Properties

First published Thu Sep 23, 1999; substantive revision Mon Nov 24, 2025

Properties are those entities that can bepredicated ofthings or, in other words,attributed to them. Thus,properties are often calledpredicables. Other terms for themare “attributes”, “qualities”,“features”, “characteristics”,“types”. Properties are also ways things are, entitiesthat thingsexemplify orinstantiate. For example,if we say that this is a leaf and is green, we are attributing thepropertiesleaf andgreen to it, and, if thepredication is veridical, the thing in question exemplifies theseproperties. Hence, properties can also be characterized asexemplifiables, with the controversial exception of thosethat cannot be instantiated, e.g., some would say,round andsquare. It is typically assumed that no other entities can bepredicated and exemplified (Aristotle,Categories, 2a). Forexample, ordinary objects like apples and chairs cannot be predicatedof, and are not exemplified by, anything.

The nature and existence of properties have always been central andcontroversial issues in philosophy since its origin, and interest inthem keeps flourishing, as Allen’s (2016) and Maurin’s(2022) introductory texts well testify (see also surveys orcollections such as Loux 1972; Oliver 1996; Mellor & Oliver 1997;Koons & Pickavance 2017; Marmodoro & Mayr 2019; Fisher &Maurin 2021). At least since Plato, who called them“ideas” or “forms”, properties are viewed asuniversals, i.e., as capable, (in typical cases) of beinginstantiated by different objects, “shared” by them, as itwere; consequently, in contrast withparticulars, orindividuals, of being somehow at once in different places.^[1] For example, if there are two potatoes each of which weighs 300grams, the propertyweighing 300 grams is instantiated by twoparticulars and is therefore multi-located. According to a differentconception, however, properties are themselves particulars, thoughabstract ones. As so conceived, properties are nowadayscommonly calledtropes, and are the subject of another entry. Here we shall focus onproperties as universals. Relations, e.g.,loving andbetween, can also be considered properties: they arepredicable, exemplifiable, and viewable as universals. Accordingly, weuse “property” as a generic term that also covers them,unless the context suggests otherwise. However, we shall considertheir peculiarity only to a minimum extent, since they are discussedin detail in the entry onrelations.

In §1 we consider some most fundamental themes, including themain motivations and arguments for including properties in one’sontology. In §2 we deal with the central topic of what it is forproperties to be exemplified. In §3 we tackle the intertwinedissues of the existence and identity conditions for properties. In§4 we consider different senses in which properties may bestructurally complex. In §5 we survey the frequent appeal toproperties in current metaphysics of science. In §6 we concludewith a review of formal accounts of properties and their applicationsin natural language semantics and the foundations of mathematics,which provide additional motivations for an ontological commitment toproperties.

1. Properties: Basic Ideas

There are some crucial terminological and conceptual distinctions thatare typically made in talking of properties. There are also varioussorts of reasons that have been adduced for the existence ofproperties and different traditional views about whether and in whatsense properties should be acknowledged. We shall focus on suchmatters in the following subsections.

1.1 How We Speak of Properties

Properties areexpressed, as meanings, bypredicates. In the past “predicate” was oftenused as synonym of “property”, but nowadays predicates arelinguistic entities, typically contrasted withsingularterms, i.e., simple or complex noun phrases such as“Daniel”, “this horse” or “the Presidentof France”, which can occupy subject positions in sentences andpurport todenote, orrefer to, a single thing.Following Frege, predicates areverbal phrases such as“is French” or “drinks”. Alternatively,predicates aregeneral terms such as “French”,with the copula “is” (or verbal inflection) taken toconvey an exemplification link (P. Strawson 1959; Bergmann 1960). Weshall conveniently use “predicate” in both ways.Predicates arepredicated of singular terms, therebygenerating sentences such as “Daniel is French”. In thefamiliar formal language of first-order logic, this would be rendered,say, as “\(F(d)\)”, thus representing the predicate with acapital letter. The set or class of objects to which a predicateveridically applies is often called theextension of thepredicate, or of the corresponding property. This property, incontrast, is called theintension of the predicate, i.e., itsmeaning. This terminology traces back to the Middle Age and in thelast century has led to the habit of calling sets and propertiesextensional andintensional entities, respectively.Extensions and intensions can hardly be identified; this isimmediately suggested by paradigmatic examples of co-extensionalpredicates that appear to differ in meaning, such as “has aheart”, and “has kidneys” (see§3.1).

Properties can also be referred to by singular terms, or so it seems.First of all, there are singular terms, e.g., “beinghonest” or “honesty”, that result from thenominalization 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 morecontroversially, there are demonstratives, such as “that shadeof red”, deployed while pointing to a red object (Heal1997).

Frege (1892) and Russell (1903) had different opinions regarding theontological import of nominalization. According to the former,nominalized predicates stand for a “correlate” of the“unsaturated” entity that the predicate stands for (inFrege’s terminology they are a “concept correlate”and a “concept”, respectively). According to the latter,who speaks of “inextricable difficulties” in Frege’sview (Russell 1903: §49), they stand for exactly the same entity.Mutatis mutandis, they similarly disagreed about othersingular terms that seemingly refer to properties. The ontologicaldistinction put forward by Frege is mainly motivated by the fact thatgrammar indeed forbids the use of predicates in subject position. Butthis hardly suffices for the distinction and it is dubious that othermotivations can be marshalled (Parsons 1986). We shall thus take forgranted Russell’s line here, although many philosophers supportFrege’s view or at least take it very seriously(Castañeda 1976; Cocchiarella 1986a; Landini 2008; Trueman2021).

1.2 Arguments for Properties

Properties are typically invoked to explain phenomena of philosophicalinterest. The most traditional task for which properties have beenappealed to is to provide a solution to the so-called “one overmany problem” via a corresponding “one over manyargument”. This traces back at least to Socrates and Plato(e.g.,Phaedo, 100 c-d) and keeps being rehearsed (Russell1912: ch. 9; Butchvarov 1966; Armstrong 1978a: ch. 7; Loux & Crisp2017: ch.1). The problem is that certain things are many, they arenumerically different, and yet they are somehow one: they appear to besimilar, in a way that suggests a uniform classification, their beinggrouped together into a single class. For example, some objects havethe same shape, certain others have the same color, and still othersthe same weight. Hence, the argument goes, something is needed toexplain this phenomenon and properties fill the bill: the objects inthe first group, say, all have the propertyspherical, thosein the secondred, and those in the thirdweighing 200grams.

Relatedly, properties have been called for to explain our use ofgeneral terms. How is it, e.g., that we apply “spherical”to those balls over there and refuse to do it for the nearby bench? Itdoes not seem to be due to an arbitrary decision concerning where, orwhere not, to stick a certain label. It seems rather the case that therecognition of a certain property in some objects but not in otherselicits the need for a label, “spherical”, which is thenused for objects having that property and not for others. Propertiesare 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 theuse of general terms and predicates in itself does not involve anontological commitment to entities corresponding to them, since it isby deploying singular terms that we purport to refer to something (seealso Sellars 1960). However, as noted, predicates can be nominalizedand thus occur as singular terms. Hence, even if one agrees withQuine, nominalized predicates still suggest the existence ofproperties as their referents, at least to the extent that the use ofnominalized predicates cannot be paraphrased away (Loux & Crisp2017: 21–29; and see §§3–4 of the entry onplatonism in metaphysics).

After Quine (1948), quantification is the landmark of ontologicalcommitment. We can thus press the point even more by noting that wemake claims and construct arguments that appear to involvequantification over properties, with quantifiers reaching (i) overpredicate positions, or even (ii) over both subject and predicatepositions (Castañeda 1976).^[2] As regards (i) consider: this apple is red; this tomato is red;hence, there is something that this apples is and this tomato also is.As for (ii), consider: wisdom is more important than beauty; Mary iswise and Elisabeth is beautiful; hence, there is something that Maryis which is more important than Mary is. Especially in the wake ofWilliamson’s work (2013), there is nowadays a growing trend ofhigher-order metaphysics (Fritz & Jones 2024). According to it,such a quantification over predicate positions is crucially assumed indoing metaphysics, although its ontological import regarding theexistence of properties has been questioned (Cameron 2019; Liggins2021).

Quantification over properties seems ubiquitous not just in ordinarydiscourse but in science as well. For example, the inverse square lawfor dynamics and the reduction of temperature to mean molecular energycan be taken to involve quantification over properties such as masses,distances and temperatures: the former tells us that the attractionforce between any two bodies depends on such bodies’ havingcertain masses and being at a certain distance, and the latter informsus that the fact that a sample of gas has a given temperature dependson its having such and such mean kinetic energy.

Swoyer (1999: §3.2) considers these points within a long list ofarguments that have been, or can be, put forward to motivate anontological commitment to properties. He touches on topics such asa priori knowledge, change, causation, measurement, laws ofnature, intensional logic, natural language semantics, numbers (weshall cover some of this territory in§5 and§6).

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

1.3 Traditional Views about the Existence of Universals

Do universals really exist? There are three long-standing answers tothis question:realism,nominalism, andconceptualism.

According to realists, universals exist as mind-independent entities.Intranscendent realism, put forward by Plato, they existeven if uninstantiated and are thus “transcendent” or“ante res” (“before the things”). Inimmanent realism, defended by Aristotle in opposition to hismaster, they are “immanent” or “inrebus” (“in things”), as they exist only ifinstantiated by objects. Contemporary notable supporters are Russell(1912) for the former and Armstrong (1978a) for the latter (for recenttakes on this old dispute see the essays by Loux, Van Inwagen, Loweand Galluzzo in Galluzzo & Loux 2015). Transcendentism is ofcourse a less economical position and elicits epistemological worriesregarding our capacity to graspante res universals and theirrelationships with their bearers (Benacerraf 1973, Field 1989,Himelright 2023). Nevertheless, such worries may be countered invarious ways (cf. §5 of the entry onplatonism in metaphysics; Bealer 1982: 19–20; 1998: §2; Linsky & Zalta 1995).Uninstantiated properties may well have work to do in capturing theintuitive idea that there are unrealized possibilities, in partakingin certain laws of nature, in accounting for merely possiblemanifestations of unmanifested dispositions and in dealing withcognitive content (see§3). A good summary of pro-transcendentist arguments and immanentistrejoinders is offered by Allen (2016: §2.3). Alvarado (2020),Berman (2020) and Tugby (2022) offer full-length defenses oftranscendentism. Finally, immanentism has been recently targeted by anargument concluding that it gives rise to circles of grounding when itcomes to the relationship between immanent universals and the statesof affairs in which such universals partake: roughly, the existence ofthe former grounds but is also grounded in the existence of the latter(see Alvarado 2020, Costa 2021, Raven 2022, Costa & Giordani 2024.For replies, see Imaguire 2021 and Giordani & Tremolanti2022).

Nominalists eschew mind-independent universals. They may either resortto tropes in their stead, or acceptpredicate nominalism,which tries to make it without mind-independent properties at all, bytaking the predicates themselves to do the classifying job thatproperties are supposed to do. This is especially indigestible to arealist, since it seems to put the cart before the horse by makinglanguage and mind responsible for the similarities we find in the richvarieties of things surrounding us. Some even say that this involvesan 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 withconcepts, understood as non-linguistic mind-dependent entities,typically functioning as meanings of predicates. The mind-dependenceof concepts however makes conceptualism liable to the same kind ofcart/horse worry just voiced above in relation to predicate nominalism.^[3]

The arguments considered in§1.2 constitute the typical motivation for realism, which is the stancethat we take for granted here. They may be configured as an abductiveinference to the best explanation (Swoyer 1999). Thus, of course, theyare not foolproof, and in fact nominalism is still a popular view,which is discussed in detail in the entry onnominalism in metaphysics, as well as in the entry ontropes. Conceptualism appears to be less common nowadays, although it stillhas supporters (cf. Cocchiarella 1986a: ch. 3; 2007), and it is worthnoting that empirical research onconcepts is flourishing.

1.4 Properties in Propositions and States of Affairs

We have talked above in a way that might give the impression thatpredication is an activity thatwe perform, e.g., when we sayor think that a certain apple is red. Although some philosophers mightthink of it in this way, predication, orattribution, mayalso be viewed as a special link that connects a property to a thingin a way that gives rise to aproposition, understood as a complex featuring the property and the thing (orconcepts of them) as constituents with different roles: the latteroccurs in the propositionas logical subject orargument, as is often said, and the formerasattributed to such an argument. If the proposition is true (thepredication is veridical), the argument exemplifies the property, viz.the former is an instance of the latter. The idea that propertiesyield propositions when appropriately connected to an argumentmotivates Russell’s (1903) introduction of the term “propositional function” to speak of properties. We take for granted here that predication isunivocal. However, according to some neo-Meinongian philosophers,there are two modes of predication, sometimes characterized as“external” and “internal” (Castañeda1972; Rapaport 1978; Zalta 1983; see entry onnonexistent objects). Zalta (1983) traces back the distinction to Mally and uses“exemplification” to characterize the former and“encoding” to characterize the latter. Roughly, the ideais that non-existent objects may encode properties that existentobjects exemplify. For instance,winged is exemplified bythat bird over there and is encoded by the winged horse.

It is often assumed nowadays that, when an object exemplifies aproperty, there is a further, complex, entity, a fact or state ofaffairs (Bergmann 1960; Armstrong 1997, entries onfacts andstates of affairs), having the property (qua attributed) and the object(qua argument) as constituents (this compositional conceptionis not always accepted; see, e.g., Bynoe 2011 for a dissenting voice).Facts are typically taken to fulfill the theoretical roles oftruthmakers (the entities that make true propositions true, see entryontruthmakers) and causalrelata (the entities connected by causalrelations; see §1 of the entry onthe metaphysics of causation). Not all philosophers, however, distinguish between propositions andstates 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 otherentities can have: they can occur in propositions and facts both asarguments and as attributed (Russell 1903: §48). For example, intruly saying that this apple is red and that red is a color, weexpress a proposition whereinred occurs as attributed, andanother proposition whereinred occurs as argument.Correspondingly, there are two facts withred in both roles,respectively. This duplicity grounds the common distinction betweendifferentorders ortypes of properties:first-order ones are properties of things that are notthemselves predicables;second-order ones are properties offirst-order properties; and so on. Even though the formal andontological issues behind this terminology are controversial, it iswidely used and is often connected to the subdivision betweenfirst-order and higher-order logics (see, e.g., Thomason 1974; Oliver1996; Williamson 2013; entry ontype theory). It originates from Frege’s and Russell’s logicaltheories, especially Russell’s type theory, wherein distinctionsof types and orders are rigidly regimented in order to circumvent thelogical paradoxes (see§6).

1.5 Relations

A relation is typically attributed to a plurality of objects. Thesejointly instantiate the relation in question, if theattribution is veridical. In this case, therelata (asarguments) and the relation (as attributed) are constituents of astate of affairs. Depending on the number of objects that it canrelate, 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 andbetween 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 theyare predicated of one, and of no, object, respectively, and may thenbe 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; Orilia1999; Van Inwagen 2004; 2015; Zalta 1983). This terminology is alsoapplied to predicates and sentences; for example, the predicate“between” is triadic, and the sentence “Peter isbetween Tom and May” is zero-adic. Accordingly, standardfirst-order logic employs predicates with a fixed degree, typicallyindicated by a superscript, e.g., \(P^1\), \(Q^2\), \(R^3\), etc.

In natural language, however, many predicates appear to bemultigrade orvariably polyadic; i.e., they can beused with different numbers of arguments, as they can be true ofvarious numbers of things. For example, we say “John is liftinga 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 ofinference, called “argument deletion”, which also suggeststhat many predicates thatprima facie could be assigned acertain fixed degree are in fact multigrade. For example, “Johnis eating a cake” suggests that “is eating” isdyadic, but since, by argument deletion, it entails “John iseating”, one could conclude that it is also monadic and thusmultigrade. Often one can resist the conclusion that there aremultigrade predicates. For example, it could be said that “Johnis eating” is simply short for “John is eatingsomething”. But it seems hard to find a systematic andconvincing strategy that allows us to maintain that natural languagepredicates have a fixed degree. This has motivated the construction oflogical languages that feature multigrade predicates in order toprovide a more appropriate formal account of natural language (Grandy1976; Graves 1993; Orilia 2000a). Since natural language predicatesappear to be multigrade, one may be tempted to take the properties andrelations that they express to also be multigrade, and the metaphysicsof science may lend support to this conclusion (Mundy 1989).

Seemingly, relations are not jointly instantiatedsimpliciter;how the instantiation occurs also playsa role. This comes to the fore in particular with non-symmetricrelations such asloving. For example, if John loves Mary,thenloving is jointly instantiated by John and Mary in acertain way, whereas if it is Mary who loves John, thenloving is instantiated by John and Mary in another way.Accordingly, relations pose a special problem: explicating thedifference between facts, such asAbelard loves Eloise andEloise loves Abelard, that at leastprima facieinvolve exactly the same constituents, namely a non-symmetric relationand two other items (loving, Abelard, Eloise). Such facts areoften said to differ in “relational order” or in the“differential application” of the non-symmetric relationin question, and the problem then is that of characterizing what thisrelational order or differential application amounts to.

Russell (1903: §218) attributed an enormous importance to thisissue and attacked it repeatedly. Despite this, it has been prettymuch neglected until the end of last century, with only few othersconfronting it systematically (e.g., Bergmann 1992; Hochberg 1987).However, Fine (2000) has forcefully brought it again on theontological agenda and proposed a novel approach that has receivedsome attention. Fine identifiesstandard andpositionalist views (analogous to two approaches defended byRussell at different times (1903; 1984); cf. Orilia 2008). Accordingto the former, typically dubbeddirectionalism, relations areintrinsically endowed with a “direction”, which allows usto distinguish, e.g.,loving andbeing loved:Abelard loves Eloise andEloise loves Abelarddiffer, because they involve two relations that differ in direction(e.g., the former involvesloving and the latterbeingloved). According to the latter, relations have different“positions” that can somehow hostrelata:Abelard loves Eloise andEloise loves Abelarddiffer, because the two positions of the very samelovingrelation are differently occupied (by Abelard and Eloise in one caseand by Eloise and Abelard in the other case). Fine goes on to proposeand endorse an alternative, “anti-positionalist”standpoint, according to which relations have neither direction norpositions. The literature on this topic keeps growing and there arenow various proposals on the market, including new versions ofpositionalism (Orilia 2014, forthcoming; Donnelly 2016, 2021; Dixon2018), directionalism (Dixon forthcoming), andprimitivism,according to which differential application cannot be analyzed(MacBride 2014).

Russell (1903: ch. 26) also had a key role in leading philosophers toacknowledge that at least some relations, in particularspatio-temporal ones, areexternal, i.e., cannot be reducedto monadic properties, or the mere existence, of therelata,in contrast to internal relations that can be so reduced. This was abreakthrough after a long tradition tracing back at least to Aristotleand the Scholastics wherein there seems to be hardly any place forexternal relations (see entry onmedieval theories of relations).^[4]For an extensive overview of the ontology of relations, seePaolini Paoletti 2024.

1.6 Universalsversus Tropes

According to some philosophers, universals and tropes may coexist inone ontological framework (see, e.g., Lowe 2006 for a well-knowngeneral system of this kind, and Orilia 2006a, for a proposal based onempirical data from quantum mechanics). However, nowadays they aretypically seen as alternatives, with the typical supporter ofuniversals (“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 takeadvantage of states of affairs. Both parties may agree, say, thatthere are two red apples, \(a\) and \(b\). They will immediatelydisagree, however, for the universalist will add that

there are two distinct states of affairs,that a is redandthat b is red,
such states are similar in having the universalred asconstituent, and
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 asconstituents and rather urge that there are two distinct tropes, theredness of \(a\) and the redness of \(b\), which play a theoreticalrole analogous to the one that the universalist would invoke for suchstates of affairs. Hence, tropists claim that tropes can be causalrelata (D. Williams 1953) and truthmakers (Mulligan, Simons, &Smith 1984).

Tropes are typically taken to besimple, i.e., without anysubconstituent (see §2.2 of the entry ontropes). Their playing the role of states of affairs with universals asconstituents depends on this: universals combine two functions, onlyone of which is fulfilled by tropes. On the one hand, universals arecharacterizers, inasmuch as they characterize concreteobjects. On the other hand, they are alsounifiers, to theextent that different concrete objects may be characterized by thevery same universal, which is thus somehow shared by all of them; whenthis is the case, there is, according to the universalist, anobjective similarity among the different objects (see§1.2). In contrast, tropes are only characterizers, for, at least astypically understood, they cannot be shared by distinct concreteobjects. Given its dependency on one specific object, say, the apple\(a\), a trope can do the work of a state of affairs with \(a\) asconstituent. But for tropes to play this role, the tropist will haveto pay a price and introduce additional theoretical machinery toaccount for objective similarities among concrete objects. To thisend, she will typically resort to the idea that there are objectiveresemblances among tropes, which can then be grouped together inresemblance classes. These resemblance classes play the role ofunifiers for the tropist. Hence, from the tropist’s point ofview “property” is ambiguous, since it may stand for thecharacterizers (tropes) or for the unifiers (resemblance classes) (cf.§6.5 of the entry onmental causation). Similarly, “exemplification” and related words may beregarded as ambiguous insofar as they can be used either to indicatethat an object exemplifies a certain trope or to indicate that theobject relates to a certain resemblance class by virtue ofexemplifying a trope in that class.^[6]

1.7 Kinds of Properties

Many important and often controversial distinctions among differentkinds of properties have been made throughout the whole history ofphilosophy until now, often playing key roles in all sorts of disputesand arguments, especially in metaphysics. Here we shall briefly reviewsome of these distinctions and others will surface in the followingsections. More details can be found in other more specialized entries,to which we shall refer.

Locke influentially distinguished betweenprimary andsecondary qualities; the former are objective features ofthings, such as shapes, sizes and weights, whereas the latter aremind-dependent, e.g.,colors,tastes,sounds, andsmells. This contrastwas already emphasized by the Greek atomists and was revived in moderntimes by Galileo, Descartes, and Boyle.

At least since Aristotle, theessential properties of anobject have been contrasted with itsaccidental properties;the object could not exist without the former, whereas it could failto have the latter (see entry onessential vs. accidental properties). Among essential properties, some acknowledgeindividualessences (also called “haecceities” or“thisnesses”), which univocally characterize a certainindividual. 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 ifSocrates does not exist. In contrast, Plantinga (1974) views them ascapable of existing without the individuals of which they areessences, e.g.,Socratizing, which could have existed even ifSocrates had not existed. See§5.2 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 encodeprinciples of individuation and persistence that allow us toobjectively count objects. For example, there is a fact of the matterregarding how many things in this room instantiatebeing adesk andbeing a cat. On the other hand, non-sortalproperties such asred orwater do not allow us tocount in a similarly obvious way. This distinction is often appealedto in contemporary metaphysics (P. Strawson 1959: ch. 5, §2;Armstrong 1978a: ch. 11, §4), where, in contrast, the traditionalone betweengenus andspecies plays a relatively small role. The latter figured conspicuously inAristotle and in much subsequent philosophy inspired by him. We canview a genus as a property more general than a corresponding speciesproperty, in a hierarchically relative manner. For example,beinga mammal is a genus relative to the speciesbeing ahuman, but it is a species relative to the genusbeing ananimal. The possession of a property calleddifferentiais appealed to in order to distinguish between different speciesfalling under a common genus; e.g., as the tradition has it, thedifferentia for being human isbeing rational (Aristotle,Categories, 3a). Similar hierarchies of properties, howeverwithout anything like differentiae, come with the distinction ofdeterminables and determinates, which appears to be more prominent in current metaphysics. Colorproperties provide typical examples of such hierarchies, e.g., withred andscarlet 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 propertiesthemselves, exemplify properties, or, in different terminology,bear,have orpossess properties. Reversingorder, we can also say that propertiescharacterize, orinhere in, the items that exemplify them. There is then avery general phenomenon of exemplification to investigate, which hasbeen labeled in various ways, as the variety of terms of art justdisplayed testifies. All such terms have often been given specialtechnical senses in the rich array of different explorations of thisterritory since Ancient and Medieval times up to the present age (see,e.g., Lowe 2006: 77). These explorations can hardly be disentangledfrom the task of providing a general ontological picture with its owncategorial distinctions. In line with what most philosophers donowadays, we choose “exemplification”, or, equivalently,“instantiation” (and their cognates), to discuss thisphenomenon in general and to approach some different accounts thathave been given of it in recent times. This sweeping use of theseterms is to be kept distinct from the more specialized uses of themthat will surface below (and to some extent have already surfacedabove) in describing specific approaches by different philosopherswith their own terminologies.

2.1 Monist vs. Pluralist Accounts

We have taken for granted that there is just one kind ofexemplification, applying indifferently to different categories ofentities. Thismonist option may indeed be considered thedefault one. A typical recent case of a philosopher who endorses it isArmstrong (1997). He distinguishes three basic categories,particulars, properties or relations, and states of affairs, and takesexemplification as cutting across them: properties and relations areexemplified not only by particulars, but by properties or relationsand states of affairs as well. But some philosophers arepluralist: they distinguish different kinds ofexemplification, in relation to categorial distinctions in theirontology.

One may perhaps attribute different kinds of exemplification to theabove-considered Meinongians in view of the different sorts ofpredication that they admit (see, e.g., Monaghan’s (2011)discussion of Zalta’s theory). A more typical example of thepluralist alternative is however provided by Lowe (2006), whodistinguishes “instantiation”,“characterization” and “exemplification” inhis account of four fundamental categories: objects, and threedifferent sorts of properties, namely kinds (substantial universals),attributes and modes (tropes).^[7] To illustrate, Fido is a dog insofar as itinstantiates thekinddog, \(D\), which in turn ischaracterized bythe attribute of barking, \(B\). Hence, when Fido is barking, itexemplifies \(B\)occurrently by virtue of beingcharacterized 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 \(B\) (seeGorman 2014 for a critical discussion of this sort of view).

2.2 Compresence and Partial Identity

Most philosophers, whether tacitly or overtly, appear to takeexemplification as primitive and unanalyzable. However, on certainviews of particulars, it might seem that exemplification is reduced tosomething more fundamental.

A well-known such approach is thebundle theory, which takesparticulars to be nothing more than “bundles” ofuniversals connected by a special relation, commonly calledcompresence, after Russell (1948: Pt. IV, ch. 8)^[8]. Despite well-known problems (Van Cleve 1985), this view, orapproaches 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 thisperspective, that a particular exemplifies a property amounts to theproperty’s being compresent with the properties that constitutethe bundle with which the particular in question is identified. Itthus looks as if exemplification is reduced to compresence.Nevertheless, compresence itself is presumably jointly exemplified bythe properties that constitute a given bundle, and thus at most thereis a reduction, to compresence, of exemplificationby aparticular (understood as a bundle), and not an elimination ofexemplification in general.

Another, more recent, approach is based onpartial identity.Baxter (2001) and, inspired by him, Armstrong (2004), have proposedrelated assays of exemplification, which seem to analyze it in termsof such partial identity. These views have captured some interest andtriggered discussions (see, e.g., Mumford 2007; Baxter’s (2013)reply to critics and Baxter’s (2018) rejoinder to Brown2017).

Baxter (2001) relies on the notion ofaspect and on therelativization of numerical identity tocounts. In his view,both particulars and propertieshave aspects, which can besimilar to distinct aspects of other particulars or properties. Thenumerical identity of aspects is relative to standards for counting,counts, which group items in count collections: aspects ofparticulars in theparticular collection, and aspects ofuniversals, in theuniversal collection. There can then be across-count identity, which holds between an aspect in theparticular collection, and an aspect in the universal collection,e.g., the aspectsHume as human andhumanity as had byHume. In this case, the universal and the particular in question(humanity and Hume, in our example) arepartially identical.Instantiation, e.g., that Hume instantiates humanity, then amounts tothis partial identity of a universal and a particular. One may havethe feeling, as Baxter himself worries (2001: 449), that in thisapproach instantiation has been traded for something definitely moreobscure, such as aspects and an idiosyncratic view of identity. It canalso be suspected that particulars’ and properties’having aspects is presupposed in this analysis, where thishaving is a relation rather close to exemplificationitself.

Armstrong (2004) tries to do without aspects. At first glance it seemsas if he analyzes exemplification, for he takes the exemplification ofa property (a universal) by a particular to be a partial identity ofthe property and the particular; as he puts it (2004: 47), “[i]tis not a mere mereological overlap, as when two streets intersect, butit is a partial identity”. However, when we see more closelywhat this partial identity amounts to, the suspicion arises that itpresupposes exemplification. For Armstrong appears to identify aparticular via the properties that itinstantiates andsimilarly a property via the particulars thatinstantiate it.So that we may identify a particular, \(x\), via a collection ofpropertiesqua instantiated by \(x\), say \(\{F_x,\) \(G_x,\)\(H_x,\) …, \(P_x,\) \(Q_x,\) \(\ldots\};\) and a property,\(P\), via a collection of particularsqua instantiating\(P\), say \(\{P_a, P_b , \ldots ,P_x, P_y , \ldots \}\). By puttingthings in this way, we can then say that a particular is partiallyidentical to a property when the collection that identifies theparticular has an element in common with the collection thatidentifies the property. To illustrate, the \(x\) and the \(P\) of ourexample are partially identical because they have the element \(P_x\)in common. Now, the elements of these collections are neitherpropertiestout court nor particularstout court,which led us to talk of propertiesqua instantiated andparticularsqua instantiating.^[9] But this of course presupposes instantiation. Moreover, there is theunwelcome consequence that the world becomes dramatically lesscontingent than we would have thought at first sight, for neither aconcrete particular nor a property can exist without it being the casethat the former has the properties it happens to have, and that thelatter is instantiated by the same particulars that actuallyinstantiate it; we get, as Mumford puts it (2007: 185), “a majornew kind of necessity in the world”.

2.3 Bradley’s Regress

One important motivation, possibly the main one, behind attempts atanalysis such as the ones we have just seen is the worry to avoid theso-calledBradley’s regress regarding exemplification (Baxter 2001: 449; Mumford 2007: 185),which goes as follows. Suppose that the individual \(a\) has theproperty \(F\). For \(a\) to instantiate \(F\) it must belinkedto \(F\) by a (dyadic) relation of instantiation, \(I_1\). Butthis requires a further (triadic) relation of instantiation, \(I_2\),that connects \(I_1, F\) and \(a\), and so on without end. At eachstage a further connecting relation is required, and thus it seemsthat nothingever gets connected to anything else (it is notclear to what extent Bradley had this version in mind; for referencesto analogous regresses prior to Bradley’s, see Gaskin 2008: ch.5, §70).

This regress has traditionally been regarded as vicious (see, e.g.,Bergmann 1960), although philosophers such as Russell (1903: §55)and Armstrong (1997: 18–19) have argued that it is not. In doingso, however, they seem to take for granted the fact that \(a\) has theproperty \(F\) (pretty much as in thebrute fact approach;see below) and go on to see \(a\)’s and \(F\)’sinstantiating \(I_1\) as a further fact that is merelyentailed 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 standardresponse to the regress. But those who see the regress as viciousassume that the various exemplification relations are introduced in aneffort toexplain 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 ofentailments is beside the point (cf. Loux & Crisp 2017:31–35; Vallicella 2002).

It should be noted, however, that this perspective suggests adistinction between an “internalist” and an“externalist” version of the regress (in the terminologyof Orilia 2006a). In the former, at each stage we postulate a newconstituent of the fact, or state of affairs, \(s\), that existsinsofar as \(a\) has the property \(F\), and there is viciousnessbecause \(s\) can never be appropriately characterized.^[10] In the latter, at each stage we postulate a new, distinct, state ofaffairs, whose existence is required by the existence of the state ofaffairs of the previous stage. This amounts to admitting infiniteexplanatory and metaphysical dependence chains. However, according toOrilia (2006b: §7), since no decisive arguments against suchchains exist, the externalist regress should not be viewed as vicious(for criticisms, see Maurin 2015 and Allen 2016: §2.4.1; for asimilar view about predication, see Gaskin 2008).

A typical attempt to avoid the regress has been the proposal thatinstantiation is not a relation, or at least not a normal one. Incriticizing Bradley’s argument, Broad (1933: 85) discusses themetaphor of two objects glued to the opposite ends of a bit of string;it is, he notes, as if Bradley required some additional ingredient tofasten each object and each end to the glue itself, whereas in factthe glue just glues without further intermediaries. Similarly, onemight think, instantiation is asui generis linkage thatneeds no intermediaries. In this vein, Strawson (1959) speaks of anonrelational tie and Bergmann (1960) of anexus.For Grossmann (1992: 55-56), all relations, including exemplification,are like glue: they connect without any intermediary super-glue.Alternatively, some have rejected instantiation altogether. Accordingto Frege, it is not needed, because properties have “gaps”that can be filled, and according to a reading of Wittgenstein’sTractatus, because objects and properties can be connectedlike links in a chain. However, both strategies are problematic, asargued by Vallicella (2002). His basic point is that, if \(a\) hasproperty \(F\), we need an ontological explanation of why \(F\) and\(a\) happen to be connected in such a way that \(a\) has \(F\) as oneof its properties (unless \(F\) is a property that \(a\) hasnecessarily). But none of these strategies can provide thisexplanation. For example, the appeal to gaps is pointless: \(F\) has agap whether or not it is filled by \(a\) (for example, it could befilled in by another object), and thus the gap cannot explain the factthat \(a\) 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 bytaking a state of affairs, say \(x\)’s being \(P\), as capableby itself of holding together its constituents, i.e., the object \(x\)and the property \(P\) (see also Perovic 2016). Thus, there is no needto invoke a relation of exemplification linking \(x\) and \(P\) inorder to explain how \(x\) and \(P\) succeed in giving rise to aunitary item, namely the state of affairs in question. There seems tobe a circularity here for it appears that we want to explain how anobject and a property come to be united in a state of affairs byappealing to the result of this unification, namely the state ofaffairs itself. But perhaps this view can be interpreted as simply theidea that states of affairs should be taken for granted in aprimitivist fashion without seeking an explanation of their unity byappealing to exemplification or otherwise; this is thebrute factapproach, as we may call it (for supporters, see Van Inwagen(1993: 37) and Oliver (1996: 33); for criticisms and a possibledefense, see Vallicella 2002, and Orilia 2016, respectively).

Lowe (2006) has tried to tackle Bradley’s regress within hispluralist approach to exemplification. In his view, characterization,instantiation and exemplification are “formal” and thusquite different from garden-variety relations suchgiving orloving. This guarantees that these three relations escapeBradley’s regress (Lowe 2006: 30, 80, 90).^[11] Let us illustrate how, by turning back to the Fido example of§2.1. What a mode instantiates and what it characterizes belong to itsessence. In other words, a mode cannot exist without instantiating theattribute it instantiates and characterizing the object itcharacterizes. Hence, mode \(b\), by simply existing, instantiatesattribute \(B\) and characterizes Fido. Moreover, sinceexemplification (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 \(b\) characterizesFido, that \(b\) instantiates \(B\), and that Fido exemplifies \(B\)(i.e., is barking), all of which are made true by \(b\). Hence, thereis no need to postulate as truthmakers states of affairs withconstituents, Fido and \(b\), related by characterization, or \(b\)and \(B\), related by exemplification, or Fido and \(B\), related byexemplification. This, in Lowe’s opinion, eschewsBradley’s regress, since this arises precisely because we appealto states of affairs with constituents in need of a glue thatcontingently keep them together. Nevertheless, there is no loss ofcontingency in Lowe’s world picture, for an object need not becharacterized by the modes that happen to characterize it. Thus, forexample, mode \(b\) might have failed to exist and there could havebeen a Fido silence mode in its stead, in which case the propositionthat Fido is barking would have been false and the proposition thatFido is silent would have been true. One may wonder however what makesit the case that a certain mode is a mode of just a certain object andnot of another one, say another barking dog. Even granting that it isessential for \(b\) to be a mode of Fido, rather than of another dog,it remains true that it isof Fido, rather than of the otherdog, and one may still think that thisbeing of is also glueof some sort, perhaps with a contingency inherited from thecontingency of \(b\) (which might have failed to exist). The suspicionthen is that the problem of accounting for the relation between a modeand an object has replaced the Armstrongian one of what makes it thecase that a universal \(P\) and an object \(x\) make up the state ofaffairs\(x\)’s being \(P\). But the former problem,one may urge, is no less thorny than the latter, and someuniversalists like Armstrong may consider uneconomical Lowe’sacceptance of modes in addition to universals (for accounts ofBradley’s regress analogous to Lowe’s, but within athoroughly tropist ontology, see Section 3.2 of the entry ontropes. See also Hakkarainen & Keinänen 2023, for a criticalanalysis).

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

2.4 Self-exemplification

Presumably, properties exemplify properties. For example, ifproperties are abstract objects, as is usually thought, then seeminglyevery property exemplifies abstractness. But then we should also grantthat there is self-exemplification, i.e., a property exemplifyingitself. For example, abstractness is itself abstract and thusexemplifies itself. Self-exemplification however has raised severeperplexities at least since Plato.

Plato appears to hold thatall properties exemplifythemselves, when he claims that forms participate in themselves. Thisclaim is crucially involved in his so-calledthird manargument, which led him to worry that his theory of forms isincoherent (Parmenides, 132 ff.). As we see matters now, itis not clear why we should hold that all properties exemplifythemselves (Armstrong 1978a: 71); for instance, people are honest, buthonesty itself is not honest (see, however, the entry onPlato’sParmenides, and Marmodoro forthcoming).

Nowadays, a more serious worry related to self-exemplification isRussell’s famous paradox, constructed as regarding the propertyof non-self-exemplification, which appears to exemplify itself iff itdoes not, thus defying the laws of logic, at least classical logic.The discovery of his paradox (and then the awareness of relatedpuzzles) led Russell to introduce a theory of types, which institutesa total ban on self-predication by a rigid segregation of propertiesinto a hierarchy oftypes (more on this in§6.1). The account became more complex and rigid, as Russell moved fromsimple toramified type theory, which involves adistinction oforders within types (see entries ontype theory andRussell’s paradox, and, for a detailed reconstruction of how Russell reacted to theparadox, Landini 1998).

In type theory all properties are, we may say,typed. Thisapproach has never gained unanimous consensus and its many problematicaspects are well-known (see, e.g., Fitch 1952: Appendix C; Bealer1989). Just to mention a few, the type-theoretical hierarchy imposedon properties appears to be highly artificial and multipliespropertiesad infinitum (e.g., since presumably propertiesare abstract, for any property \(P\) of type \(n\), there is anabstractness of type \(n+1\) that \(P\) exemplifies). Moreover, manycases of self-exemplification are innocuous and common. For example,the property ofbeing a property is itself a property, so itexemplifies itself. Accordingly, many recent proposals are type-free(see§6.1) and thus view properties asuntyped, capable of beingself-predicated, sometimes veridically. An additional motivation tomove 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 contingentassumption, namely that someone, say John, is thinking just about theproperty of being a property \(P\) such that John is thinking ofsomething that is not exemplified by \(P\).

3. Existence and Identity Conditions for Properties

Quine (1957 [1969: 23]) famously claimed that there should be noentity without identity. His paradigmatic case concerns sets: two ofthem are identical iff they have exactly the same members. Since thenit has been customary in ontology to search for identity conditionsfor given categories of entities and to rule out categories for wantof identity conditions (against this, see Lowe 1989). Quine startedthis trend precisely by arguing against properties and this hasstrictly intertwined the issues of which properties there are and oftheir identity conditions.^[13]

3.1 From Extensionality to Hyperintensionality

In an effort to provide identity conditions for properties, one couldmimic those for sets, or equate the former with the latter (as inclass nominalism; see note 3), and provide the followingextensionalist identity conditions: two properties areidentical iff they are co-extensional. This criterion can hardly work,however, since there are seemingly distinct properties with the sameextension, such ashaving a heart andhavingkidneys, and even wildly different properties such asspherical andweighing 2 kilos could by accident beco-extensive.

One could then try the followingintensional identityconditions: two properties are identical iff they areco-intensional, i.e., necessarily co-extensional, where thenecessity in question is logical necessity. This guarantees thatspherical andweighing 2 kilos are different even ifthey happen to be co-extensional. Following this line, one may takeproperties to beintensions, understood as, roughly,functions that assign extensions (sets of objects) to predicates atlogically possible worlds. Thus, for instance, the predicates“has a heart” and “has kidneys” stand fordifferent intensions, for even if they have the same extension in theactual world they have different extensions at worlds where there arecreatures with heart and no kidneys or vice versa. This approach isfollowed by Montague (1974) in his pioneering work in natural languagesemantics, and in a similar way by Lewis (1986b), who reducesproperties to sets of possible objects in his modal realism,explicitly committed to possible worlds and merepossibiliainhabiting them. Most philosophers find this commitment unappealing.Moreover, one may wonder how properties can do their causal work ifthey are conceived of in this way (for further criticisms see Bealer1982: 13–19, and 1998: §4; Egan 2004). However, thecriterion of co-intensionality may be accepted without also buying thereduction of properties to sets ofpossibilia (Bealer viewsthis as the identity condition for his conception 1 properties; seebelow). Still, co-intensionality must face two challenges coming fromopposite 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 oftemperature to mean kinetic energy, could suggest that some propertiesare identical even if not co-intensional. For example, one may acceptthathaving absolute temperature of 300K ishaving meanmolecular kinetic energy of \(6.21 \times 10^{-21}\) (Achinstein1974: 259; Putnam (1970: §1) and Causey (1972) speak of“synthetic identity” and “contingentidentity”, respectively). Rather than logical necessity, it isnomological necessity, necessity on the basis of the causallaws of nature, that becomes central in this line of thought.Following it, some have focused on thecausal andnomological roles of properties, i.e., roughly, the causesand effects of their being instantiated, and their involvement in lawsof nature, respectively. They have thus advancedcausal ornomic criteria. According to them, two properties are identicaliff they have the same causal (Achinstein 1974: §XI; Shoemaker1980) or nomological role (Swoyer 1982; Kistler 2002). This line hasbeen influential, as it connects to the “puredispositionalism” discussed in§5.2. There is however a suspicion of circularity here, since causal andnomological roles may be viewed as higher-order properties (see entryondispositions, §3).

On the other hand, once matters of meaning and mental content aretaken into account, co-intensionality might seem too weak, for itmakes 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 logicalnecessity, even, for example,being triangular andbeingtrilateral are identical. However, one could insist that“trilateral” and “triangular” appear to havedifferent meanings, which somehow involve the different geometricalpropertieshaving a side, andhaving an angle,respectively. And ifbeing triangular were really identicaltobeing trilateral, from the fact that John believes that acertain object has the former property, one should be able to inferthat John also believes that such an object has the latter property.Yet, John’s ignorance may make this conclusion unwarranted. Inthe light of this, borrowing a term from Cresswell (1975), one maymove from intensional tohyperintensional identityconditions, according to which two properties, such as trilateralityand triangularity, may be different even if they are co-intensional.In order to implement this idea, Bealer (1982) take two properties tobe identical iff they have the same analysis, i.e., roughly, theyresult from the same ultimate primitive properties and the samelogical operations applied to them (see also Menzel 1986; 1993). Zalta(1983; 1988) has developed an alternative available to those who admittwo modes of predication (see§1.4): two properties are identical iff they are (necessarily) encoded bythe same objects.

Hyperintensional conditions make of course for finer distinctionsamong entities than the other criteria we considered. Accordingly, theformer are often called “fine-grained” and the latter“coarse-grained”, and the same denominations arecorrespondingly reserved for the entities that obey the conditions inquestion. Coarse- or fine-grainedness is a relative matter. Theextensional criterion is more coarse-grained than the intensional orcausal/nomological ones. And hyperintensional conditions themselvesmay be more or less fine-grained: properties could be individuatedalmost as finely as the predicates expressing them, to the point that,e.g., evenbeing \(P\) and \(Q\) andbeing \(Q\) and\(P\) are kept distinct, but one may also envisage less stringentconditions that make for the identification of properties of that sort(Bealer 1982: 54). It is conceivable, however, that one could belogically obtuse to the point of believing that something has acertain property without believing that it has a trivially equivalentproperty. Thus, to properly account for mental content,maximal hyperintensionality appears to be required, and it isin fact preferred by Bealer. Even so, the paradox ofanalysis raises a serious issue. One could say, for example, thatbeing acircle isbeing a locus of points equidistant from apoint, since the latter provides the analysis of the former. Inreply, Bealer distinguishes between “being a circle” asdesignating a simple “undefined” property, which is ananalysandum distinct from theanalysans,being alocus of points equidistant from a point, and “being acircle” as designating a complex “defined” property,which is indeed identical to thatanalysans. Orilia (1999:§5.5) similarly distinguishes between the simpleanalysans and the complexanalysandum, withoutadmitting that expressions for properties such as “being acircle” could be ambiguous in the way Bealer suggests. Oriliarather argues that the “is” of analysis does not expressidentity but a weaker relation, which is asymmetrical in thatanalysans andanalysandum play different roles init. The matter keeps being discussed. Rosen (2015) appeals togrounding to characterize the different role played by theanalysans. 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 betweenconception 1 properties,orqualities, andconception 2 properties, orconcepts (understood as mind-independent). With a differentand now widespread terminology, Lewis (1983, 1986b) followed suit,speaking of asparse and anabundant conception ofproperties. According to the former, there are relatively fewproperties, namely just those responsible for the objectiveresemblances and causal powers of things; they cut nature at itsjoints, and science is supposed to individuate themaposteriori. According to the latter, there are immensely manyproperties, corresponding to all meaningful predicates we couldpossibly imagine and to all sets of objects, and they can be assumeda priori. (It should be clear that “few” and“many” are used in a comparative sense, for the number ofsparse properties may be very high, possibly infinite). To illustrate,the sparse conception admits properties currently accepted byempirical science such ashaving negative charge orhaving spin up and rejects those that are no longer supportedsuch ashaving a certain amount of caloric, which featured ineighteenth century chemistry; in contrast, the abundant conception mayacknowledge the latter property and all sorts of other properties,strange as they may be, e.g.,negatively charged or disliked bySocrates,round and square or Goodman’s (1983)notoriousgrue andbleen. Lewis (1986b: 60) tries tofurther characterize the distinction by taking the sparse propertiesto beintrinsic, rather thanextrinsic (e.g.,being 6 feet tall, rather thanbeing taller thanTom), andnatural, where naturalness is something thatadmits of degrees (for example, he says, masses and charges areperfectly natural, colors are less natural andgrue andbleen are paradigms of unnaturalness). Much work on suchissues has been done since then (see entries onintrinsic vs. extrinsic properties andnatural properties).

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

At one end of the spectrum, there is the most extreme version of thesparse conception,minimalism, which accepts all of theseprinciples:

there are only coarse-grained properties,
they exist only if instantiated and thus are contingentbeings,
they are all instantiated by things in space-time (setting asidethose instantiated by other properties),
they are fundamental and thus their existence must be sanctionedby microphysics.

This approach is typically motivated by physicalism andepistemological qualms regarding transcendent universals. Thebest-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 lessminimalist versions of the sparse conception. For example, some haveurged uninstantiated properties to account for features of measurement(Mundy 1987), vectors (Bigelow & Pargetter 1991: 77), or naturallaws (Tooley 1987), and some even that there are all the propertiesthat can be possibly exemplified, where the possibility in question iscausal or nomic (Cocchiarella 2007: ch. 12). In line with positionstraditionally found in emergentism (see§5.1), Schaffer (2004) proposes that there are sparse properties asfundamental, and in addition, as grounded on them, the properties thatneed be postulated at all levels of scientific explanations, e.g.,chemical, biological and psychological ones. Even Armstrong goesbeyond 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). Allthese positions appear to be primarily concerned with issues in themetaphysics of science (see§5) and typically display too short a supply of properties to deal withmeaning and mental content, and thus with natural language semanticsand the foundations of mathematics (see§6). However, minimalists might want to resort to concepts, understood asmind-dependent entities, in dealing with such issues (e.g., along thelines proposed in Cocchiarella 2007).

Matters of meaning and mental content are instead what typicallymotivate 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; Castañeda 1976; Jubien1989; Lewis 1986b; Orilia 1999; Zalta 1988; Van Inwagen 2004):properties are fine-grained necessary entities that exist even ifuninstantiated, or even impossible to be instantiated. In its mostextreme version, maximalism adopts identity conditions thatdifferentiate properties as much as possible, but more moderateversions can be obtained by slightly relaxing such conditions. Viewsof this sort are hardly concerned with physicalist constraints or thelike and rather focus on the explanatory advantages ofhyperintensionality. These may well go beyond the typical motivationsconsidered above: Nolan (2014) argues that hyperintensionality isincreasingly important in metaphysics in dealing with issues such ascounter-possible conditionals, explanation, essences, grounding,causation, confirmation and chance.

Rather than choosing between the sparse and abundant conceptions, thevery promoters of this distinction have opted in different ways for adualism of properties, according to which there areproperties of both kinds. Lewis endorses abundant properties asreduced to sets ofpossibilia, and sparse properties eitherviewed as universals, and corresponding to some of the abundantproperties (1983), or as themselves sets of possibilia (1986b: 60).Bealer (1982) proposes a systematic account wherein qualities are thecoarse-grained properties that “provide the world with itscausal and phenomenal order” (1982: 183) and concepts are thefine-grained properties that can function as meanings and asconstituents of mental contents. He admits a stock of simpleproperties, which are both qualities and concepts (1982: 186),wherefrom complex qualities and complex concepts are differentlyconstructed: on the one hand, bythought-building operations,which give rise to fine-grained qualities; on the other hand, bycondition-building operations, which give rise tocoarse-grained qualities. Orilia (1999) has followed Bealer’slead in also endorsing both coarse-grained qualities and fine-grainedconcepts, without however identifying simple concepts and simplequalities. In Orilia’s account concepts are never identical toqualities, but may correspond to them; in particular, the identitystatements of intertheoretic reductions should be taken to express thefact 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 presencemay however be widespread. For example, Putnam’s (1970: §1)distinction betweenpredicates andphysicalproperties, and Armstrong’s above-mentioned recognition ofsecond-class properties may be seen as forms of dualism.

4. Complex Properties

It is customary to distinguish betweensimple andcomplex properties, even though some philosophers take allproperties to be simple (Grossmann 1983: §§58–61). Theformer are not characterizable in terms of other properties, areprimitive 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 notobvious that there are simple properties, since one may imagine thatall properties are analyzable into constituentsad infinitum(Armstrong 1978b: 67). Even setting this aside, to provide examples isnot easy. Traditionally, determinate colors are cited, but nowadaysmany would rather appeal to fundamental physical properties such ashaving a certain electric charge. It is easier to provideputative examples of complex properties, once some other propertiesare taken for granted, e.g.,blue and spherical orblueor non-spherical. Theselogically compound properties,which involve logical operations, will be considered in§4.1. Next, in§4.2 we shall discuss other kinds of complex properties, calledstructural (after Armstrong 1978b), which are eliciting agrowing interest, as a recent survey testifies (Fisher 2018). Theircomplexity has to do with the subdivisions of their instances intosubcomponents. Typical examples come from chemistry, e.g.,H₂O andmethane understood as propertiesof molecules.

It should be noted that it is not generally taken for granted thatcomplex properties literally have parts or constituents. Somephilosophers take this line (Armstrong 1978a: 36–39, 67ff.;Bigelow & Pargetter 1989; Orilia 1999), whereas others demur, andrather think that talk in terms of structures with constituents ismetaphorical and dependent on our reliance on structured terms such as“blue and spherical” (Bealer 1982; Cocchiarella 1986a;Swoyer 1998: §1.2).

4.1 Logically Compound Properties

At leastprima facie, our use of complex predicates suggeststhat there are corresponding complex properties involving all sorts oflogical operations. Thus, one can envisageconjunctiveproperties such asblue and spherical, negative propertiessuch asnon-spherical,disjunctive properties suchasblue or non-spherical,existentially oruniversally quantified properties such aslovingsomeone orloved by everyone,reflexiveproperties such asloving oneself, etc. Moreover, one couldadd to the list properties with individuals as constituents, which arethen denied the status of apurely qualitative property,e.g.,taller than Obama.

It is easy to construct complex predicates. But whether there reallyare corresponding properties of this kind is a much more difficult andcontroversial issue, tightly bound to the sparse/abundant distinction.In the sparse conception the tendency is to dispense with them. Thisis understandable since in this camp one typically postulatespropertiesin rebus for empirical explanatory reasons. Butthen, if we explain some phenomena by attributing properties \(F\) and\(G\) to an object, while denying that it does not exemplify \(H\), itseems of no value to add that it also has, say,F and G,F or H, andnon-H. Nevertheless, Armstrong, theleading supporter of sparseness, has defended over the years a mixedstance without disjunctive and negative properties, but withconjunctive ones. Armstrong’s line has of course its opponents(see, e.g., Paolini Paoletti 2014; 2017a; 2020a; Zangwill 2011 arguesthat there are negative properties, but they are less real thanpositive ones). On the other hand, in the abundant conception, allsorts of logically compound properties are acknowledged. Since thefocus now is on meaning and mental content, it appears natural topostulate such properties to account for our understanding of complexpredicates and of how they differ from simple ones. Even here,however, there are disagreements, when it comes to problematicpredicates such the “does not exemplify itself” ofRussell’s paradox. Some suggest that in this case one shoulddeny there is a corresponding property, in order to avoid the paradox(Van Inwagen 2006). However, we understand this predicate and thus itseemsad hoc that the general strategy of postulating aproperty fails. It seems better to confront the paradox with othermeans (see§6).

4.2 Structural Properties

Following Armstrong (1978b: 68–71; see also Fisher 2018:§2), a structural property \(F\) is typically viewed as auniversal, and can be characterized thus:

an object exemplifying \(F\), say \(x\), must have“relevant” proper parts that do not exemplify \(F\);
there must be some other “relevant” properties somehowinvolved in \(F\) that are not exemplified by the object inquestion;
the relevant proper parts must rather exemplify one or the otherof the relevant properties involved in \(F\).

For instance, a molecule \(w\) exemplifyingH₂Ohas as parts two atoms \(h_1\) and \(h_2\) and an atom \(o\) that donot exemplifyH₂O; this property involveshydrogen andoxygen, which are not exemplified by\(w\); \(h_1\) and \(h_2\) exemplifyhydrogen, and \(o\)exemplifiesoxygen. Forrelational structuralproperties, there is the further condition that

a certain “relevant” relation that links the relevantproper parts must also be involved.

H₂O is a case in point: the relevant (chemical)relation isbonding, which links the three atoms in question.Another, often considered, example ismethane as a structuralproperty of methane-molecules (CH₄), whichinvolves abonding relation between atoms and the jointinstantiation ofhydrogen (by four atoms) andcarbon(by one atom).Non-relational structural properties do notrequire condition (iv). For instance,mass 1 kg involves manyrelevant properties of the kindmass n kg, for \(n \lt 1\),which are instantiated by relevant proper parts of any object that is1 kg in mass, and these proper parts are not linked by any relevantrelation (Armstrong 1978b: 70).

In most approaches, including Armstrong’s, the relevantproperties and the relevant relation, if any, are conceived of asparts, or constituents, of the structural property in question, andthus their being involved in the latter is a parthood relation. Forinstance,hydrogen,oxygen andbonding areconstituents ofH₂O. Moreover, the composition ofstructural properties is isomorphic to that of the complexesexemplifying them. These two theses characterize what Lewis (1986a)calls the “pictorial conception”. It is worth noting thatstructural properties differ from conjunctive properties such ashuman and musician, in that the constituents of the latter(i.e.,human andmusician) are instantiated bywhatever entity exemplifies the conjunctive property (i.e., a humanbeing who is also a musician), whereas the constituents of astructural property (e.g.,hydrogen,oxygen andbonding) are not instantiated by the entities thatinstantiate it (e.g.,H₂O molecules).

Structural properties have been invoked for many reasons. Armstrong(1978b: ch. 22) appeals to them to explain the resemblance ofuniversals belonging to a common class, e.g., lengths and colors.Armstrong (1988; 1989) also treats physical quantities and numbersthrough structural properties. The former involve, as in the abovemass 1 kg example, smaller quantities such asmass 0.1kg andmass 0.2 kg (this view is criticized by Eddon2007). The latter are internal proportion relations between structuralproperties (e.g.,being a nineteen-electrons aggregate) andunit-properties (e.g.,being an electron). Moreover,structural properties have been appealed to in treatments of laws ofnature (Armstrong 1989; Lewis 1986a), some natural kinds (Armstrong1978b, 1997; Hawley & Bird 2011), possible worlds (Forrest 1986),ersatz times (Parsons 2005), emergent properties(O’Connor & Wong 2005), linguistic types (Davis 2014), onticstructural realism (Psillos 2012). However, structural universals havealso 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 ofComposition of classicalmereology. 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 differentstructures, e.g., butane and isobutane(C₄H₁₀). Here,butane andisobutane are different structural universals. Yet, theyarise from the same universals recurring the same number of times. Letus dub this “isomer problem.”

Two further problems may be pointed out. First, even molecules withthe same number and types of atoms and the same structures may vary invirtue of their spatial orientation (Kalhat 2008). This phenomenon isknown aschirality. How can structural properties account forit? Secondly, the composition of structural properties is restricted:not every collection of properties gives rise to a structuralproperty. This violates another principle of classical mereology: thePrinciple of Unrestricted Composition.

Lewis dismisses alternatives to the pictorial conception, such as the“linguistic conception” (in which structural universalsare set-theoretic constructions out of simple universals) and the“magic conception” (in which structural universals are notcomplex and are primitively connected to the relevant properties).Moreover, he also rejects the possibility of there beingamphibians, i.e., particularized universals lying betweenparticulars and full-fledged universals. Amphibians would solve themultiple recurrence problem, as they would be identical with multiplerecurrences of single universals. For example, inmethane,there would be four amphibians ofhydrogen.

To reply to Lewis’ challenges, two main strategies have beenadopted on behalf of the pictorial conception: a non-mereologicalcomposition strategy, according to which structural properties do notobey the principles of classical mereology, and a mereology-friendlystrategy. Let us start with the former.

Armstrong (1986) admits that the composition of structural universalsis non-mereological. In his 1997, he emphasizes that states of affairsdo not have a mereological kind of composition, and views universals,including structural ones, asstates of affairs-types which,as such, are not mereological in their composition. Structuraluniversals themselves turn out to be state of affairs-types of aconjunctive sort (1997: 34 ff.). To illustrate, let us go back to theabove example withH₂O exemplified by the watermolecule \(w\). It involves the conjunction of these states ofaffairs:

atom \(h_1\) being hydrogen,
atom \(h_2\) being hydrogen,
atom \(o\) being oxygen,
\(h_1\), \(h_2\), and \(o\) being bonded.

This conjunction of states of affairs provides an example of a stateof affairs-type identifiable with the structural universalH₂O.

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

As regards the mereology-friendly strategy, it has been suggested thatstructural properties actually include extra components accounting forstructures, so that the Principle of Uniqueness of Composition issafe. Consideringmethane, Pagès (2002) claims thatsuch a structural property is composed of the propertiescarbon andhydrogen, abonding relation anda peculiar first-order, formal relation between the atoms. Accordingto Kalhat (2008), the extra component is a second-order arrangementrelation between the first-order properties. Ordering formalrelations—together with causally individuatedproperties—are also invoked by McFarland (2018).

At the intersection of these strategies, Bigelow and Pargetter (1989,1991) claim that structural universals are internally relationalproperties that supervene on first-order properties and relations andon second-order proportion relations. Second-order proportionrelations are relations such ashaving four times as. Theyrelate first-order properties and relations. For example, inmethane, the relationhaving four times as relatestwo conjunctive properties: (i)hydrogen and being part of thismolecule; (ii)carbon and being part of thismolecule.

Campbell (1990) solves the multiple recurrence problem and the isomerproblem by appealing to tropes. Inmethane, there aremultiple hydrogen tropes. Inbutane andisobutane,distinct tropes entertain distinct structural relations.

Despite Lewis’ dismissal, a number of theories appeal toamphibians (i.e., particularized universals lying between particularsand full-fledged universals) or, more precisely, to entities similarto amphibians. Fine (2017) suggests that structural properties shouldbe treated by invoking arbitrary objects (Fine 1985). Davis’(2014) account of linguistic types as structural properties confrontsthe problem of the multiple occurrence of one type, e.g.,“dog”, in another (e.g., “every dog likes anotherdog”) (Wetzel 2009). In general, the idea that types areproperties of tokens is a natural one, although there are alsoalternative views on the matter (see entry ontypes and tokens). Be this as it may, Davis proposes that types cannot occur withinfurther types as tokens: they can only occur assubtypes.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 insymmetric ones. The approach could be extended to properties such asmethane, which would be taken to have four distincthydrogen subtypes.

5. Properties in the Metaphysics of Science

When it comes to the metaphysical underpinnings of scientifictheories, properties play a prominent role: it appears that sciencecan hardly be done without appealing to them. This adds up to the casefor realism about properties and to our understanding of them. Weshall illustrate this here first by dwelling on some miscellaneoustopics and then by focusing on a debate regarding the very nature ofthe properties invoked in science, namely whether or not they areessentially dispositional. Roughly, an object exemplifies adispositional property, such assoluble orfragile, by having a power or disposition to act or beingacted upon in a certain way in certain conditions. For example, aglass’ disposition ofbeing fragile consists in itspossibly shattering in certain conditions (e.g., if struck with acertain force). In contrast, something exemplifies acategorical property, e.g.,made of salt orspherical, by merely being in a certain way (see entry ondispositions).

5.1 Miscellaneous Topics

Many predicates in scientific theories (e.g., “being agene” and “being a belief”) are functionallydefined. Namely, their meaning is fixed by appealing to some functionor set of functions (e.g., encoding and transmitting geneticinformation). To make sense of functions, properties are needed.First, functions can be thought of as webs of causal relationshipsbetween properties or property-instances. Secondly, predicates such as“being a gene” may be taken to refer tohigher-level properties, or at least to something that hassuch properties. In this case, the property ofbeing a genewould be the higher-level property ofpossessing some furtherproperties (e.g., biochemical ones)that play the relevantfunctions (i.e., encoding and transmitting genetic information).Alternatively,being a gene would be a property thatsatisfies the former higher-level property. Thirdly, the ensuingrealization relationships between higher-level and function-playingproperties can only be defined by appealing to properties (see entryonfunctionalism).

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

Entities in nature are typified bynatural kinds. These are mostly thought of as properties or property-like entitiesthat 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 asproperties, specifiable in terms of a magnitude, a certain number, anda unit measure, e.g., kg or meter, which can itself be specified interms of properties (Mundy 1987; Swoyer 1987). Following Eddon’s(2013) overview, it is possible to distinguish two main strands here:relational andmonadic properties theories (see alsoDasgupta 2013). In relational theories (Bigelow & Pargetter 1988;1991), quantities arise from proportion relations, which may in turnbe related by higher-order proportion relations. Consider a certainquantity, e.g.,mass 3 kg. Here thethrice as massiveas proportion relation is relevant, which holds of certain pairsof objects \(a\) and \(b\) such that \(a\) is thrice as massive as\(b\). By virtue of such relational facts, the first members of suchpairs have a mass of 3 kg. Another relational approach is by Mundy(1988), who claims thatmass 3 kg is a relation between(ordered pairs of) objects and numbers. For example,mass 3kg is a relation holding between the ordered pair \(\langle a,b\rangle\) (assuming that \(a\) is thrice as massive as \(b\), and notthe other way round) and the number 3; or—if relations havebuilt-in relational order—between \(a, b\) and 3. Knowles (2015)holds a similar view, wherein physical quantities are relations thatobjects bear to numbers. According to the monadic properties approach,quantities are intrinsic properties of objects (Swoyer 1987; Armstrong1988; 1989). As we saw, Armstrong develops such a view by takingquantities to be structural properties. Related metaphysical inquiriesconcern the dimensions of quantities (Skow 2017) and the status offorces (Massin 2009).

Lastly, properties play a prominent role in two well-known accounts ofthelaws of nature: thenomological necessitation account andlawdispositionalism. The former has been expounded by Tooley (1977;1987), Dretske (1977) and Armstrong (1983). Roughly, followingArmstrong, a law of nature consists in a second-order and externalnomological necessitation relation \(\rN\), contingently holdingbetween first-order universals \(P\) and \(Q\): \(\rN(P, Q)\). Such ahigher-order fact necessitates certain lower-order regularities, i.e.,all the objects that have \(P\) also have \(Q\) (by nomologicalnecessity, in virtue of \(\rN(P, Q))\). Law dispositionalisminterprets laws of nature by appealing to dispositional properties.According to it, laws of nature derive from the essence of suchproperties. This implies that laws of nature hold with metaphysicalnecessity: whenever the dispositions are in place, the relevant lawsmust be in place too (Cartwright 1983; Ellis 2001; Bird 2007;Chakravartty 2007; Fischer 2018; see also Schrenk 2017 and Dumsday2019). (For criticisms, see, e.g., van Fraassen 1989 as regards theformer approach, and McKitrick 2018 and Paolini Paoletti 2020b, asregards the latter).

5.2. Essentially Categorical vs. Essentially Dispositional Properties

There is a domestic dispute among supporters of properties in themetaphysics of science, regarding the very nature of such properties(or at least the fundamental ones). We may distinguish two extremeviews,pure dispositionalism andpurecategoricalism. According to the former, all properties areessentially dispositional (“dispositional”, forbrevity’s sake, from now on), since they are nothing more thancausal powers; theircausative roles, i.e., which effectstheir instantiations can cause, exhaust their essences. According topure categoricalism, all properties are essentially categorical (inbrief, “categorical”, in the following), because theircausative roles are not essential to them. If anything is essential toa property, it is rather a non-dispositional and intrinsic aspectcalled “quiddity” (after Armstrong 1989), which need notbe seen as an additional entity over and above the property itself(Locke 2012, in response to Hawthorne 2001). Between such views therelie a number of intermediate positions.

Pure dispositionalism has been widely supported in the last fewdecades (Mellor 1974; 2000; Shoemaker 1984: ch. 10 and 11 [in terms ofcausal roles]; Mumford 1998; 2004; Bird 2005; 2007; Chakravartty 2007;Whittle 2008; see Tugby 2013 for an attempt to argue from this sort ofview to a Platonist conception of properties). There are three mainarguments in favor of it. First, pure dispositionalism easily accountsfor the natural necessity of the laws of nature, insofar as such anecessity just derives from the essence of the dispositionalproperties involved. Secondly, dispositional properties can be easilyknown as they really are, because it is part of their essence thatthey affect us in certain ways. Thirdly, at the (presumablyfundamental) micro-physical level, properties are only describeddispositionally, 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-cutdistinction between dispositional and would-be non-dispositionalproperties (Cross 2005). Secondly, it seems that the essence ofcertain properties does not include, or is not exhausted by, theircausative roles: qualia, inert and epiphenomenal properties,^[14] structural and geometrical properties, spatio-temporal properties(Prior 1982; Armstrong 1999; for some responses, see Mellor 1974 andBird 2007; 2009). Thirdly, there can besymmetrical causativeroles. Consider three distinct properties \(A\), \(B\) and \(C\) suchthat \(A\) can cause \(B\), \(B\) can cause \(A\), \(A\) and \(B\) cantogether cause \(C\) and nothing else characterizes \(A\) and \(B\).\(A\) and \(B\) have the same causative role. Therefore, in puredispositionalism, they turn out to be identical, against thehypothesis (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 roleof a dispositional property \(P\) “points towards” furtherproperties \(S\), \(T\), etc.qua possible effects. Theessential causative roles of the latter “point towards”still other properties, and so on. The first regress concerns theidentity of \(P\), which is never fixed, as it depends on the identityof further properties, which depend for their identity on still otherproperties, and so on (Lowe 2006; Barker 2009; 2013). The secondregress concerns the knowability of \(P\) (Swinburne 1980). \(P\) isonly knowable through its possible effects (i.e., the instantiation of\(S\), \(T\), etc.), included in its causative role. Yet, suchpossible effects are only knowable through their possible effects, andso on. The third regress concerns the actuality of \(P\) (Armstrong1997). \(P\)’s actuality is never reached, since \(P\) isnothing but the power to give rise to \(S\), \(T\), etc., which arenothing 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 onlycontingently associated to a property. Therefore, on purecategoricalism, a property can possibly have distinct causative roles,which allows it to explain—among other things—the apparentcontingency of causative roles and the possibility of recombining aproperty with distinct causative roles. Its supporters include Lewis(1986b, 2009), Armstrong (1999), Schaffer (2005), and more recentlyLivanios (2017), who provides further arguments based on themetaphysics of science. Kelly (2009) and Smith (2016) may be added tothe list, although they take roles to be non-essential and necessary.More generally, it is one thing to hold that a property is essentiallydispositional—which is incompatible with pure categoricalism. Itis another thing to hold that a property is essentially categoricalbut is necessarily “tied” to a certain causative role,e.g., because the latter derives from/is necessitated by the essenceof that property. This position is compatible with purecategoricalism. On this option, see Kimpton-Nye (2018), Yates (2018a),Coates (2021), Lenart (2021), Tugby (2021) and (2022).

However, pure categoricalism falls prey to two sorts of difficulties.First, the contingency of causative roles has some unpalatableconsequences: unbeknownst to us, two distinct properties can swaptheir roles; they can play the same role at the same time; the samerole can be played by one property at one time and by another propertyat a later time; an “alien” property can replace afamiliar one by playing its role (Black 2000). Secondly, and moregenerally, we are never able to know which properties play whichroles, nor are we able to know the intrinsic nature of suchproperties. This consequence should be accepted with “Ramseyanhumility” (Lewis 2009; see also Langton 1998 for a related,Kantian, sort of humility) or it should be countered in the same wayas we decide to counter any broader version of skepticism (Schaffer2005). On this issue, see also Whittle (2006); Locke (2009); Kelly(2013); Yates (2018b).

Let us now turn to some intermediate positions.

According todualism (Ellis 2001; Molnar 2003), there areboth dispositional and categorical properties. Dualism is meant tocombine 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 thepowerful qualities view, every property isboth dispositional and categorical (or qualitative). The main problemhere is how to characterize the distinction and relation of these two“sides”. Martin (2008) and Heil (2003), (2012) suggestthat they are two distinct ways of partially considering one and thesame property, whereas Mumford (1998) explores the possibility ofseeing them as two distinct ways of conceptualizing the property inquestion. Heil claims that the qualitative and the dispositional sidesneed to be identified with one another and with the whole property,thus accepting an identity theory of powerful qualities (see alsoMartin 2008, G. Strawson 2008, Jaworski 2016, N. Williams 2019).Jacobs (2011) holds that the qualitative side consists in thepossession of some qualitative nature by the property, whereas thedispositional side consists in that property being (part of) asufficient truthmaker for certain counterfactuals. Dispositional andqualitative sides may also be seen as essential or non-essentialhigher-order properties of properties, as supervenient andontologically innocent aspects of properties (Giannotti 2021), or asconstituents of the essence of properties (Taylor 2018). Moreover, thequalitative side may—though need not—be seen as morefundamental than the dispositional side, e.g., because the latterderives from the former (see Azzano 2021, Coates 2021, 2023,Kimpton-Nye 2021). For an exhaustive critical overview of theseoptions, see Livanios (2024). Livanios (2024) develops a version ofthe powerful qualities view in the neighborhood of categoricalismcalled “powerful categoricalism”. In general, the powerfulqualities view is between Scylla and Charybdis. If it reifies thedispositional and qualitative sides, it runs the risk of implying somesort of dualism. If it insists on the identity between them, it runsthe 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 & Mönnich 1989: 133). In the next subsection we shalloutline how they work. In the two subsequent subsections we shallbriefly consider their deployment in natural language semantics andthe foundations of mathematics, which can be taken to provide furtherreasons 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, inparticular variables that are meant to range over properties and thatcan be quantified over. This can be achieved in two ways. Either(option 1; Cocchiarella 1986a) the terms standing for properties arepredicates or (option 2; Bealer 1982) such terms are subject termsthat can be linked to other subject terms by a special predicate thatis meant to express a predication relation (let us use“pred”) pretty much as in standard set theory a specialpredicate, “\(\in\)”, is used to express the membershiprelation. To illustrate, given the former option, an assertion such as“there is a property that both John and Mary have” can berendered 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 1993for further discussion).

Whatever option one follows, in spelling out such theories onetypically postulates a rich realm of properties. Traditionally, thisis 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 acorresponding \(n\)-adic property. Following option 1, it goes asfollows:

\[\tag{CP} \exists R^n \forall x_1 \ldots \forall x_n(R^n(x_1 , \ldots ,x_n) \leftrightarrow A). \]

Alternatively, one can use a variable-binding operator, \(\lambda\),that, given an open wff, generates a term (called a “lambdaabstract”) that is meant to stand for a property. This way toproceed is more flexible and is followed in the most recent versionsof property theory. We shall thus stick to it in the following. Toillustrate, we can apply “\(\lambda\)” to the openformula, “\((R(x) \amp S(x))\)” to form the one-placecomplex predicate “\([\lambda x~(R(x) \amp S(x))]\)”; if“\(R\)” denotesbeing red and “\(S\)”denotesbeing square, then this complex predicate denotes thecompound, conjunctive propertybeing 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 forloves, this complex predicatedenotes the compound propertyloving someone (whereas“\([\lambda y~ \exists x(L(x, y))]\)” would denotebeing loved by someone). To ensure that lambda abstractsdesignate the intended property, one should assume a “principleof lambda conversion”. Given option 1, it can be statedthus:

\[\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 fromsimultaneously replacing each \(x_i\) in \(A\) with \(t_i\) (for \(1\le i \le n)\), provided \(t_i\) is free for \(x_i\) in \(A\)). Forexample, given this principle, \([\lambda x~ (R(x) \amp S(x))](j)\) isthe case iff \((R(j) \amp S(j))\) is also the case.

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

However, for reasons sketched in§2.4, type theory is hardly satisfactory. Accordingly, many type-freeversions of property theory have been developed over the years and noconsensus on what the right strategy is appears to be in sight. Ofcourse, without type-theoretical constraints, given \((\lambda\)-conv)and classical logic (CL), paradoxes such as Russell’simmediately 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 orpredicates may (but need not) denote properties (Swoyer 1998), formalcounterparts of (complex) predicates like “being a property thatdoes not exemplify itself” (formally, “\([\lambda x~{\sim}x(x)]\)”) could exist in the object language withoutdenoting properties; from this perspective, Russell’s paradoxmerely shows that such predicates do not stand for properties(similarly, according to Schnieder 2017, it shows that it is notpossible that a certain property exists). But we would like to havegeneral criteria to decide when a predicate stands for a property andwhen it does not. Moreover, one may wonder what gives these predicatesany significance at all if they do not stand for properties. There arethen motivations for building type-free property theories in which allpredicates stand for properties. We can distinguish two main strandsof them: those that weaken CL and those that circumscribe\((\lambda\)-conv) (some of the proposals to be mentioned below areformulated in relation to set theory, but can be easily translatedinto proposals for property theory).

An early example of the former approach was offered in a 1938 paper bythe Russian logician D. A. Bochvar (Bochvar 1938 [1981]), where theprinciple of excluded middle is sacrificed as a consequence of theadoption of what is now known as Kleene’s weak three-valuedscheme. An interesting recent attempt based on giving up excludedmiddle is Field 2004. A rather radical alternative proposal is toembrace a paraconsistent logic and give up the principle ofnon-contradiction (Priest 1987). A different way of giving up CL is byquestioning its structural rules and turn to a substructural logic, asin Mares & Paoli (2014). The problem with all these approaches iswhether their underlying logic is strong enough for all the intendedapplications of property theory, in particular to natural languagesemantics 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 suchas ZFC, minus extensionality, as if they were about properties ratherthan sets (Schock 1969; Bealer 1982; Jubien 1989). The problem withthis is that these axioms, understood as talking about sets, can bemotivated by the iterative conception of sets, but they seem ratherad hoc when understood as talking about properties(Cocchiarella 1985). An alternative can be found in Cocchiarella1986a, where \((\lambda\)-conv) is circumscribed by adapting toproperties the notion of stratification used by Quine for sets. Thisapproach is however subject to a version of Russell’s paradoxderivable from contingent but intuitively possible facts (Orilia 1996)and to a paradox of hyperintensionality (Bozon 2004) (see Landini 2009and 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’sand Belnap’s theory of circulardefinitions.

As noted in§2.4, Russell’s ramified type theory involves a hierarchy of orderswithin types. Plate (forthcoming) proposes to keep the orders withoutthe types. This results in a formal approach to properties verydifferent from the ones considered above. It involves fundamentalproperties, which, like particulars, are of the lowest order, and thennon-fundamental properties of higher orders.

Independently of the paradoxes (Bealer & Mönnich 1989: 198ff.), there is the issue of providing identity conditions forproperties, specifying when it is the case that two properties areidentical. If one thinks of properties as meanings of natural languagepredicates and tries to account for intensional contexts, one will beinclined to assume rather fine-grained identity conditions, possiblyeven allowing that \([\lambda x~(R(x) \amp S(x))]\) and \([\lambdax~(S(x) \amp R(x))]\) are distinct (see Fox & Lappin 2015 for anapproach based on operational distinctions among computablefunctions). On the other hand, if one thinks of properties as causallyoperative entities in the physical world, one will want to providerather coarse-grained identity conditions. For instance, one mightrequire that \([\lambda x~ A]\) and \([\lambda x~ B]\) are the sameproperty iff it is physically necessary that \(\forall x(A\leftrightarrow B)\). Bealer (1982) tries to combine the twoapproaches (see also Bealer & Mönnich 1989)^[15].

6.2 Semantics and Logical Form

The formal study of natural language semantics started with Montagueand gave rise to a flourishing field of inquiry (see entry onMontague semantics). The basic idea in this field is to associate to natural languagesentences wffs of a formal language, in order to represent sentencemeanings in a logically perspicuous manner. The association reflectsthe compositionality of meanings: different syntactic subcomponents ofsentences correspond systematically to syntactic subcomponents of thewffs; subcomponents of wffs thus represent the meanings of thesubcomponents of the sentences. The formal language eschewsambiguities and has its own formal semantics, which grants thatformulas have logical properties and relations, such as logical truthand entailments, so that in particular certain sequences of formulascount as logically valid arguments. The ambiguities we normally findin natural language sentences and the entailment relations that linkthem are captured by associating ambiguous sentences to differentunambiguous wffs, in such a way that when a natural language argumentis felt to be valid there is a corresponding sequence of wffs thatcount as a logically valid argument. In order to achieve all this,Montague appealed to a higher-order logic. To see why this wasnecessary, 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 syntacticsimilarity of the three sentences (they all have the samesubject-predicate form), and the validity of the argument, Montagueassociates (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). Theselambda-abstracts occur in predicate position, as predicates ofpredicates, so that (1a)–(3a) can be read, respectively, as:every Greek is instantiated bybeing mortal;thepresident of Greece is instantiated bybeing Greek;the president of Greece is instantiated bybeingmortal. Given lambda-conversion plus quantifier and propositionallogic, the argument is valid, as desired. It should be noted thatlambda abstracts such as these can be taken to stand for peculiarproperties of properties, classifiable asdenoting concepts(after Russell 1903; see Cocchiarella 1989). One may then say thenthis approach to semantics makes a case for the postulation ofdenoting concepts, in addition to the more obvious and general factthat 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 wesaw in§3.1, Montague took them to be intensions as set-theoreticallycharacterized in terms of possible worlds. Moreover, he took them tobe typed, since, to avoid logical paradoxes, he relied on type theory.After Montague, these two assumptions have been typically taken forgranted in natural language semantics, though with attempts to recoverhyperintensionality somehow (Cresswell 1985) in order to capture thesemantics of propositional attitudes verbs such as“believe”, affected by the phenomena regarding mentalcontent hinted at in§3.1. However, the development of type-free property theories has suggestedthe radically different road of relying on them to provide logicalforms for natural language semantics (Chierchia 1985; Chierchia &Turner 1988; Orilia 2000b; Orilia & Landini 2019). This allows oneto capture straightforwardly natural language inferences that appearto presuppose type-freedom, as they feature quantifiers that bindsimultaneously subject and predicate positions (recall the example of§1.2). Moreover, by endowing the selected type-free property theory withfine-grained identity conditions, one also accounts for propositionalattitude verbs (Bealer 1989). Thus, we may say that this line makes acase for properties understood as untyped and highly fine-grained.

6.3 Foundations of Mathematics

Since the systematization in the first half of last century, whichgave rise to paradox-free axiomatizations ofset theory such as ZFC, sets are typically taken for granted in the foundationsof mathematics and it is well known that they can do all the worksthat numbers can do. This has led to the proposal of identifyingnumbers with sets. Russell’s type theory was an alternative thatrather relied on properties (viewed as propositional functions), inbacking up a logicist reduction of mathematics to logic. In essence,the idea was that properties can do all the work that sets aresupposed to do, thus making the latter dispensable. Hence, Russellspoke of his approach as a “no-classes” theory of classes(see Landini 2011: 115, and §2.4 of the entry onRussell’s logical atomism; cf. Bealer 1982: 111–119, and Jubien 1989 for followers of thisline). Following this line, numbers are then seen as properties ratherthan sets.

The Russellian approach did not encounter among mathematicians asuccess comparable to that of set theory. Nevertheless, from anontological point of view, it appears to be more economical in itsrelying on properties, to the extent that the latter are needed anyway for all sorts of explanatory jobs, reviewed above, which sets,qua extensional entities, can hardly perform. As we haveseen, type theory is problematic. However, type-free property theoriescan come to the rescue by replacing typed properties with untyped onesin the foundations of mathematics. And in fact the advocates of suchtheories have often proposed to recover the logicist program, inparticular by identifying natural numbers with untyped properties ofproperties (Bealer 1982; Cocchiarella 1986b; Orilia 2000b). (See alsoentry onlogicism and neologicism).

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Other Internet Resources

“Properties” by Sophie Allen,Internet Encyclopedia of Philosophy.
“The World of Universals”, Bertrand Russell,Problems of Philosophy, Chapter 9.
“Universals”, by Mary McLeod and Eric Rubenstein,Internet Encyclopedia ofPhilosophy.
“The Theory of Abstract Objects”, by Edward N. Zalta, Metaphysics Research Lab, StanfordUniversity.

Acknowledgments

This entry was first authored by Chris Swoyer. In 2011 and 2016Francesco Orilia proceeded to major updates by still relying on theoriginal text by Swoyer, who accordingly remained a co-author. In the2020 update, we performed a major restructuring and rewriting in orderto cover new territory while aiming at the same time at a more conciseentry. Hence, we did not use the original text and Swoyer is no longera co-author. However, we are indebted to him for many ideas and choiceof topics, and we gratefully acknowledge this. We also wish to thankAnna Marmodoro, Jessica Wilson and an anonymous referee for theirdetailed and helpful comments on the 2020 update. Help, advice anduseful comments on previous updates were provided by Donald Baxter,Christopher von Bülow, Laura Celani, Michele Paolini Paoletti,Gideon Rosen, Edward Zalta.

The 2020 update was supported by the Italian Ministry of Education,University and Research through the PRIN 2017 program “TheManifest Image and the Scientific Image”, prot.2017ZNWW7F_004.

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