## 1. What is Nominalism?




The word 'Nominalism', as used by contemporary
philosophers in the Anglo-American tradition, is ambiguous. In one
sense, its most traditional sense deriving from the Middle Ages, it
implies the rejection of *universals*. In another, more modern
but equally entrenched sense, it implies the rejection of *abstract
objects*. To say that these are distinct senses of the word
presupposes that *universal* and *abstract object* do not
mean the same thing. And in fact they do not. For although different
philosophers mean different things by *universal*, and likewise
by *abstract object*, according to widespread usage a universal
is something that can be instantiated by different entities and an
abstract object is something that is neither spatial nor temporal.




Thus there are (at least) two kinds of Nominalism, one that
maintains that there are no universals and one that maintains that
there are no abstract
 objects.[1]
Realism about universals is the doctrine that
there are universals, and Platonism is the doctrine that there are
abstract objects.




But Nominalism is not simply the rejection of universals or abstract
objects. For if that were the case, a nihilist, someone who believed
that there are no entities at all, would count as a nominalist.
Similarly, someone who rejected universals or abstract objects but were
agnostic about the existence of particulars or concrete objects would
count as a nominalist. Given how the term 'Nominalism' is
used in contemporary philosophy, such philosophers would not be
nominalists. The word 'Nominalism' carries an implication
that the corresponding doctrine asserts that everything is particular
or concrete, and that this is not vacuously true.




Thus one kind of Nominalism asserts that there are particular
objects and that everything is particular, and the other asserts that
there are concrete objects and that everything is concrete.




As noted above, the two forms of nominalism are independent. The
possibility of being a nominalist in one sense but not in the other has
been exemplified in the history of philosophy. For instance, David
Armstrong (1978; 1997) is a believer in universals, and so he is not a
nominalist in the sense of rejecting universals, but he believes that
everything that exists is spatiotemporal, and so he *is* a
nominalist in the sense of rejecting abstract objects. And there are
those who, like Quine at a certain point of his philosophical
development (1964; 1981), accept sets or classes and so are not
nominalists in the sense of rejecting abstract objects and yet reject
universals and so are nominalists in the sense of rejecting
universals.




Thus Nominalism, in both senses, is a kind of anti-realism. For one
kind of Nominalism denies the existence, and therefore the reality, of
universals and the other denies the existence, and therefore the
reality, of abstract objects. But what does Nominalism claim with
respect to the entities alleged by some to be universals or abstract
objects, e.g. properties, numbers, propositions, possible worlds? Here
there are two general options: (a) to deny the existence of the alleged
entities in question, and (b) to accept the existence of these entities
but to argue that they are particular or concrete.




Sometimes Nominalism is identified with those positions exemplifying
strategy (a). But this seems to be based on the thought that what makes
a position nominalist is the rejection of properties, numbers,
propositions, etc. In this entry, however, I shall understand
Nominalism in a broader way, namely as encompassing positions
implementing strategies (a) or (b) above. For Nominalism has nothing
against properties, numbers, propositions, possible worlds, etc.,
*as such*. What Nominalism finds uncongenial in entities like
properties, numbers, possible worlds and propositions is that they are
supposed to be universals or abstract objects. Thus the mere rejection
of properties, numbers, possible worlds, propositions, etc., does not
make one a nominalist - to be a nominalist one needs to reject
them because they are supposed to be universals or abstract
objects. Michael Jubien, for instance, rejects propositions, but he
admits properties and relations construed Platonistically; his reasons
for rejecting propositions have nothing to do with their alleged
abstract character (Jubien 2001: 48-54). It would be odd to call
Jubien a nominalist about propositions.




Thus according to my usage in this entry, acceptance of the existence
of properties, propositions, possible worlds and numbers is compatible
with being a nominalist. What is required of nominalists who accept
the existence of numbers, properties, possible worlds and propositions
is that they think of them as particulars or concrete
objects.[2] And
rejecting properties, propositions, possible worlds, numbers, and any
other items is not sufficient for being a Nominalist about them: to be
a Nominalist one must reject them on account of their being universal
or abstract objects.


## 2. Abstract objects and universals


### 2.1 Abstract Objects




What is an abstract object? There is no standard definition of the
phrase. Perhaps the most common conception of abstract objects is that
of non-spatiotemporal and causally inert objects. Often the requirement
that abstract objects are causally inert is not an independent
condition but is derived from the requirement that abstract objects are
not spatiotemporal since it is assumed that only spatiotemporal
entities can enter in causal relations.




But this conception of abstract objects has been criticised. Games
and languages are supposedly abstract and yet they are temporal
entities, since they come into being at a certain point in time, and
some of them develop and change in time (Hale 1987, 49). Defining
abstract objects simply as causally inert objects also presents
problems (see, for example, the entry on 
 abstract objects).




There have been other proposals as to how to characterise abstract
objects. One approach defines abstract objects as those the
understanding of whose names involves a recognition that the named
object is in the range of a certain functional expression (Dummett
1973, 485). It has also been thought that an abstract object is one
that either couldn't possibly exist or couldn't possibly be concrete,
depending on whether you interpret the predicate 'E!'
(used to formally represent the definition of 'abstract')
as an existence predicate or a concreteness predicate (Zalta 1983, 60,
50-52). On another conception of
abstract objects these are objects that cannot exist separately from
other entities (Lowe 1995, 
 514).[3] 
 (For a discussion of the various ways
of characterising the abstract/concrete distinction see Burgess and Rosen
1997, 13-25.)




There are thus several alternative conceptions of abstract objects.
But in what follows I shall take abstract objects to be those that are
non-spatiotemporal and causally inert. This is because what motivates
Nominalism (in one of its senses) is basically the rejection of
non-spatiotemporal and causally inert objects. That is, the nominalist
sees trouble with abstract objects simply because he sees trouble with
non-spatiotemporal, causally inert objects. That this is so can be seen
from the fact that nominalist theories are often motivated by
empiricist or naturalist views, which find no place for
non-spatiotemporal, causally inert 
 objects.[4] 
 Thus, for example, one of
the main problems with mathematical objects -- a subclass of
abstract objects -- from a nominalist point of view is that it is
not easy to see how we can come to have knowledge or form reliable
beliefs about them and refer to them, since there are no causal
relations between them and us. But this presupposes that what makes
abstract objects problematic is their causal inertness. And the source
of their causal inertness might be their lack of spatiotemporality.




The characterisation of abstract objects as non-spatiotemporal and
causally inert objects might be thought unsatisfactory to the extent
that it tells us only what they are not, but not what they are. But
this is not a problem for the nominalist. The business of the
nominalist is to reject such objects, not to characterise them in a
positive way. And for the purposes of rejecting abstract objects, their
characterisation as non-spatiotemporal, causally inert objects is a
reasonably clear characterisation (at least as clear as the notions of
spatiotemporal object, causation, causal power, and related ones
are).




Historically the distinction between abstract and concrete objects
has been thought of as exclusive and exhaustive. But the exhaustiveness
of the distinction has recently been questioned. Linsky and Zalta argue
that while abstract objects are necessarily abstract, there are objects
which are not concrete but could have been concrete. These objects are
non-concrete in virtue of being non-spatiotemporal and causally inert
but they are not abstract since they could have been concrete (Linsky
and Zalta 1994). Since Nominalism rejects abstract objects because of
their non-spatiotemporality and their causal inertness, Nominalism also
rejects non-concrete objects.


### 2.2 Universals




The nominalist about universals rejects universals -- but what
are they? The distinction between particulars and universals is usually
taken to be both exhaustive and exclusive, but whether there is such a
distinction is 
 controversial.[5] 
 The distinction can be drawn in terms of
a relation of instantiation: we can say that something is a universal
if and only if it can be instantiated (whether
it can be instantiated by particulars or universals) -- otherwise
it is a particular. Thus while both particulars and universals can
instantiate entities, only universals can be instantiated. If
*whiteness* is a universal then every white thing is an instance
of it. But the things that are white, e.g. Socrates, cannot have any
 instances.[6]




Realists about universals typically think that properties (e.g.
whiteness), relations (e.g. betweenness), and kinds (e.g. gold) are
universals. Where do universals exist? Do they exist in the things that
instantiate them? Or do they exist outside them? To maintain the second
option is to maintain an *ante rem* realism about universals. If
universals exist outside their instances then it is plausible to
suppose that they exist outside space and time. If so, assuming their
consequent causal inertness, universals are abstract objects. To
maintain that universals exist in their instances is to maintain an
*in re* realism about universals. If universals exist in their
instances, and their instances exist in space or time, then it is
plausible to think that universals exist in space or time, in which
case they are concrete. In this case universals can be multiply
located, i.e. they can occupy more than one place at the same time, for
*in re* universals are wholly located at each place they occupy
(thus if there is whiteness *in re*, then such a thing can be
six meters apart from itself).




Thus, both on *ante rem* and *in re* realism about
universals, universals enjoy a relation with space very different from
that apparently enjoyed by ordinary objects of experience like houses,
horses and men. For such particulars are located in space and time and cannot
be located in more than one place at the same time. But universals are
either not located in space or else they can occupy more than one place
at the same time.


## 3. Arguments against abstract objects and universals




Are there general arguments against abstract objects? There are
some, although it must be said that some of the most famous deniers of
abstract objects have not always based their rejection on arguments.
This is the case, for instance, of Goodman and Quine who, in their
*Steps toward a Constructive Nominalism*, base their rejection
of mathematical abstract objects on a basic intuition (1947, 105).




One argument against postulating abstract objects is based on
Ockham's razor. According to this principle one should not
multiply entities or kinds of entities unnecessarily. Thus if one can
show that certain concrete objects can perform the theoretical roles
usually associated with abstract objects, one should refrain from
postulating abstract objects. The effectiveness of this kind of appeal
to Ockham's razor is, of course, conditional upon our having been
shown that concrete objects can play the theoretical roles associated
with abstract objects. But if every theoretical role played by
*abstracta* can be played by *concreta* and vice versa,
then one needs a further reason why one should postulate
*concreta* only rather than *abstracta* only. Sometimes
the only evidence for the existence of the *abstracta* in
question is that they perform the theoretical role in question. In that
case one can use the principle that one should not postulate *ad
hoc* entities or kinds of entities unnecessarily (Rodriguez-Pereyra
2002, 210-16). That is, one should not postulate, if possible,
entities for which there is no independent evidence, i.e. entities for
the existence of which the only evidence available is that they
satisfactorily perform a certain theoretical role.




Another common and widely discussed argument against abstract
objects is an epistemological argument. The argument is grounded in the
thought that given that abstract objects are causally inert, it is
difficult to understand how we can have knowledge or reliable belief
about them. Sometimes a similar argument is advanced according to which
the problem with Platonism is that, given the causal inertness of
abstract objects, it cannot explain how linguistic or mental reference
to abstract objects is possible (see Benacerraf 1973 and Field 1989,
25-7). Admittedly these arguments do not conclusively establish
Nominalism but, if they work, they show an explanatory *lacuna*
in Platonism. The challenge for the Platonist is to explain how
knowledge of and reference to abstract objects is possible. Most of the
debate with respect to this argument has concentrated on the particular
application of the argument to the case of mathematical objects (for
more on this debate see the entry on
 Platonism in metaphysics 
 and Burgess and Rosen 1997, pp.
35-60.)




Another, now less common, argument against Platonism, is that its
ontology is unintelligible. Sometimes the unintelligibility of abstract
objects is linked to their lack of clear and intelligible conditions of
identity. But it is not the abstractness of abstract objects that makes
them lack clear identity conditions, since some abstract objects, like
sets, have clear and intelligible conditions of identity. But the
identity conditions for sets are intelligible only if the notion of a
set is intelligible. Some, like Goodman, are apparently unable to
understand how different entities can be composed out of the same
ultimate constituents. But, again, it is not in virtue of being
abstract, i.e. non-spatiotemporal and causally inert, that sets violate
Goodman's principle on composition. For there could be
*simple* abstract objects.




Many of these arguments and motivations for the rejection of
abstract objects are also arguments and motivations for rejecting
non-spatiotemporal *ante rem* universals. But Ockham's razor can
also be used against universals conceived of as spatiotemporal
entities, provided it can be shown that particulars can play the
theoretical roles normally assigned to *in re* universals. For even if
they are spatiotemporal, universals are nevertheless a distinctive kind
of entity.




There are other, more specific arguments against universals. One is
that postulating such things leads to a vicious infinite regress. For
suppose there are universals, both monadic and relational, and that
when an entity instantiates a universal, or a group of entities
instantiate a relational universal, they are linked by an instantiation
relation. Suppose now that *a* instantiates the universal
*F*. Since there are many things that instantiate many
universals, it is plausible to suppose that instantiation is a
relational universal. But if instantiation is a relational universal,
when *a* instantiates *F*, *a*,
*F* and the instantiation relation are linked by an instantiation
relation. Call this instantiation relation *i*2 (and suppose
it, as is plausible, to be distinct from the instantiation relation
(*i*1) that links *a* and *F*). Then
since *i*2 is also a universal, it looks as
if *a*,
*F*, *i*1 and *i*2
will have to be linked by another instantiation
relation *i*3, and so on *ad
infinitum*. (This argument has its source in Bradley 1893,
27-8.)




Whether this regress shows some sort of incoherence in realism about
universals or is merely uneconomical is a debatable issue. The realist
about universals can, however, maintain that the regress is illusory,
for instance by maintaining that although particulars instantiate
universals, this involves no relation between them (Armstrong 1997,
 118).[7]




Other arguments against universals are based on the principles that
there cannot be necessary connections between wholly distinct
existences and that no two things can be composed of exactly the same
 parts.[8] 
 Consider the universal *methane*. A molecule
instantiates *methane* if and only if it consists of four
hydrogen atoms bonded to a single carbon atom. Thus, necessarily,
*methane* is instantiated only if *carbon* is
instantiated. But this seems to be a necessary connection between two
wholly distinct entities, the universals *methane* and
*carbon*. One answer here is that *methane* and
*carbon* are not wholly distinct universals since the universal
*carbon* is a component or a part of the universal
*methane*, the other parts being the universal *hydrogen*
and the relational universal *bonded*. The problem here is that
a molecule instantiates *butane* if and only if it consists of a
chain of four carbon atoms, with the adjacent ones bonded, and the end
carbon atoms are bonded to three hydrogen atoms each, while the middle
carbon atoms are bonded to two hydrogen atoms each (thus the formula
for butane is
CH3-CH2-CH2-CH3). So, if
butane is not to be necessarily connected to wholly distinct
universals, one should say that *carbon*, *hydrogen* and
*bonded* are the parts of *butane*. But then
*methane* and *butane* are composed of exactly the same
parts. So it looks as if structural universals (i.e. universals like
*methane* and *butane*, such that whatever instantiates
them must consist of parts instantiating certain universals and
standing in certain relations to each other) offend either against the
principle that there are no necessary connections between wholly
distinct existences or the principle that no two entities can be
composed of exactly the same parts (see Lewis 1986b for further
discussion).




This, in itself, is not an argument against universals *per
se* but only against *structural* universals. Even so, if a
theory of universals must postulate states of affairs, as Armstrong
thinks it must, then the argument can be made to work against
universals in general. For the state of affairs that *Rab*
(where *R* is any non-symmetrical relation) necessitates that
*b* exists, which seems to be a necessary connection between
wholly distinct existences. And saying that *a*, *b* and
*R* are parts of the state of affairs that *Rab* means
trouble if one thinks that no two entities can be composed of exactly
the same parts, for the distinct state of affairs that *Rba*
would also be composed of *a*, *b* and *R*. There
are two things the defender of universals can do: (a) to accept simple,
non-structural universals but reject both structural universals and
states of affairs; (b) to accept that some entities can be composed of
exactly the same parts (provided they are related in different ways).
(b) seems to be more popular among realists about universals. (See
Armstrong 1986, Forrest 1986b and Armstrong 1997, 31-38, for
further discussion.)


## 4. Varieties of Nominalism


### 4.1 Nominalism about universals




Given that nominalists about universals believe only in particulars,
there are two strategies that they might implement regarding the
question of the alleged existence of allegedly universal entities like
properties and relations. One strategy is to reject the existence of
such entities. Another strategy is to accept that such entities exist
but to deny that they are universals. Both strategies have been
implemented in the history of philosophy. One way to implement these
strategies is to provide nominalistically acceptable paraphrases or
analyses of sentences that appear (a) to be true and (b) imply the
existence of universals. Another way, more fashionable nowadays, is to
give a nominalistic account of the truthmakers for sentences that are
apparently made true by universals.




What follows is a brief review of the main nominalistic positions of
this sort, and of some of the problems they face. For the sake of
brevity I shall illustrate the positions only with respect to
properties. The extension to kinds and relations is straightforward and
only occasionally do I say what a certain theory says about
relations.




Properties are entities that are meant to play different theoretical
roles. For instance, one role they are meant to play is that of being
the semantic values of predicates. Another role is that of accounting
for similarity and the causal powers of things. But there is no reason
why these different roles should be played by one and the same kind of
entity. When philosophers nowadays discuss the issue of universals they
normally think of properties as entities that account for the
similarity and causal powers of things. Properties in this sense are
sometimes called *sparse* properties, as opposed to
*abundant* properties (the distinction between sparse and
abundant properties comes from Lewis 1983). Sparse properties are those
which would be sufficient to account for the similarity and causal
powers of things, and to characterise them completely and without
redundancy. In what follows it is assumed, for the sake of example,
that properties like *being square* and *being scarlet*
count as sparse.




The question that realists and nominalists about universals try to
answer is: What makes *F*-things *F*
(where "*F*" is a sparse property
predicate)? For instance, what makes a square thing square? For the
realist about universals if something is square, this is in virtue of
the thing instantiating the universal squareness. In general, for the
realist about universals, things have the sparse properties they do in
virtue of instantiating universals.




How do nominalists answer this question? A popular nominalist theory
of properties is so-called Trope Theory, which has been held by Donald
Williams (1953), Keith Campbell (1990), and Douglas Ehring (2011) among others. Trope theory
does not reject the existence of properties, but takes properties to be
certain entities usually called 'tropes'. Tropes are
particulars, in the same sense in which individual people and
individual apples are particulars. Thus when there is a scarlet apple
the scarletness of the apple is not a universal but a particular
scarletness, the scarletness of *this* apple, which exists
exactly where and when *this* apple is 
 scarlet.[9] 
Such a
particular scarletness is a trope. The apple is scarlet not in virtue
of instantiating a universal but in virtue of possessing a scarlet
trope.




But what makes scarlet tropes scarlet tropes? One possible answer
here is that scarlet tropes are scarlet tropes because they resemble
each other, where resemblance is not explained in terms of
instantiating some same universal. Of course crimson tropes also
resemble each other. What makes a trope scarlet is that it resembles
these tropes (the scarlet ones) as opposed to resembling those ones
(the crimson ones).




Another answer would be that scarlet tropes form a primitive natural
class (this view has been forcefully defended by Ehring 2011:
175-241). But whether or not what makes scarlet tropes scarlet tropes
is that they resemble each other, scarlet tropes do resemble each
other. And the fact that they do raises an important problem. This is
the problem of the resemblance regress. Suppose
that *a*, *b* and *c* are scarlet apples. If so,
each one has its own scarlet trope: call
them *sa*, *sb*, and
*sc*. Since *sa*,
*sb*, and *sc* are scarlet tropes,
every two of them resemble each other. But then there are three
resemblance tropes as well: the resemblance between
*sa* and *sb*, the resemblance
between *sa* and *sc*, and the
resemblance between *sb* and *sc*.
But these resemblance tropes, since they are resemblance tropes,
resemble each other. So there are 'second-order'
resemblance tropes: the resemblance between the resemblance between
*sa* and *sb* and the resemblance
between *sa* and *sc*, the
resemblance between the resemblance between *sa* and
*sb* and the resemblance between
*sb* and *sc*, and the resemblance
between the resemblance between *sa* and
*sc* and the resemblance between
*sb* and *sc*. But these
'second-order' resemblance tropes resemble each other. So
there are 'third-order' resemblance tropes, and so on
*ad infinitum*.




There are some ways out for the trope theorist. One solution is to
argue that the regress is not vicious at all and that at most it
represents an increment in the number of entities (not *kinds*
of entities) postulated by the theory. Another solution is to deny the
existence of resemblance tropes and make do only with resembling tropes
(for further discussion see Daly 1997 and Maurin 2002,
96-115).




There are other forms of nominalism about universals, two of which
are Predicate Nominalism and Concept Nominalism. The realist about
universals admits that the predicate 'scarlet' applies to a
scarlet thing. But he says that the predicate 'scarlet'
applies to it in virtue of its being scarlet, which is nothing else
than its instantiating the universal scarletness. Similarly he says
that the thing in question falls under the concept *scarlet* in
virtue of being scarlet, which is nothing else than the thing
instantiating the universal *scarletness*. But for Predicate
Nominalism there is nothing like scarletness. According to this theory
a thing is scarlet in virtue of the fact that the predicate
'scarlet' applies to it. Similarly, according to Concept
Nominalism (or Conceptualism), there is nothing like scarletness and a
thing is scarlet in virtue of its falling under the concept
 *scarlet*.[10]
 These two views entail that if there were no
speakers or thinkers, things would not be scarlet. If only because of
this many would feel inclined towards another view, called *Ostrich
 Nominalism*.[11]
 This view, held by Quine, among others, maintains
that there is nothing in virtue of which our thing is scarlet: it just
is scarlet (Devitt 1980, 97). But many think that being scarlet cannot
be a metaphysically ultimate fact, but that there must be something in
virtue of which scarlet things are scarlet.




Another theory is Mereological Nominalism, according to which the
property of being scarlet is the aggregate of scarlet things, and for
which something is scarlet in virtue of being a part of the aggregate
of scarlet 
 things.[12]
 An aggregate, or mereological sum, is a
particular. But the theory faces a difficulty with so-called extensive
properties like mass and shape. Not every part of the aggregate of
square things is square since, for instance, not every sum of squares
is itself square, and not every part of a square is itself square. So
it is false that square things are square in virtue of being parts of
the aggregate of square things.




A better theory in the same spirit is *Class Nominalism*, a
version of which was maintained by Lewis (1983). Whether abstract or
not, classes are particular on this 
 view.[13]
 According to Class
Nominalism properties are classes of things, and so the property of
*being scarlet* is the class of all and only scarlet
 things.[14]





One problem with this theory is that no two classes can have the
same members, while it does not seem that properties with the same
instances need be the same. So there is no guarantee that the
identification of properties with classes is correct. And even if
correct, the identification is clearly not necessarily correct.
Furthermore, if every *F* is a *G* and
vice versa, the theory forces us to say that what makes something
*F* is the same as what makes it *G*. But
while every *F* might be a *G* and vice
versa, it does not follow that what makes things *F* is
the same as what makes them *G*.




One solution to this is to embrace a version of Modal Realism, for
instance David Lewis', according to which other possible worlds
exist and contain things of the same kinds as the things in the actual
world (see Lewis 1986a). Then properties get identified with classes
whose members need not belong to the same possible world. Thus the
property of scarlet things is the class of things that are scarlet in
any possible 
 world.[15]
 And even if every actual *F* is a
*G* and vice versa, since not every possible
*F* is a *G* or vice versa, what makes
something *F*, namely belonging to the class of actual
and possible *F*s, is not the same as what makes it
*G*. The theory denies that there are and there could be
necessarily coextensive properties.




Another version of Nominalism is Resemblance Nominalism. According
to this theory, it is not that scarlet things resemble one another
because they are scarlet, but what makes them scarlet is that they
resemble one another. Thus what makes something scarlet is that it
resembles the scarlet things. Similarly, what makes square things
square is that they resemble one another, and so what makes something
square is that it resembles the square things. Resemblance is
fundamental and primitive and so either there are no properties or the
properties of a thing depend on what things it resembles.




Thus on one version of the theory a property like *being
scarlet* is a certain class whose members satisfy certain definite
resemblance conditions. On another version of the theory there are no
properties, but what makes scarlet things scarlet is that they satisfy
certain resemblance conditions.




What are these resemblance conditions? Sometimes the resemblance
conditions include some that must be satisfied, not by the things in
question (e.g. not by the scarlet things), but by things suitably
related to them. For instance, in the version of Resemblance Nominalism
developed in Rodriguez-Pereyra 2002, what makes scarlet things scarlet
is that they resemble each other, that there is a degree of resemblance
*d* such that no two scarlet things, and no two
*n*th-order pairs (two-membered unordered classes) whose
ur-elements are scarlet things, resemble each other to a degree less
than *d*, and that the class of scarlet things is or fails to be
included in certain other classes defined in terms of resemblance
conditions like the ones just mentioned (see Rodriguez-Pereyra 2002,
156-98, for details). Of course the crimson things also resemble
each other and they also meet the other conditions having to do with
resemblance degrees and their class being or failing to be included in
certain other classes. But this does not mean that what makes something
scarlet is what makes something crimson: what makes a scarlet thing
scarlet is that it resembles *these* things (i.e. the scarlet
ones), which happen to satisfy the stated conditions having to do with
resemblance degrees and their class being or failing to be included in
certain other classes while what makes a crimson thing crimson is that
it resembles *those* things (i.e. the crimson ones), which also
happen to satisfy the stated conditions having to do with resemblance
degrees and their class being or failing to be included in certain
other classes.




The resemblance nominalist ontology is an ontology of resembling
particulars like horses, atoms, houses, stars, men (and classes). But
the resemblance nominalist does not reify resemblance. Thus that
*a* and *b* resemble each other does not require that
there are three entities there: *a*, *b* and a third,
relational entity that is their resemblance. The only entities
involved in that situation are *a* and *b*. In this
respect, Resemblance Nominalism resembles Ostrich Nominalism. The
difference is that whereas the latter admits many sorts of basic facts
involving only particulars - '*a* is
scarlet', '*b* is an electron' - the
former admits only basic facts of the form '*a*
resembles *b* to such and such a degree'.




Like Class Nominalism, Resemblance Nominalism faces the problem
about the identity of coextensive properties, and the solution is the
same, namely to adopt some version of Modal Realism according to which
merely possible particulars are as real as actual ones. Thus (part of)
what makes a certain apple scarlet is that it resembles all scarlet
things, including merely possible scarlet things.




Russell (1912, 96-7) and others think that Resemblance
Nominalism faces the resemblance regress. But this regress presupposes
that resemblances are entities that can resemble one another. Since
Resemblance Nominalism does not reify resemblances, the regress does
not arise (see Rodriguez-Pereyra 2002, 105-23, for further
discussion).




Finally, there is Causal Nominalism, according to which what makes it
true that *a is F* is that *a* would stand in certain
causal relations given certain circumstances. In other words, the
claim is that for *a* to be *F* is for the theory that
which charts out the functional role of *F*-particulars to be
true of *a* (Whittle 2009, 246). *F*-particulars will
resemble each other in realising the same functional role, but this
does not collapse Causal Nominalism into Resemblance Nominalism, since
such resemblances are not what explains why *a* is *F*,
but a consequence of what explains that, namely the fact that such
particulars realise a certain functional role (Whittle 2009:
255). Similar reasons might also suggest that Causal Nominalism does
not collapse into any of the other nominalisms. But it has been argued
that to be thoroughly nominalistic, Causal Nominalism owes a
nominalistic account of what it is for different particulars to
realise the same functional role, and such an account can only be in
terms of any of the nominalisms distinguished above, in which case
Causal Nominalism collapses into some other form of nominalism (Tugby
2013).




Which one of these theories is the best has to be decided by
comparing how they score with respect to certain theoretical virtues,
like accommodating firm and stable intuitions and common sense
opinions, avoiding the unnecessary multiplication of entities, reducing
the number of undefined primitive concepts, etc.


### 4.2 Nominalism about abstract objects


#### 4.2.1 Nominalism about propositions




Most theories of propositions take them to be abstract or imply that
they are. One can divide theories of propositions into those that take
them to be structured entities and those that take them to be
unstructured entities. Each conception comprises a family of
theories.




The most popular conceptions of unstructured propositions are those
that take them to be either sets of possible worlds or functions from
possible worlds to truth-values (Lewis 1986a, 53; Stalnaker 1987, 2).
On these theories a proposition is the set of possible worlds in which
it is true, or a function that has the value *True* when it
takes as argument a world where the proposition is true and has the
value *False* when it takes as argument a world where the
proposition is false.




But sets are, *prima facie*, abstract objects. So it looks as
if those who take propositions to be sets of possible worlds should
count as platonists about propositions. But some people, like Lewis
(1986a, 83) and Maddy (1990, 59), believe that sets of spatiotemporally
located members are spatiotemporally located where and when their
members are, in which case sets of spatiotemporally located members are
 concrete.[16]
 But since it lacks any members, the empty set is not
spatiotemporally located. And since there are necessarily false
propositions, that is, propositions that are true in no possible world,
it is plausible, on this conception of propositions, to identify these
propositions with the empty set. So some propositions (at least one)
seem to be abstract objects. And functions also seem to be abstract
objects. And the truth values *True* and *False* seem to
be abstract objects as well. So these accounts of propositions as sets
of possible worlds or functions from possible worlds to truth values,
if they are to be nominalistic accounts of propositions, require some
consistent and plausible nominalistic account of pure sets, functions
and truth values as concrete objects.




There are other theories of propositions that take them to be
unstructured entities. George Bealer has a conception of unstructured
propositions according to which they are *sui generis*
irreducible intensional entities. His propositions can exist even if
the objects they are about do not exist and they can be actual even if
the objects they are about are not actual (Bealer 2006, 232-4).
Such propositions are abstract objects.




Among conceptions of propositions as structured entities one can
distinguish, roughly, between Russellian and Fregean versions. Both the
Russellian and the Fregean conceptions of propositions are
*families* of theories. In general Fregean theories will take a
proposition to be a complex entity with a particular structure whose
constituents are *senses*. But senses are abstract objects. And
if, as seems plausible, a complex entity whose constituents are
abstract objects must be an abstract object itself (how could an object
be in space or time when its constituents exist neither in space nor
time?), then, on this account, propositions are abstract objects.




According to the Russellian conception of propositions, a
proposition is a complex entity with a particular structure whose
constituents are particulars and/or properties and/or 
 relations.[17]
 Are
propositions of this sort abstract objects? If all the particulars are
concrete then perhaps propositions are concrete objects, even if
properties and relations are abstract. For one may say that
propositions are where and when the particulars which are their
constituents are. But this sounds arbitrary. Why not say that
propositions are where their constituent properties and relations are,
that is, nowhere? In any case, that particulars (and even properties
and relations) are concrete does not immediately settle the matter
whether propositions in the sense of complexes of particulars and
properties and/or relations are abstract objects. For what kind of
complex entities are propositions? Sometimes they are considered to be
*ordered* *sets*. If this is what propositions are, then
the nominalist needs a satisfactory nominalistic account of ordered
sets. If propositions are another kind of complex entity, then the
nominalist about propositions must make sure that objects of that kind
are concrete.




One nominalist option is to show that the roles associated with
propositions (e.g. being truth-bearers and objects of propositional
attitudes) are actually played by concrete objects. One common thought
here is to propose that sentences play the roles associated by
propositions. This strategy is exemplified by Quine. In *Word and
Object* he proposes eternal sentences as truth-bearers (Quine 1960,
208). Eternal sentences are better as truth-bearers than other
sentences in being true or false independently of time, place, speaker
and the like. But they are as bad as other sentences in admitting of
variation in truth value from one language to another (Quine 1969,
 142).[18]
 But note that from the point of view of a nominalist about
abstract objects, there is a much worse problem with eternal sentences,
namely that they may be abstract objects. They may be abstract objects
because they are sentence *types*, and a type may be an abstract
object, for instance if one takes them to be sets or abstract
universals (admittedly one might attempt to take them to be
non-abstract universals).




The alternative is to take concrete token sentences (utterances or
written inscriptions) as the objects that play the roles normally
associated with propositions. One problem here is that only a finite
number of sentences ever get uttered. And so some find it difficult to
make sense of general logical laws, e.g. the law that any two
falsehoods form a false disjunction, since the disjunction may not get
uttered or written (Quine 1969, 143). (One possible solution might be
to reformulate the law so as to say that if the disjunction of
*P* and *Q* exists, it is false if and only if
*P* and *Q* are false.)




In this area, as in many others, a nominalist strategy is to supply
a nominalistically acceptable paraphrase of sentences that appear to
posit abstract entities. That is, there are certain sentences that seem
to be true and whose truth seems to entail that there are propositions.
The nominalist can then paraphrase those sentences into others which
allegedly mean the same and whose truth seems to entail only the
existence of, say, token sentences. For example, 'Seneca said
that man is a rational animal' is true and seems to entail that
there is a proposition, namely what Seneca said. But according to
Scheffler's inscriptionalism, on which that-clauses are treated
as single predicates of concrete inscriptions, to say that Seneca said
that man is a rational animal is simply to say that Seneca produced a
that-man-is-a-rational-animal inscription (Scheffler 1954, 84).




So we have a sentence whose truth apparently entails the existence
of propositions and an alleged paraphrase that apparently entails the
existence of concrete inscriptions only. Assuming that they do have the
same meaning (in which case both sentences entail exactly the same),
why think that the apparent ontological commitments (i.e. those
entities the truth of a sentence appears to entail) of the nominalistic
paraphrase are the real ontological commitments of both the paraphrase
and the original sentence? The fact that the original sentence and its
paraphrase are semantically equivalent does not give any reason to
think that the real ontological commitments of both are the apparent
ontological commitments of the paraphrase rather than those of the
original sentence. (This point has its source in Alston 1958,
9-10.) What the nominalist must do is to argue that the
paraphrase reveals and makes apparent the real meaning of the original
sentence, so that the apparent commitments of the paraphrase are the
real commitments of both paraphrase and original sentence.




Another nominalist option is to deny that there are propositions and
any entities that play their theoretical roles. If so, apparently true
sentences that entail the existence of propositions are false. Thus
this kind of Nominalism about propositions is a sort of fictionalism,
called *semantic fictionalism* (Balaguer 
 1998).[19]
 Thus a
sentence like 'Nestor believed that the gods do not give men all
things at the same time' is not true on this account because (a)
'that'-clauses (like 'that the gods do not give men
all things at the same time') are referential singular terms, (b)
if anything is the referent of 'that the gods do not give men all
things at the same time', this is a proposition, and (c) there
are no propositions. Thus talk about propositions is a fiction, since
there aren't any, but it is a useful fiction since it is a
descriptive aid that allows us to make it easier to say what we want to
say about the world and it allows us to represent the structure of
certain parts of the world -- for instance the logico-linguistic
structure of propositions can be used to represent the empirical
structure of belief states (Balaguer 1998, 817-18).


#### 4.2.2 Nominalism about possible worlds




The word 'Nominalism' is not very often used to refer to
any stance with respect to possible worlds. But since some philosophers
take possible worlds to be abstract objects, a nominalist about
possible worlds will be, for the purposes of this section, someone who
thinks that possible worlds are not abstract objects, and this will
include those who believe that there are no possible worlds (but not
those who simply do not believe that they 
 exist).[20]





The question about the nature of possible worlds is a hotly debated
topic. Some, for instance Alvin Plantinga, think that possible worlds
are states of affairs that are both possible and maximal. A maximal
state of affairs is one that includes or precludes every state of
affairs -- where a state of affairs *S* includes a state
of affairs *S*\* if and only if it is not possible that *S*
obtain and *S*\* fail to obtain, and *S* precludes *S*\*
if and only if it is not possible that both obtain (Plantinga 1974,
45; 2003a, 107; 2003b,
 194).[21] 
 According to Plantinga possible but not necessary
states of affairs can obtain and can fail to obtain. Those states of
affairs that obtain are actual. The actual world includes every actual
state of affairs (Plantinga 2003a, 107; 2003b, 195). Merely possible
states of affairs and worlds exist but do not obtain (Plantinga 2003a,
107; 2003b, 195). States of affairs, and therefore possible worlds,
are thought of as abstract objects by Plantinga. Indeed even the
actual world is an abstract object for Plantinga, since it has no
center of mass, it is neither a concrete object nor a mereological sum
of concrete objects and, like the state of affairs of Ford's being
ingenious, has no spatial parts at all (2003a, 107).




For Stalnaker possible worlds are ways the world might have been and
such ways are properties (2003, 7). All these ways the world might have
been actually exist but only one of them is instantiated -- the
way the world actually is. He naturally takes these properties to be
abstract objects (2003, 
 32).[22]
 A view like this has been further
developed by Peter Forrest, who proposes certain properties that he
calls *natures* (certain conjunctions of natural non-relational
properties) to play the role played by possible worlds. These natures
are, for the most part, uninstantiated properties (1986a, 15). It is
natural to think that they are abstract 
 objects.[23]





Another option is to take possible worlds as maximally consistent
sets of propositions. R. M. Adams (1974) sketched such a theory. If
propositions are abstract objects, then on this theory possible worlds
are abstract objects. But there are other options open. Adams suggests
that someone might, *a la* Leibniz, take propositions to
be thoughts in the mind of God. But if so, and if God is in time and
therefore concrete, then presumably his thoughts also are. And if we
assume that sets of spatiotemporally located entities are
spatiotemporally located (because they are wherever and whenever their
members are), then sets of concrete objects are concrete. Thus sets of
thoughts of a concrete deity are concrete.




Another option would be to take possible worlds as sets of spacetime
points and think of each such set as representing the possibility that
all and only the points in it are occupied (the view is proposed as an
illustration in Cresswell 1972, 
 136).[24]
 This assumes, as Cresswell
notes, that all properties of things are determined by the properties
of certain basic entities whose properties can all be expressed in
terms of the spacetime points they occupy. If sets of spacetime points can be
seen as concrete then this might be a way of taking possible worlds as
concrete. This view derives from certain passages by Quine, where he
develops the idea that every distribution of space points could be
taken as a possible world momentary state (1969, 148). But to avoid
certain difficulties (some having to do with ontological economy,
others having to do with the notion of a point and the relativity of
position), Quine proposes to bypass spacetime points and takes possible
worlds as certain *sets* of *number* quadruples (Quine
1969, 151). To be nominalistically acceptable this account of possible
worlds would need to be accompanied by a nominalistically acceptable
treatment of sets and numbers.




All the previously mentioned accounts of possible worlds are
actualist in the sense that they take actual existence and existence
*simpliciter* to coincide. One of the most developed
nominalistic accounts of possible worlds, that of David Lewis, is not
actualist but possibilist: according to Lewis to exist
*simpliciter* is one thing and to be actual is another. For
Lewis 'actual' is an indexical predicate so that from the
point of view of each world only that world is actual and none of the
others are. Thus, unlike Plantinga, Adams, and Stalnaker, Lewis does not
take every possible world to exist actually.




For Lewis possible worlds are maximal sums of spatiotemporally
related objects. A sum of spatiotemporally related objects is maximal
if and only if nothing that is not part of the sum is spatiotemporally
related to any part of the sum in question. Since sums of
spatiotemporally related objects are sums of concrete objects, and sums
of concrete objects are concrete objects, Lewisian possible worlds are
concrete 
 objects.[25]
,
[26]





Another theory of possible worlds has been developed
by David Armstrong. Armstrong has an actualist combinatorialist theory
of possibility, according to which what is possible is determined by
appropriate combinations of actual elements (particulars and
universals). The basic notion in Armstrong's theory of
possibility and possible worlds is that of an atomic state of affairs.
A state of affairs brings together a particular and a universal (if
the universal is a property), or some particulars and a universal (if
the universal is a 
 relation).[27]





These elements (particulars and universals) define a range of
combinations, some of them actualised, some not. These combinations
must respect the form of states of affairs (thus *Aristotle's
being wise* is an actualised combination, *Aristotle's
being a general* is an unactualised combination, and
*wisdom's being Aristotle* does not respect the form of
states of affairs and so does not fall in the range of combinations
defined by particulars and universals). The possible atomic states of
affairs are the combinations of particulars and universals which
respect the form of states of affairs. The merely possible atomic
states of affairs are the *re*combinations of particulars and
universals, i.e. those combinations which do not actually occur, like
*Aristotle's being a 
 general*.[28]
 Possible worlds are, for
Armstrong, conjunctions of possible atomic states of affairs (1989, 47,
 48).[29]





Armstrong's combinatorialism is actualist in the sense that
everything that exists actually exists. But he does not identify his
merely possible states of affairs and merely possible worlds with
actually existing entities. So merely possible states of affairs and
worlds do not actually exist and, therefore, given Armstrong's
actualism, do not exist at all (Armstrong 1989, 49).




Armstrong's rejection of possible worlds is not exactly a nominalistic
stance about them since his opposition to them is not based on their
alleged abstract character. In believing that possible worlds do not
exist, Armstrong is rather a kind of fictionalist about possible
worlds, and so he calls himself (1989, 49). But if one believes that
possible worlds do not exist, and so one is a fictionalist about
possible worlds in this sense, one can also be a fictionalist about
possible worlds in a different sense, namely the sense of
so-called *modal fictionalism*. According to modal fictionalism
sentences with an apparent quantification over possible worlds must be
understood as quantification within the scope of a story prefix (Rosen
1990, 332). Let *PW* be a theory that postulates possible
worlds. 'According to PW' is then a story
prefix.[30] Thus
the modal fictionalist says that when he utters 'There is a
possible world where there are blue swans' what he is really
saying is that according to PW there is a world where there are blue
swans (Rosen 1990, 332). But since quantification within a story
prefix is not existentially committing, the modal fictionalist can
utter things like 'Since there might have been blue swans, there
is a possible world where there are blue swans' without
committing himself to possible
 worlds.[31]





Now, from the point of view of a nominalist, adoption of modal
fictionalism must be coupled with some sort of nominalistically
acceptable account of stories, or theories, or representations in
general. For accepting something like 'According to PW there are
worlds where there are blue swans' seems to commit one to PW, and
PW is a theory, and so one seems thereby committed to theories. But
theories seem to be abstract objects. So the fictionalist nominalist
needs a nominalist account of theories. If, for instance, theories are
sets of propositions, a nominalist account of sets and propositions
would do as a nominalistic account of 
 theories.[32]