection between relations and those items in relational situations
that actually do the relating, but also the advent of a new--and,
we might add, fairly harmless--form of
 anti-realism.[47]




In the end, therefore, it is fair to say that the fourteenth
century marks a shifting of medieval paradigms both with respect to
the proper analysis of relational situations and with respect to the
proper characterization of relations. It is important to add, however, that
these shifts cannot be fully explained in terms of developments within
the medieval discussion of relations, but are instead part-and-parcel
of broader theoretical shifts in medieval accounts of the relationship
between mind, language, and reality--shifts which are closely
associated with the rise of late-medieval nominalism
 generally.[48]