[Editor’s Note: The following new entry by Sam Cowling and Daniel Giberman replaces theformer entryon this topic by the previous author.]

Nominalism is an exclusionary thesis in ontology. It asserts thatthere are no entities of certain sorts. Precisely which entities itexcludes depends on the relevant variety of nominalism, but nominalisttheses typically deny the existence of universals or abstractentities. For those who accept nominalism, a central challenge inmetaphysics is to make sense of phenomena that anti-nominalisttheories explain via universals or abstract entities. For instance,anti-nominalists often rely upon universals to explain similarity,propositions to explain linguistic meaning, and numbers to explainmathematical truth. Nominalists who reject the existence of suchentities must therefore find a credible way to explain how things canbe similar without sharing universals, have linguistic meaning withoutexpressing propositions, or be truths of mathematics without theexistence of any mathematical entities.

Since nominalists seek to provide metaphysical theories withoutpositing universals or abstract entities, the resulting nominalisttheories often require distinctive resources and face significantcompetition in the form of rival anti-nominalist theories. It is alsocontroversial whether the nominalist prohibitions against universalsor abstract entities are justifiable. This entry surveys arguments fornominalism, opposing anti-nominalist arguments for the existence ofuniversals or abstract entities, and a variety of nominalist theoriesthat eschew commitment to such entities.

Before proceeding, a note on terminology and scope is needed.Throughout this entry, we use “anti-nominalism” and itscognates to denote theories that accept the existence of universals orabstract entities. The term “anti-nominalism” avoids thehistorical connotations of the commonly used term“platonism.” The history of nominalism in metaphysics isextensive and resists concise summary except to say that the realityof universals, numbers, propositions, and related entities has longbeen controversial. This entry sets aside this lengthy history tofocus on the contemporary metaphysical debates regarding nominalism.(See the entry on the medieval problem of universals.) Finally,“nominalism aboutFs” is typically understood asthe thesis thatFs do not exist, and to deny the existence ofa category of entities is to adopt a metaphysical position regardingsuch entities. ‘Nominalism in metaphysics’ might thereforeseem a redundant title for this entry since any version of nominalismmust be within the realm of metaphysics. Our focus in what follows is,however, with nominalism as a view that informs metaphysical debatesregarding universals as well as putative abstracta such as properties,propositions, and types. While we will touch upon certain argumentsfrom the philosophy of mathematics that have been influential inshaping the metaphysical debate over nominalism, and while there is noclear-cut divide between metaphysics and the philosophy ofmathematics, our focus is on the status of nominalism in the formerrather than the latter.

• What is Nominalism in Metaphysics?
  • 1.1 Nominalism as the Rejection of Universals
  • 1.2 Nominalism as the Rejection of Abstract Entities
• 2. Arguments for Nominalism in Metaphysics
  • 2.1 Causal Arguments for Nominalism
  • 2.2 Other Arguments for Nominalism
• 3. Arguments against Nominalism in Metaphysics
• 4. Nominalist Strategies
  • 4.1 General Nominalist Strategies
  • 4.2 Nominalist Accounts of Qualitative Natures
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

What is Nominalism in Metaphysics?

Contemporary debates regarding nominalism are troubled byterminological obstacles. Typically, nominalism is understood as oneof two theses: (1)nominalism about universals, which deniesthat such entities exist and holds instead that all entities areparticulars or individuals; (2)nominalism about abstractentities, which denies that such entities exist and holds insteadthat there are only concrete entities. On some views, (2) entails (1)since universals are properly categorized as abstract rather thanconcrete. For example, some views take universals to be eternal,non-spatiotemporal, necessarily existing entities and thereforecategorize universals as abstract rather than concrete (see Bealer1993 and Jubien 2001). Others reject this view about the location andnecessity of universals and instead hold universals to be concreteentities. In light of the resulting complexity, we will, at key pointsin what follows, separately examine nominalism about universals andnominalism about abstract entities since the debate over thesenominalist theses can differ significantly. For instance, a familiarmotivation for nominalism about universals is a metaphysical concernabout the possibility of entities being wholly shareable by distinctmaterial objects. In contrast, objections against abstract entitieslike sets or propositions are unlikely to concern shareability.Instead, arguments for nominalism about abstract entities are oftenmotivated by the fact that such entities are alleged to exist outsidespace or time and are therefore mysterious or unknowable. Despitethese differences, nominalist opposition to universals or abstractentities often significantly overlaps.

Since the rejection of abstract entities and the rejection ofuniversals are related but separable concerns, we begin by discussingeach of these two strands of nominalism. In each case, we assume thatnominalism is a thesis regarding theexistence of entities.An alternative approach to understanding nominalism holds it toconcernfundamentality rather than existence. On thisalternative way of understanding nominalism, it is the denial thatuniversals or abstract entities are fundamental entities. Conversely,anti-nominalism, when so understood, is the thesis that such entitiesare fundamental. Although matters of fundamentality and metaphysicaldependence are highly relevant to the debate over nominalism, weassume here that nominalism is specifically a thesis regardingexistence rather than fundamentality. (On fundamentality andontological disagreement, see Cameron 2008, Schaffer 2009, Imaguire2018, Schulte 2019, and Declos 2021.)

Alongside the above nominalist theses regarding universals andabstract entities, an influential thesis labelled as“nominalism” is defended by Nelson Goodman and draws uponpreceding work in Leonard and Goodman 1940. Goodman’s version ofnominalism asserts that all entities are “individuals” inthe specific sense of being mereological atoms or constructed viamereology. As Goodman (1956: 25) notes, this version of nominalism ispotentially compatible with the existence of certain sets or classes,but rejects any ontology of sets or classes which admits “thecomposition of more than one entity out of exactly the same entitiesby membership.” For this reason, Goodman’s nominalism isincompatible with standard versions of set theory which accept theconstruction or generation of sets in violation of classicalextensional mereology (e.g., a hierarchy of pure sets constructed froma single empty or null set). (On this version of nominalism, seeCohnitz and Rossberg 2006, Oliver 1993, Eberle 1970, and Dummett1956.) Despite receiving less attention than nominalism regardinguniversals or abstract entities, Goodman’s nominalism iscomprehensive in its development and historically influential,receiving attention in Lewis 1991 (38–41).

This entry largely leaves aside discussion of views that find faultwith both nominalism and anti-nominalism on metaontological grounds.For instance, we will not examine the relationship of nominalism withviews like those of Carnap (1950) that reject the standardinterpretation of ontological claims like “Numbers exist”or “There are no universals” assumed by most nominalistsand anti-nominalists. For metaontological challenges to the standardconception of the debate over nominalism, see Balaguer 1998, Chalmers2009, and Thomasson 2005.

1.1 Nominalism as the Rejection of Universals

Nominalism about universals is the thesis that there are no universals(see Armstrong 1978). While this sounds simple enough, the notion of auniversal is complicated. To begin, we can introduce a separate notionof aqualitative nature, where such natures, if they exist,are responsible for the world’s qualitative variation andsameness. Consider, for example, a red apple and a green avocado. Ifthere are qualitative natures, then the apple and the avocado havedistinct color-relevant qualitative natures, one relevant to rednessand one relevant to greenness. They also enjoy distincttexture-relevant qualitative natures, one relevant to smoothness andthe other to bumpiness. Further examples are easy to generate, and, ifthere are qualitative natures, then there is potentially one for everypossible qualitative classification.

Suppose that there are such entities as qualitative natures. Then theapple and the avocado each has a color-relevant one. Moreover, theapple’s color-relevant qualitative nature isnumerically distinct from the avocado’s. Now consider asecond apple of exactly the same shade of red as the first apple. Bystipulation, the first apple’s color-relevant qualitative natureisqualitatively identical to the second apple’scolor-relevant qualitative nature, insofar as both apples resembleexactly in color. But is the first apple’s color-relevantqualitative nature alsonumerically identical to that of thesecond apple? Or are qualitative natures the kind of entity thatcannot be in more than one spatial location or “inhere” inmore than one object at the same time?

This final question is helpful for understanding the debate overuniversals: can a single qualitative nature be in more than onelocation (or inhere in more than one object or enjoy more than oneinstance) at the same time? If one answers affirmatively, then onelikely accepts universals. If one answers negatively, then theresulting view will likely fall into one of two further camps. First,if one accepts the presupposition that thereare qualitativenatures but denies that they can be in more than one location orobject or instance at a time, one likely rejects universals whileaccepting what are usually calledtropes. A trope is aparticular (i.e., non-universal) qualitative nature. That is, thewhole qualitative nature is not in multiple places or shared bymultiple disjoint individuals at the same time. For most theorists,tropes are specific to their location, or even to their bearer(Williams 1953, Campbell 1990, Maurin 2002, and Fisher 2020; see theentry on tropes). Others allow wholly multi-located tropes (Ehring2011), or shared tropes, so long as one of the sharing bearers is apart of the other, as when an apple and its skin share a redness trope(Giberman 2022b). Alternatively, if one denies the presupposition thatthere are any qualitative natures, one likely rejects both universalsand tropes and instead accepts what we will callstrictnominalism about properties. Strict nominalists deny that anykind of special metaphysical posits are needed to explain theworld’s qualitative variety. Instead, strict nominalism holdsthat the phenomenon of qualitative variety is reducible to facts aboutentities other than qualitative natures, for example, to materialobjects, sets of material objects, mereological sums of materialobjects, or linguistic-cognitive facts (see Quine 1948, Sellars 1963,Lewis 1983, Effingham 2020).

When nominalism is understood as the rejection of universals, it isregularly assumed that trope theory (and, of course, strictnominalism) qualify as nominalist views. Since universals are whollyshareable entities (the whole universal may be shared by disjointbearers), the case for nominalism about universals is often premisedupon this controversial feature. Arguments against nominalism aboutuniversals frequently concern distinctive properties relevant to theirputative shareability that are ascribed to them by competing accountsof universals—e.g., their alleged lack of spatiotemporallocation or the alleged possibility of being wholly located inmultiple places at the same time. Nominalists about universals oftenseek to avoid these commitments when assigning entities the generaltheoretical role of “property.” Nominalism-friendlycandidates for that role include tropes; sets, pluralities, or fusionsof tropes; and sets, pluralities, or fusions of material objects. Suchentities are compatible with nominalism about universals, since theyavoid commitment to the existence of wholly multiply located ornon-spatial qualitative natures. They are, however, potentiallyincompatible with nominalism about abstract entities depending uponthe assumed understanding of “abstract entity.” Forexample, theories that accept nominalism regarding universals in partby identifying properties with sets of tropes risk violatingnominalism about abstract entities by virtue of accepting theexistence of sets, which are standardly categorized as abstractentities (see, e.g., Ehring 2011).

Despite the unwelcome complexity, understanding the debate overnominalism and universals requires carefully distinguishing universalsfrom non-universals (or ‘particulars’), withoutidentifying the concept of a universal with the concept of aqualitative nature. This leaves open the difficult question of whatfeatures distinguish universals from non-universals. Various proposalshave been advanced. Universals have been claimed to be distinctiveinsofar as they arepredicable (i.e., they can be true orfalse of something: Carmichael 2010),multiply located (i.e.,they can be wholly located in distinct regions of space at the sametime: Armstrong 1978, Lewis 1986: 83, Giberman 2022a),multiplyinstantiable (i.e., they can be had by distinct entities: Hoffmanand Rosenkrantz 2003),subject to the principle of the identity ofindiscernibles (Wisdom 1934, Williams 1986, Ehring 2004, 2011),being the fundamental explanans of qualitative similarity(Armstrong 1978) orserving as the semantic value of qualitativepredication (van Inwagen 2004). Despite these theoretical optionsfor distinguishing universals from particulars, there is no consensusas to which, if any, of the suggested distinguishing criteria ofuniversality listed above is the correct one. For instance, while someposit universals and assert that they are multiply located, rivaltheories of universals hold that they areante rem entitiesthat lack any spatiotemporal location whatsoever. This has led somephilosophers to doubt that there is any such criterion (Ramsey 1925,MacBride 2005).

In summary, precisely characterizing nominalism about universals iscontroversial, since the precise characterization of universals iscontroversial. Clarity can be gained by focusing on the concept ofqualitative nature, when understood as a concept neutral betweentheories that accept universals and theories that accept tropes. As wehave also seen, nominalism regarding universals may sometimes be usedin the “strict” sense above, according to which there areno such entities as qualitative natures regardless of whether they areuniversals, tropes, or various constructions from such entities. Thereare many interesting versions of strict nominalism in the literature,including some that have been newly developed or recently defended.Section 4 surveys several nominalist strategies for addressing themetaphysics of qualitative natures.

1.2 Nominalism as the Rejection of Abstract Entities

In their influential discussion of nominalism, Quine and Goodman(1947: 105) begin as follows: “We do not believe in abstractentities. No one supposes that abstract entities—classes,relations, properties, etc.—exist in space-time; but we meanmore than this. We renounce them altogether.” While therespective views of Quine and Goodman would subsequently change, thisearly discussion helped popularize a conception of nominalism that isnot tied to the rejection of universals but instead to the moregeneral rejection ofabstract entities where this category isusually taken to include numbers, classes, properties, propositions,and several other kinds of entities such as linguistic types,biological kinds, and fictional characters. Although Quine and Goodmanexplicitly refuse to provide a positive argument against abstractentities in their paper, ontological suspicions regarding suchentities are regularly encountered: abstract entities seem markedlyunlike ordinary concrete entities, and, in the case of classes ornumbers, a commitment to their existence is a commitment to infinitelymany imperceptible entities outside space and perhaps outside time.Overviews of the case for nominalism and metaphysical worries aboutabstract entities can be found in Burgess and Rosen 1997 (3–66),Szabó 2003, Cowling 2017, and Liggins 2023. Also see the entryon abstract objects.

The abstract-concrete distinction that separates entities andstructures the debate over this version of nominalism is often held tobe an exclusive and exhaustive distinction that partitions all ofreality (see Hale 1988 and Grossman 1992: 7). Mathematical entitieslike the number seven or the Pythagorean theorem serve as paradigmexamples of abstract entities and ordinary objects like tables and tapshoes are paradigm examples of concrete entities. In turn, entitiesare often assumed to have their abstract or concrete statuspermanently and necessarily (see Melia 2008a: 146, but cf. Williamson2010 and Linsky and Zalta 1996). Although the concept of an abstract(or concrete) entity can be adopted as a primitive notion,philosophers have more usually sought to provide an analysis of whatit is for an entity to be abstract (or concrete). And, given theassumed exclusiveness and exhaustiveness of the distinction, ananalysis of either the conceptabstract entity orconcrete entity would suffice for defining the complementaryconcept. (On alleged features of the abstract-concrete distinction,see Cowling 2017: 70–74.)

Lewis (1986: 81–86) summarizes several approaches forunderstanding the abstract-concrete distinction and applies each tohis modal realist theory of possible worlds. His influentialdiscussion makes evident that there are a considerable variety ofnon-equivalent conceptions of the abstract-concrete distinction. Ofthese, two have been especially influential and warrant briefsummary.

According to thespatiotemporal approach, the distinctionbetween abstract and concrete entities should be understood in termsof spatial or spatiotemporal location: abstract entities lacklocations, while concrete entities have them. With regard to ourparadigm examples, this approach renders a plausible verdict: numbersdo not seem to be anywhere (even if inscriptions that name them are),and pieces of furniture surely seem to be located in spacetime. As wemove beyond paradigm cases, tough questions multiply quickly. Arespacetime regions located at themselves, or are they abstract? Is theset that has Obama as a member, {Obama}, located where Obama is ornowhere at all? What of views on which universals are wholly locatedwhere they are instantiated? And, when we turn to more controversialcases, it is far from clear that this approach offers any usefulguidance. For instance, while Hamlet andHamlet do not seemto have spatial locations, they do seem to have come into existencearound 1600. Might they have temporal but not spatial locations? Andwhat would that fact mean for their status as abstract or concreteentities? Additional challenges for this approach revolve around thecase of souls or deities that are purported to exist outside space ortime. On most views, such entities are not plausibly included in thecategory of abstract entity, especially given relationships they arethought to bear to concrete reality. (On spatial and spatiotemporalapproaches, see Hale 1988, Lowe 1995, and Hoffman and Rosenkrantz2003.)

According to a competingcausal approach, the distinctionbetween abstract and concrete entities should be understood in termsof causal interaction: abstract entities do not causally interact withanything, but concrete entities do (or at least can). Tables andchairs are created, destroyed, reflect photons, and stub toes; numbersdo none of these things. Like the spatiotemporal approach, the causalapproach faces difficult questions when we seek answers regarding theabstract/concrete status of non-paradigm cases. If musical works andworks of fiction are created by humans, then how could they beabstract entities given this causal approach? If properties areabstract entities, then how can properties likebeing fragileoccupy causal roles in our theories? Could there be what Forrest(1982) calls anepiphenomenalon, a concrete entity whichessentially stands in no causal relations whatsoever? Some of thesequestions clearly require us to provide a fully-fledged theory ofcausal interaction, but others seem to suggest that theabstract-concrete division is not ultimately a causal one. (See entryon objects.)

Alternative approaches for analyzing the abstract-concrete distinctiondraw upon the above features—e.g., by taking abstract entitiesto beessentially non-spatiotemporal—or requirealtogether different resources—e.g., by holding abstractentities to lack certain kinds of qualitative or intrinsic properties(see Cowling 2017: 74–92, and Hoffman and Rosenkrantz 2003:46–52). There remains no uncontroversial way to draw theabstract-concrete distinction, but the consequences of nominalismregarding abstract entities depend directly upon the correct accountof the distinction. For example, if abstract entities are held to bedistinctively non-spatiotemporal, nominalism proves equivalent to thethesis that reality is entirely spatiotemporal.

As Quine and Goodman’s remark above makes clear, the truth ofnominalism about abstract entities promises to have a significantimpact on a wide range of metaphysical projects. For instance, ifthere are no such things as linguistic types and there are onlyspecific concrete inscriptions, how can we provide general theories ofsemantics or syntax? After all, our talk about the word“moose” and its meaning appear to be about the type“moose”, not any specific token. This worry points towarddifficult questions about what exactly types are and whether they areabstract rather than concrete. (On types, see Wetzel 2009, Dodd 2000,and entry ontypes and tokens.) If nominalism about abstract entities does require that we do withouttypes, how can we truthfully speak of the common housecat, the NanaimoBar, or the foxtrot? Similar issues have been raised regarding a broadvariety of entities, each of which have been alleged to be abstractrather than concrete: numbers, sets, functions, directions,propositions, meanings, properties, kinds, possibilities, possibleworlds, fictional works, fictional characters, musical works, events,states of affairs, and many more besides.

If the just-listed entities are indeed abstract, then nominalism aboutabstract entities would seem to be of profound philosophicalsignificance. Aesthetic theories cannot truly ascribe properties tofictional or musical works. Ethical theories cannot evaluate the moralproperties of events, states of affairs, or types of events.Epistemological theories cannot correctly describe our justificationfor believing propositions or accurately enumerate possibilitiesinvolving skeptical scenarios. And so on across philosophy. To besure, some philosophers are content to assume anti-nominalism andother philosophers hold the fate of their preferred theories to besomehow independent of the debate over abstract entities. But mostnominalists take the nonexistence of abstract entities to havefar-reaching implications for the kinds of theories we can reasonablyaccept. Precisely which theories are ruled out by nominalism will, asnoted above, depend upon the correct account of the abstract-concretedistinction.

Since the abstract-concrete distinction is controversial, nominalistswho rely upon it to formulate their metaphysical theories face animportant challenge. They must provide a principled defense fordrawing the abstract-concrete distinction in the first place. This isbecause it is open to metaphysicians to reject the distinction as amandatory commitment and deny that it usefully or accuratelycategorizes entities. For example, in evaluating the view thatordinary objects are set-theoretic constructions, Sider (2013: 287)asserts that the abstract-concrete distinction is not a mandatorytheoretical commitment but is instead eliminable if it proves to betheoretically useless or unhelpfully obscure. If our balance ofevidence suggests we are best served to dispense with thisdistinction, nominalists would seem to be obliged to abandon theircentral thesis or instead to reformulate it, perhaps as a series ofspecific prohibitions against positing numbers, properties,propositions, and so on.

2. Arguments for Nominalism in Metaphysics

Why accept nominalism and thereby reject the existence of universalsor abstract entities? In this section, we consider the case for eachversion of nominalism, beginning with the case of nominalism aboutabstract entities. According to Goodman and Quine (1943: 105), theircommitment to this form of nominalism “is based on aphilosophical intuition that cannot be justified by appeal to anythingmore ultimate.” Few philosophers have been content to acceptthis foundationalist rationale for nominalism about abstract entities.(Quine and Goodman subsequently revised their increasingly divergentviews on nominalism. See Goodman 1956 and Quine 2008.) In place of adogmatic insistence on the truth of nominalism, a range of argumentshave been mounted that either identify significant problems withanti-nominalism or purport to show that nominalist alternatives areultimately preferable.

2.1 Causal Arguments for Nominalism

Perhaps the most influential arguments in defense of nominalism arethose premised upon the allegedly non-causal nature of paradigmaticabstract entities like numbers and sets. Thelocus classicusis Benacerraf 1973, which poses an epistemic challenge foranti-nominalist theories: to account for justified beliefs aboutmathematical entities in the absence of causal interaction betweenbelievers and mathematical entities. This challenge stems from theapparently non-causal nature of abstract entities and an allegedlycausal dimension of epistemic justification. As Benacerraf remarks,“a typical ‘standard’ account (at least in the caseof number theory or set theory) will depict truth conditions in termsof conditions on objects whose nature, as normally conceived, placesthem beyond the reach of the better understood means of humancognition (e.g., sense perception and the like)” (p. 668).Benacerraf’s initial presentation of the epistemic argumentremains influential; however, Field (1980, 1989) offers an alternativeformulation and defense of the causal-epistemic argument in support ofnominalism. Field’s version of the argument is premised, notupon a causal constraint on justification, but on an allegedlyinsurmountable problem for anti-nominalists: to explain plausibly thejustified mathematical beliefs we take mathematicians and others topossess in the absence of a causal relationship between epistemicagents and mathematical reality. (On causal-epistemic arguments, seeCheyne 1998, Nutting 2020, Liggins 2010, Sjölin Wirlingforthcoming.)

Causal-epistemic arguments hinge upon whether anti-nominalists canaccount for our knowledge of abstract entities if such entities areindeed non-causal and therefore not epistemically accessible throughordinary means. Subsequent discussions of anti-nominalist epistemologyhave sought to provide a credible account of our awareness of abstractentities and our justified beliefs regarding them (see Chudnoff 2012,Bengson 2015, Lewis 1986: 108–115, and Clarke-Doane 2017). Assome have noted, a parallel issue arises if we assume causalinteraction is necessary forsemantic relations likereference between speakers and abstract entities like numbers. Thisparallel causal-semantic version of the above argument againstabstract entities is more usually mentioned than defended, however(see Jubien 1977, Lear 1977, and Davies 2019).

A different causation-based argument for nominalism relies upon ametaphysical thesis, sometimes called the Eleatic Principle orAlexander’s Dictum, according to which entities must stand incausal relations or be causally active in order to exist (see Oddie1982 and Colyvan 1998). Granted this controversial thesis as well asthe essentially non-causal nature of abstract entities, nominalismwould seem to follow. Versions of anti-nominalism that claim fictionalcharacters, musical works, types, and other apparently createdentities are abstract are significantly less likely to grant thatabstract entities are essentially non-causal, given the causalcharacter of creation. Anti-nominalist views about these kinds ofentities are diverse, however, with some asserting they are abstractyet, unlike mathematical entities, nevertheless able to stand incausal relations (see Caplan and Matheson 2006, Mag Uidhir 2013, andFriedell 2020).

2.2 Other Arguments for Nominalism

Other arguments against abstract entities seek to show that nominalisttheories are stronger, better justified, or otherwise more virtuousthan anti-nominalist competitors. These arguments are dependent uponthe specific content of nominalist and anti-nominalist rivals, but themost familiar arguments are premised upon ontologicalparsimony—roughly speaking, nominalists believe in fewerentities than anti-nominalists and hold that, other things beingequal, we ought to prefer simpler theories to less simplealternatives. This parsimony-driven defense of nominalism assumes thatparsimony is a theoretical virtue, which makes belief in certaintheories better justified (see Parsons 1976, Nolan 1999, and Bricker2020). For this line of argument to succeed, dispensing withontological commitment to abstract entities must not require excessivetheoretical vices like undue complexity in formulation, additionalprimitive notions, or other significant theoretical costs.Additionally, as Melia (2008b) notes, the option of dispensing withabstract entities is only one way to secure ontological simplicity.For example, an alternative would be to dispense with concreteentities and develop a Pythagorean metaphysics of exclusively abstractentities. This illustrates how the evaluation of theories occurs in acomplex and interconnected way where additional virtues likeconservativeness—roughly, conformity with extantcommitments—play an under-recognized role.

Another family of arguments sometimes held to favor nominalism aboutabstract entities stems from worries about indeterminacy ornon-uniqueness that result from positing abstract entities in ourmathematical, semantic, or metaphysical theories. As Benacerraf (1965)argues, some anti-nominalist accounts of mathematics face a seriouschallenge: how to choose from among incompatible yet equally plausiblecandidate reductions of arithmetical notions likenaturalnumber to set theory. For example, should we accept the accountthat identifies the number two with the set {{0}} in one sequence orwith {0,{0}} in a different yet equally viable sequence? Roughlyspeaking, anti-nominalists seem required to either privilege a uniqueset-theoretic reduction without principled reasons or instead acceptthat certain theoretical notions likenatural number aresurprisingly indeterminate in meaning. As subsequent commentators havenoted, a parallel non-uniqueness problem arises when providing atheory of propositions or properties in terms of set theory.Additional versions of the problem have been alleged to arise whenanti-nominalists accept ontologically rich conceptions of properties(see Moore 1999, Caplan and Tillman 2013, Melia 1992). In addition,Field (1980: ii) suggests that worries about the “desirabilityof eliminating certain sorts of ‘arbitrariness’ or‘conventional choice’” are quite general and serveas important motivations for nominalism. The non-uniqueness problemhas prompted the development of certain structuralist approaches tothe philosophy of mathematics. While some of these are nominalist incharacter, other anti-nominalist versions hold structures to beabstract entities (see Chihara 2004, Shapiro 1997, Resnik 1981,Hellman 1989, and the entry on structuralism in the philosophy ofmathematics).

Despite the fact that some philosophers hold our thought and talkabout possibilities to require a commitment to possibilities asabstract entities, modal considerations have also been used to supportthe case for nominalism. For example, Humeans about modality rejectnecessary connections between distinct entities, and, for mostanti-nominalists, commitment to properties, propositions, ormathematical entities specifically require such connections. (OnHumeanism, see Lewis 1986, Wilson 2015, Nolan 1996.) For instance, ona standard anti-nominalist view, the truth of certain claims aboutindividuals and properties is analyzed in terms of the instantiationof properties. For instance, if May is human, then, according to theproposed anti-nominalist analysis, May must instantiatebeinghuman. Such an analysis, in turn, entails additional claims aboutproperties—e.g., that, if May is human, then May instantiatesthe propertyhaving some property or other and thatbeinghuman instantiates the propertybeing a property.Anti-nominalist analyses of this sort often connect properties withpropositions. So, in our present example, if May is human, then thepropositionthat May is human must exist and instantiate thepropertybeing true. Necessary connections among propertiesand propositions are standard features of these kinds of metaphysicalanalyses and, for this reason, a commitment to Humeanism is in tensionwith those theories committed to abstract entities that stand in suchmetaphysically necessary relationships.

Modality-based arguments against anti-nominalist theories have alsobeen advanced with weaker assumptions and without a dependence uponHumeanism. For instance, Dorr (2008) notes that anti-nominalists areobliged to accept a vast number of “brute necessities” ifabstract entities are to play their required theoretical role. (UnlikeHumeanism, this stricture against brute necessities is potentiallyconsistent with necessary connections between entities.) A prohibitionagainst admitting brute necessities into our theories has thereforebeen cited as a reason to reject anti-nominalism. Brute necessitiesalso play a key role in an argument Lewis (1986: 174–191) offersagainst certain anti-nominalist theories of possibleworlds—views he describes as forms of “magicalersatzism.” Whether Lewis’s argument against magicalersatzism can be deployed against some or perhaps all other abstractentities remains unclear, but van Inwagen (1986) discusses theapplication of this modality-driven argument to the case of sets (seeNolan 2020 and Himelright 2023). An additional modal challenge foranti-nominalism arises in explaining the distinctive necessaryexistence of abstract entities–e.g., precisely why they arenecessary rather than contingent beings. See Rosen 2002 on whetheranti-nominalism is more plausibly viewed as a contingent or necessarythesis.

Immediately after their bold proclamation that the justification ofnominalism requires no further support, Quine and Goodman suggest thatthe case for nominalism can be “fortified” by otherconsiderations. Specifically, they say that “what seems to bethe most natural principle for abstracting classes or properties leadsto paradoxes” and “[e]scape from these paradoxes canapparently be effected only by recourse to alternative rules whoseartificiality and arbitrariness arouse suspicion that we are lost in aworld of make-believe” (1943: 105). This remark makes salientthe threat of paradox that looms in the development of formal theoriesregarding mathematical entities like classes as well as properties,propositions, and others (see entry onRussell’s Paradox). Nominalist alternatives to set theory premised upon mereology arenotable responses to such paradoxes (for discussion, see entry onStanisław Leśniewski). In addition to the worry that specific theories regarding abstractentities like classes or properties might succumb to paradox, there isthe further worry that the interaction of certain independentlyplausible anti-nominalist theories might lead to paradox—e.g.,Kaplan’s Paradox, which involves set theory and its interactionwith the metaphysics of propositions and possibilities (Kaplan 1995).While worries about paradox might militate against specific theoriesof sets, properties, propositions, and so on, there is littlecompelling evidence that it is the abstractness of the relevantentities that makes them endemically or unavoidably paradoxical. Noris the formal character of the relevant theories plausibly thought tobe the unique source of difficulty. For instance, F.H. Bradley’sinfluential regress argument threatens to show that certainconceptions of properties, facts, or states of affairs yield vicious,infinite explanatory regresses, but Bradley-style arguments have beendeployed against anti-nominalist theories that are largely informal intheir presentation and development (see Bradley 1893 and the entry onBradley’s regress). It seems, then, the allegedly“fortifying” considerations mentioned by Quine and Goodmando little to constitute a general case against the existence ofabstract entities.

3. Arguments against Nominalism in Metaphysics

Arguments for anti-nominalism purport to establish the existence ofabstract entities or universals. In doing so, they raise (at least)two kinds of questions. The first kind of question concernsontological commitments—roughly, whether numbers,universals, fictional works, and so on exist. The second kind ofquestion concernscategorealcommitments—roughly, whether or not the relevantentities are correctly categorized as abstract rather than concrete.We considered the latter issue in Section One and noted thecontroversy about how to distinguish between abstract and concreteentities. (A parallel issue was also seen to arise for nominalismregarding universals in distinguishing universals from particulars.)Since answers to these ontological and categoreal questions areseparable, nominalists can reject anti-nominalist alternatives ineither of two ways: they can deny theontological component,which asserts that entities of a certain sort exist, or they can denythecategoreal component, according to which the relevantentities are abstract or universal. This section sets aside potentialcategoreal disagreements and surveys arguments offered in defense ofthe existence of abstract entities and universals.

There are many ways to partition the field of anti-nominalistarguments. One way to do so begins by dividingsemanticanti-nominalist arguments, which proceed on the basis of facts aboutmeaning and reference, fromalethic anti-nominalismarguments, which are premised upon the truth of claims that apparentlyinvolve abstract entities. A direct semantic argument foranti-nominalism asserts that sentences like ‘Seven is an evennumber’ is meaningful and that terms like “the PythagoreanTheorem”, “Superman”, or “Piano Sonata inB Minor” refer to or express abstract entities and, giventhese semantic facts, the existence of abstract entities follows as aconsequence. Such arguments have the virtue of not presupposingspecific claims about, say, mathematics, but nevertheless depend uponthe relevant semantic theses about reference or meaning and theirconnection to ontology. More frequently, however, anti-nominalistsdevelop their case for abstract entities on alethic grounds byobserving the acceptability and apparent truth of the following kindsof claims:

(1)

Seven is a prime number.

(2)

The Pythagorean Theorem is more familiar than its converse.

(3)

Superman was created after Liszt’sPiano Sonata in BMinor.

Alethic arguments proceed from the truth of sentences like(1)–(3) plus an additional principle that links the truth ofthese claims to the existence of the abstract entities that are theirapparent subject matter. A variety of principles have been proposed toplay this linking role and thereby support the anti-nominalistconclusion. We will briefly note three broad approaches to providingsuch a principle, which we will group as Fregean, truthmaker-based,and Quinean.

The Fregean approach invokes a thesis about the linguistic category ofterms like “seven”, “The Pythagorean Theorem”,and “Superman” in sentences like (1)–(3) and notesthat in such cases these expressions function as singular terms. So,given the truth of such claims, the existence of the number seven, thePythagorean Theorem, and Superman is held to follow. This broadapproach, inspired by Frege (1884), takes questions about theexistence of entities to be properly answered by determining whetherthey are the referents of singular terms in true sentences. Discussingthe argument for the existence of numbers in Frege 1884, Hale (1996:438) summarizes this argument schema as follows: “statements ofthe kind in question… are to be regarded as involvingexpressions… functioning as singular terms; thus, since thefunction of a singular term is precisely to effect reference to anobject, the truth of those statements requires the existence of theobjects to which the expressions refer.” The existence of thenumber seven is therefore a consequence of the truth of (1) and thelinguistic category of the term “seven” as it occurswithin (1). Importantly, this approach holds that the notion of objectis conceptually connected with the semantics and syntax of certainexpressions and that truth is a guide to ontological fact. (On Fregeanarguments for abstract entities, see Wright 1983, Rosen 1993, andMacBride 2003.)

Truthmaker-based approaches discern the ontological consequences ofsentences like (1)–(3) by asking what entity (or entities) serveas their truthmaker(s). On this approach, the truthmaking relation isa distinctive one that holds between sentences or propositions, on theone hand, and entities like facts, states of affairs, or objects onthe other. (On truthmaker theory, see Armstrong 2004 and the entry ontruthmakers. On truthmaking and nominalism, see Armstrong 1997: 115, Melia 2005,and Asay 2020: 175–197.) The extraction of ontologicalcommitments via truthmaker theory crucially depends upon the detailsof the relevant theory of truthmakers and truthmaking rather than thesemantics or syntax of claims like (1)–(3). For instance, sometruthmaker theorists hold the sentence ‘Socrates is ahuman’ has as its truthmaker a fact or state of affairsinvolving Socrates and the propertybeing human. Contrastingtruthmaker approaches instead hold that neither facts, states ofaffairs, or properties are required by the truth of ‘Socrates ishuman’; it requires instead tropes or perhaps only the existenceof Socrates as he actually is as a truthmaker. Along with differinggeneral approaches to truthmaking, the precise truthmaker posited fora sentence is regularly a matter of controversy. The truthmakerapproach for defending anti-nominalism will hold that, withoutabstract entities like seven, the Pythagorean Theorem, and Superman,we cannot plausibly account for the truth of claims like(1)–(3). If, however, truthmaker theorists can supply an accountof truthmaking that identifies viable, nominalist-friendly truthmakersfor claims like (1)–(3), nominalists are in a position to resista truthmaker-based argument for anti-nominalism.

The Quinean approach supplements the truth of claims (1)–(3)with a criterion of ontological commitment, which provides a methodfor determining what entities must exist if such claims are true.Roughly speaking, the Quinean criterion takes the ontologicalcommitments of a sentence to be the values of the variables that fallwithin the scope of the quantifiers of the sentence when translatedinto first-order logic as part of a theory. Where truthmaker-basedarguments depend upon a background theory of truthmaking, arguments ofthis sort hinge upon a distinctive role for quantifiers and variablesas means of discerning the commitments of theories (see Quine 1948 andthe relevant sections of the entry on ontological commitment). Forinstance, the sentence “Socrates is human” would betranslated as “∃x Socratizes(x) & Is-Human(x)”and thus any theory in which “Socrates is human” occurs isontologically committed to something that is Socrates by virtue of thevariable bound by the existential quantifier ranging over a domainthat includes such an entity. The Quinean criterion for ontologicalcommitment provides a highly specific answer to questions regardingthe ontological commitments of sentences; however, the Quineancriterion leaves open which theories to accept as well as the properregimentation of theories and their constituent sentences. As we willsee below, some nominalist responses to Quinean arguments proceed bydenying that claims like (1)–(3) are properly regimented in waysthat require quantification over putatively abstract entities.

In developing arguments for anti-nominalism, Quine’s criterionof ontological commitment is widely adopted; however, Quine pairs hisdeployment of the criterion with a more general commitment to thethesis of ontological naturalism, according to which our best, maturescientific theories are the proper arbiters of what sentences we oughtto accept. In turn, these theories are our best guides to what exists.The conjunction of Quine’s criterion and ontological naturalismalong with the apparent indispensability of mathematical entities forour best physical theories is the basis of what is often-called theQuine-Putnam indispensability argument for mathematical realism. Thisargument has been the subject of intense interest in the philosophy ofmathematics (see Quine 1970, Putnam 1971, Colyvan 2001 and entry onIndispensability Arguments in the Philosophy of Mathematics). Many who reject ontological naturalism nevertheless rely uponQuine’s criterion in order to determine what the ontologicalcommitments of true sentences might be. Quine’s criterion istherefore a premise in various arguments for different kinds ofabstract entities. For instance, Quine’s criterion would seem tosuggest that, despite the fact that (3) is not part of any scientifictheory, its truth requires the existence of fictional characters andmusical works provided that one accepts that (3) ought to regimentedin a manner that requires quantification over entities that have theproperty ofbeing Superman orbeing Liszt’s PianoSonata in B Minor.

The arguments above proceed from the truth of claims that apparentlyexpress or refer to abstract entities. Other anti-nominalist argumentsfocus on the explanatory role of abstract entities or universals astheoretical posits.Explanatory or abductive arguments foranti-nominalism aim to show that theories committed to the existenceof abstract entities provide the best or perhaps the only viableexplanations of certain phenomena. Generally speaking, anti-nominalisttheories enjoy an advantage in simplicity of formulation even if theyrequire the existence of abstract entities. In contrast, nominalisttheories often seek to replace or translate claims like (1)–(3)into claims that make no reference to (and therefore require noquantification over) abstract entities. Focus on explanatoryconsiderations in ontology has brought questions about theoreticalvirtues and their correct application to the forefront of metaphysicaldebate, while raising broader concerns about the (dis)analogiesbetween metaphysical posits like properties or propositions andscientific posits like electrons or genes. (On theoretical virtues,see Schindler 2018 and Nolan 1999.)

Explanatory anti-nominalist arguments are wide-ranging. For example,arguments for the existence of propositions allege that theorieswithout such entities are incapable of suitably explaining semanticphenomena like meaning, linguistic practices like translation, orcognitive phenomena like belief and doubt (see Church 1951). Parallelarguments for anti-nominalist views about possible worlds hold that wecannot satisfactorily explain our modal reasoning or the content ofour counterfactual thought and talk without positing such entities.Additional arguments hold that our best theories of laws of naturerequire commitment to universals and relations among universals, whichsome hold to be abstract entities (see, e.g., Armstrong 1982 andTooley 1977). Other anti-nominalist arguments in the philosophy of artclaim that we cannot account for the phenomena of musical appreciationor authentic performance without commitment to the existence ofmusical works (see Dodd 2000). In advancing an argument forproperties, propositions, and other entities, the remarks of Church(1951: 104) are representative of explanatory anti-nominalist efforts:“To those who find forbidding the array of abstract entities andprinciples concerning them which is here proposed, I would say thatthe problems which give rise to [an anti-nominalist theory] aredifficult and a simpler theory is not known to be possible.”Although propositions, properties, and possible worlds are familiarposits of philosophical theories, the breadth and power of explanatoryanti-nominalist arguments is significantly increased if the pervasivereference to and theorizing about entities like linguistic types,biological kinds, and chemical properties likemalleabilityis best understood as abstract discourse. On such a view, ourcommitments to abstract entities are exceptionally broad in scope.(For a remarkable catalogue of the deployment of types in variouscontexts, see Wetzel 2009: 3–21.)

4. Nominalist Strategies

Those who accept nominalism about abstract entities and hold thatentities like propositions, fictional entities, and numbers would beproperly categorized as abstract are in the difficult situation ofhaving to do without them. Nominalists of this kind must thereforefind a way to make sense of the apparent meaningfulness and apparenttruth of our thought and talk as in claims like (1)–(3) above.This section provides an overview of what can be seen as the positivecomponent of nominalist metaphysical theories.

We divide our survey of nominalist strategies by first consideringgeneral strategies—roughly, strategies that areapplicable to a variety of domains that apparently concern abstractentities like mathematical entities, propositions, musical works,types, or other kinds. We then considerproperty-theoreticstrategies, which are specifically aimed at accounting forproperty-theoretic phenomena like similarity, predication, and othermatters linked to qualitative natures. Although some generalstrategies can be applied to the property-theoretic domain, nominalistapproaches within metaphysics have often focused specifically on thecase of properties. For this reason, a variety of distinctiveproposals regarding our thought and talk about properties have beendeveloped. The aim of the remaining two sections is therefore tointroduce a very broad range of nominalist theories withinmetaphysics.

4.1 General Nominalist Strategies

Suppose that we accept nominalism about abstract entities and alsohold that some category of entities,Fs, would be properlycategorized as abstract. It follows that, on our view, there are noFs. How, then, can we make sense of the apparentmeaningfulness, truth, and usefulness of our thought and talk aboutFs? As we noted in Section Three, claims such as(1)–(3) are often used to motivate a commitment to abstractentities. So, if nominalism about abstract entities is to prove anattractive metaphysical position, nominalists are obliged to explainhow these claims might be (or merely appear to be) meaningful, true,and useful, given the nonexistence of their ostensible subjectmatter.

General nominalist strategies seek to provide an account of what wewill call “abstract discourse”—roughly, theories orclaims such as (1)–(3) that apparently refer to and quantifyover entities like numbers, theorems, properties, fictionalcharacters, and so on. Given their commitment to abstract entities,anti-nominalists are well positioned to treat abstract discourse asmeaningful, intended literally, true, as well as ontologicallycommittal. In contrast, nominalists must reject some component of theanti-nominalist stance toward abstract discourse. In catalogingvarious general nominalist strategies, we will leave aside certainapproaches that eschew abstract entities, but involve even broaderdisagreements with anti-nominalists—e.g., idealism, solipsism,and certain forms of skepticism. (On anti-nominalism and idealism, seeHofweber 2016.)

Perhaps the most radical nominalist view of abstract discourse isnon-cognitivism, according to which claims like (1)–(3)are without ordinary semantic content. Such a view enjoys littleplausibility if we focus solely on the commonalities between abstractdiscourse and ordinary discourse. This is because (1)–(3) seemunexceptional when compared with parallel claims about concreteentities. There are, however, a range of instrumentalist viewsregarding our discourse about unobservable entities within scientifictheories that, if otherwise defensible, would provide a precedent foraccommodating abstract discourse. On these views, claims such as(1)–(3) have no descriptive semantic function and serve insteadas “instruments” for systematizing our descriptive claimsabout concrete entities. Instrumentalist views of this kind have beenspecifically developed with regard to mathematical discourse and theavoidance of ontological commitment to mathematical entities (seeBurgess 1983 and Leng 2010).

For nominalists who accept that abstract discourse has semanticcontent yet warrants a kind of non-literal construal, a differentstrategy is to deny that claims like (1)–(3) are typically usedto assert their apparent semantic content and instead communicatemetaphorical or figurative information. Such views acknowledge merelyapparent commitment to numbers or propositions in claims like(1)–(3) but recast them as “representational aids”rather than entities named by singular terms. The taxonomy ofstrategies that rely on this approach is controversial. Moreover,several views about the content of abstract discourse may turn out toundermine the possibility of literally denying the existence ofabstract entities, since attempts to assert the thesis of nominalismwould be surreptitiously non-literal. Figuralist and fictionalistapproaches of this sort, which seek to explain the representationalcontribution of abstract discourse as broadly metaphorical, remainactively debated. (For discussion, see Yablo 2002a, 2002b, 2005, andthe entries on fictionalism and fictionalism in the philosophy ofmathematics.)

For nominalists who grant that abstract discourse is meaningful andintended literally, a remaining alternative is to deny its truth. Thisstrategy sets nominalism about abstract entities in apparent conflictwith common sense, since nominalists must plausibly explain away therobust intuition that at least some claims like (1)–(3) aretrue.

An influential strategy for nominalists who take abstract discourse tobe meaningful and intended literally is to hold that our alethicintuitions about claims like (1)–(3) are best explained inparallel to our intuitions that Sherlock Holmes is a detective or thatGarfield is a cat. All such claims are falsesimpliciter yettrue according to familiar and contextually salient fictions. (See,e.g., Field 1980 on mathematical entities and Rosen 1990 on possibleworlds.) Although it is not true that Sherlock Holmes is a detective,the following claim is held to be true: according to the SherlockHolmes stories, Sherlock Holmes is a detective. Once made explicit,speakers are inclined to recognize the distinction between these kindsof claims and the corresponding difference in their truth values.Moreover, for nominalists who accept this sort of fictionalistapproach, the denial of claims like (1)–(3) can be paired withthe claim that there are contextually salient “fictions”like arithmetic, property theory, or realism about propositions, anyof which can be used to explain our apparent confidence in the truthof (1)–(3). Furthermore, the development and adoption of thesefictions is explained by their expedience or value in reasoning aboutconcrete entities. For instance, the development of a robust fictionregarding numbers or about a plurality of abstract possible worlds isof value precisely because it allows us to speak and reason in conciseor powerful ways about a world of exclusively concrete entities.

In developing general nominalist strategies, efforts to provide“paraphrases” of abstract discourse play a significant butcontroversial role. For example, some nominalists have proposed thatwe ought to reject the following sort of claim as false given itsontological commitment to the number two:

(4)

Two is the number of solar gas giants.

At the same time, the nominalist who pursues this paraphrase strategyholds the following sentence, (5), to be true and lacking theontological commitment of (4) to the abstract entity, the numbertwo:

(5)

There is a solar gas giantx and a solar gas gianty andx is distinct fromy and every solargas giant is either identical tox or identical toy.

The final component of this paraphrase strategy holds that ourontological commitments are determined by nominalist-friendlysentences like (5) that serve as “replacements” of claimslike (4). Importantly, if (4) and its replacement (5) are supposed tobe synonymous, it is unclear how only one could be true or why thenominalist is obliged to accept only the ontological commitments of(5) rather than (4). If, however, the proffered replacement for (4)need not be synonymous, the nominalist who pursues this strategy mustexplain precisely why it is a satisfactory replacement that obviatesthe ontological commitments of (4). For these and other reasons, therole of paraphrase in ontological debates, especially those regardingabstract entities, remains a controversial topic in nominalistmetaphysics (see Alston 1958, Keller 2015, von Solodkoff 2014, Melia2000).

Burgess and Rosen (1997, 2005) draw an influential distinction amongnominalist strategies which is useful for understanding the structureand aims of various approaches. They dividehermeneuticnominalists who hold that “nominalist disbelief in numbersand their ilk is in the fullest sense compatible with belief incurrent mathematics and science” (p. 7) fromrevolutionarynominalists who take as their goal “reconstruction orrevision: the production of novel mathematical and scientific theoriesto replace current theories” (p. 6). (On this distinction, seeHellman 2001.) This distinction draws out a key question for would-benominalists: how can we reconcile the nominalist rejection of abstractentities with the role abstract discourse plays in our best scientifictheories? Broadly speaking, Burgess and Rosen’s revolutionarynominalist approaches seek to produce “novel theories byassigning novel meanings to the words of current theories” inorder to “permit the nominalist to speak and write just likeeveryone else while doing mathematics or science” while, at thesame time, providing a nominalistic reinterpretation of such claims“while doing philosophy” (p. 6). In contrast, theirhermeneutic nominalist does not aim to produce anything novel by wayof reconstrual or reinterpretation. Instead, the hermeneuticnominalist claims that nominalist theories are ultimately “ananalysis of what really ‘deep down’ words of currenttheories have meant all along, despite appearances ‘on thesurface’.” Producing comprehensive and satisfactorynominalist paraphrases is therefore a way to demonstrate that theapparent commitment of mathematics and science to abstract entities ismerely apparent. Each stance takes the production ofnominalist theories that avoid ontological commitment to abstractentities as significant; however, they differ over what the aim ofthese “nominalistic construals” ought to be (Burgess 1998,Leng 2010, Colyvan 2006).

Nominalists who accept what is often called Meinongianism aim topreserve abstract discourse and its truth but reject certain thesesregarding language and ontology (see Routley 1980 [2018]: 14; andPriest 2005: vii, 134–155). For those who accept the Quineanview of ontological commitment, the domains of what there is and ofwhat exists are coextensive, so, of necessity, there are no objectsthat do not exist (see the entry onontological commitment). In opposition, Meinongians assert that there are objects that do notexist (see Meinong 1904 and Parsons 1980, and the entry onnonexistent objects). When implemented by nominalists, the Meinongian approach permits thepositing of nonexistent entities as referents of singular terms andthe values of variables and, in turn, upholds the truth of claims like(1)–(3) even while denying that abstract entities actuallyexist. The Meinongian tradition has received comparatively limiteddiscussion as a specific tool for would-be nominalists. This isperhaps because nonexistent objects are subject to some of the samemetaphysical objections as abstract entities.

It also remains contentious how to distinguish Meinongian approachesfrom other views that reject the Quinean criterion of ontologicalcommitment. (For discussion, see Lewis 1990.) For example, Azzouni(2004, 2010) defends a non-Meinongian version of nominalism premisedupon the rejection of Quinean views regarding ontological commitment.On the resulting view, only concrete objectsreally existdespite the apparent truth of claims within mathematics and fictionand the apparent reference to and quantification over numbers andfictional characters. In developing this view, Azzouni defends adistinction between the “quantifier commitments” and“existential commitments” of a theory. While claims like(1)–(3) are true and have numbers, theorems, and charactersamong their quantifier commitments, it is a further question whethersuch entities exist and what, if any, existential commitments andhence ontological consequences such claims have.

Related approaches posit distinct meanings of quantifiers and cognateexpressions, only some of which are privileged for the purposes ofontology. So, if our concerns are metaphysical in kind (e.g., when wedebate the existence of abstract entities), nominalists hold thatclaims like (1)–(3) prove false on this privilegedinterpretation. However, in ordinary contexts, claims like(1)–(3) are true and this is because of anon-ontologically-committing meaning associated with the relevantquantifiers and cognate expressions. Roughly speaking, when(1)–(3) are ordinarily used they incur no metaphysicalconsequences and are therefore consistent with the truth ofnominalism. But, if asserted in metaphysical contexts and thusexpressing their privileged, ontologically committal meanings, theyare incompatible with nominalism and entail the truth ofanti-nominalism (Hofweber 2005, Sider 2011, Dorr 2008, Korman2015).

Our final general nominalist strategy requires no distinctive semanticmachinery and seeks to uphold abstract discourse as meaningful, true,intended literally, and committed to no abstract entities. It is,however, the least specific in content and is closer to a promissorynote in theory development or a disposition towards certainmetaphysical options. According to what we will callconcretism, the nominalist simply holds that, on the finalanalysis, any of the putatively abstract entities relevant to claimslike (1)–(3) will turn out to be a concrete entity. Concretismobviously requires plausibly developing the relevant theories ofnumbers, properties, musical works, and so on. Generally speaking,however, the prospects for these views are likely to depend uponhaving a sufficiently expansive ontology of concrete entities or asufficiently rich ideology of theoretical concepts. For example, modalrealists like Lewis (1986) have sought to analyze discourse aboutpossibilities via concrete entities in the form of a vast ontology ofconcrete, merely possible worlds. Subsequent efforts have sought toshow that this ontology of concrete entities is sufficient for alsoproviding a nominalist theory of some mathematical entities in termsof sums and pluralities of concrete entities and, potentially in turn,providing an account of properties and propositions (see Nolan 2002:151–174). How concretism, when paired with realism about merelypossible entities, ultimately compares to anti-nominalism is an openquestion.

4.2 Nominalist Accounts of Qualitative Natures

The preceding section considered how nominalists about abstractentities might develop theories in the absence of ontologicalcommitment to numbers, propositions, fictionalia, and other putativeabstracta. As we have noted, nominalists about universals are morespecific in their exclusionary thesis: they reject universals, butthey may nevertheless accept that there are properties and, on someviews, the properties they posit the existence of are also abstractentities—e.g., tropes or sets of individuals. The debate betweendefenders of universals and competing theorists of qualitative naturesis wide-ranging, but in this section we examine those versions ofnominalism that reject qualitative natures altogether. As noted inSection One, we take such views to be versions of what we call“strict nominalism.”

Strict nominalists face an especially broad metaphysical challenge.The reason why is clear: what we will call “propertydiscourse” is immensely wide-ranging. For it is intuitive tosuppose that every object either qualitatively resembles or variesfrom every other. Correspondingly, everything seems to be a potentialsubject of descriptive classification. By contrast, it is certainlynot intuitive to hold that every object is, say,mathematical, non-actual, fictional, or musical. Thus, while noteverything is such that our discourse about it prima facie indicatesthat there are numbers or (mere) possibilities or creative works,plausiblyeverything is such that our discourse about itindicates that there are qualitative natures.

The key task for the strict nominalist, which will be the focus ofthis section, is to develop a strategy for explaining propertydiscourse without invoking qualitative natures. The issue at handconcerns not just first-order predicative discourse like “Thatapple is red”, but also discourse about similarity betweenobjects (“That apple and my house are both red”),explanatory discourse (“I bought the apples becausethey’re red”), and higher-order discourse about propertiesthemselves (“Red is a color”), which can get increasinglycomplicated when descriptive designation and gradient similarity areinvoked (“The color of Bob’s favorite fruit is more likethe color of my most expensive possession than it is like the color ofyours”). The challenge of accounting for such discourse has beencited by a variety of pro-qualitative-nature theorists as the basisfor assessing the prospects of strict nominalism (see, for example,Jackson 1977, Armstrong 1978, Campbell 1990, and Yi 2018).

Of course, the breadth of the challenge does not by itself tell uswhether it can be met, and strict nominalists have generated a richvariety of proposals for how to do so. Below we introduce severalspecies of strict nominalism about properties, each of which eschewcommitment to qualitative natures, and then briefly note thecomplexity involved in pairing them with the general nominaliststrategies. The objective will be to supply an introductory glimpse ofthe issues, rather than an exhaustive guide.

Strict nominalisms about properties includeaustere,class,predicate,concept,resemblance,metalinguistic,causal,mereological,reistic,second-order, andgrounding nominalism. The taxonomy of these positions iscomplicated, since some occupants of this theoretical space partiallyoverlap and the relevant literature’s terminology is notuniform. The present survey is organized around the guiding questionof whether the form of nominalism at hand supplies a stock of entitieseligible to populate the extension of predicates like“property”, “feature”, and“attribute” although none of these views acceptqualitative natures, understood as sui generis posits required toexplain variety and resemblance (e.g. universals, tropes, orconstructions therefrom).

We count five strict nominalisms as furnishing candidate members forthe extension of “property”, while still eschewingqualitative natures. These are class, mereological, reistic, causal,and grounding nominalisms, respectively.

Class nominalism holds that the metaphysical structure ofclass membership is necessary and sufficient to explain thequalitative status of concrete particular objects, like apples andfire engines. (We assume here that “classes” and“sets” can be used interchangeably.) It says thatproperties are nothing more than classes of their instances, where“instances” of properties are concrete particular objects,not tropes. For example, the class nominalist holds that there isredness, but that it just is the class of red objects, or—oncertain versions of the view—the class of metaphysicallypossible red objects (see Quinton 1957, Lewis 1983).

Other strict nominalisms focus not on classes of ordinary concreteparticular objects, but on individual non-ordinary objects. Accordingtomereological nominalism, properties are mereologicalfusions of their instances with “instances” understood asconcrete particular objects, rather than tropes. On such a view,redness is the fusion of all (or all metaphysically possible (red))stop signs, ripe strawberries, Santa suits, etc. (Zemach 1982,Effingham 2020). According toreisticnominalism,which is based on work by Franz Brentano, qualitative variation ismetaphysically governed by spatially overlapping concrete particularobjects of a much finer qualitatively classifiable grain than ordinaryobjects like apples (see Kriegel 2015 and the entry onreism). For example, the view distinguishes the-apple-qua-red fromthe-apple-qua-round, and holds that the existence of the former (butnot the latter) explains that the apple is red.

Causal nominalism is the thesis that the existence and natureof properties is determined by particular concrete objects’causal profiles (where causal profiles are not reified in any propertyrealist sense), as opposed to those causal profiles being somehowexplained by primitive qualitative natures (Whittle 2009; see alsoTugby 2013). Precisely what kind of entity this view takes as theextension of “property” is an interesting question. Whatis clear is that the identity of an arbitrary property is determinedby particular concrete objects’ causal profiles.

Grounding nominalism bases the nominalistic project onfundamentality rather than existence, contending that properties arederivative entities grounded in comparatively-more-fundamentalconcrete particular individuals (see Imaguire 2018 and Schulte 2019).Whether (and in what ways) the view thereby differs substantially fromother forms of strict nominalism is an interesting open question. Onone precisification, the relevant grounded entities just arequalitative natures, albeit non-fundamental ones, but otherconceptually possible versions identify derivative properties withconstructions from concrete particular objects.

Among those versions of strict nominalism that do not posit entitiesthat serve as the extension of “property” and similarpredicates are austere, predicate, concept, and metalinguisticnominalisms.

The key idea behindaustere nominalism—sometimes called‘ostrich’ nominalism (originally a pejorative jab byArmstrong 1978)—is that no underlying metaphysical structure isneeded in order to explain that individuals are qualitativelyclassifiable, for example, that a stop sign or an apple is red, orthat both are (equally) red (see Quine 1948, Devitt 1980, Van Cleve1994, Parsons 1999, Imaguire 2014). In the standard austere case, thisis in part because no metaphysical explanation whatsoever ofqualitative classification is thought to be needed.

Predicate andconcept nominalism assert that objectswarrant qualitative classification just in case—andbecause—certain predicates or concepts, respectively,apply to them, where the relevant application conditions do not invokeany mind-independent entities that might play the role of a“property” (Searle 1969). The chief difference betweenthese two versions of nominalism is that predicates are linguisticwhile concepts are mental. But both versions are typically taken totreat the metaphysics of properties as a mind-dependent issue, sincepredicates and concepts are both mind-dependent. (Note: Searle (1969)says that thereare properties; however, he treats this asequivalent to holding that predicates have meanings.)

Metalinguistic nominalism (here following the terminology ofLoux 2002) says that discourse seeming to refer to properties is bestconstrued metalinguistically, as being about language itself (Carnap1934, Sellars 1963). Roughly, an utterance of ‘the apple isred’ says not that the apple has some non-linguistic qualitativestatus, but rather that it (or perhaps the description ‘theapple’) is related in a certain way to the predicate‘red.’ One useful upshot is an interesting paraphrasescheme for higher-order talk of properties, whereby, for example,‘red is a color’ becomes a claim about ‘red’being a color predicate. Predicate nominalism and metalinguisticnominalism overlap in certain ways, and the former can potentially beconstrued as a species of the latter although versions of these views(e.g. Searle 1969 and Sellars 1963) differ in significant details.

Resemblance and second-order nominalism are more complicated withrespect to whether they provide entities as semantic values of“property.” Resemblance nominalism says that qualitativeclassification is governed by primitive resemblance facts (even ifthere is, strictly speaking, no resemblance relation), rather thanresemblance being explanatorily dependent on qualitative natures(Rodriguez-Pereyra 2002). Thus the view can be developed to allow forentities that fall under the extension of “property”,namely, classes of concrete particular objects that satisfy therelevant resemblance conditions. Alternatively, since the view makesno special theoretical appeal to such classes, it can also bedeveloped to eschew properties and “property” talkentirely.

Finally, second-order nominalism treats predication as its centralconcern in the metaphysics of properties and examines it through thelens of second-order quantification into predicate position (Jones2018, Trueman 2020). (Note: “is” takes its predicative usein locutions like “x is F” throughout thisparagraph.) Second-order nominalism says that if some first-orderindividualx is F (for some predicate “F”), thenthere is something Y, in the second-order sense of “thereis”, such that: (i)x is Y, and (ii) whenever any firstorder individualz is Y,z is F (Jones 2018). Theview is a version of strict nominalism insofar as it eschewsqualitative natures, but it may be considered a kind of“nominalist realism”, following Jones 2018, depending onhow one reads the existential commitments of second-order quantifiers.It may also count as allowing entities eligible to populate theextensions of some “property”-like higher-orderpredicates. It is a further question whether any such predicate may bemore (or less) metaphysically perspicuous than English predicates like“property” and “attribute.”

As the above survey indicates, the emphasis and details of generalnominalist strategies and property-specific versions of nominalism canvary substantially. Assessing the relationships between these kinds ofviews remains an area of significant further study for nominalists. Itis tempting, for instance, to view strict nominalist theories whichoffer extensions for “property” like classes ormereological sums as well-positioned to pursue the“replacement” of discourse apparently about qualitativenatures in ontologically leaner terms. Alternatively, more radicalversions of strict nominalism, according to which there are noentities within the extension of “property”, may be easierto reconcile with general nominalist strategies on which much of ourqualitative discourse is non-literal or figurative in character. Weshould be careful not to assume that these positions line up soconveniently, however. Quine, for example, was certainly not afictionalist, figuralist, or non-cognitivist about qualitativepredication, despite being the leading austere nominalist (Quine1980). Among other things, the prospects for any such views willdepend on the posited connections and affinities between thetheoretical roles claimed for properties and other putative abstractalike propositions. (On unified conceptions of abstracta, see Zalta1983 and Fine 1977.)

On the flipside, we should also exercise caution before simplyassuming that, say, class or mereological nominalists cannot be orought not to be, say, fictionalists or figuralists about propertydiscourse. While to our knowledge actual class and mereologicalnominalistsdo take talk of “properties”literally to be about classes or fusions of objects, one could developa version of those metaphysical theses that treated classes orfusions, respectively, rather like fictionalists might treat theconcrete materials constitutive of tokens of novels or films—thepages and the celluloid. On this imagined view, property discourse isfictional, and the pertinent materials we distinctively interact within learning and engaging in the fiction are classes (or fusions) ofconcrete individual objects.

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• Rodriguez-Pereyra, Gonzalo, “Nominalism in Metaphysics”,Stanford Encyclopedia of Philosophy (Spring 2025 Edition),Edward N. Zalta & Uri Nodelman (eds.), URL = <https://plato.stanford.edu/archives/spr2025/entries/nominalism-metaphysics/>. [This was the previous entry on this topic in theStanfordEncyclopedia of Philosophy – see theversion history.]

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Acknowledgments

The authors would like to thank Ben Caplan and an anonymous reviewerfor their helpful comments. This entry replaces a previous entry byGonzalo Rodriguez-Pereyra. The authors gratefully acknowledge Dr.Rodriguez-Pereyra’s support and have used his entry as aninstructive guide for compiling the present version.