## 1. Properties: Basic Ideas



There are some crucial terminological and conceptual distinctions that
are typically made in talking of properties. There are also various
sorts of reasons that have been adduced for the existence of
properties and different traditional views about whether and in what
sense properties should be acknowledged. We shall focus on such
matters in the following subsections.


### 1.1 How We Speak of Properties



Properties are *expressed*, as meanings, by
*predicates*. In the past "predicate" was often
used as synonym of "property", but nowadays predicates are
linguistic entities, typically contrasted with *singular
terms*, i.e., simple or complex noun phrases such as
"Daniel", "this horse" or "the President
of France", which can occupy subject positions in sentences and
purport to *denote*, or *refer to*, a single thing.
Following Frege, predicates are *verbal phrases* such as
"is French" or "drinks". Alternatively,
predicates are *general terms* such as "French",
with the copula "is" (or verbal inflection) taken to
convey an exemplification link (P. Strawson 1959; Bergmann 1960). We
shall conveniently use "predicate" in both ways.
Predicates are *predicated* of singular terms, thereby
generating sentences such as "Daniel is French". In the
familiar formal language of first-order logic, this would be rendered,
say, as "\(F(d)\)", thus representing the predicate with a
capital letter. The set or class of objects to which a predicate
veridically applies is often called the *extension* of the
predicate, or of the corresponding property. This property, in
contrast, is called the *intension* of the predicate, i.e., its
meaning. This terminology traces back to the Middle Age and in the
last century has led to the habit of calling sets and properties
*extensional* and *intensional* entities, respectively.
Extensions and intensions can hardly be identified; this is
immediately suggested by paradigmatic examples of co-extensional
predicates that appear to differ in meaning, such as "has a
heart", and "has kidneys" (see
 SS3.1).



Properties can also be referred to by singular terms, or so it seems.
First of all, there are singular terms, e.g., "being
honest" or "honesty", that result from the
*nominalization* of predicates, such as "is honest"
or "honest" (some think that "being \(F\)" and
"\(F\)-ness" stand for different kinds of property
(Levinson 1991). Further, there are definite descriptions, such as
"Mary's favorite property". Finally, though more
controversially, there are demonstratives, such as "that shade
of red", deployed while pointing to a red object (Heal
1997).



Frege (1892) and Russell (1903) had different opinions regarding the
ontological import of nominalization. According to the former,
nominalized predicates stand for a "correlate" of the
"unsaturated" entity that the predicate stands for (in
Frege's terminology they are a "concept correlate"
and a "concept", respectively). According to the latter,
who speaks of "inextricable difficulties" in Frege's
view (Russell 1903: SS49), they stand for exactly the same entity.
*Mutatis mutandis*, they similarly disagreed about other
singular terms that seemingly refer to properties. The ontological
distinction put forward by Frege is mainly motivated by the fact that
grammar indeed forbids the use of predicates in subject position. But
this hardly suffices for the distinction and it is dubious that other
motivations can be marshalled (Parsons 1986). We shall thus take for
granted Russell's line here, although many philosophers support
Frege's view or at least take it very seriously
(Castaneda 1976; Cocchiarella 1986a; Landini 2008).


### 1.2 Arguments for Properties



Properties are typically invoked to explain phenomena of philosophical
interest. The most traditional task for which properties have been
appealed to is to provide a solution to the so-called "one over
many problem" via a corresponding "one over many
argument". This traces back at least to Socrates and Plato
(e.g., *Phaedo*, 100 c-d) and keeps being rehearsed (Russell
1912: ch. 9; Butchvarov 1966; Armstrong 1978a: ch. 7; Loux 1998:
ch.1). The problem is that certain things are many, they are
numerically different, and yet they are somehow one: they appear to be
similar, in a way that suggests a uniform classification, their being
grouped together into a single class. For example, some objects have
the same shape, certain others have the same color, and still others
the same weight. Hence, the argument goes, something is needed to
explain this phenomenon and properties fill the bill: the objects in
the first group, say, all have the property *spherical*, those
in the second *red*, and those in the third *weighing 200
grams*.



Relatedly, properties have been called for to explain our use of
general terms. How is it, e.g., that we apply "spherical"
to those balls over there and refuse to do it for the nearby bench? It
does not seem to be due to an arbitrary decision concerning where, or
where not, to stick a certain label. It seems rather the case that the
recognition of a certain property in some objects but not in others
elicits the need for a label, "spherical", which is then
used for objects having that property and not for others. Properties
are thus invoked as meanings of general terms and
predicates (Plato, *Phaedo*, 78e; Russell 1912: ch.
9). In contrast with this, Quine (1948; 1953 [1980: 11, 21, 131] )
influentially argued that the use of general terms and predicates in
itself does not involve an ontological commitment to entities
corresponding to them, since it is by deploying singular terms that we
purport to refer to something (see also Sellars 1960). However, as
noted, predicates can be nominalized and thus occur as singular terms.
Hence, even if one agrees with Quine, nominalized predicates still
suggest the existence of properties as their referents, at least to
the extent that the use of nominalized predicates cannot be
paraphrased away (Loux 2006: 25-34; entry on
 platonism in metaphysics, SS4).



After Quine (1948), quantificational idiom is the landmark of
ontological commitment. We can thus press the point even more by
noting that we make claims and construct arguments that appear to
involve quantification over properties, with quantifiers reaching (i)
over predicate positions, or even (ii) over both subject and predicate
positions (Castaneda
 1976).[2]
 As regards (i), consider: this apple is red; this tomato is red;
hence, there is something that this apple is and this tomato also is.
As for (ii), consider: wisdom is more important than beauty; Mary is
wise and Elisabeth is beautiful; hence, there is something that Mary
is which is more important than something that Elisabeth is.



Quantification over properties seems ubiquitous not just in ordinary
discourse but in science as well. For example, the inverse square law
for dynamics and the reduction of temperature to mean molecular energy
can be taken to involve quantification over properties such as masses,
distances and temperatures: the former tells us that the attraction
force between any two bodies depends on such bodies' having
certain masses and being at a certain distance, and the latter informs
us that the fact that a sample of gas has a given temperature depends
on its having such and such mean kinetic energy.



Swoyer (1999: SS3.2) considers these points within a long list of
arguments that have been, or can be, put forward to motivate an
ontological commitment to properties. He touches on topics such as
*a priori* knowledge, change, causation, measurement, laws of
nature, intensional logic, natural language semantics, numbers (we
shall cover some of this territory in
 SS5
 and
 SS6).



Despite all this, whether, and in what sense, properties should be
admitted in one's ontology appears to be a perennial issue,
traditionally shaped as a controversy about the existence of
universals.


### 1.3 Traditional Views about the Existence of Universals



Do universals really exist? There are three long-standing answers to
this question: *realism*, *nominalism*, and
*conceptualism*.



According to realists, universals exist as mind-independent entities.
In *transcendent* realism, put forward by Plato, they exist
even if uninstantiated and are thus "transcendent" or
"*ante res*" ("before the things"). In
*immanent* realism, defended by Aristotle in opposition to his
master, they are "immanent" or "*in
rebus*" ("in things"), as they exist only if
instantiated by objects. Contemporary notable supporters are Russell
(1912) for the former and Armstrong (1978a) for the latter (for recent
takes on this old dispute see the essays by Loux, Van Inwagen, Lowe
and Galluzzo in Galluzzo & Loux 2015). Transcendentism is of
course a less economical position and elicits epistemological worries
regarding our capacity to grasp *ante res* universals.
Nevertheless, such worries may be countered in various ways (cf. entry
on
 platonism in metaphysics, SS5;
 Bealer 1982: 19-20; 1998: SS2; Linsky & Zalta 1995),
and uninstantiated properties may well have work to do, particularly
in capturing the intuitive idea that there are unrealized
possibilities and in dealing with cognitive content (see
 SS3).
 A good summary of pro-transcendentist arguments and immanentist
rejoinders is offered by Allen (2016: SS2.3). See also Costa
forthcoming for a new criticism of immanentism, based on the
notion of grounding.



Nominalists eschew mind-independent universals. They may either resort
to tropes in their stead, or accept *predicate nominalism*,
which tries to make it without mind-independent properties at all, by
taking the predicates themselves to do the classifying job that
properties are supposed to do. This is especially indigestible to a
realist, since it seems to put the cart before the horse by making
language and mind responsible for the similarities we find in the rich
varieties of things surrounding us. Some even say that this involves
an idealist rejection of a mind-independent world (Hochberg 2013).
Conceptualists also deny that there are mind-independent universals,
and because of this they are often assimilated to nominalists. Still,
they can be distinguished insofar as they replace such universals with
concepts, understood as non-linguistic mind-dependent entities,
typically functioning as meanings of predicates. The mind-dependence
of concepts however makes conceptualism liable to the same kind of
cart/horse worry just voiced above in relation to predicate
 nominalism.[3]



The arguments considered in
 SS1.2
 constitute the typical motivation for realism, which is the stance
that we take for granted here. They may be configured as an abductive
inference to the best explanation (Swoyer 1999). Thus, of course, they
are not foolproof, and in fact nominalism is still a popular view,
which is discussed in detail in the entry on
 nominalism in metaphysics,
 as well as in the entry on
 tropes.
 Conceptualism appears to be less common nowadays, although it still
has supporters (cf. Cocchiarella 1986a: ch. 3; 2007), and it is worth
noting that empirical research on concepts is flourishing.


### 1.4 Properties in Propositions and States of Affairs



We have talked above in a way that might give the impression that
predication is an activity that *we* perform, e.g., when we say
or think that a certain apple is red. Although some philosophers might
think of it in this way, predication, or *attribution*, may
also be viewed as a special link that connects a property to a thing
in a way that gives rise to a
 proposition,
 understood as a complex
featuring the property and the thing (or concepts of them) as
constituents with different roles: the latter occurs in the
proposition *as logical subject* or *argument*, as is
often said, and the former *as attributed* to such an argument.
If the proposition is true (the predication is veridical), the
argument exemplifies the property, viz. the former is an instance of
the latter. The idea that properties yield propositions when
appropriately connected to an argument motivates Russell's
(1903) introduction of the term
"propositional
function" to speak of properties. We
take for granted here that predication is univocal. However, according
to some neo-Meinongian philosophers, there are two modes of
predication, sometimes characterized as "external" and
"internal" (Castaneda 1972; Rapaport 1978; Zalta
1983; see entry on
 nonexistent objects).
 Zalta (1983) traces back the distinction to Mally and uses
"exemplification" to characterize the former and
"encoding" to characterize the latter. Roughly, the idea
is that non-existent objects may encode properties that existent
objects exemplify. For instance, *winged* is exemplified by
that bird over there and is encoded by the winged horse.



It is often assumed nowadays that, when an object exemplifies a
property, there is a further, complex, entity, a fact or state of
affairs (Bergmann 1960; Armstrong 1997, entries on
 facts
 and
 states of affairs),
 having the property (*qua* attributed) and the object
(*qua* argument) as constituents (this compositional conception
is not always accepted; see, e.g., Bynoe 2011 for a dissenting voice).
Facts are typically taken to fulfill the theoretical roles of
truthmakers (the entities that make true propositions true, see entry
on
 truthmakers)
 and causal *relata* (the entities connected by causal
relations; see entry on
 the metaphysics of causation, SS1).
 Not all philosophers, however, distinguish between propositions and
states of affairs; Russell (1903) acknowledges only propositions and,
for a more recent example, so does Gaskin (2008).



It appears that properties can have a double role that no other
entities can have: they can occur in propositions and facts both as
arguments and as attributed (Russell 1903: SS48). For example, in
truly saying that this apple is red and that red is a color, we
express a proposition wherein *red* occurs as attributed, and
another proposition wherein *red* occurs as argument.
Correspondingly, there are two facts with *red* in both roles,
respectively. This duplicity grounds the common distinction between
different *orders* or *types* of properties:
*first-order* ones are properties of things that are not
themselves predicables; *second-order* ones are properties of
first-order properties; and so on. Even though the formal and
ontological issues behind this terminology are controversial, it is
widely used and is often connected to the subdivision between
first-order and higher-order logics (see, e.g., Thomason 1974; Oliver
1996; Williamson 2013; entry on
 type theory).
 It originates from Frege's and Russell's logical
theories, especially Russell's type theory, wherein distinctions
of types and orders are rigidly regimented in order to circumvent the
logical paradoxes (see
 SS6).


### 1.5 Relations



A relation is typically attributed to a plurality of objects. These
*jointly* instantiate the relation in question, if the
attribution is veridical. In this case, the *relata* (as
arguments) and the relation (as attributed) are constituents of a
state of affairs. Depending on the number of objects that it can
relate, a relation is usually taken to have a number of
"places" or a "degree" ("adicity",
"arity"), and is thus called "dyadic"
("two-place"), "triadic"
("three-place"), etc. For example, *before* and
*between* are dyadic (of degree 2) and triadic (of degree 3),
respectively. In line with this, properties and propositions are
"monadic" and "zero-adic" predicables, as they
are predicated of one, and of no, object, respectively, and may then
be seen as limiting cases of relations (Bealer 1982, where properties,
relations and propositions are suggestively grouped under the acronym
"PRP;" Dixon 2018; Menzel 1986; 1993; Swoyer 1998; Orilia
1999; Van Inwagen 2004; 2015; Zalta 1983). This terminology is also
applied to predicates and sentences; for example, the predicate
"between" is triadic, and the sentence "Peter is
between Tom and May" is zero-adic. Accordingly, standard
first-order logic employs predicates with a fixed degree, typically
indicated by a superscript, e.g., \(P^1\), \(Q^2\), \(R^3\), etc.



In natural language, however, many predicates appear to be
*multigrade* or *variably polyadic*; i.e., they can be
used with different numbers of arguments, as they can be true of
various numbers of things. For example, we say "John is lifting
a table", with "lifting" used as dyadic, as well as
"John and Mary are lifting a table", with
"lifting" used as triadic. Moreover, there is a kind of
inference, called "argument deletion", which also suggests
that many predicates that *prima facie* could be assigned a
certain fixed degree are in fact multigrade. For example, "John
is eating a cake" suggests that "is eating" is
dyadic, but since, by argument deletion, it entails "John is
eating", one could conclude that it is also monadic and thus
multigrade. Often one can resist the conclusion that there are
multigrade predicates. For example, it could be said that "John
is eating" is simply short for "John is eating
something". But it seems hard to find a systematic and
convincing strategy that allows us to maintain that natural language
predicates have a fixed degree. This has motivated the construction of
logical languages that feature multigrade predicates in order to
provide a more appropriate formal account of natural language (Grandy
1976; Graves 1993; Orilia 2000a). Since natural language predicates
appear to be multigrade, one may be tempted to take the properties and
relations that they express to also be multigrade, and the metaphysics
of science may lend support to this conclusion (Mundy 1989).



Seemingly, relations are not jointly instantiated
*simpliciter*; *how* the instantiation occurs also plays
a role. This comes to the fore in particular with non-symmetric
relations such as *loving*. For example, if John loves Mary,
then *loving* is jointly instantiated by John and Mary in a
certain way, whereas if it is Mary who loves John, then
*loving* is instantiated by John and Mary in another way.
Accordingly, relations pose a special problem: explicating the
difference between facts, such as *Abelard loves Eloise* and
*Eloise loves Abelard*, that at least *prima facie*
involve exactly the same constituents, namely a non-symmetric relation
and two other items (*loving*, Abelard, Eloise). Such facts are
often said to differ in "relational order" or in the
"differential application" of the non-symmetric relation
in question, and the problem then is that of characterizing what this
relational order or differential application amounts to.



Russell (1903: SS218) attributed an enormous importance to this
issue and attacked it repeatedly. Despite this, it has been pretty
much neglected until the end of last century, with only few others
confronting it systematically (e.g., Bergmann 1992; Hochberg 1987).
However, Fine (2000) has forcefully brought it again on the
ontological agenda and proposed a novel approach that has received
some attention. Fine identifies *standard* and
*positionalist* views (analogous to two approaches defended by
Russell at different times (1903; 1984); cf. Orilia 2008). According
to the former, relations are intrinsically endowed with a
"direction", which allows us to distinguish, e.g.,
*loving* and *being loved*: *Abelard loves
Eloise* and *Eloise loves Abelard* differ, because they
involve two relations that differ in direction (e.g., the former
involves *loving* and the latter *being loved*).
According to the latter, relations have different
"positions" that can somehow host *relata*:
*Abelard loves Eloise* and *Eloise loves Abelard*
differ, because the two positions of the very same *loving*
relation are differently occupied (by Abelard and Eloise in one case
and by Eloise and Abelard in the other case). Fine goes on to propose
and endorse an alternative, "anti-positionalist"
standpoint, according to which relations have neither direction nor
positions. The literature on this topic keeps growing and there are
now various proposals on the market, including new versions of
positionalism (Orilia 2014; Donnelly 2016; Dixon 2018), and
*primitivism*, according to which differential application
cannot be analyzed (MacBride 2014).



Russell (1903: ch. 26) also had a key role in leading philosophers to
acknowledge that at least some relations, in particular
spatio-temporal ones, are *external*, i.e., cannot be reduced
to monadic properties, or the mere existence, of the *relata*,
in contrast to internal relations that can be so reduced. This was a
breakthrough after a long tradition tracing back at least to Aristotle
and the Scholastics wherein there seems to be hardly any place for
external relations (see entry on
 medieval theories of relations).[4]


### 1.6 Universals *versus* Tropes



According to some philosophers, universals and tropes may coexist in
one ontological framework (see, e.g., Lowe 2006 for a well-known
general system of this kind, and Orilia 2006a, for a proposal based on
empirical data from quantum mechanics). However, nowadays they are
typically seen as alternatives, with the typical supporter of
universals ("universalist") trying to do without tropes
(e.g., Armstrong 1997) and the typical supporter of tropes
("tropist") trying to dispense with universals (e.g.,
Maurin
 2002).[5]
 In order to clarify how differently they see matters, we may take
advantage of states of affairs. Both parties may agree, say, that
there are two red apples, \(a\) and \(b\). They will immediately
disagree, however, for the universalist will add that


1. there are two distinct states of affairs, *that a is red*
and *that b is red*,
2. such states are similar in having the universal *red* as
constituent, and
3. they differ insofar as the former has \(a\) as constituent,
whereas the latter has \(b\).



The tropist will reject these states of affairs with universals as
constituents and rather urge that there are two distinct tropes, the
redness of \(a\) and the redness of \(b\), which play a theoretical
role analogous to the one that the universalist would invoke for such
states of affairs. Hence, tropists claim that tropes can be causal
relata (D. Williams 1953) and truthmakers (Mulligan, Simons, &
Smith 1984).



Tropes are typically taken to be *simple*, i.e., without any
subconstituent (see Section 2.2 of the entry on
 tropes).
 Their playing the role of states of affairs with universals as
constituents depends on this: universals combine two functions, only
one of which is fulfilled by tropes. On the one hand, universals are
*characterizers*, inasmuch as they characterize concrete
objects. On the other hand, they are also *unifiers*, to the
extent that different concrete objects may be characterized by the
very same universal, which is thus somehow shared by all of them; when
this is the case, there is, according to the universalist, an
objective similarity among the different objects (see
 SS1.2).
 In contrast, tropes are only characterizers, for, at least as
typically understood, they cannot be shared by distinct concrete
objects. Given its dependency on one specific object, say, the apple
\(a\), a trope can do the work of a state of affairs with \(a\) as
constituent. But for tropes to play this role, the tropist will have
to pay a price and introduce additional theoretical machinery to
account for objective similarities among concrete objects. To this
end, she will typically resort to the idea that there are objective
resemblances among tropes, which can then be grouped together in
resemblance classes. These resemblance classes play the role of
unifiers for the tropist. Hence, from the tropist's point of
view "property" is ambiguous, since it may stand for the
characterizers (tropes) or for the unifiers (resemblance classes) (cf.
entry on
 mental causation, SS6.5).
 Similarly, "exemplification" and related words may be
regarded as ambiguous insofar as they can be used either to indicate
that an object exemplifies a certain trope or to indicate that the
object relates to a certain resemblance class by virtue of
exemplifying a trope in that
 class.[6]


### 1.7 Kinds of Properties



Many important and often controversial distinctions among different
kinds of properties have been made throughout the whole history of
philosophy until now, often playing key roles in all sorts of disputes
and arguments, especially in metaphysics. Here we shall briefly review
some of these distinctions and others will surface in the following
sections. More details can be found in other more specialized entries,
to which we shall refer.



Locke influentially distinguished between *primary* and
*secondary* qualities; the former are objective features of
things, such as shapes, sizes and weights, whereas the latter are
mind-dependent, e.g., colors, *tastes*,
*sounds*, and *smells*. This contrast was already
emphasized by the Greek atomists and was revived in modern times by
Galileo, Descartes, and Boyle.



At least since Aristotle, the *essential* properties of an
object have been contrasted with its *accidental* properties;
the object could not exist without the former, whereas it could fail
to have the latter (see entry on
 essential vs. accidental properties).
 Among essential properties, some acknowledge *individual
essences* (also called "haecceities" or
"thisnesses"), which univocally characterize a certain
individual. Adams (1979) conceives of such properties as involving,
via the identity relation, the very individual in question, e.g.,
Socrates: *being identical to Socrates*, which cannot exist if
Socrates does not exist. In contrast, Plantinga (1974) views them as
capable of existing without the individuals of which they are
essences, e.g., *Socratizing*, which could have existed even if
Socrates had not existed. See
 SS5.2
 on the issue of the essences of properties themselves.



*Sortal* properties are typically
expressed by count nouns like "desk" and "cat"
and are taken to encode principles of individuation and persistence
that allow us to objectively count objects. For example, there is a
fact of the matter regarding how many things in this room instantiate
*being a desk* and *being a cat*. On the other hand,
non-sortal properties such as *red* or *water* do not
allow us to count in a similarly obvious way. This distinction is
often appealed to in contemporary metaphysics (P. Strawson 1959: ch.
5, SS2; Armstrong 1978a: ch. 11, SS4), where, in contrast, the
traditional one between *genus* and
 *species*
 plays a relatively small role. The latter figured conspicuously in
Aristotle and in much subsequent philosophy inspired by him. We can
view a genus as a property more general than a corresponding species
property, in a hierarchically relative manner. For example, *being
a mammal* is a genus relative to the species *being a
human*, but it is a species relative to the genus *being an
animal*. The possession of a property called *differentia*
is appealed to in order to distinguish between different species
falling under a common genus; e.g., as the tradition has it, the
differentia for being human is *being rational* (Aristotle,
*Categories*, 3a). Similar hierarchies of properties, however
without anything like differentiae, come with the distinction of
 determinables and determinates,
 which appears to be
more prominent in current metaphysics. Color properties provide
typical examples of such hierarchies, e.g., with *red* and
*scarlet* as determinable and determinate, respectively.


## 2. Exemplification



We saw right at the outset that objects exemplify, or instantiate,
properties. More generally, items of all sorts, including properties
themselves, exemplify properties, or, in different terminology,
*bear*, *have* or *possess* properties. Reversing
order, we can also say that properties *characterize*, or
*inhere in*, the items that exemplify them. There is then a
very general phenomenon of exemplification to investigate, which has
been labeled in various ways, as the variety of terms of art just
displayed testifies. All such terms have often been given special
technical senses in the rich array of different explorations of this
territory since Ancient and Medieval times up to the present age (see,
e.g., Lowe 2006: 77). These explorations can hardly be disentangled
from the task of providing a general ontological picture with its own
categorial distinctions. In line with what most philosophers do
nowadays, we choose "exemplification", or, equivalently,
"instantiation" (and their cognates), to discuss this
phenomenon in general and to approach some different accounts that
have been given of it in recent times. This sweeping use of these
terms is to be kept distinct from the more specialized uses of them
that will surface below (and to some extent have already surfaced
above) in describing specific approaches by different philosophers
with their own terminologies.


### 2.1 Monist vs. Pluralist Accounts



We have taken for granted that there is just one kind of
exemplification, applying indifferently to different categories of
entities. This *monist* option may indeed be considered the
default one. A typical recent case of a philosopher who endorses it is
Armstrong (1997). He distinguishes three basic
categories, particulars, properties or relations, and states of
affairs, and takes exemplification as cutting across them: properties
and relations are exemplified not only by particulars, but by
properties or relations and states of affairs as well. But some
philosophers are *pluralist*: they distinguish different kinds
of exemplification, in relation to categorial distinctions in their
ontology.



One may perhaps attribute different kinds of exemplification to the
above-considered Meinongians in view of the different sorts of
predication that they admit (see, e.g., Monaghan's (2011)
discussion of Zalta's theory). A more typical example of the
pluralist alternative is however provided by Lowe (2006), who
distinguishes "instantiation",
"characterization" and "exemplification" in
his account of four fundamental categories: objects, and three
different sorts of properties, namely kinds (substantial universals),
attributes and modes
 (tropes).[7]
 To illustrate, Fido is a dog insofar as it *instantiates* the
kind *dog*, \(D\), which in turn is *characterized* by
the attribute of barking, \(B\). Hence, when Fido is barking, it
*exemplifies* \(B\) *occurrently* by virtue of being
characterized by a barking mode, \(b\), that instantiates \(B\); and,
when Fido is silent, it exemplifies \(B\) *dispositionally*,
since \(D\), which Fido instantiates, is characterized by
\(B\) (see Gorman 2014 for a critical discussion of this sort of
view).


### 2.2 Compresence and Partial Identity



Most philosophers, whether tacitly or overtly, appear to take
exemplification as primitive and unanalyzable. However, on certain
views of particulars, it might seem that exemplification is reduced to
something more fundamental.



A well-known such approach is the *bundle theory*, which takes
particulars to be nothing more than "bundles" of
universals connected by a special relation, commonly called
*compresence*, after Russell (1948: Pt. IV, ch.
 8)[8].
 Despite well-known problems (Van Cleve 1985), this view, or
approaches in its vicinity, keep having supporters (Casullo 1988;
Curtis 2014; Dasgupta 2009; Paul 2002; Shiver 2014; J. Russell 2018;
see Sider 2020, ch. 3, for a recent critical analysis). From this
perspective, that a particular exemplifies a property amounts to the
property's being compresent with the properties that constitute
the bundle with which the particular in question is identified. It
thus looks as if exemplification is reduced to compresence.
Nevertheless, compresence itself is presumably jointly exemplified by
the properties that constitute a given bundle, and thus at most there
is a reduction, to compresence, of exemplification *by a
particular* (understood as a bundle), and not an elimination of
exemplification in general.



Another, more recent, approach is based on *partial identity*.
Baxter (2001) and, inspired by him, Armstrong (2004), have proposed
related assays of exemplification, which seem to analyze it in terms
of such partial identity. These views have captured some interest and
triggered discussions (see, e.g., Mumford 2007; Baxter's (2013)
reply to critics and Baxter's (2018) rejoinder to Brown
2017).



Baxter (2001) relies on the notion of *aspect* and on the
relativization of numerical identity to *counts*. In his view,
both particulars and properties *have* aspects, which can be
similar to distinct aspects of other particulars or properties. The
numerical identity of aspects is relative to standards for counting,
*counts*, which group items in count collections: aspects of
particulars in the *particular* collection, and aspects of
universals, in the *universal* collection. There can then be a
*cross-count identity*, which holds between an aspect in the
particular collection, and an aspect in the universal collection,
e.g., the aspects *Hume as human* and *humanity as had by
Hume*. In this case, the universal and the particular in question
(humanity and Hume, in our example) are *partially identical*.
Instantiation, e.g., that Hume instantiates humanity, then amounts to
this partial identity of a universal and a particular. One may have
the feeling, as Baxter himself worries (2001: 449), that in this
approach instantiation has been traded for something definitely more
obscure, such as aspects and an idiosyncratic view of identity. It can
also be suspected that particulars' and properties'
*having* aspects is presupposed in this analysis, where this
*having* is a relation rather close to exemplification
itself.



Armstrong (2004) tries to do without aspects. At first glance it seems
as if he analyzes exemplification, for he takes the exemplification of
a property (a universal) by a particular to be a partial identity of
the property and the particular; as he puts it (2004: 47), "[i]t
is not a mere mereological overlap, as when two streets intersect, but
it is a partial identity". However, when we see more closely
what this partial identity amounts to, the suspicion arises that it
presupposes exemplification. For Armstrong appears to identify a
particular via the properties that it *instantiates* and
similarly a property via the particulars that *instantiate* it.
So that we may identify a particular, \(x\), via a collection of
properties *qua* instantiated by \(x\), say \(\{F\_x,\) \(G\_x,\)
\(H\_x,\) ..., \(P\_x,\) \(Q\_x,\) \(\ldots\};\) and a property,
\(P\), via a collection of particulars *qua* instantiating
\(P\), say \(\{P\_a, P\_b , \ldots ,P\_x, P\_y , \ldots \}\). By putting
things in this way, we can then say that a particular is partially
identical to a property when the collection that identifies the
particular has an element in common with the collection that
identifies the property. To illustrate, the \(x\) and the \(P\) of our
example are partially identical because they have the element \(P\_x\)
in common. Now, the elements of these collections are neither
properties *tout court* nor particulars *tout court*,
which led us to talk of properties *qua* instantiated and
particulars *qua*
 instantiating.[9]
 But this of course presupposes instantiation. Moreover, there is the
unwelcome consequence that the world becomes dramatically less
contingent than we would have thought at first sight, for neither a
concrete particular nor a property can exist without it being the case
that the former has the properties it happens to have, and that the
latter is instantiated by the same particulars that actually
instantiate it; we get, as Mumford puts it (2007: 185), "a major
new kind of necessity in the world".


### 2.3 Bradley's Regress



One important motivation, possibly the main one, behind attempts at
analysis such as the ones we have just seen is the worry to avoid the
so-called Bradley's regress
regarding
exemplification (Baxter 2001: 449; Mumford 2007: 185), which goes as
follows. Suppose that the individual \(a\) has the property \(F\). For
\(a\) to instantiate \(F\) it must be *linked to* \(F\) by a
(dyadic) relation of instantiation, \(I\_1\). But this requires a
further (triadic) relation of instantiation, \(I\_2\), that connects
\(I\_1, F\) and \(a\), and so on without end. At each stage a further
connecting relation is required, and thus it seems that nothing
*ever* gets connected to anything else (it is not clear to what
extent Bradley had this version in mind; for references to analogous
regresses prior to Bradley's, see Gaskin 2008: ch. 5,
SS70).



This regress has traditionally been regarded as vicious (see, e.g.,
Bergmann 1960), although philosophers such as Russell (1903: SS55)
and Armstrong (1997: 18-19) have argued that it is not. In doing
so, however, they seem to take for granted the fact that \(a\) has the
property \(F\) (pretty much as in the *brute fact approach*;
see below) and go on to see \(a\)'s and \(F\)'s
instantiating \(I\_1\) as a further fact that is merely
*entailed* by the former, which in turn entails \(a\)'s,
\(F\)'s and \(I\_1\)'s instantiating \(I\_2\), and so on.
This way of looking at the matter tends to be regarded as a standard
response to the regress. But those who see the regress as vicious
assume that the various exemplification relations are introduced in an
effort to *explain* the very existence of the fact that \(a\)
has the property \(F\). Hence, from their explanatory standpoint,
taking the fact in question as an unquestioned ground for a chain of
entailments is beside the point (cf. Loux 1998: 31-36;
Vallicella 2002).



It should be noted, however, that this perspective suggests a
distinction between an "internalist" and an
"externalist" version of the regress (in the terminology
of Orilia 2006a). In the former, at each stage we postulate a new
constituent of the fact, or state of affairs, \(s\), that exists
insofar as \(a\) has the property \(F\), and there is viciousness
because \(s\) can never be appropriately
 characterized.[10]
 In the latter, at each stage we postulate a new, distinct, state of
affairs, whose existence is required by the existence of the state of
affairs of the previous stage. This amounts to admitting infinite
explanatory and metaphysical dependence chains. However, according to
Orilia (2006b: SS7), since no decisive arguments against such
chains exist, the externalist regress should not be viewed as vicious
(for criticisms, see Maurin 2015 and Allen 2016: SS2.4.1; for a
similar view about predication, see Gaskin 2008).



A typical line for those convinced that the regress is vicious has
consisted in proposing that instantiation is not a relation, or at
least not a normal one. Some philosophers hold that it is a *sui
generis* linkage that hooks things up without intermediaries.
Peter Strawson (1959) calls it a *non-relational tie* and
Bergmann (1960) calls it a *nexus*. Broad (1933: 85) likened
instantiation to glue, which *just* sticks two sheets of paper
together, without needing anything additional; similarly,
instantiation *just* relates. An alternative line has been to
reject instantiation altogether. According to Frege, it is not needed,
because properties have "gaps" that can be filled, and
according to a reading of Wittgenstein's *Tractatus*,
because objects and properties can be connected like links in a chain.
However, both strategies are problematic, as argued by Vallicella
(2002). His basic point is that, if \(a\) has property \(F\), we need
an ontological explanation of why \(F\) and \(a\) happen to be
connected in such a way that \(a\) has \(F\) as one of its properties
(unless \(F\) is a property that \(a\) has necessarily). But none of
these strategies can provide this explanation. For example, the appeal
to gaps is pointless: \(F\) has a gap whether or not it is filled by
\(a\) (for example, it could be filled in by another object), and thus
the gap cannot explain the fact that \(a\) has \(F\) as one of its
properties.



Before turning to exemplification as partial identity, Armstrong
(1997: 118) has claimed that Bradley's regress can be avoided by
taking a state of affairs, say \(x\)'s being \(P\), as capable
by itself of holding together its constituents, i.e., the object \(x\)
and the property \(P\) (see also Perovic 2016). Thus, there is no need
to invoke a relation of exemplification linking \(x\) and \(P\) in
order to explain how \(x\) and \(P\) succeed in giving rise to a
unitary item, namely the state of affairs in question. There seems to
be a circularity here for it appears that we want to explain how an
object and a property come to be united in a state of affairs by
appealing to the result of this unification, namely the state of
affairs itself. But perhaps this view can be interpreted as simply the
idea that states of affairs should be taken for granted in a
primitivist fashion without seeking an explanation of their unity by
appealing to exemplification or otherwise; this is the *brute fact
approach*, as we may call it (for supporters, see Van Inwagen
(1993: 37) and Oliver (1996: 33); for criticisms and a possible
defense, see Vallicella 2002, and Orilia 2016, respectively).



Lowe (2006) has tried to tackle Bradley's regress within his
pluralist approach to exemplification. In his view, characterization,
instantiation and exemplification are "formal" and thus
quite different from garden-variety relations such *giving* or
*loving*. This guarantees that these three relations escape
Bradley's regress (Lowe 2006: 30, 80,
 90).[11]
 Let us illustrate how, by turning back to the Fido example of
 SS2.1.
 What a mode instantiates and what it characterizes belong to its
essence. In other words, a mode cannot exist without instantiating the
attribute it instantiates and characterizing the object it
characterizes. Hence, mode \(b\), by simply existing, instantiates
attribute \(B\) and characterizes Fido. Moreover, since
exemplification (the occurrent one, in this case) results from
"composing" characterization and instantiation,
\(b\)'s existence also guarantees that Fido exemplifies \(B\).
According to Lowe, we thus have some truths, that \(b\) characterizes
Fido, that \(b\) instantiates \(B\), and that Fido exemplifies \(B\)
(i.e., is barking), all of which are made true by \(b\). Hence, there
is no need to postulate as truthmakers states of affairs with
constituents, Fido and \(b\), related by characterization, or \(b\)
and \(B\), related by exemplification, or Fido and \(B\), related by
exemplification. This, in Lowe's opinion, eschews
Bradley's regress, since this arises precisely because we appeal
to states of affairs with constituents in need of a glue that
contingently keep them together. Nevertheless, there is no loss of
contingency in Lowe's world picture, for an object need not be
characterized by the modes that happen to characterize it. Thus, for
example, mode \(b\) might have failed to exist and there could have
been a Fido silence mode in its stead, in which case the proposition
that Fido is barking would have been false and the proposition that
Fido is silent would have been true. One may wonder however what makes
it the case that a certain mode is a mode of just a certain object and
not of another one, say another barking dog. Even granting that it is
essential for \(b\) to be a mode of Fido, rather than of another dog,
it remains true that it is *of* Fido, rather than of the other
dog, and one may still think that this *being of* is also glue
of some sort, perhaps with a contingency inherited from the
contingency of \(b\) (which might have failed to exist). The suspicion
then is that the problem of accounting for the relation between a mode
and an object has replaced the Armstrongian one of what makes it the
case that a universal \(P\) and an object \(x\) make up the state of
affairs *\(x\)'s being \(P\)*. But the former problem,
one may urge, is no less thorny than the latter, and some
universalists like Armstrong may consider uneconomical Lowe's
acceptance of tropes in addition to universals (for accounts of
Bradley's regress analogous to Lowe's, but within a
thoroughly tropist ontology, see Section 3.2 of the entry on
 tropes.)



As it should be clear from this far from exhaustive survey,
Bradley's regress deeply worries ontologists and the attempts to
tame it keep
 flowing.[12]


### 2.4 Self-exemplification



Presumably, properties exemplify properties. For example, if
properties are abstract objects, as is usually thought, then seemingly
every property exemplifies abstractness. But then we should also grant
that there is self-exemplification, i.e., a property exemplifying
itself. For example, abstractness is itself abstract and thus
exemplifies itself. Self-exemplification however has raised severe
perplexities at least since Plato.



Plato appears to hold that *all* properties exemplify
themselves, when he claims that forms participate in themselves. This
claim is crucially involved in his so-called *third man
argument*, which led him to worry that his theory of forms is
incoherent (*Parmenides*, 132 ff.). As we see matters now, it
is not clear why we should hold that all properties exemplify
themselves (Armstrong 1978a: 71); for instance, people are honest, but
honesty itself is not honest (see, however, the entry on
 Plato's *Parmenides*,
 and Marmodoro forthcoming).



Nowadays, a more serious worry related to self-exemplification is
Russell's famous paradox, constructed as regarding the property
of non-self-exemplification, which appears to exemplify itself iff it
does not, thus defying the laws of logic, at least classical logic.
The discovery of his paradox (and then the awareness of related
puzzles) led Russell to introduce a theory of types, which institutes
a total ban on self-predication by a rigid segregation of properties
into a hierarchy of *types* (more on this in
 SS6.1).
 The account became more complex and rigid, as Russell moved from
*simple* to *ramified* type theory, which involves a
distinction of *orders* within types (see entries on
 type theory
 and
 Russell's paradox,
 and, for a detailed reconstruction of how Russell reacted to the
paradox, Landini 1998).



In type theory all properties are, we may say, *typed*. This
approach has never gained unanimous consensus and its many problematic
aspects are well-known (see, e.g., Fitch 1952: Appendix C; Bealer
1989). Just to mention a few, the type-theoretical hierarchy imposed
on properties appears to be highly artificial and multiplies
properties *ad infinitum* (e.g., since presumably properties
are abstract, for any property \(P\) of type \(n\), there is an
abstractness of type \(n+1\) that \(P\) exemplifies). Moreover, many
cases of self-exemplification are innocuous and common. For example,
the property of *being a property* is itself a property, so it
exemplifies itself. Accordingly, many recent proposals are type-free
(see
 SS6.1)
 and thus view properties as *untyped*, capable of being
self-predicated, sometimes veridically. An additional motivation to
move in this direction is a new paradox proposed by Orilia and Landini
(2019), which affects simple type theory. It is
"contingent" in that it is derived from a contingent
assumption, namely that someone, say John, is thinking just about the
property of being a property \(P\) such that John is thinking of
something that is not exemplified by \(P\).


## 3. Existence and Identity Conditions for Properties



Quine (1957 [1969: 23]) famously claimed that there should be no
entity without identity. His paradigmatic case concerns sets: two of
them are identical iff they have exactly the same members. Since then
it has been customary in ontology to search for identity conditions
for given categories of entities and to rule out categories for want
of identity conditions (against this, see Lowe 1989). Quine started
this trend precisely by arguing against properties and this has
strictly intertwined the issues of which properties there are and of
their identity
 conditions.[13]


### 3.1 From Extensionality to Hyperintensionality



In an effort to provide identity conditions for properties, one could
mimic those for sets, or equate the former with the latter (as in
class nominalism; see note 3), and provide the following
*extensionalist* identity conditions: two properties are
identical iff they are co-extensional. This criterion can hardly work,
however, since there are seemingly distinct properties with the same
extension, such as *having a heart* and *having
kidneys*, and even wildly different properties such as
*spherical* and *weighing 2 kilos* could by accident be
co-extensive.



One could then try the following *intensional* identity
conditions: two properties are identical iff they are
*co-intensional*, i.e., necessarily co-extensional, where the
necessity in question is logical necessity. This guarantees that
*spherical* and *weighing 2 kilos* are different even if
they happen to be co-extensional. Following this line, one may take
properties to be *intensions*, understood as, roughly,
functions that assign extensions (sets of objects) to predicates at
logically possible worlds. Thus, for instance, the predicates
"has a heart" and "has kidneys" stand for
different intensions, for even if they have the same extension in the
actual world they have different extensions at worlds where there are
creatures with heart and no kidneys or vice versa. This approach is
followed by Montague (1974) in his pioneering work in natural language
semantics, and in a similar way by Lewis (1986b), who reduces
properties to sets of possible objects in his modal realism,
explicitly committed to possible worlds and mere *possibilia*
inhabiting them. Most philosophers find this commitment unappealing.
Moreover, one may wonder how properties can do their causal work if
they are conceived of in this way (for further criticisms see Bealer
1982: 13-19, and 1998: SS4; Egan 2004). However, the
criterion of co-intensionality may be accepted without also buying the
reduction of properties to sets of *possibilia* (Bealer views
this as the identity condition for his conception 1 properties; see
below). Still, co-intensionality must face two challenges coming from
opposite fronts.



On the one hand, from the perspective of empirical science,
co-intensionality may appear too strong as a criterion of identity.
For the identity statements of scientific reductions, such as that of
temperature to mean kinetic energy, could suggest that some properties
are identical even if not co-intensional. For example, one may accept
that *having absolute temperature of 300K* is *having mean
molecular kinetic energy of \(6.21 \times 10^{-21}\)* (Achinstein
1974: 259; Putnam (1970: SS1) and Causey (1972) speak of
"synthetic identity" and "contingent
identity", respectively). Rather than logical necessity, it is
*nomological* necessity, necessity on the basis of the causal
laws of nature, that becomes central in this line of thought.
Following it, some have focused on the *causal* and
*nomological* roles of properties, i.e., roughly, the causes
and effects of their being instantiated, and their involvement in laws
of nature, respectively. They have thus advanced *causal or
nomic* criteria. According to them, two properties are identical
iff they have the same causal (Achinstein 1974: SSXI; Shoemaker
1980) or nomological role (Swoyer 1982; Kistler 2002). This line has
been influential, as it connects to the "pure
dispositionalism" discussed in
 SS5.2.
 There is however a suspicion of circularity here, since causal and
nomological roles may be viewed as higher-order properties (see entry
on
 dispositions, SS3).



On the other hand, once matters of meaning and mental content are
taken into account, co-intensionality might seem too weak, for it
makes a property \(P\) identical to any logically equivalent property,
e.g., assuming classical logic, *\(P\) and (\(Q\) or not
\(Q\))*. And with a sufficiently broad notion of logical
necessity, even, for example, *being triangular* and *being
trilateral* are identical. However, one could insist that
"trilateral" and "triangular" appear to have
different meanings, which somehow involve the different geometrical
properties *having a side*, and *having an angle*,
respectively. And if *being triangular* were really identical
to *being trilateral*, from the fact that John believes that a
certain object has the former property, one should be able to infer
that John also believes that such an object has the latter property.
Yet, John's ignorance may make this conclusion unwarranted. In
the light of this, borrowing a term from Cresswell (1975), one may
move from intensional to *hyperintensional* identity
conditions, according to which two properties, such as trilaterality
and triangularity, may be different even if they are co-intensional.
In order to implement this idea, Bealer (1982) take two properties to
be identical iff they have the same analysis, i.e., roughly, they
result from the same ultimate primitive properties and the same
logical operations applied to them (see also Menzel 1986; 1993). Zalta
(1983; 1988) has developed an alternative available to those who admit
two modes of predication (see
 SS1.4):
 two properties are identical iff they are (necessarily) encoded by
the same objects.



Hyperintensional conditions make of course for finer distinctions
among entities than the other criteria we considered. Accordingly, the
former are often called "fine-grained" and the latter
"coarse-grained", and the same denominations are
correspondingly reserved for the entities that obey the conditions in
question. Coarse- or fine-grainedness is a relative matter. The
extensional criterion is more coarse-grained than the intensional or
causal/nomological ones. And hyperintensional conditions themselves
may be more or less fine-grained: properties could be individuated
almost as finely as the predicates expressing them, to the point that,
e.g., even *being \(P\) and \(Q\)* and *being \(Q\) and
\(P\)* are kept distinct, but one may also envisage less stringent
conditions that make for the identification of properties of that sort
(Bealer 1982: 54). It is conceivable, however, that one could be
logically obtuse to the point of believing that something has a
certain property without believing that it has a trivially equivalent
property. Thus, to properly account for mental content,
*maximal* hyperintensionality appears to be required, and it is
in fact preferred by Bealer. Even so, the paradox of analysis
raises a serious issue.
One could say, for example, that *being a circle* is *being
a locus of points equidistant from a point*, since the latter
provides the analysis of the former. In reply, Bealer distinguishes
between "being a circle" as designating a simple
"undefined" property, which is an *analysandum*
distinct from the *analysans*, *being a locus of points
equidistant from a point*, and "being a circle" as
designating a complex "defined" property, which is indeed
identical to that *analysans*. Orilia (1999: SS5.5)
similarly distinguishes between the simple *analysans* and the
complex *analysandum*, without admitting that expressions for
properties such as "being a circle" could be ambiguous in
the way Bealer suggests. Orilia rather argues that the
"is" of analysis does not express identity but a weaker
relation, which is asymmetrical in that *analysans* and
*analysandum* play different roles in it. The matter keeps
being discussed. Rosen (2015) appeals to grounding to characterize the
different role played by the *analysans*. Dorr (2016) provides
a formal account of the "is" used in identity statements
involving properties, according to which it stands for a symmetric
"identification" relation.


### 3.2 The Sparse and the Abundant Conceptions



Bealer (1982) distinguishes between *conception 1* properties,
or *qualities*, and *conception 2* properties, or
*concepts* (understood as mind-independent). With a different
and now widespread terminology, Lewis (1983, 1986b) followed suit,
speaking of a *sparse* and an *abundant conception* of
properties. According to the former, there are relatively few
properties, namely just those responsible for the objective
resemblances and causal powers of things; they cut nature at its
joints, and science is supposed to individuate them *a
posteriori*. According to the latter, there are immensely many
properties, corresponding to all meaningful predicates we could
possibly imagine and to all sets of objects, and they can be assumed
*a priori*. (It should be clear that "few" and
"many" are used in a comparative sense, for the number of
sparse properties may be very high, possibly infinite). To illustrate,
the sparse conception admits properties currently accepted by
empirical science such as *having negative charge* or
*having spin up* and rejects those that are no longer supported
such as *having a certain amount of caloric*, which featured in
eighteenth century chemistry; in contrast, the abundant conception may
acknowledge the latter property and all sorts of other properties,
strange as they may be, e.g., *negatively charged or disliked by
Socrates*, *round and square* or Goodman's (1983)
notorious *grue* and *bleen*. Lewis (1986b: 60) tries to
further characterize the distinction by taking the sparse properties
to be *intrinsic*, rather than *extrinsic* (e.g.,
*being 6 feet tall*, rather than *being taller than
Tom*), and *natural*, where naturalness is something that
admits of degrees (for example, he says, masses and charges are
perfectly natural, colors are less natural and *grue* and
*bleen* are paradigms of unnaturalness). Much work on such
issues has been done since then (see entries on
 intrinsic vs. extrinsic properties
 and
 natural properties).



Depending on how sparse or abundant properties are, we can have two
extreme positions and other more moderate views in between.



At one end of the spectrum, there is the most extreme version of the
sparse conception, *minimalism*, which accepts all of these
principles:


1. there are only coarse-grained properties,
2. they exist only if instantiated and thus are contingent
beings,
3. they are all instantiated by things in space-time (setting aside
those instantiated by other properties),
4. they are fundamental and thus their existence must be sanctioned
by microphysics.



This approach is typically motivated by physicalism and
epistemological qualms regarding transcendent universals. The
best-known contemporary supporter of minimalism is Armstrong (1978a,b,
1984). Another minimalist is Swoyer (1996).



By dropping or mitigating some of the above principles, we get less
minimalist versions of the sparse conception. For example, some have
urged uninstantiated properties to account for features of measurement
(Mundy 1987), vectors (Bigelow & Pargetter 1991: 77), or natural
laws (Tooley 1987), and some even that there are all the properties
that can be possibly exemplified, where the possibility in question is
causal or nomic (Cocchiarella 2007: ch. 12). In line with
positions traditionally found in emergentism (see
 SS5.1),
 Schaffer (2004) proposes that there are sparse properties as
fundamental, and in addition, as grounded on them, the properties that
need be postulated at all levels of scientific explanations, e.g.,
chemical, biological and psychological ones. Even Armstrong goes
beyond minimalist strictures, when in his later work (1997)
distinguishes between "first-class properties"
(universals), identified from a minimalist perspective, and
"second-class properties" (supervening on universals). All
these positions appear to be primarily concerned with issues in the
metaphysics of science (see
 SS5)
 and typically display too short a supply of properties to deal with
meaning and mental content, and thus with natural language semantics
and the foundations of mathematics (see
 SS6).
 However, minimalists might want to resort to concepts, understood as
mind-dependent entities, in dealing with such issues (e.g., along the
lines proposed in Cocchiarella 2007).



Matters of meaning and mental content are instead what typically
motivate the views at the opposite end of the spectrum. To begin with,
*maximalism*, i.e., the abundant conception in all its glory
(Bealer 1982; 1993; Carmichael 2010; Castaneda 1976; Jubien
1989; Lewis 1986b; Orilia 1999; Zalta 1988; Van Inwagen 2004):
properties are fine-grained necessary entities that exist even if
uninstantiated, or even impossible to be instantiated. In its most
extreme version, maximalism adopts identity conditions that
differentiate properties as much as possible, but more moderate
versions can be obtained by slightly relaxing such conditions. Views
of this sort are hardly concerned with physicalist constraints or the
like and rather focus on the explanatory advantages of
hyperintensionality. These may well go beyond the typical motivations
considered above: Nolan (2014) argues that hyperintensionality is
increasingly important in metaphysics in dealing with issues such as
counter-possible conditionals, explanation, essences, grounding,
causation, confirmation and chance.



Rather than choosing between the sparse and abundant conceptions, the
very promoters of this distinction have opted in different ways for a
*dualism of properties*, according to which there are
properties of both kinds. Lewis endorses abundant properties as
reduced to sets of *possibilia*, and sparse properties either
viewed as universals, and corresponding to some of the abundant
properties (1983), or as themselves sets of possibilia (1986b: 60).
Bealer (1982) proposes a systematic account wherein qualities are the
coarse-grained properties that "provide the world with its
causal and phenomenal order" (1982: 183) and concepts are the
fine-grained properties that can function as meanings and as
constituents of mental contents. He admits a stock of simple
properties, which are both qualities and concepts (1982: 186),
wherefrom complex qualities and complex concepts are differently
constructed: on the one hand, by *thought-building operations*,
which give rise to fine-grained qualities; on the other hand, by
*condition-building operations*, which give rise to
coarse-grained qualities. Orilia (1999) has followed Bealer's
lead in also endorsing both coarse-grained qualities and fine-grained
concepts, without however identifying simple concepts and simple
qualities. In Orilia's account concepts are never identical to
qualities, but may correspond to them; in particular, the identity
statements of intertheoretic reductions should be taken to express the
fact that two different concepts correspond to the same quality.
Despite its advantages in dealing with a disparate range of phenomena,
dualism has not gained any explicit consensus. Its implicit presence
may however be widespread. For example, Putnam's (1970: SS1)
distinction between *predicates* and *physical
properties*, and Armstrong's above-mentioned recognition of
second-class properties may be seen as forms of dualism.


## 4. Complex Properties



It is customary to distinguish between *simple* and
*complex* properties, even though some philosophers take all
properties to be simple (Grossmann 1983: SSSS58-61). The
former are not characterizable in terms of other properties, are
primitive and unanalyzable and thus have no internal structure,
whereas the latter somehow have a structure, wherein other properties,
or more generally entities, are parts or constituents. It is not
obvious that there are simple properties, since one may imagine that
all properties are analyzable into constituents *ad infinitum*
(Armstrong 1978b: 67). Even setting this aside, to provide examples is
not easy. Traditionally, determinate colors are cited, but nowadays
many would rather appeal to fundamental physical properties such as
*having a certain electric charge*. It is easier to provide
putative examples of complex properties, once some other properties
are taken for granted, e.g., *blue and spherical* or *blue
or non-spherical*. These *logically compound* properties,
which involve logical operations, will be considered in
 SS4.1.
 Next, in
 SS4.2
 we shall discuss other kinds of complex properties, called
*structural* (after Armstrong 1978b), which are eliciting a
growing interest, as a recent survey testifies (Fisher 2018). Their
complexity has to do with the subdivisions of their instances into
subcomponents. Typical examples come from chemistry, e.g.,
*H2O* and *methane* understood as properties
of molecules.



It should be noted that it is not generally taken for granted that
complex properties literally have parts or constituents. Some
philosophers take this line (Armstrong 1978a: 36-39, 67ff.;
Bigelow & Pargetter 1989; Orilia 1999), whereas others demur, and
rather think that talk in terms of structures with constituents is
metaphorical and dependent on our reliance on structured terms such as
"blue and spherical" (Bealer 1982; Cocchiarella 1986a;
Swoyer 1998: SS1.2).


### 4.1 Logically Compound Properties



At least *prima facie*, our use of complex predicates suggests
that there are corresponding complex properties involving all sorts of
logical operations. Thus, one can envisage *conjunctive*
properties such as *blue and spherical, negative* properties
such as *non-spherical*, *disjunctive* properties such
as *blue or non-spherical*, *existentially* or
*universally quantified* properties such as *loving
someone* or *loved by everyone*, *reflexive*
properties such as *loving oneself*, etc. Moreover, one could
add to the list properties with individuals as constituents, which are
then denied the status of a *purely qualitative* property,
e.g., *taller than Obama*.



It is easy to construct complex predicates. But whether there really
are corresponding properties of this kind is a much more difficult and
controversial issue, tightly bound to the sparse/abundant distinction.
In the sparse conception the tendency is to dispense with them. This
is understandable since in this camp one typically postulates
properties *in rebus* for empirical explanatory reasons. But
then, if we explain some phenomena by attributing properties \(F\) and
\(G\) to an object, while denying that it does not exemplify \(H\), it
seems of no value to add that it also has, say, *F and G*,
*F or H*, and *non-H*. Nevertheless, Armstrong, the
leading supporter of sparseness, has defended over the years a mixed
stance without disjunctive and negative properties, but with
conjunctive ones. Armstrong's line has of course its opponents
(see, e.g., Paolini Paoletti 2014; 2017a; 2020a; Zangwill 2011 argues
that there are negative properties, but they are less real than
positive ones). On the other hand, in the abundant conception, all
sorts of logically compound properties are acknowledged. Since the
focus now is on meaning and mental content, it appears natural to
postulate such properties to account for our understanding of complex
predicates and of how they differ from simple ones. Even here,
however, there are disagreements, when it comes to problematic
predicates such the "does not exemplify itself" of
Russell's paradox. Some suggest that in this case one should
deny there is a corresponding property, in order to avoid the paradox
(Van Inwagen 2006). However, we understand this predicate and thus it
seems *ad hoc* that the general strategy of postulating a
property fails. It seems better to confront the paradox with other
means (see
 SS6).


### 4.2 Structural Properties



Following Armstrong (1978b: 68-71; see also Fisher 2018:
SS2), a structural property \(F\) is typically viewed as a
universal, and can be characterized thus:


1. an object exemplifying \(F\), say \(x\), must have
"relevant" proper parts that do not exemplify \(F\);
2. there must be some other "relevant" properties somehow
involved in \(F\) that are not exemplified by the object in
question;
3. the relevant proper parts must rather exemplify one or the other
of the relevant properties involved in \(F\).



For instance, a molecule \(w\) exemplifying *H2O*
has as parts two atoms \(h\_1\) and \(h\_2\) and an atom \(o\) that do
not exemplify *H2O*; this property involves
*hydrogen* and *oxygen*, which are not exemplified by
\(w\); \(h\_1\) and \(h\_2\) exemplify *hydrogen*, and \(o\)
exemplifies *oxygen*. For *relational* structural
properties, there is the further condition that


4. a certain "relevant" relation that links the relevant
proper parts must also be involved.



*H2O* is a case in point: the relevant (chemical)
relation is *bonding*, which links the three atoms in question.
Another, often considered, example is *methane* as a structural
property of methane-molecules (*CH4*), which
involves a *bonding* relation between atoms and the joint
instantiation of *hydrogen* (by four atoms) and *carbon*
(by one atom). *Non-relational* structural properties do not
require condition (iv). For instance, *mass 1 kg* involves many
relevant properties of the kind *mass n kg*, for \(n \lt 1\),
which are instantiated by relevant proper parts of any object that is
1 kg in mass, and these proper parts are not linked by any relevant
relation (Armstrong 1978b: 70).



In most approaches, including Armstrong's, the relevant
properties and the relevant relation, if any, are conceived of as
parts, or constituents, of the structural property in question, and
thus their being involved in the latter is a parthood relation. For
instance, *hydrogen*, *oxygen* and *bonding* are
constituents of *H2O*. Moreover, the composition of
structural properties is isomorphic to that of the complexes
exemplifying them. These two theses characterize what Lewis (1986a)
calls the "pictorial conception". It is worth noting that
structural properties differ from conjunctive properties such as
*human and musician*, in that the constituents of the latter
(i.e., *human* and *musician*) are instantiated by
whatever entity exemplifies the conjunctive property (i.e., a human
being who is also a musician), whereas the constituents of a
structural property (e.g., *hydrogen*, *oxygen* and
*bonding*) are not instantiated by the entities that
instantiate it (e.g., *H2O* molecules).



Structural properties have been invoked for many reasons. Armstrong
(1978b: ch. 22) appeals to them to explain the resemblance of
universals belonging to a common class, e.g., lengths and colors.
Armstrong (1988; 1989) also treats physical quantities and numbers
through structural properties. The former involve, as in the above
*mass 1 kg* example, smaller quantities such as *mass 0.1
kg* and *mass 0.2 kg* (this view is criticized by Eddon
2007). The latter are internal proportion relations between structural
properties (e.g., *being a nineteen-electrons aggregate*) and
unit-properties (e.g., *being an electron*). Moreover,
structural properties have been appealed to in treatments of laws of
nature (Armstrong 1989; Lewis 1986a), some natural kinds (Armstrong
1978b, 1997; Hawley & Bird 2011), possible worlds (Forrest 1986),
*ersatz* times (Parsons 2005), emergent properties
(O'Connor & Wong 2005), linguistic types (Davis 2014), ontic
structural realism (Psillos 2012). However, structural universals have
also been questioned on various grounds, as we shall now see.



Lewis (1986a) raises two problems for the pictorial conception. First,
it is not clear how one and the same universal (e.g.,
*hydrogen*) can recur more than once in a structural universal.
Let us dub this "multiple recurrence problem". Secondly,
structural universals violate the Principle of Uniqueness of
Composition of classical
 mereology.
 According to this principle, given a certain collection of parts,
there is only one whole that they compose. Consider isomers, namely,
molecules that have the same number and types of atoms but different
structures, e.g., butane and isobutane
(*C4H10*). Here, *butane* and
*isobutane* are different structural universals. Yet, they
arise from the same universals recurring the same number of times. Let
us dub this "isomer problem."



Two further problems may be pointed out. First, even molecules with
the same number and types of atoms and the same structures may vary in
virtue of their spatial orientation (Kalhat 2008). This phenomenon is
known as *chirality*. How can structural properties account for
it? Secondly, the composition of structural properties is restricted:
not every collection of properties gives rise to a structural
property. This violates another principle of classical mereology: the
Principle of Unrestricted Composition.



Lewis dismisses alternatives to the pictorial conception, such as the
"linguistic conception" (in which structural universals
are set-theoretic constructions out of simple universals) and the
"magic conception" (in which structural universals
are not complex and are primitively connected to the relevant
properties). Moreover, he also rejects the possibility of there being
*amphibians*, i.e., particularized universals lying between
particulars and full-fledged universals. Amphibians would solve the
multiple recurrence problem, as they would be identical with multiple
recurrences of single universals. For example, in *methane*,
there would be four amphibians of *hydrogen*.



To reply to Lewis' challenges, two main strategies have been
adopted on behalf of the pictorial conception: a non-mereological
composition strategy, according to which structural properties do not
obey the principles of classical mereology, and a mereology-friendly
strategy. Let us start with the former.



Armstrong (1986) admits that the composition of structural universals
is non-mereological. In his 1997, he emphasizes that states of affairs
do not have a mereological kind of composition, and views universals,
including structural ones, as *states of affairs-types* which,
as such, are not mereological in their composition. Structural
universals themselves turn out to be state of affairs-types of a
conjunctive sort (1997: 34 ff.). To illustrate, let us go back to the
above example with *H2O* exemplified by the water
molecule \(w\). It involves the conjunction of these states of
affairs:


1. atom \(h\_1\) being hydrogen,
2. atom \(h\_2\) being hydrogen,
3. atom \(o\) being oxygen,
4. \(h\_1\), \(h\_2\), and \(o\) being bonded.



This conjunction of states of affairs provides an example of a
state of affairs-type identifiable with the structural universal
*H2O*.



In pursuit of the non-mereological composition strategy, Forrest
(1986; 2006; 2016) relies on logically compound properties, in
particular (in his 2016) conjunctive, existentially quantified and
reflexive ones. Bennett (2013) argues that entities are part of
further entities by occupying specific slots within the latter.
Parthood slots are distinct from each other. Therefore, which entity
occupies which slot matters to the composition of the resultant
complex entity. Hawley (2010) defends the possibility of there being
multiple composition relations. In a more general way, McDaniel (2009)
points out that the relation of structure-making does not obey some of
the principles of classical mereology. Mormann (2010) argues that
distinct categories (to be understood as in
 category theory)
 come together with distinct parthood and composition relations.



As regards the mereology-friendly strategy, it has been suggested that
structural properties actually include extra components accounting for
structures, so that the Principle of Uniqueness of Composition is
safe. Considering *methane*, Pages (2002) claims that
such a structural property is composed of the properties
*carbon* and *hydrogen*, a *bonding* relation and
a peculiar first-order, formal relation between the atoms. According
to Kalhat (2008), the extra component is a second-order arrangement
relation between the first-order properties. Ordering formal
relations--together with causally individuated
properties--are also invoked by McFarland (2018).



At the intersection of these strategies, Bigelow and Pargetter (1989,
1991) claim that structural universals are internally relational
properties that supervene on first-order properties and relations and
on second-order proportion relations. Second-order proportion
relations are relations such as *having four times as*. They
relate first-order properties and relations. For example, in
*methane*, the relation *having four times as* relates
two conjunctive properties: (i) *hydrogen and being part of this
molecule*; (ii) *carbon and being part of this
molecule*.



Campbell (1990) solves the multiple recurrence problem and the isomer
problem by appealing to tropes. In *methane*, there are
multiple hydrogen tropes. In *butane* and *isobutane*,
distinct tropes entertain distinct structural relations.



Despite Lewis' dismissal, a number of theories appeal to
amphibians (i.e., particularized universals lying between particulars
and full-fledged universals) or, more precisely, to entities similar
to amphibians. Fine (2017) suggests that structural properties should
be treated by invoking arbitrary objects (Fine 1985). Davis'
(2014) account of linguistic types as structural properties confronts
the problem of the multiple occurrence of one type, e.g.,
"dog", in another (e.g., "every dog likes another
dog") (Wetzel 2009). In general, the idea that types are
properties of tokens is a natural one, although there are also
alternative views on the matter (see entry on
 types and tokens).
 Be this as it may, Davis proposes that types cannot occur within
further types as tokens: they can only occur as *subtypes*.
Therefore, subtypes lie, like amphibians, between types and tokens.
Subtypes are individuated by their positions in asymmetric wholes,
whereas they are primitively and non-qualitatively distinct in
symmetric ones. The approach could be extended to properties such as
*methane*, which would be taken to have four distinct
*hydrogen* subtypes.


## 5. Properties in the Metaphysics of Science



When it comes to the metaphysical underpinnings of scientific
theories, properties play a prominent role: it appears that science
can hardly be done without appealing to them. This adds up to the case
for realism about properties and to our understanding of them. We
shall illustrate this here first by dwelling on some miscellaneous
topics and then by focusing on a debate regarding the very nature of
the properties invoked in science, namely whether or not they are
*essentially* dispositional. Roughly, an object exemplifies a
*dispositional* property, such as *soluble* or
*fragile*, by having a power or disposition to act or being
acted upon in a certain way in certain conditions. For example, a
glass' disposition of *being fragile* consists in its
possibly shattering in certain conditions (e.g., if struck with a
certain force). In contrast, something exemplifies a
*categorical* property, e.g., *made of salt* or
*spherical*, by merely being in a certain way (see entry on
 dispositions).


### 5.1 Miscellaneous Topics



Many predicates in scientific theories (e.g., "being a
gene" and "being a belief") are functionally
defined. Namely, their meaning is fixed by appealing to some function
or set of functions (e.g., encoding and transmitting genetic
information). To make sense of functions, properties are needed.
First, functions can be thought of as webs of causal relationships
between properties or property-instances. Secondly, predicates such as
"being a gene" may be taken to refer to
*higher-level* properties, or at least to something that has
such properties. In this case, the property of *being a gene*
would be the higher-level property of *possessing some further
properties* (e.g., biochemical ones) *that play the relevant
functions* (i.e., encoding and transmitting genetic information).
Alternatively, *being a gene* would be a property that
satisfies the former higher-level property. Thirdly, the ensuing
realization relationships between higher-level and function-playing
properties can only be defined by appealing to properties (see entry
on
 functionalism).



More generally, many ontological
dependence and reduction
relationships primarily concern properties: type-identity (Place 1956;
Smart 1959; Lewis 1966) and token-identity (Davidson 1980) (see entry
on
 the mind/brain identity theory),
 supervenience (Horgan 1982; Kim 1993; entry on
 supervenience);
 realization (Putnam 1975; Wilson 1999; Shoemaker 2007), 
scientific reduction
(Nagel 1961). Moreover,
a non-reductive relation such as ontological emergence (Bedau 1997;
Paolini Paoletti 2017b) is typically characterized as a relation
between emergent properties
and more fundamental
properties. Even mechanistic explanations often appeal to properties
and relations in characterizing the organization, the components, the
powers and the phenomena displayed by mechanisms (Glennan & Illari
2018; entry on
 mechanisms in science).



Entities in nature are typified by
 natural kinds.
 These are mostly thought of as properties or property-like entities
that carve nature at its joints (Campbell, O'Rourke, & Slater
2011) at distinct layers of the universe: microphysical (e.g.,
*being a neutron*), chemical (e.g., *being gold*),
biological (e.g., *being a horse*).



Physical quantities such as mass or length are typically treated as
properties, specifiable in terms of a magnitude, a certain number, and
a unit measure, e.g., kg or meter, which can itself be specified in
terms of properties (Mundy 1987; Swoyer 1987). Following Eddon's
(2013) overview, it is possible to distinguish two main strands here:
*relational* and *monadic properties* theories (see also
Dasgupta 2013). In relational theories (Bigelow & Pargetter 1988;
1991), quantities arise from proportion relations, which may in turn
be related by higher-order proportion relations. Consider a certain
quantity, e.g., *mass 3 kg*. Here the *thrice as massive
as* proportion relation is relevant, which holds of certain pairs
of objects \(a\) and \(b\) such that \(a\) is thrice as massive as
\(b\). By virtue of such relational facts, the first members of such
pairs have a mass of 3 kg. Another relational approach is by Mundy
(1988), who claims that *mass 3 kg* is a relation between
(ordered pairs of) objects and numbers. For example, *mass 3
kg* is a relation holding between the ordered pair \(\langle a,
b\rangle\) (assuming that \(a\) is thrice as massive as \(b\), and not
the other way round) and the number 3; or--if relations have
built-in relational order--between \(a, b\) and 3. Knowles (2015)
holds a similar view, wherein physical quantities are relations that
objects bear to numbers. According to the monadic properties approach,
quantities are intrinsic properties of objects (Swoyer 1987; Armstrong
1988; 1989). As we saw, Armstrong develops such a view by taking
quantities to be structural properties. Related metaphysical inquiries
concern the dimensions of quantities (Skow 2017) and the status of
forces (Massin 2009).



Lastly, properties play a prominent role in two well-known accounts of
the
 laws of nature:
 the *nomological necessitation account* and *law
dispositionalism*. The former has been expounded by Tooley (1977;
1987), Dretske (1977) and Armstrong (1983). Roughly, following
Armstrong, a law of nature consists in a second-order and external
nomological necessitation relation \(\rN\), contingently holding
between first-order universals \(P\) and \(Q\): \(\rN(P, Q)\). Such a
higher-order fact necessitates certain lower-order regularities, i.e.,
all the objects that have \(P\) also have \(Q\) (by nomological
necessity, in virtue of \(\rN(P, Q))\). Law dispositionalism
interprets laws of nature by appealing to dispositional properties.
According to it, laws of nature "flow" from the essence of
such properties. This implies that laws of nature hold with
metaphysical necessity: whenever the dispositions are in place, the
relevant laws must be in place too (Cartwright 1983; Ellis 2001; Bird
2007; Chakravartty 2007; Fischer 2018; see also Schrenk 2017 and
Dumsday 2019). (For criticisms, see, e.g., van Fraassen 1989 as
regards the former approach, and McKitrick 2018 and Paolini Paoletti
2020b, as regards the latter).


### 5.2. Essentially Categorical vs. Essentially Dispositional Properties



There is a domestic dispute among supporters of properties in the
metaphysics of science, regarding the very nature of such properties
(or at least the fundamental ones). We may distinguish two extreme
views, *pure dispositionalism* and *pure
categoricalism*. According to the former, all properties are
essentially dispositional ("dispositional", for
brevity's sake, from now on), since they are nothing more than
causal powers; their *causative roles*, i.e., which effects
their instantiations can cause, exhaust their essences. According to
pure categoricalism, all properties are essentially categorical (in
brief, "categorical", in the following), because their
causative roles are not essential to them. If anything is essential to
a property, it is rather a non-dispositional and intrinsic aspect
called "quiddity" (after Armstrong 1989), which need not
be seen as an additional entity over and above the property itself
(Locke 2012, in response to Hawthorne 2001). Between such views there
lie a number of intermediate positions.



Pure dispositionalism has been widely supported in the last few
decades (Mellor 1974; 2000; Shoemaker 1984: ch. 10 and 11 [in terms of
causal roles]; Mumford 1998; 2004; Bird 2005; 2007; Chakravartty 2007;
Whittle 2008; see Tugby 2013 for an attempt to argue from this sort of
view to a Platonist conception of properties). There are three main
arguments in favor of it. First, pure dispositionalism easily accounts
for the natural necessity of the laws of nature, insofar as such a
necessity just "flows" from the essence of the
dispositional properties involved. Secondly, dispositional properties
can be easily known as they really are, because it is part of their
essence that they affect us in certain ways. Thirdly, at the
(presumably fundamental) micro-physical level, properties are only
described dispositionally, which is best explained by their being
dispositional (Ellis & Lierse 1994; N. Williams 2011).



Nevertheless, pure dispositionalism is affected by several problems.
First, some authors believe that it is not easy to provide a clear-cut
distinction between dispositional and would-be non-dispositional
properties (Cross 2005). Secondly, it seems that the essence of
certain properties does not include, or is not exhausted by, their
causative roles: qualia, inert and epiphenomenal
 properties,[14]
 structural and geometrical properties, spatio-temporal properties
(Prior 1982; Armstrong 1999; for some responses, see Mellor 1974 and
Bird 2007; 2009). Thirdly, there can be *symmetrical* causative
roles. Consider three distinct properties \(A\), \(B\) and \(C\) such
that \(A\) can cause \(B\), \(B\) can cause \(A\), \(A\) and \(B\) can
together cause \(C\) and nothing else characterizes \(A\) and \(B\).
\(A\) and \(B\) have the same causative role. Therefore, in pure
dispositionalism, they turn out to be identical, against the
hypothesis (Hawthorne 2001; see also Contessa 2019). Fourthly,
according to some, pure dispositionalism falls prey to (at least)
three distinct regresses (for a fourth regress, see Psillos 2006).
Such regresses arise from the fact that the essential causative role
of a dispositional property \(P\) "points towards" further
properties \(S\), \(T\), etc. *qua* possible effects. The
essential causative roles of the latter "point towards"
still other properties, and so on. The first regress concerns the
identity of \(P\), which is never fixed, as it depends on the identity
of further properties, which depend for their identity on still other
properties, and so on (Lowe 2006; Barker 2009; 2013). The second
regress concerns the knowability of \(P\) (Swinburne 1980). \(P\) is
only knowable through its possible effects (i.e., the instantiation of
\(S\), \(T\), etc.), included in its causative role. Yet, such
possible effects are only knowable through their possible effects, and
so on. The third regress concerns the actuality of \(P\) (Armstrong
1997). \(P\)'s actuality is never reached, since \(P\) is
nothing but the power to give rise to \(S\), \(T\), etc., which are
nothing but the powers to give rise to further properties, and so on.
For responses to these regresses, see Molnar 2003; Marmodoro 2009;
Bauer 2012; McKitrick 2013; see also Ingthorsson 2013.



Pure categoricalism seems to imply that causative roles are only
contingently associated to a property. Therefore, on pure
categoricalism, a property can possibly have distinct causative roles,
which allows it to explain--among other things--the apparent
contingency of causative roles and the possibility of recombining a
property with distinct causative roles. Its supporters include Lewis
(1986b, 2009), Armstrong (1999), Schaffer (2005), and more recently
Livanios (2017), who provides further arguments based on the
metaphysics of science. Kelly (2009) and Smith (2016) may be added to
the list, although they take roles to be non-essential and necessary
(for further options, see also Kimpton-Nye 2018, Yates 2018a, Coates
forthcoming and Tugby forthcoming).



However, pure categoricalism falls prey to two sorts of difficulties.
First, the contingency of causative roles has some unpalatable
consequences: unbeknownst to us, two distinct properties can swap
their roles; they can play the same role at the same time; the same
role can be played by one property at one time and by another property
at a later time; an "alien" property can replace a
familiar one by playing its role (Black 2000). Secondly, and more
generally, we are never able to know which properties play which
roles, nor are we able to know the intrinsic nature of such
properties. This consequence should be accepted with "Ramseyan
humility" (Lewis 2009; see also Langton 1998 for a related,
Kantian, sort of humility) or it should be countered in the same way
as we decide to counter any broader version of skepticism (Schaffer
2005). On this issue, see also Whittle (2006); Locke (2009); Kelly
(2013); Yates (2018b).



Let us now turn to some intermediate positions.



According to *dualism* (Ellis 2001; Molnar 2003), there are
both dispositional and categorical properties. Dualism is meant to
combine the virtues of pure dispositionalism and pure categoricalism.
It then faces the charge of adopting a less parsimonious ontology,
since it accepts two classes of properties rather than one, i.e.,
dispositional and categorical ones.



According to the *identity theory* (Heil 2003; 2012; Jaworski
2016; Martin 2008; G. Strawson 2008; see also N. Williams 2019), every
property is both dispositional and categorical (or qualitative). The
main problem here is how to characterize the distinction and relation
of these two "sides". Martin and Heil suggest that they
are two distinct ways of partially considering one and the same
property, whereas Mumford (1998) explores the possibility of seeing
them as two distinct ways of conceptualizing the property in question.
Heil claims that the qualitative and the dispositional sides need to
be identified with one another and with the whole property. Jacobs
(2011) holds that the qualitative side consists in the possession of
some qualitative nature by the property, whereas the dispositional
side consists in that property being (part of) a sufficient truthmaker
for certain counterfactuals. Dispositional and qualitative sides may
also be seen as essential, higher-order properties of properties, as
supervenient and ontologically innocent aspects of properties
(Giannotti 2019), or as constituents of the essence of properties
(Taylor 2018). In general, the identity theory is between Scylla and
Charybdis. If it reifies the dispositional and qualitative sides, it
runs the risk of implying some sort of dualism. If it insists on the
identity between them, it runs the opposite risk of turning into a
pure dispositionalist theory (Taylor 2018).


## 6. Formal Property Theories and their applications



Formal property theories are logical systems that aim at formulating
"general noncontingent laws that deal with properties"
(Bealer & Monnich 1989: 133). In the next subsection we shall
outline how they work. In the two subsequent subsections we shall
briefly consider their deployment in natural language semantics and
the foundations of mathematics, which can be taken to provide further
reasons for the acknowledgment of properties in one's ontology,
or at least of certain kinds of properties.


### 6.1 Logical Systems for Properties



These systems allow for terms corresponding to properties, in
particular variables that are meant to range over properties and that
can be quantified over. This can be achieved in two ways. Either
(option 1; Cocchiarella 1986a) the terms standing for properties are
predicates or (option 2; Bealer 1982) such terms are subject terms
that can be linked to other subject terms by a special predicate that
is meant to express a predication relation (let us use
"pred") pretty much as in standard set theory a special
predicate, "\(\in\)", is used to express the membership
relation. To illustrate, given the former option, an assertion such as
"there is a property that both John and Mary have" can be
rendered as


\[\notag \exists P(P(j) \amp P(m)).\]



Given the second option, it can be rendered as


\[\notag \exists x(\textrm{pred}(x, j) \amp \textrm{pred}(x, m)).\]



(The two options can be combined as in Menzel 1986; see Menzel 1993
for further discussion).



Whatever option one follows, in spelling out such theories one
typically postulates a rich realm of properties. Traditionally, this
is done by a so-called comprehension principle which, intuitively,
asserts that, for any well-formed formula ("wff") \(A\),
with \(n\) free variables, \(x\_1 , \ldots ,x\_n\), there is a
corresponding \(n\)-adic property. Following option 1, it goes as
follows:


\[\tag{CP} \exists R^n \forall x\_1 \ldots \forall x\_n(R^n(x\_1 ,
\ldots ,x\_n) \leftrightarrow A). \]



Alternatively, one can use a variable-binding operator, \(\lambda\),
that, given an open wff, generates a term (called a "lambda
abstract") that is meant to stand for a property. This way to
proceed is more flexible and is followed in the most recent versions
of property theory. We shall thus stick to it in the following. To
illustrate, we can apply "\(\lambda\)" to the open
formula, "\((R(x) \amp S(x))\)" to form the one-place
complex predicate "[\(\lambda x (R(x) \amp S(x))\)]"; if
"\(R\)" denotes *being red* and "\(S\)"
denotes *being square*, then this complex predicate denotes the
compound, conjunctive property *being red and square*.
Similarly, we can apply the operator to the open formula
"\(\exists y(L(x, y))\)" to form the one-place predicate
"[\(\lambda x \exists y(L(x, y))\)]"; if
"\(L\)" stands for *loves*, this complex predicate
denotes the compound property *loving someone* (whereas
"[\(\lambda y \exists x(L(x, y))\)]" would denote
*being loved by someone*). To ensure that lambda abstracts
designate the intended property, one should assume a "principle
of lambda conversion". Given option 1, it can be stated
thus:


\[\tag{\(\lambda\)-conv} [\lambda x\_1\ldots x\_n A](t\_1 , \ldots
,t\_n) \leftrightarrow A(x\_1 /t\_1 , \ldots ,x\_n /t\_n).\]



\(A(x\_1 /t\_1 , \ldots ,x\_n /t\_n)\) is the wff resulting from
simultaneously replacing each \(x\_i\) in \(A\) with \(t\_i\) (for \(1
\le i \le n)\), provided \(t\_i\) is free for \(x\_i\) in \(A\)). For
example, given this principle, [\(\lambda x (R(x) \amp S(x))](j)\) is
the case iff \((R(j) \amp S(j))\) is also the case.



Standard second-order logic allows for predicate variables bound by
quantifiers. Hence, to the extent that these variables are taken to
range over properties, this system could be seen as a formal theory of
properties. Its expressive power is however limited, since it does not
allow for subject terms that stand for properties. Thus, for example,
one cannot even say of a property \(F\) that \(F = F\). This is a
serious limitation if one wants a formal tool for a realm of
properties whose laws one is trying to explore. Standard higher order
logics beyond the second order obviate this limitation by allowing for
predicates in subject position, provided that the predicates that are
predicated of them belong to a higher type. This presupposes a grammar
in which predicates are assigned types of increasing levels, which can
be taken to mean that the properties themselves, for which the
predicates stand for, are arranged into a hierarchy of types. Thus,
such logics appropriate one version or another of the type theory
concocted by Russell to tame his own paradox and related conundrums.
If a predicate can be predicated of another predicate only if the
former is of a type higher than the latter, then self-predication is
banished and Russell's paradox cannot even be formulated.
Following this line, we can construct a type-theoretical formal
property theory. The simple theory of types, as presented, e.g., in
Copi (1971), can be seen as a prototypical version of such a property
theory (if we neglect the principle of extensionality assumed by
Copi). The type-theoretical approach keeps having supporters. For
example, it is followed in the property theories embedded in
Zalta's (1983) theory of abstract objects and more recently in
the metaphysical systems proposed by Williamson (2013) and Hale
(2013).



However, for reasons sketched in
 SS2.4,
 type theory is hardly satisfactory. Accordingly, many type-free
versions of property theory have been developed over the years and no
consensus on what the right strategy is appears to be in sight. Of
course, without type-theoretical constraints, given \((\lambda\)-conv)
and classical logic (CL), paradoxes such as Russell's
immediately follow (to see this, consider this instance of
\((\lambda\textrm{-conv}): [\lambda x {\sim}x(x)]([\lambda x
{\sim}x(x)]) \leftrightarrow{\sim}[\lambda x {\sim}x(x)]([\lambda x
{\sim}x(x)])\)). In formal systems where abstract singular terms or
predicates may (but need not) denote properties (Swoyer 1998), formal
counterparts of (complex) predicates like "being a property that
does not exemplify itself" (formally, "[\(\lambda x
{\sim}x(x)\)]") could exist in the object language without
denoting properties; from this perspective, Russell's paradox
merely shows that such predicates do not stand for properties
(similarly, according to Schnieder 2017, it shows that it is not
possible that a certain property exists). But we would like to have
general criteria to decide when a predicate stands for a property and
when it does not. Moreover, one may wonder what gives these predicates
any significance at all if they do not stand for properties. There are
then motivations for building type-free property theories in which all
predicates stand for properties. We can distinguish two main strands
of them: those that weaken CL and those that circumscribe
\((\lambda\)-conv) (some of the proposals to be mentioned below are
formulated in relation to set theory, but can be easily translated
into proposals for property theory).



An early example of the former approach was offered in a 1938 paper by
the Russian logician D. A. Bochvar (Bochvar 1938 [1981]), where the
principle of excluded middle is sacrificed as a consequence of the
adoption of what is now known as Kleene's weak three-valued
scheme. An interesting recent attempt based on giving up excluded
middle is Field 2004. A rather radical alternative proposal is to
embrace a paraconsistent logic and give up the principle of
non-contradiction (Priest 1987). A different way of giving up CL is by
questioning its structural rules and turn to a substructural logic, as
in Mares and Paoli (2014). The problem with all these approaches is
whether their underlying logic is strong enough for all the intended
applications of property theory, in particular to natural language
semantics and the foundations of mathematics.



As for the second strand (based on circumscribing \((\lambda\)-conv)),
it has been proposed to read the axioms of a standard set theory such
as ZFC, minus extensionality, as if they were about properties rather
than sets (Schock 1969; Bealer 1982; Jubien 1989). The problem with
this is that these axioms, understood as talking about sets, can be
motivated by the iterative conception of sets, but they seem rather
*ad hoc* when understood as talking about properties
(Cocchiarella 1985). An alternative can be found in Cocchiarella
1986a, where \((\lambda\)-conv) is circumscribed by adapting to
properties the notion of stratification used by Quine for sets. This
approach is however subject to a version of Russell's paradox
derivable from contingent but intuitively possible facts (Orilia 1996)
and to a paradox of hyperintensionality (Bozon 2004) (see Landini 2009
and Cocchiarella 2009 for a discussion of both). Orilia (2000; Orilia
& Landini 2019) has proposed another strategy for circumscribing
\((\lambda\)-conv), based on applying to exemplification Gupta's
and Belnap's theory of circular definitions.



Independently of the paradoxes (Bealer & Monnich 1989: 198
ff.), there is the issue of providing identity conditions for
properties, specifying when it is the case that two properties are
identical. If one thinks of properties as meanings of natural language
predicates and tries to account for intensional contexts, one will be
inclined to assume rather fine-grained identity conditions, possibly
even allowing that [\(\lambda x(R(x) \amp S(x))\)] and [\(\lambda
x(S(x) \amp R(x))\)] are distinct (see Fox & Lappin 2015 for an
approach based on operational distinctions among computable
functions). On the other hand, if one thinks of properties as causally
operative entities in the physical world, one will want to provide
rather coarse-grained identity conditions. For instance, one might
require that [\(\lambda x A\)] and [\(\lambda x B\)] are the same
property iff it is physically necessary that \(\forall x(A
\leftrightarrow B)\). Bealer (1982) tries to combine the two
approaches (see also Bealer & Monnich
 1989)[15].


### 6.2 Semantics and Logical Form



The formal study of natural language semantics started with Montague
and gave rise to a flourishing field of inquiry (see entry on
 Montague semantics).
 The basic idea in this field is to associate to natural language
sentences wffs of a formal language, in order to represent sentence
meanings in a logically perspicuous manner. The association reflects
the compositionality of meanings: different syntactic subcomponents of
sentences correspond systematically to syntactic subcomponents of the
wffs; subcomponents of wffs thus represent the meanings of the
subcomponents of the sentences. The formal language eschews
ambiguities and has its own formal semantics, which grants that
formulas have logical properties and relations, such as logical truth
and entailments, so that in particular certain sequences of formulas
count as logically valid arguments. The ambiguities we normally find
in natural language sentences and the entailment relations that link
them are captured by associating ambiguous sentences to different
unambiguous wffs, in such a way that when a natural language argument
is felt to be valid there is a corresponding sequence of wffs that
count as a logically valid argument. In order to achieve all this,
Montague appealed to a higher-order logic. To see why this was
necessary, one may focus on this valid argument:



(1)
every Greek is mortal;
(2)
the president of Greece is Greek;


therefore,



(3)
the president of Greece is mortal.


To grant compositionality in a way that respects the syntactic
similarity of the three sentences (they all have the same
subject-predicate form), and the validity of the argument, Montague
associates (1)-(3) to formulas such as these:


\[ \tag{1a} [\lambda F \forall x(\mathrm{G}(x) \rightarrow
F(x)](\mathrm{M}); \] \[ \tag{2a} [\lambda F \exists x(\forall y(P(x)
\rightarrow x = y) \amp F(x))](G); \] \[ \tag{3a} [\lambda F \exists
x(\forall y(P(x) \rightarrow x = y) \amp F(x))](\mathrm{M}). \]



The three lambda abstracts in (1a)-(3a) represent, respectively,
the meanings of the three noun phrases in (1)-(3). These
lambda-abstracts occur in predicate position, as predicates of
predicates, so that (1a)-(3a) can be read, respectively, as:
*every Greek* is instantiated by *being mortal*; *the
president of Greece* is instantiated by *being* Greek;
*the president of Greece* is instantiated by *being
mortal*. Given lambda-conversion plus quantifier and propositional
logic, the argument is valid, as desired. It should be noted that
lambda abstracts such as these can be taken to stand for peculiar
properties of properties, classifiable as *denoting concepts*
(after Russell 1903; see Cocchiarella 1989). One may then say then
this approach to semantics makes a case for the postulation of
denoting concepts, in addition to the more obvious and general fact
that it grants properties as meanings of natural language predicates
(represented by symbols of the formal language).



This in itself says nothing about the nature of such properties. As we
saw in
 SS3.1,
 Montague took them to be intensions as set-theoretically
characterized in terms of possible worlds. Moreover, he took them to
be typed, since, to avoid logical paradoxes, he relied on type theory.
After Montague, these two assumptions have been typically taken for
granted in natural language semantics, though with attempts to recover
hyperintensionality somehow (Cresswell 1985) in order to capture the
semantics of propositional attitudes verbs such as
"believe", affected by the phenomena regarding mental
content hinted at in
 SS3.1.
 However, the development of type-free property theories has suggested
the radically different road of relying on them to provide logical
forms for natural language semantics (Chierchia 1985; Chierchia &
Turner 1988; Orilia 2000b; Orilia & Landini 2019). This allows one
to capture straightforwardly natural language inferences that appear
to presuppose type-freedom, as they feature quantifiers that bind
simultaneously subject and predicate positions (recall the example of
 SS1.2).
 Moreover, by endowing the selected type-free property theory with
fine-grained identity conditions, one also accounts for propositional
attitude verbs (Bealer 1989). Thus, we may say that this line makes a
case for properties understood as untyped and highly fine-grained.


### 6.3 Foundations of Mathematics



Since the systematization in the first half of last century, which
gave rise to paradox-free axiomatizations of set
theory such as ZFC, sets are
typically taken for granted in the foundations of mathematics and it
is well known that they can do all the works that numbers can do. This
has led to the proposal of identifying numbers with sets.
Russell's type theory was an alternative that rather relied on
properties (viewed as propositional functions), in backing up a
logicist reduction of mathematics to logic. In essence, the idea was
that properties can do all the work that sets are supposed to do, thus
making the latter dispensable. Hence, Russell spoke of his approach as
a "no-classes" theory of classes (see Landini 2011: 115,
and SS2.4 of the entry on
 Russell's logical atomism;
 cf. Bealer 1982: 111-119, and Jubien 1989 for followers of this
line). Following this line, numbers are then seen as properties rather
than sets.



The Russellian approach did not encounter among mathematicians a
success comparable to that of set theory. Nevertheless, from an
ontological point of view, it appears to be more economical in its
relying on properties, to the extent that the latter are needed any
way for all sorts of explanatory jobs, reviewed above, which sets,
*qua* extensional entities, can hardly perform. As we have
seen, type theory is problematic. However, type-free property theories
can come to the rescue by replacing typed properties with untyped ones
in the foundations of mathematics. And in fact the advocates of such
theories have often proposed to recover the logicist program, in
particular by identifying natural numbers with untyped properties of
properties (Bealer 1982; Cocchiarella 1986b; Orilia 2000b). (See also
entry on
 logicism and neologicism).