## 1. Perfect naturalness and degrees of naturalness



Naturalness comes in degrees. But an important part of Lewis's
picture, also dominant in the subsequent literature, is that it has a
*maximal* degree. Some properties are *perfectly*
natural. They are all equally natural, and no properties are more
natural than them. Lewis also calls these "fundamental"
properties. And the connotations of the word 'fundamental'
are apt. According to Lewis, there is a good sense in which one would
have *exhaustively* described the world if one said which
things (or \(n\)-tuples of things) instantiate each perfectly natural
property, and in which one would have *failed* to exhaustively
describe the world if one merely provided this information for some
but not all of the perfectly natural properties. As we will see in
 SS4.4,
 Lewis cashes out these thoughts in modal terms: all truths are
*necessitated* by the truths about the distribution of all
perfectly natural properties, but not by the truths about the
distribution of only some of the perfectly natural properties. Others,
like Sider
 (2011),
 have explored quite different ways of cashing them out.



Giving examples of perfectly natural properties is not supposed to be
easy. Views that identify certain properties as perfectly natural are
bold hypotheses about the fundamental structure of reality, and thus
bound up with many central debates in metaphysics. According to Lewis,
the methods of physics have a central role to play in the project of
identifying the perfectly natural properties. Until we have the kind
of final theory that many physicists are aiming for, we should not
hope to be able to list the actually-instantiated perfectly natural
 properties.[2]
 But looking to physics in this way is already taking a stand on many
controversial questions in philosophy. Those inclined towards dualism
in the philosophy of mind (see the entries on
 dualism
 and
 physicalism)
 may hold that some *mental* properties--for example,
consciousness, or specific phenomenal states, or relations like
acquaintance, judgment or knowledge--are perfectly natural;
indeed, this might with some plausibility be regarded as definitive of
dualism. Similarly, those inclined towards "non-naturalist
realism" in metaethics (see the entry on
 moral non-naturalism)
 may hold that some *normative* or *evaluative*
properties, such as goodness, betterness, or moral permissibility, are
perfectly natural.



Perfect naturalness is generally understood to be definable in terms
of relative naturalness: a property is perfectly natural just in case
no property is more natural than it. But Lewis also held that relative
naturalness could be defined in terms of perfect naturalness: \(F\) is
more natural than \(G\) just in case \(F\) has a shorter and less
complex definition in terms of perfectly natural properties than \(G\)
(Lewis
 1986b,
 61). Here, a "definition" of a property can be understood
as an expression referring to that property in a formal language which
only has simple predicates expressing perfectly natural properties (in
addition to certain logical resources which allow complex predicates
to be constructed out of these simple
 ones).[3]
 This claim has been controversial even among naturalness enthusiasts.
One influential worry (raised in Sider
 1995)
 is that many of the properties we are interested in, and between
which we need to draw distinctions of naturalness (e.g. greenness
and grueness), will only have *infinite* definitions in terms
of the perfectly natural properties, so that they will all come out
equally
 natural.[4]
 Nevertheless, length and complexity of definitions has been broadly
accepted at least as a heuristic guide. For example, someone who held
that *adjacency* and *chargedness* were perfectly
natural would be expected to hold that *being adjacent to something
charged* is more natural than *being adjacent to an uncharged
thing that is adjacent to something charged*.


## 2. History



The history of philosophical theorising about properties and
property-like entities (concepts, predicates, terms, forms,
universals, qualities...) contains many contrasts that seem
broadly similar to the contrast between more and less natural
properties, or between perfectly natural properties and the rest. A
few examples:


* In a much-cited passage in the *Phaedrus* (265e), Plato
says giving a good definition involves "dividing things...
where the natural joints are, and not trying to break any part, after
the manner of a bad carver".
* Elsewhere (*Parmenides* 130a-e), Plato suggests that
some but not all predicates correspond to Forms: there are Forms of
justice, beauty, and goodness but no Forms of hair, mud, or dirt. (See
the entry on
 Plato's Parmenides)
* Aristotle (*De Interpretatione* 11: 20b15-22)
distinguishes between terms with "unity", like
'white', 'walking', 'man',
'animal', and 'rational animal', and other
terms that lack unity, like 'walking white man'. He also
makes a further distinction between genuine terms (which include both
those with and without unity) and 'indefinite names' like
'non-walking' and 'non-man'.
* In classical and medieval Indian philosophy, the
Nyaya-Vaisesika tradition distinguishes "genuine
universals" (*jati*) from "imposed" or
"nominal" properties (*upadhi*). The
tenth-century logician Udayana lists six "impediments"
which can be used to show that a word does not stand for a
*jati*: for example, no two *jati* are
coextensive, and if two *jati* share an instance, then one
is such that every instance of it is also an instance of the other.
(For more details and citations, see the entry on
 analytic philosophy in Early Modern India.)
* There is a long tradition (stretching back to Plato and Aristotle)
according to which some properties (terms, concepts, forms,...)
are *simple* (primitive, indefinable, unanalysable), and while
the remainder are complex--some more complex than
others--and in some sense "definable" from the simple
ones. One author who is particularly interested in this contrast, and
who conceives of it in a way that makes it clearly one of metaphysics
as opposed to mere human psychology, is Leibniz. He argues that there
must be absolutely simple terms for it to be possible to conceive of
anything at all (Mates
 1986,
 58). He has no settled view as regards what these absolutely simple
terms are, although he does at one point (Leibniz
 1680-85)
 suggest a list of "simpler" terms in terms of which all
others can be defined.
* The founding figures of analytic philosophy continue this
tradition. Moore
 (1903,
 SS1.7) argues that 'good' denotes something
"simple and unanalysable", whereas 'horse'
denotes something complex. (See the entry on
 G. E. Moore.)
 Bertrand Russell
 (1918,
 111) claims that the "ultimate simples, out of which the world
is built" include simple particulars, simple qualities, and
simple relations. (See the entry on
 Russell's logical atomism.)
 Later, Ramsey
 (1925)
 argues for eliminating all complex properties, so that most
predicates will not correspond to properties at all.
* In David Armstrong's theory of universals (Armstrong
 1978),
 which Lewis credits as the main inspiration for his work on
naturalness, many predicates do not correspond to universals at all;
moreover, some universals are more complex than others, and some may
indeed be perfectly simple.



The tradition of theorising about "simple" and
"complex" properties is a particularly important
antecedent for the contemporary use of 'natural', for two
reasons. First, one can ask both whether a property is more or less
simple than some other property, and whether it is perfectly simple.
Second, whereas it is supposed to be relatively easy to tell in many
cases that a property is complex, identifying perfectly simple
properties is generally not supposed to be easy: such claims need to
be argued for as part of some ambitious metaphysical theory. The
central difference between this traditional use of
'simple' and the contemporary use of 'natural'
is that the former use is bound up with a battery of assumed analogies
between properties and other kinds of entities--concrete objects,
or linguistic expressions--which the latter use does not require.
The claim that a property is complex is understood to entail that in
some sense it has other properties as "parts", from which
it is "put together"; the perfectly simple properties are
exactly those that lack parts--the analogues of letters or words
in the realm of linguistic expressions. The paradigm for
"parthood" is the relation between the conjunction of two
properties and each of those two properties: for example, *being
red* and *being square* would be taken to be parts of
*being red and square*. Indeed, for most of the history of this
tradition this seems to be the only clearly recognised form of
complexity; insofar as properties like *not being red and
square* are recognised at all, they seem to be excluded from the
domain to which distinctions of simplicity are to be applied. Only
relatively late, in the works of authors like Moore and Russell, do we
find a picture where conjunction is merely one of several different
operations for building complex properties, others including
disjunction, existential quantification, etc. The contemporary use of
'natural' is plausibly seen as a cleaned-up version of
this older tradition, purged of the additional assumptions that might
be read into the mereological terminology. It is compatible with the
thesis of "intensionalism", according to which necessarily
coextensive properties are identical; Lewis was in fact an influential
defender of this thesis. Given intensionalism, the fact that a
property can be derived from other properties by logical operations
like conjunction cannot rule out its being perfectly natural, since
even a perfectly natural property \(F\) will be identical to the
conjunction of \(F\)-or-\(G\) with \(F\)-or-not-\(G\), for any other
property \(G\). However the contemporary use is also compatible with
finer-grained views about the identity of properties, and it would be
natural for proponents of such views to hold that no perfectly natural
property is a conjunction, disjunction, existential quantification,
etc. Such generalizations will be considered further in
 SS4.10
 below.


## 3. Related uses of 'natural'



'Natural' is sometimes used in ways that are closely
related to our target use, but applied to things other than
properties. This section will survey some of these uses and discuss
how they might relate to the target use.


### 3.1 Natural classes



As part of his notorious "modal realism" (see SS6 of
the entry on
 David Lewis),
 Lewis identified properties with sets: for example, the property of
being a talking donkey is just the set of all talking donkeys (a set
which Lewis, surprisingly, held to be nonempty). Few philosophers have
accepted this view. The dominant view is that distinct properties can
have exactly the same instances, although distinct sets never have
exactly the same members. Many are nevertheless happy to apply
'natural' to sets (or classes) as well as properties: for
example, the class of all green things would be counted as more
natural than the class of grue things.



It seems plausible that there should be some intimate connection
between this notion of naturalness for classes and naturalness for
properties; but it is not obvious how it should be spelled out. Since
there seem to be differences in naturalness among coextensive
properties--for example, *being charged* seems more
natural than *being charged and not a talking
donkey*--identifying a property's degree of naturalness
with that of its extension (the class of its instances) is not a
promising approach. More plausibly, one might identify the degree of
naturalness of a class with the *maximum* degree of naturalness
of the properties whose extension it is, or with some kind of
aggregate of the degrees of naturalness of these properties. However,
this may still be too simple, since there is a case that the
naturalness of other properties need to be taken into account. For
example, if there is a very natural equivalence relation (reflexive,
symmetric, transitive relation) with only a few equivalence classes,
we might want to count those equivalence classes as highly natural
even if they are not the extensions of any highly natural monadic
properties.


### 3.2 Plural naturalness.



Especially since Boolos
 (1984),
 it is widely accepted that there is an important distinction between
plural quantification ('there are some things such
that...them...') and singular quantification over sets
('there is a set of things such that...it...'),
and between applying a predicate collectively to *some things*
('these people could lift a piano') and applying a
predicate to a set ('this set of people could lift a
piano') (see the entry on
 plural quantification).[5]
 'Natural' is sometimes used as a plural predicate (Dorr
 2007;
 Rosen
 2015).
 It would be tendentious to equate the claim that some things are
collectively natural with the claim that they form a natural class:
for example, the things that are not charged seem collectively rather
natural, although they presumably do not form a class (since if they
did, it would have to include itself as a member). It is more
plausible to go the other way, analysing 'natural class'
as 'class whose members are collectively natural'. The
task of explaining plural naturalness in terms of the naturalness of
properties raises similar difficulties to that of explaining the
naturalness of classes.


### 3.3 Natural propositions and concepts



Once one is accustomed to comparing the naturalness of properties, it
is natural to want to extend such comparisons to propositions. For
example, the proposition that some things are green should count as
more natural than the proposition that some things are grue. While the
use of 'natural' to apply to propositions is rare, talk of
propositions--or of proposition-like entities like
*hypotheses*, *theories*, *accounts*,
*proposals*, etc.--as being more or less *simple*
is ubiquitous, in science as well as philosophy, and arguably amounts
to the same thing.



A few authors identify propositions with properties: for example,
Lewis
 (1986b,
 53-4) holds that the proposition that some swans are blue is
*being a world containing a blue swan*. Others (e.g. Bealer
 1982)
 reject this identification, but still treat propositions as
property-like: as binary relations stand to the number 2 and monadic
properties stand to 1, propositions stand to 0. Given either of these
views, the notion of a natural proposition should raise no new
 difficulties.[6]



This way of thinking of propositions is not universal, however. For
many authors, a central role of propositions is to play a certain role
in the theory of mental states and/or linguistic meaning (see the
entry on
 propositions).
 This role is often taken to motivate an account of propositions as
more "finely individuated" than properties. For example,
some would distinguish the proposition that there are vixens from the
proposition that there are female foxes, on the grounds that someone
could believe one but not the other, while also identifying the
property of being a vixen with the property of being a female fox.
Many who hold such views *also* recognise a coarser-grained
domain of entities--sometimes called 'states of
affairs' (see the entry on
 states of affairs)--that
 stand to 0 as \(n\)-ary properties stand to \(n\). Some also
recognise fine-grained entities, sometimes called *concepts*,
that stand to positive integers as propositions stand to 0. For
example, they may say that the concept *being a vixen* is
distinct from the concept *being a female fox*, although they
correspond to the same property. In this setting, it is less obvious
how one might understand comparisons of naturalness for propositions
and concepts, assuming one wants to allow for differences in
naturalness between distinct propositions corresponding to the same
state of affairs, or between distinct concepts corresponding to the
same property. The question how such comparisons might be conceived,
and whether they have anything to do with naturalness for properties,
will depend on issues in the theory of mental and linguistic content
that are beyond the scope of this article.


### 3.4 'Natural' as a higher-order predicate



We have been assuming an "abundant" theory of properties,
on which we can take for granted that there are such properties as
*being negatively charged and not part of a fish*, *being
grue*, and so on. But one can easily slip into inconsistency in
talking like this. The obvious generalization is the following
schema:



*Naive Comprehension*
  

For any \(x\), \(\phi(x)\) if and only if \(x\) instantiates *being
an \(x\) such that \(\phi(x)\)* 



Unfortunately, this is inconsistent (given classical logic) because of
Russell's paradox: it entails that *being an \(x\) such that
\(x\) does not instantiate \(x\)* instantiates itself if and only
if it doesn't instantiate itself, and hence that it both does
and does not instantiate itself. So any "abundant" theory
of properties needs to be crafted with care.



Lewis combined the identification of properties with classes with a
standard mathematical theory of sets and classes (Lewis
 1991),
 on which some things--such as the classes--are too numerous
to form a class. His view thus provides for such properties as
*being grue*, but provides no good candidate referents for
terms like 'being a property', 'being instantiated
by all triangles', 'identity', 'being more
natural than', etc. Intuitive claims like 'identity is
more natural than *being as tall as*' are thus hard to
accommodate.



One possible reaction to this situation is to regiment 'is
perfectly natural', 'is more natural than', etc. not
as ordinary predicates--expressions that combine with names or
ordinary variables to form sentences--but as *higher-order
predicates*: expressions that combine directly with ordinary
predicates to form sentences, just as ordinary predicates combine with
names and pronouns. In such a regimentation, we can form well-formed
formulae like 

\[x = y \land \text{AsTallAs}(x,y) \land
\text{MoreNatural}(=, \text{AsTallAs})
\]

 where the symbols '=' and
'AsTallAs' occur without equivocation both as predicates
taking the variables '\(x\)' and '\(y\)' as
arguments, and as the arguments of the higher-order predicate
'MoreNatural'. In neither case are they understood as
names for properties, or as names at all. And in neither case is there
anything like quotation going on: \(`\text{MoreNatural}(=,
\text{AsTallAs})\rsquo\) is no more *about* the expressions
'=' and 'AsTallAs' than 'Every man is
mortal' is about the expressions 'man' and 'is
mortal'.



If one admits this as intelligible, one might also admit uses of
'natural' and 'more natural than' where they
combine with meaningful expressions that are not ordinary predicates
to make sentences. For example, one might allow a comparative
naturalness predicate that combines with two *binary sentential
operators*, making 'MoreNatural(and, or)' well-formed.
Again, this is not supposed to be a claim about the *words*
'and' and 'or'. One might pronounce it as
'Conjunction is more natural than disjunction', but this
may be misleading, given that the English word
'conjunction' seems much more like a name than like a
binary sentential operator. 



A willingness to allow 'natural' to play these
multifarious syntactic roles is a central development in Sider
 (2011).
 (Sider signals this by using the word 'structural'
instead of 'perfectly natural'.) As well as defending the
intelligibility and importance of questions posed in such terms, Sider
defends certain specific claims of the relevant sort: e.g. that
either Structural(and) or Structural(or); that either
Structural([?]) or Structural([?]); that
Structural(Structural); and that it is not the case that
Structural(necessarily).



Dorr and Hawthorne
 (2013)
 advocate a similar flexibility about the syntax of naturalness
claims. They suggest that these claims should be regimented within in
a higher-order logic, where for any syntactic categories
("types") \(\alpha\_1,\ldots ,\alpha\_n\), there is a type
\(\langle\alpha\_1,\ldots ,\alpha\_n\rangle\), expressions of which can
combine with expressions of types \(\alpha\_1,\ldots ,\alpha\_n\) to
form sentences. Using \(e\) for the type of names, an ordinary monadic
predicate is of type \(\langle e\rangle\), so a higher-order predicate
taking such a predicate as an argument is of type \(\langle\langle
e\rangle\rangle\). Sentences (or, more generally, formulas) can be
identified with expressions of type \(\langle\rangle\); so binary
operators like \(\wedge\) ('and') and \(\vee\)
('or') are of type
\(\langle\langle\rangle,\langle\rangle\rangle\). The proposal is to
introduce, for each type \(\alpha\), expressions
'PerfectlyNatural\(\_\alpha\)' of type
\(\langle\alpha\rangle\), 'MoreNatural\(\_\alpha\)' of type
\(\langle\alpha ,\alpha\rangle\), and so on. We can thus formulate
questions like 'Is it the case that MoreNatural
\(\_{\langle\langle\rangle,\langle\rangle\rangle}(\text{and},
\text{or})\)?' Standard higher order languages also admit
variables and quantifiers of all types, so we can also say things like
'\(\neg\exists x^{\langle\langle \rangle
,\langle\rangle\rangle}\text{MoreNatural}\_{\langle\langle\rangle,\langle\rangle\rangle}(x,\wedge)?\)'--'No
binary operation is more natural than conjunction'.



There are some differences between Sider's approach and this
higher-order approach. One is that Sider does not think of
'Structural' as *ambiguous*, despite its ability to
occupy different syntactic roles. One upshot is that for him there is
just a single question 'Structural(Structural)?' where a
higher-order approach would distinguish infinitely many:
'Structural\(\_{\langle\langle
e\rangle\rangle}\)(Structural\(\_{\langle e\rangle})\)?',
'Structural\(\_{\langle\langle\langle
e\rangle\rangle\rangle}\)(Structural\(\_{\langle\langle
e\rangle\rangle})\)?', and so on. Another is that Sider allows
'Structural' to combine with linguistic items that would
not, on the higher-order approach, be meaningful units at all, such as
 variable-binders.[7]
 Indeed Sider suggests that we can make sense of attributions of
structuralness where the argument of 'Structural' is
something like a *grammatical category* rather than a
linguistic
 expression.[8]
 Most importantly, Sider does not have higher order
*quantification*, so his regimentation of 'Conjunction is
perfectly natural' does not extend to claims like
'Conjunction is the only perfectly natural binary
operator'.


### 3.5 Natural magnitudes



There are many properties that have to do with how massive things are:
they include determinate mass properties like *being 17
grams in mass*; more general properties like *having mass*;
relations between objects like *being more massive than*;
relations among properties like *being a greater mass than*;
and many more. But what about *mass* itself? Is it one of the
aforementioned properties? Some authors have suggested that the answer
is no: things like *mass* and *density*--and
perhaps also *shape*, *athletic ability*,
etc.--belong to a sui generis ontological category of
"quantities" or "magnitudes". Each of these
items is associated with certain *values* or *degrees*,
e.g. *three grams* or *three grams per cubic
centimeter*; these may belong to some further sui generis
category. An object can "have" a magnitude
"to" a degree. But the magnitude *mass* is not the
binary relation *object \(x\) has mass to degree \(y\)*.



Given this picture, it is natural to apply 'natural' to
magnitudes as well as properties. For example one might propose that
mass is more natural than density. In a late paper, Lewis
 (2009)
 talks like this (calling magnitudes 'properties that come in
degrees'), although he leaves it open whether they are identical
to properties in our sense. Weatherson
 (2006),
 taking it for granted that magnitudes are not properties, argues that
some magnitudes are perfectly natural, and suggests that there may not
be any perfectly natural
 properties.[9]




However, the background theory of magnitudes as sui generis entities
remains quite underdeveloped. For example, it is unclear what entities
can be the "values" of a magnitude; whether there are
analogues for magnitudes of logical operations like conjunction and
quantification in the realm of quantities; and whether there are
analogues for magnitudes of the comprehension principles
characteristic of "abundant" conceptions of properties. It
is thus not yet clear how much of a difference moving to a
magnitude-based framework would require in theorising about
naturalness.


### 3.6 Natural kinds



Another category whose relation to classes, properties, and concepts
is contested is that of *kinds*. To judge by how we talk, kinds
seem more "concrete" than properties: compare 'I
used to drive that kind of car' versus 'I used to drive
that property' (see Carlson
 1977).
 Further complicating the relationship is the fact that the notion
'natural kind' is the topic of a body of literature large
enough to be the topic of a separate entry
 (natural kinds).
 This literature, which traces back at least to the discussion of
"Kinds" by Mill
 (1843),
 is guided by a set of paradigms rather different from those that
guide the literature on natural properties. For example, following
Mill, it is widely denied that there is a natural kind to which all
and only *white things* belong, on the grounds that they have
nothing *else* important in common other than their color. So
even if one identified kinds with properties, there would be a case
for distinguishing the senses of 'natural' in
'natural property' and 'natural kind' (see
Tobin
 2013).
 Of course this does not rule out analysing one of the notions in
terms of the other: for example, being a very natural kind might
require being coextensive with several different sufficiently natural
properties that are probabilistically correlated given the actual laws
of nature, but not necessarily equivalent (c.f. Boyd
 1991).


## 4. Naturalness and other notions



Lewis
 (1983)
 argues for the theoretical utility of the notion of naturalness by
proposing an array of generalizations linking naturalness to other
topics, including similarity, intrinsicness, materialism, laws of
nature, and meaning. Subsequent authors have offered competing or
complementary proposals. Many proposals, like Lewis's, purport
to "analyse", or provide necessary and sufficient
conditions for, some other property partly in terms of naturalness;
some purport to analyse naturalness in terms of other properties;
others involve other kinds of generalizations. This section will
attempt a survey.


### 4.1 Similarity



Introductions to our topic often emphasise a connection to similarity.
Whereas any two objects, no matter how similar or dissimilar, share
infinitely many properties and are divided by infinitely many
properties, sharing of *natural* properties "makes for
similarity" (Lewis
 1983,
 347;
 1986b,
 60), while division by natural properties "makes for
dissimilarity". These vague generalizations suggest two
projects: that of analysing naturalness in terms of similarity, and
that of analysing similarity in terms of naturalness.



The former project has important historical antecedents. Starting with
a two-place predicate intended to express a specific notion of
similarity (between total experiential states), Carnap
 (1928)
 defines a "quality circle"--roughly, a natural
class--as a class any two members of which are similar, which is
not contained in any other such class. Goodman
 (1951)
 influentially objected to some consequences of this definition (see
the entry on
 Nelson Goodman,
 SS4.2): for example, it implies that whenever \(A\), \(B\), and
\(C\) are quality circles, \((A\cap B)\cup(B\cap C)\cup(C\cap A)\) is
contained in some quality circle (the "difficulty of imperfect
community"). This does not seem plausibly necessarily true,
equating 'quality circle' with 'perfectly natural
class'. 



Subsequent attempts to explain naturalness in terms of resemblance
have used richer inputs. For example, Rodriguez-Pereyra
 (2002)
 defines 'property class' in terms of a three-place
predicate '\(x\) resembles \(y\) to degree \(n\)'. Here
\(n\) is, surprisingly, a *natural* number--the intention
is that the degree of resemblance between two objects should turn out
to equal the total number of "property classes" that
contain
 both.[10]
 Rodriguez-Pereyra's conception of degrees of resemblance is
unusual, since it conflicts with the intuitive thought that everything
resembles *itself* at least as strongly as anything could
resemble anything else. On a more popular approach, the degree of
*dissimilarity* between two objects \(x\) and \(y\) is
represented by a non-negative real number \(d(x,y)\) satisfying the
following "pseudometric space" axioms:

 

\[\begin{align}
\tag{i} d(x,x) &= 0 \\
\tag{ii} d(x,y) &= d(y,x) \\
\tag{iii} d(x,z) &\le d(x,y)+d(y,z).
\end{align}\]


This suggests several possible strategies for defining a notion of
naturalness for classes. One could take into account facts about the
"shape" of a class: for example, whether it is spherical
(such that for some \(x\) and \(y\), \(z\) belongs to it just in case
\(d(x,z)\le d(x,y)\)), or convex (such that whenever \(x\) and \(z\)
belong to it and \(d(x,z)=d(x,y)+d(y,z)\), \(y\) belongs to
 it).[11]
 One could also consider facts about its "size",
e.g. the maximum degree of dissimilarity between any two of its
 members.[12]
 Or one could consider the extent of the gulf between a class and its
complement: the minimum degree of dissimilarity between any member and
any non-member.



If one is willing to help oneself to some notion like 'the
degree of dissimilarity between \(x\) as it is at world \(w\) and
\(y\) as it is at world \(w'\)', one could use similar
strategies to analyse naturalness for properties. However, this notion
of "transworld resemblance" seems to cry out for further
explanation. It is also worth noting that this kind of analysis looks
inhospitable to the idea that the description of the world in terms of
*perfectly* natural properties is in any sense
"complete" (see SS4.4.1 below): indeed, it would be
rather surprising for multiple properties to be *exactly* tied
for first place on any similarity-based measure of naturalness.



The opposite project, of analysing similarity in terms of naturalness,
could be carried out in several ways. Assuming we are concerned with a
relatively intuitive notion of similarity on which everything is
maximally similar to itself, and on which objects can be more or less
similar without sharing any *perfectly* natural properties, the
prospects are best if we invoke a numerical measure of naturalness.
Given such a measure, one might propose the following analysis:



*Division*
  

The degree of dissimilarity between \(x\) and \(y\) is the sum of the
degrees of naturalness of all the properties that divide \(x\) and
\(y\) (i.e., are instantiated by \(x\) but not by \(y\) or by \(y\)
but not by
 \(x\)).[13]




A nice feature of
 *Division*
 is that it automatically validates the pseudometric space axioms
(i)-(iii) above. (By contrast, if we measured the degree of
similarity of \(x\) and \(y\) by summing the degrees of naturalness of
the properties *shared* by \(x\) and \(y\), we would have to
allow some objects--namely, those that instantiate many natural
properties--to be more similar to themselves than others.) But
the approach also faces several problems:


1. It is not obvious that there is a reasonable numerical measure of
naturalness on which the relevant sums are
 well-defined.[14]
2. It is not obvious how to reconcile *Division* with the evident
vagueness and context-sensitivity of expressions like 'more
similar than'. It seems implausible that all of this variation
can be traced back to vagueness and context-sensitivity in 'more
natural than' in such a way that *Division* comes out
true in all contexts. One might appeal here to quantifier domain
restriction: perhaps, in contexts where only psychological similarity
matters, *Division* is true so long as the universal quantifier
over properties is understood as restricted to psychological
properties. However we may need more flexibility than this to allow
both for contexts where psychological similarity matters a lot but
bodily similarity matters a little and for contexts where bodily
similarity matters a lot and psychological similarity matters a
little.
3. *Division* makes it hard to connect similarity to quantitative
differences in the ways one might expect. Consider three triangles A,
B, C, where A is equilateral with side length 1m; B is isosceles with
side lengths 1m, 1m, 1.01m; and C is isosceles with side lengths 1m,
1m, 1.1m. The paradigms we began with (and the
simplicity-of-definition idea, and the connections to mental and
linguistic content discussed
 SS4.7
 below) suggest that for most values of \(x\) other than 1, *being
isosceles with one side \(x\) times as long as the other two* is
much less natural than *being equilateral*. Given this, it
looks like A and B, being divided by *being equilateral*, may
be counted by *Division* as more dissimilar than triangles B
and C are. This contrasts with a more familiar conception of
similarity in this domain, on which the degree of dissimilarity
between a 1,1,1 triangle and a 1,1,\(x\) triangle approaches 0 as
\(x\) approaches 1.
4. Problems arise even with continua of properties where there is no
independent reason to posit different degrees of naturalness. Consider
mass. If we assign all mass-interval properties like *having a mass
between 100 and 105 kg* positive naturalness scores, we will get
the disastrous result that any two objects that differ in mass are
dissimilar to the same, infinite, degree. But if we assign them all a
score of zero, it seems impossible to recover the intuitive judgment
that objects whose masses differed by only 1% could resemble more
closely than any objects whose mass differed by 10% possibly could. We
might hope to thread the needle by saying that most mass-interval
properties score zero but a few (countably many) are natural enough to
get positive scores. But this seems to require invidiously privileging
some units for mass over others, and will still make for
counterintuitive discontinuities in the similarity measure as
discussed under
 (c).[15]



Problem (a) is fairly specific to *Division*, but (b)-(d)
seem potentially challenging for other ways of cashing out the vague
slogan "natural properties make for similarity".


### 4.2 Duplication and intrinsicness



One theoretically important kind of similarity is
*duplication*, or perfect intrinsic qualitative similarity: the
relation that would hold between two peas in a pod if it weren't
for all the microscopic differences in the arrangement of their
molecules, and perhaps holds in fact between certain pairs of hydrogen
atoms or spacetime points. Lewis
 (1986b)
 proposes essentially the following definition:



*Duplication*
  

\(x\) is a duplicate of \(y\) if and only if there is a bijective
function \(f\) from the parts of \(x\) to the parts of \(y\), such
that for any any perfectly natural \(n\)-ary property \(R\) and any
sequence \(x\_1,\ldots,x\_n\) of parts of \(x\), \(x\_1,\ldots,x\_n\)
instantiate \(R\) just in case \(f(x\_1),\ldots,f(x\_n)\) instantiate
 \(R\).[16]




Those who deny that there are any perfectly natural properties (e.g.
on the grounds that for every property we can find a more natural one,
or on the grounds that some *magnitudes*
 (SS3.5)
 are more natural than any properties) will presumably reject the
right-to-left direction of
 *Duplication*,
 to avoid the absurd result that any two objects with equally many
parts are duplicates. This need not, however, put paid to the program
of defining duplication in terms of naturalness. For example, Langton
and Lewis
 (1998)
 suggest a quite different definition in terms of 'more
natural': for details, see SS3.3 in the entry on
 intrinsic vs. extrinsic properties.[17]




One worry about the left-to-right direction of
 *Duplication*
 is based on the fact that, under plausible assumptions, it entails
that duplicate objects must always have exactly the same perfectly
natural
 properties.[18]
 This is in tension with certain unorthodox views about perfect
naturalness, e.g. that specific location properties (e.g. being
*here*) or vector properties (e.g. moving in *this*
direction) are perfectly
 natural.[19]
 Proponents of such views might consider modifying
 *Duplication*
 to allow certain nontrivial permutations of perfectly natural
properties.



Duplication is a useful relation to have a grip on. Lewis uses it in
defining several other things, such as determinism (a deterministic
proposition is one that never divides possible worlds with duplicate
initial temporal segments; determinism is the proposition that the
conjunction of the laws of nature is deterministic) and intrinsicness
(a property is intrinsic just in case it never divides
duplicates).



Note that defining intrinsic properties as those that do not divide
duplicates is hopeless unless it is combined with something like
Lewis's modal realism. *Being a dog-owner* is not
intrinsic; but unless there are vast numbers of people, it is unlikely
that any dog-owner is a duplicate of a non-dog-owner. It will not help
to define the intrinsic properties as those that *could* not
divide duplicates, since *being a duplicate of a dog-owner* is
also not intrinsic. A better idea is to replace the binary relation of
duplication with a four-place relation between two objects and two
possible worlds, in which case we can say that property \(F\) is
intrinsic if and only if, whenever \(x\) as it is at \(w\) is a
duplicate of \(y\) as it is at \(w'\), \(x\) has \(F\) at \(w\) just
in case \(y\) has \(F\) at \(w'\).
 *Duplication*
 can be modified to turn it into an analysis of this four-place
relation:



*Cross-world Duplication*
  

\(x\) as it is at \(w\) is a duplicate of \(y\) as it is at \(w'\) if
and only if there is a one-to-one function \(f\) whose domain includes
everything that is part of \(x\) at \(w\), and whose range includes
everything that is part of \(y\) at \(w'\), such that for any sequence
\(x\_1,\ldots,x\_n\) of objects that are parts of \(x\) at \(w'\) and
perfectly natural \(n\)-ary relation \(R\), \(x\_1,\ldots,x\_n\)
instantiate \(R\) at \(w\) if and only if \(f(x\_1),\ldots,f(x\_n)\)
instantiate \(R\) at
 \(w'\).[20]




One objection to this package concerns *non-qualitative*
properties: properties like *being identical to Obama* and
*having Obama as a part* are not counted as intrinsic.
Francescotti
 (1999),
 Eddon
 (2011),
 and others argue that this is
 wrong.[21]
 Another objection is that the analysis does not allow an intrinsic
property to be necessarily coextensive with an extrinsic one. Those
who hold that there are distinct but necessarily extensive qualitative
properties may prefer a more discriminating account of intrinsicness
on which, for example, *being spherical* is intrinsic while
*being spherical and either a home-owner or not a home-owner*
is
 not.[22]



### 4.3 Qualitative indiscernibility and qualitativeness



Another theoretically important specialised similarity relation is
*qualitative indiscernibility*. Qualitative indiscernibility is
more demanding than duplication: whereas duplicates can differ
extrinsically, qualitatively indiscernible objects must also match in
their relations to their environments, so that in a non-symmetric
universe, no two things are qualitatively indiscernible.



*Duplication*
 can be turned into an analysis of qualitative indiscernibility by
including all objects, not just the parts of the relevant objects
\(x\) and \(y\), in the domain of the function \(f\):



*Indiscernibility*
  

\(x\) is qualitatively indiscernible from \(y\) if and only if there
is a one-to-one, onto function \(f\) whose domain and range include
all objects, such that \(f(x)=y\), and for any sequence \(x\_1,\ldots
,x\_n\) of objects and any perfectly natural \(n\)-ary property \(R\),
\(x\_1,\ldots ,x\_n\) instantiate \(R\) just in case \(f(x\_1),\ldots
,f(x\_n)\) instantiate \(R\). 



One immediate worry about the left-to-right direction of
 *Indiscernibility*
 is that according to standard set theory there *are* no
functions whose domain and range include all objects (assuming sets
count as
 "objects").[23]
 There are various possible ways around this problem. For example, one
might replace the quantifier over "functions" with a
plural quantifier over ordered pairs, or a higher-order quantifier
binding a variable in dyadic predicate
 position.[24]




One significant worry about the right-to-left direction of
 *Indiscernibility*
 arises for those who posit *higher-order* perfectly natural
properties, whose instances are properties or propositions rather than
objects. For example, someone who posited that *thinking* was a
perfectly natural relation between people and propositions, modally
independent of the perfectly natural properties instantiated only by
objects, one would likely want to rule out the possibility that there
are qualitatively indiscernible people \(x\) and \(y\) such that \(x\)
thinks that \(x\) is hungry, \(y\) does not think that \(y\) is
hungry.
 *Indiscernibility*
 does not obviously do this. 



Lewis
 (1986b)
 briefly suggests a quite different analysis of qualitative
indiscernibility: \(x\) is qualitatively indiscernible from \(y\) just
in case \(x\) and \(y\) share the same "somewhat natural"
 properties.[25]
 If we take the "somewhat natural" properties to be those
with finite definitions in terms of perfectly natural properties,
sharing of these properties will follow from indiscernibility as
defined by
 *Indiscernibility*;
 however the converse implication can fail when there are infinitely
many objects.



Once we have the notion of qualitative indiscernibility, we can use it
to analyse the notion of a *qualitative property*. Intuitively,
a qualitative property is one that is not "about" any
individual: *admiring a famous philosopher* is qualitative,
while *admiring David Lewis* is not. For Lewis, a monadic
property \(F\) is qualitative if and only if whenever \(x\) is
qualitatively indiscernible from \(y\), \(x\) has \(F\) just in case
\(y\) has \(F\). As in the case of duplication, this analysis is
tenable only given a super-abundant ontology like Lewis's.
Non-modal realists may, however, adapt the idea by using a
world-relativised relation of qualitative indiscernibility, whose
definition stands to
 *Indiscernibility*
 as
 *Cross-World Duplication*
 stands to
 *Duplication*.[26]




Like the definition of intrinsicness in terms of duplication, the
definition of qualitativeness in terms of qualitative indiscernibility
fits most naturally with a view where necessarily equivalent
properties are always identical, since it will not allow us to say,
e.g., that *being red* is qualitative but *being red and
either admiring or not admiring Lewis* is non-qualitative. For
proponents of finer-grained theories of properties who want to make
such distinctions, the strategy of analysing qualitativeness in terms
of qualitative indiscernibility seems unpromising. But the project of
analysing qualitativeness in terms of perfect naturalness remains on
the table. For example, invoking the notion of "full
definability" discussed in
 SS4.10
 below, one might define a qualitative property as one fully definable
from perfectly natural
 properties.[27]




These analyses have noteworthy consequences that could be maintained
even if the project of analysing 'qualitative' were given
up. One is that all perfectly natural properties are qualitative.
Another is that all qualitative properties supervene on the perfectly
natural properties, in the following sense:



*Qualitative Supervenience*
  

If \(f\) is a bijection from the objects existing at \(w\) to the
objects existing at \(w'\) such that for every perfectly natural
\(n\)-ary property \(R\) and objects \(x\_1,\ldots ,x\_n\), \(x\_1,\ldots
,x\_n\) instantiate \(R\) at \(w\_1\) iff \(f(x\_1),\ldots ,f(x\_n)\)
instantiate \(R\) at \(w\_2\), then for every qualitative \(m\)-ary
property \(S\) and objects \(y\_1,\ldots ,y\_m\), \(y\_1,\ldots ,y\_m\)
instantiate \(S\) at \(w\_1\) iff \(f(y\_1),\ldots ,f(y\_m)\) instantiate
\(S\) at
 \(w\_2\).[28]




*Qualitative Supervenience*
 will of course not be acceptable to those who deny that there are any
perfectly natural properties. Some specific putative counterexamples
to *Qualitative Supervenience* have also been offered. For
example, Dasgupta
 (2013)
 suggests that is possible for the same objects to instantiate the
same fundamental properties even while everything is twice as massive
as it in fact is. Since Dasgupta also seems to assume that properties
like *being one kilogram in mass* to be qualitative, these
possibilities seem to involve a failure of
 *Qualitative Supervenience*.[29]



### 4.4 Modality


#### 4.4.1 Completeness



Lewis took it as axiomatic that "truth is supervenient on
being", and explained this slogan--borrowed from Bigelow
 (1988)--as
 follows:



> 
> 
> If two possible worlds are discernible in any way at all, it must be
> because they differ in what things there are in them, or in how those
> things are. And "how things are" is fully given by the fundamental,
> perfectly natural, properties and relations that those things
> instantiate.
> 
> 
> 



Spelling this out, we get: 



*TSB*
  

For any possible worlds \(w\) and \(w'\), if \(w\) and \(w'\) agree as
regards what things there are, and for any \(n\), any perfectly
natural \(n\)-ary property \(R\), and any \(n\) objects
\(x\_1,\ldots,x\_n\) that exist at \(w\) and \(w'\), the proposition
that \(R(x\_1,\ldots,x\_n)\) is true at \(w\) if and only if it is true
at \(w'\), then exactly the same propositions are true at \(w\) and
 \(w'\).[30]




We can get across the same idea without mentioning possible worlds:




Necessarily, every true proposition is metaphysically necessitated by
some collection of true propositions each of which is either (a) the
proposition that \(R(x\_1,\ldots ,x\_n)\) or \(!R(x\_1,\ldots
,x\_n)\), for some perfectly natural \(R\) and objects \(x\_1,\ldots
,x\_n\) are any objects; (b) the proposition that \(\exists y(y=x)\),
for some object \(x\), or (c) the proposition that every object is one
of \(xx\), where \(xx\) are all the objects there
 are.[31]




TSB
 follows from
 *Qualitative Supervenience*
 (see
 SS4.3)
 together with a less controversial principle we might dub
'Truth Supervenes on the Qualitative', derived from
 TSB
 by replacing 'perfectly natural' with
'qualitative'. It is thus unsurprising that putative
counterexamples to
 *Qualitative Supervenience*
 extend to
 TSB.
 For example, if one took *being left-handed* to be a
qualitative property that need not be preserved by bijections that
preserve perfectly natural properties, one might plausibly think that
worlds could agree on which objects there are and which perfectly
natural properties they instantiate without agreeing on which ones are
left-handed. Similarly, if one posits higher-order perfectly natural
properties which do not supervene on the first-order ones, one should
have doubts about
 TSB,
 assuming one interprets
 TSB's
 talk of "objects" in a way that does not include objects.
However in that case one might hold on to a variant of
 TSB
 that gives the same status to all propositions \(R(X\_1,\ldots,X\_n)\)
where \(R\) is perfectly natural (of whatever type) and
\(X\_1,\ldots,X\_n\) are of the right types to be arguments to \(R\).




TSB
 can be understood as an attempt to pin down the inchoate thought that
the description of the world in terms of perfectly natural properties
is "complete" or "exhaustive". Sider
 (2011)
 argues that
 TSB
 fails to capture the intuitive idea of completeness, primarily on the
grounds that 'it imposes no meaningful requirement of
completeness for necessary truths'.
 SS4.10
 and
 SS4.11
 below will consider some different strategies for capturing the
thought, which are plausibly taken to be stronger than
 TSB.
 Sider's own preference is to cash out the thought in a
metalinguistic way, as the thesis that "every sentence that
contains expressions that do not carve at the joints has a
metaphysical semantics". Unfortunately there is no space to
explain the novel conception of "metaphysical semantics"
Sider is appealing to; however it is worth mentioning that Sider does
not take his thesis about metaphysical semantics to imply
 TSB.


#### 4.4.2 Independence



According to Lewis
 (1986b:
 60), "there are only just enough [perfectly natural properties]
to characterise things completely and without redundancy". This
remark suggests that the perfectly natural properties constitute a
*minimal* supervenience base for all truths: if \(R\) is a
perfectly natural property, the result of replacing 'perfectly
natural property' with 'perfectly natural property distinct
from \(R\)' in
 TSB
 and/or
 *Qualitative Supervenience*
 is
 false.[32]
 Of course one way for this to be true would be for
 TSB/*Qualitative Supervenience*
 to already be false. But there is an idea in the vicinity that is
logically independent of
 TSB
 and
 *Qualitative Supervenience*,
 namely that *no perfectly natural property supervenes on all the
rest*. If we take the relevant sense of 'supervene' to
be the one in play in
 *Qualitative Supervenience*,
 this amounts to: 



*Minimality*
  

For each perfectly natural \(n\)-ary property \(R\), there are
possible worlds \(w\) and \(w'\), objects \(x\_1,\ldots,x\_n\) existing
at \(w\), and a bijection \(f\) from objects existing at \(w\) to
objects existing at \(w'\) such that \(R(x\_1,\ldots,x\_n)\) at \(w\),
not \(R(f(x\_1),\ldots,f(x\_n))\) at \(w'\), and for every perfectly
natural \(m\)-ary property \(S\) distinct from \(R\) and objects
\(y\_1,\ldots,y\_m\) existing at \(w\), \(S(y\_1,\ldots,y\_m)\) at \(w\)
if and only if \(S(f(y\_1),\ldots,f(y\_m))\) at \(w'\).



If we take the relevant sense of 'supervene' to be the one
in play in
 *TSB*,
 we get something a bit stronger, where the worlds \(w\) and \(w'\)
are required to be ones at which the same objects exist and the
function \(f\) is required to map every object to itself. 



*Minimality*
 entails that if a binary relation \(R\) is perfectly natural, its
converse--*being an \(x\) and \(y\) such that
\(R(y,x)\)*--is not also perfectly natural. This seems oddly
arbitrary: it is tempting to think that, instead, converse relations
are always equally natural. One could avoid this objection by
weakening
 *Minimality*,
 replacing 'distinct from \(R\)' with 'distinct from
\(R\) and its converse', or more generally 'distinct from
all of the *permutations* of \(R\)'. More ambitiously,
one could develop a theory of "neutral" relations (Fine
 2000),
 understood as entities for which distinctions like the one between a
relation and its converse does not arise, and treat 'perfectly
natural' as a predicate of neutral relations.



*Minimality*
 also implies that the *negation* of a perfectly natural
property is never perfectly natural. This does not so obviously raise
worries about arbitrariness. Nevertheless, the relation between a
relation and its negation is structurally quite similar to the
relation between a relation and its converse--both seem to
involve a kind of "interdefinability"--so there are
some attractions to saying that the negation of a perfectly natural
property is always perfectly natural, and more generally that the
negation of any property is equal to it in naturalness. This has been
suggested by Plate
 (2016)
 and Dorr
 (2016).[33]




A different kind of putative counterexample to
 *Minimality*
 involves perfectly natural properties which are "rigid"
in the sense that whatever instantiates or fails to instantiate them
does so necessarily. If
 TSB
 is true and \(F\) is rigid,
 TSB
 remains true when 'perfectly natural property' is
replaced by 'perfectly natural property distinct from
\(F\)'. For example, if one held that every electron is
necessarily an electron (if it exists) and that every non-electron is
necessarily not an electron (if it exists), minimality would rule out
claiming that *being an electron* is perfectly natural. Eddon
 (2013)
 argues that certain rigid *relations among properties* are
perfectly natural, and thus make trouble for minimality: one of her
examples is the three-place *being the sum of* relation that
holds, for example, between the determinate mass properties *being
exactly 5kg in mass*, *being exactly 7kg in mass*, and
*being exactly 12kg in mass*.



Finally, some have argued against
 *Minimality*
 on the grounds that certain "higher-level" properties are
perfectly natural, despite supervening on perfectly natural
"lower-level" properties. For example, Bennett
 (2017)
 argues that the property of being a "schmoton"--a
certain kind of complex particle--might be perfectly natural,
despite supervening on other perfectly natural properties: her
argument is based on a list of criteria she takes to be jointly
sufficient for perfect naturalness, including "figuring in
fundamental laws" and being such that any two of one's
instances are
 duplicates.[34]




*Minimality*
 allows for a wide variety of interesting modal connections within the
class of perfectly natural properties, so long as they do not take the
specific form of a supervenience claim. Many authors have been
attracted towards much more sweeping claims of modal independence,
often inspired by the Humean slogan "no necessary connections
between distinct existences". Lewis
 (1986b)
 defends a "principle of recombination" (see the entry on
 possible worlds),
 which entails, for example, that any two possibly-instantiated
conjunctions of perfectly natural properties are possibly instantiated
in the same world. Depending on how the principle is precisified, it
may also entail that for any two possibly-instantiated conjunctions of
perfectly natural monadic properties \(F\) and \(G\) and any perfectly
natural binary relation \(R\) it is possible that an instance of \(F\)
bears \(R\) to an instance of
 \(G\).[35]
 Some particularly far-reaching claims of this sort go by the name of
"combinatorialism": the idea is that all logically
possible patterns in which any properties might be distributed are
possibly instantiated by any given perfectly natural
 properties.[36]
 For example, since it is logically possible for there to be a monadic
property with exactly seven instances and a binary relation that holds
between any two of those instances, each perfectly natural property
\(F\) and perfectly natural binary relation \(R\) are such that it is
metaphysically possible for \(F\) to have exactly seven instances and
\(R\) to hold between any two of
 them.[37]




If some kind of combinatorialism is true, it could be a powerful tool
for figuring out whether properties are perfectly natural, since many
collections of properties that might otherwise seem good candidates to
be perfectly natural seem to be involved in the kinds of necessary
connections which, according to combinatorialism, never obtain among
perfectly natural properties. The worry for those who want to use
arguments like this to eliminate certain candidate collections of
perfectly natural properties is that *all* the otherwise
acceptable candidate collections might turn out to involved in such
apparent necessary connections. If so we will face a choice between
giving up combinatorialism and rejecting the relevant prima facie
modal judgments. To illustrate the difficulties, we can return to the
example of mass. Insofar as one takes physics as a guide to the
perfectly natural, there is pressure to think that some perfectly
natural properties have to do with mass. There are a great many
possible candidates, including:


1. Maximally specific monadic mass properties of objects, like
*having a mass of 3.12
 grams*[38]
2. Less specific monadic mass properties of objects, like *having
a mass between 3.12 and 4.03 grams*, or just *having mass*
(Wilson
 2012)
3. Binary relations between objects and real numbers, like *being
the mass in grams of* or *being the mass in kilograms
of*
4. More-than-two place relations between objects and numbers, like
the "mass ratio" relation, *\(x\) is \(n\) times as
massive as \(y\)*
5. Binary relations among objects, like *being the same mass
as*, *being greater in mass than*, *being 3.12 times as
massive as*, or *being between 3.12 and 4.03 times as massive
as*.
6. More-than-two place relations among objects, like *\(z\) is as
massive as \(x\) and \(y\) put together*, *\(z\) is
intermediate between \(x\) and \(y\) in mass*, or *\(w\) and
\(z\) stand in the same mass-ratio as \(x\) and \(y\)*
7. Relations among properties, like *F is a maximally specific
mass property that is the sum of maximally specific mass properties G
and H* (see Mundy
 1987,
 Eddon
 2013).
8. Relations linking mass to other scales, like *the mass of
\(x\) equals the distance between \(y\) and \(z\) in units where the
speed of light and Newton's gravitational constant are both
 1*.[39]



Within each of the candidate families, one can formulate certain laws
that might plausibly be claimed to be metaphysically necessary. For
example, it is tempting to suppose that it would be metaphysically
(rather than merely nomically) impossible for an object to be both one
gram in mass and two grams in mass, or for one object to be both equal
in mass to another and twice as massive as it. But if we combine all
these judgments with combinatorialism, we will have to deny that any
of the families we have considered consists entirely of perfectly
natural
 properties.[40]
 This is not out of the question: one can imagine many ways in which
mass-theoretic properties might supervene on perfectly natural
properties we do not yet have words for, in a manner consistent with
the relevant modal judgments. However, precisely because there are so
many ways this might be achieved, this response threatens to lead to a
sort of scepticism about our ability to know what the perfectly
natural properties are, or how the properties we have words for
supervene on them. Combinatorialists thus have reason to be open to
the idea that some of the relevant laws might turn out, surprisingly,
to be metaphysically contingent (though they may still be nomologcally
necessary). 


### 4.5 Universals and tropes



Armstrong
 (1978)
 developed an influential theory of "universals" (a
category he subdivides into properties and relations) which was
distinctive for its emphasis on the idea that universals are
"sparse" by comparison with predicates: for example,
Armstrong refuses to posit disjunctions of universals. Lewis
 (1999,
 1) credits Armstrong with convincing him that the concept of
naturalness is "commonsensical and serviceable". Lewis is
open to a view that posits Armstrong-style universals in addition to
properties and relations conceived of as "abundant", and
analyses 'perfectly natural property' as 'property
corresponding to some universal', i.e. as 'property
that is the property of *instantiating \(u\)* for some
universal \(u\)'. 



This analysis has some surprising consequences when combined with
certain other commitments of Armstrong's view. For example,
Armstrong holds that any two universals have a conjunction, which he
identifies with their mereological fusion. Given the proposed
analysis, this entails that the conjunction of any two perfectly
natural properties is itself perfectly
 natural.[41]
 This conflicts with the claim that no perfectly natural property
supervenes on all the rest. If one wanted to hold on to that claim,
one might instead analyse 'perfectly natural property' as
'property corresponding to some mereologically simple
 universal'.[42]
 Note however that for Armstrong it is an open question whether there
are any mereologically simple universals, and thus whether the
properties corresponding to mereologically simple universals form a
supervenience base for everything.



Armstrong also posits a wide array of "structural"
universals: for example, given any universal property \(u\_1\) and
binary universal relation \(u\_2\), Armstrong would hold that if there
are any fusions of two non-overlapping objects individually
instantiating \(u\_1\) and jointly instantiating \(u\_2\), there is a
universal \(u\_3\) that is necessarily instantiated by all and only
such fusions. If the perfectly natural properties are those that
correspond to universals, this view will lead to a great profusion of
perfectly natural properties. This fits poorly with intuitive
judgments of relative naturalness like those elicited by our initial
examples, and threatens to disrupt several of the theoretical roles
for naturalness (Sider
 1995,
 3). Lewis
 (1986a)
 argues (not on those grounds) that theorists of universals should not
posit structural universals.



Lewis was also open to the program of analysing naturalness in terms
of *tropes*--"particularised property
instances" like *Socrates' paleness* or
*Antony's love of Cleopatra* (see the entry on
 tropes).
 Lewis suggests that given a "sparse" theory of tropes
(which would deny the existence of tropes like *Socrates's
being either pale or flushed*) and a modal realist ontology, one
could define a perfectly natural property as one such that, for some
maximal set of duplicate tropes, its instances are exactly the things
that instantiate some member of that set (in the sense of
'instantiate' on which Socrates instantiates
*Socrates' paleness*). To avoid circularity,
'duplicate' here had better not be analysed in terms of
sharing of perfectly natural properties. By contrast with the case of
universals, it is not so easy for opponents of Lewisian modal realism
to take over this account. The most straightforward approach would
replace the dyadic 'duplicate' with a four-place predicate
'\(t\_1\) in \(w\_1\) is a duplicate of \(t\_2\) in \(w\_2\)';
but treating this as primitive seems even more unsatisfactory than
treating the dyadic 'duplicate' as primitive.


### 4.6 Laws of nature



One of Lewis's signature applications of the notion of
naturalness is in the analysis of the notion of a *law of
nature*, a notion of great importance for the philosophy of
science and for the theory of causation and counterfactuals. In brief,
Lewis's proposal is that it is a law of nature that \(P\) just
in case the proposition that \(P\) belongs to every "best
system", where a system is a set of true propositions closed
under necessitation, and the "goodness" of a system is a
combination of two factors: *simplicity* and
 *strength*.[43]
 The relevant notion of simplicity can, according to Lewis, be
understood in terms of
 naturalness.[44]
 The idea is that the simplicity of a system is a matter of the
*syntactic* simplicity of the its simplest axiomatisation,
where an axiomatisation of a system is a set of sentences which
express propositions which jointly entail exactly the propositions in
the system, on some interpretation where their non-logical constants
all stand for perfectly natural
 properties.[45]




Note that, given that Lewis also analyses relative naturalness in
terms of perfect naturalness and syntactic simplicity, he could just
as well have identified the degree of simplicity of a system with the
degree of naturalness of the strongest proposition in the system
(i.e. the conjunction of all the elements of the system). This
version of the account may be preferable to those who reject the
program of analysing relative naturalness in terms of perfect
naturalness.



For discussion of the motivations for this "best system"
analysis and its discontents, see the entry on
 laws of nature.
 Especially noteworthy here are objections which specifically target
the role of *naturalness* in the analysis. The most prominent
such objection (van Fraassen
 1989;
 Loewer
 (2007);
 Cohen and Callender
 2009)
 is that the account will make it impossible to know what the laws
are, because it will allow for the possibility that even scientists
who arrive at true theories by impeccable reasoning nevertheless
go wrong in taking the axioms of these theories to be laws,
because they are mistaken in thinking that the properties expressed by the
basic predicates of these theories are highly natural. This is a hard
objection to make stick, since the form of inference involved --
from the possibility of error about some question to the conclusion
that we are not in a position to know the answer to that question
-- leads, when generalized, to a rampant scepticism.
Non-sceptical epistemologists tend to think that possibilities of
error need to meet some further condition of "closeness"
or "relevance" in order to undermine knowledge; but it is
unclear why one would think that this condition is met in the present
case. As we will see in
 SS4.7
 below, natural properties are widely held to be easier to refer to,
ceteris paribus. So, arguably, a situation where people ended up with
true scientific theories stated in a language whose basic predicates
were too unnatural for the axioms to meet the bar for lawhood would
have to be quite remote and unusual.



If one rejects not just the details but the general program of
analysing lawhood in terms of naturalness, one might instead consider
an analysis of naturalness in terms of lawhood: "natural
properties are the ones that figure in laws of nature" (Lewis
 1983,
 344). (If one is analysing *perfect* naturalness, one might
want to specify that the laws should be "basic" rather
than "derived".) A central challenge for working out this
program is to explain what 'figure in' means. To get
anything like the desired extension for 'natural', we need
an extremely fine-grained theory of propositions. For example, unless
we are comfortable with the consequence that *being either
non-\(F\) or \(G\)* is perfectly natural whenever the proposition
that everything \(F\) is \(G\) is a law, we will need to deny that
that property "figures in" that proposition. This will
plausibly require denying that the property is identical to the
proposition that everything is either non-\(F\) or \(G\), and denying
that the latter proposition is a law despite its necessary equivalence
to the former. As we will see in
 SS4.10
 below, there are formidable obstacles to developing a consistent
theory of properties and propositions that is fine-grained in the
needed
 ways.[46]
 The thesis that only perfectly natural properties "figure
in" (basic) laws of nature also faces other objections. For
example, Hicks and Schaffer
 (2017)
 argue against it on the grounds that, for example, acceleration
figures in a basic law, namely \(F=ma\), but is not perfectly
 natural.[47]



### 4.7 Mental and linguistic content



Perhaps the most distinctive of the roles naturalness plays for Lewis
is the theory of mental and linguistic content. This role is sometimes
gestured at by saying that natural properties are "reference
magnets" (an expression borrowed from Hodes
 1984):
 the more natural a property it is, the easier it is for thinkers to
end up referring to it (or expressing it or thinking about it).
Reference to natural properties is something that happens "by
default"; the less natural a property is, the more specific and
finely-tuned the conditions required to refer to it.



The most widely-discussed way of fleshing out this picture takes the
form of a theory of linguistic meaning which Lewis
 (1983, 1984) presents as a sort
of toy theory, not really true but close enough to the truth for certain dialectical purposes.
 This theory takes the form of an account of what it is for an
*interpretation function* to be *admissible*
("correct", "intended") for a given
population. An interpretation function is a mapping from linguistic
expressions to semantic values of appropriate kinds (e.g. mapping
predicates to properties and sentences to propositions, or to
functions from contexts to propositions). According to the account, an
interpretation function is admissible for a population just in case it
achieves the best overall balance of "use" and
"eligibility". An interpretation function's
"use" (or "charity") score is higher the more it maps sentences that
members of that population are strongly and stably disposed to assert to true
propositions. An interpretation function's
"eligibility" score is higher the more natural the
semantic values it assigns to words (and the simpler its rules for
assigning semantic values to complex expressions). Sometimes there
will be ties for first place, so that multiple interpretation
functions will be admissible for a population. In such cases--and
perhaps also in cases of near-ties--the population's
language contains *semantic
 indeterminacy*.[48]




Lewis's reason for focusing on this toy theory involves a debate
with Putnam
 (1980).
 Putnam had argued that the only admissible interpretations of a
population in possession of an "empirically ideal" theory
are interpretations under which that theory is true (see entry
 challenges to metaphysical realism,
 SS3.5). As Lewis reconstructs Putnam's argument, his
premise is a theory similar to the toy theory except that
"use" is the only admissibility-determining factor. Lewis
offers the use-plus-eligibility theory as a minimal fix, and as an illustration of the
way in which the concept of naturalness can be a helpful resource in
formulating reductive theories of content which avoid strange
 consequences.[49]



Lewis's actual theory of content is more complex, and has to be
reconstructed from several sources. It has three stages, in each of
which naturalness plays some role. The first is an account of what it
is for a person to have certain beliefs and desires, or to be more
exact, a certain credence function (mapping properties to real numbers
in the unit interval) and valuation function (mapping maximally
specific properties to real
 numbers).[50]
 The second is an account of what it is for a certain
"Language" (function from sentences to propositions) to be
"used by" (admissible for) a given population. The third
stage extends the second stage to interpretation-functions that map
arbitrary expressions (not just sentences) to semantic values of the
appropriate type. At each stage, the resources employed include those
analysed at the previous stage.



Lewis
 (1983)
 presents the following account of the first stage: for \(C\) and
\(V\) to be, respectively, the prior credence and value function of a
certain agent \(a\) is for \(C\) to be a function from properties to
real numbers in the unit interval, and \(V\) to be a function from
maximally specific properties to real numbers, which together achieve
an optimal balance of *fit*--being such that \(a\) is
disposed, given evidence \(E\), to perform some action with a maximal
expected value of \(V\) in \(C(\bullet\mid E)\)--and certain
other desiderata. One of these other desiderata (and the only one
Lewis mentions) is eligibility: "the properties the subject
supposedly believes or desires or intends himself to have"
should not be too severely unnatural (1992: 375). A difficulty with
this formulation is that it is not obvious how to extract "the
properties the subject supposedly believes or desires or intends
himself to have" from a given \(C\) and \(V\). After all, any
\(C\) that obeys the axioms of the probability calculus will have to
assign values close to 1 to properties expressed by arbitrarily long
formulae (Dorr and Hawthorne
 2013,
 28; Schwarz
 2014,
 25). Also, insofar as "fit" is just a matter of the
subject's behavioural dispositions, the account requires
revision in the light of the anti-behaviourist views expressed in
Lewis's other writings on the philosophy of mind, for example
Lewis
 1980.[51]




At the second stage (sentence-level linguistic meaning), Lewis
 (1969;
 1975) had developed an analysis on which a Language is used
by a population just in case certain *conventions* prevail in
that population, for example a convention not to utter a sentence
unless one believes the proposition to which it is mapped by that
Language; semantic indeterminacy arises when the conventions are less than fully specific. Claims about conventions were in turn analysed in terms of
patterns of credences and preferences among the members of the
population. At the third stage (subsentential meaning), Lewis
 (1975)
 proposed that an interpretation function (or "grammar")
is admissible for a population just in case it is a "best"
interpretation whose restriction to sentences is a Language the
population uses. This paper suggests that the relevant notion of
"bestness" may be lacking in clarity and objectivity, and
that there will typically be many different best grammars, with the
result that semantic indeterminacy at the subsentential level is
pervasive. After embracing naturalness in Lewis
 1983,
 Lewis would presumably have cashed out the "bestness"
relevant for the third stage in naturalness-theoretic terms, and
consequently been less pessimistic about this part of the program, although
he never spells this out explicitly.
Lewis (1992)
 also modifies his account of the second stage, appealing to
naturalness to address a worry raised by Hawthorne
 (1990).
 The worry was that the original theory would assign incorrect
interpretations to certain sentences where--e.g. because they are
too long--hearing someone assert them would not tend to increase
hearers' confidence in the proposition that is in fact the
correct interpretation. The fix gives some admissibility-constituting
weight to eligibility (naturalness), to allow for extrapolation from
the core cases covered by the convention-constituting pattern of
credences and preferences.[52]



The final package at stages two and three is thus structurally similar
to the toy "charity plus eligibility" theory, in that admissibility of an interpretation for a
population is a matter of achieving the best overall score on several
factors, one of which is eligibility; the key difference is that the
other factor is a matter not of mapping sentences people are disposed to accept to
truths, but rather of mapping all sentences, accepted or not, to propositions for which the
appropriate conventions prevail. The final package also plausibly yields
many of the characteristic predictions of the toy theory, e.g. that we should 
expect widespread semantic indeterminacy in domains where no
candidate meaning is much more natural than the rest (see, e.g., Sider 2001).[53]
 On the other hand, some prominent
objections to the toy theory do not carry over to the final package.
For example, Hawthorne
 (2007),
 picking up on a case due to Williams
 (2007a),
 points out that the toy theory entails that it would be very easy for
an Earth-bound population to end up speaking a language all of whose
names denote things on some distant planet, since if the planet was a
sufficiently good match for Earth as we take it to be, such an
interpretation could do better on the score of charity than an
Earth-centric one, without any sacrifice in eligibility. But in the
final package, the shifted interpretation can easily be rejected on
the grounds that we do not use the Language it determines, since we do
not have, or expect one another to have, specific beliefs about how
things are on distant planets. This is true in turn because some
relations we bear to \(\langle C,V\rangle\) pairs where \(C\) is
opinionated about how things are nearby and \(V\) cares about such
things are more natural than any relations we bear to pairs that are
unopinionated and indifferent about nearby matters but opinionated and
concerned about distant planets.



The rich body of theory just described has been highly
controversial, both in detail and in its overall
 architecture.[54]
 However, the general thought that more natural properties are,
ceteris paribus, easier to refer to than less natural properties can
be considered separately from any program for analysing the notions of
mental or linguistic content. Looking for a relatively noncommittal
thesis in this vicinity, Dorr and Hawthorne
 (2013)
 suggest cashing out 'easy to refer to' in terms of a
physical measure over nomic possibilities, of the sort that features
in sciences like statistical mechanics (for more details see Dorr and
Hawthorne
 2014).
 The thought is that ceteris paribus, the more natural a property is,
the greater the measure of the set of nomically possible worlds where
it is referred to (or where it is determinately referred to, or
thought about, or something like that). For example: if there are
intelligent beings elsewhere in the universe, there is a far greater
chance that they have a word that means *triangle* than that
they have a word that means *three-to-five sided figure none of
whose sides is more than one and half times as long as any other*.
This generalization is not tied up with any general theory of mental
or linguistic content, although it does rest on the assumption that
mental and semantic facts nomically supervene on physical
 facts.[55]



### 4.8 Ampliative inference



Goodman
 (1954)
 noticed an epistemologically interesting contrast between the
proposition that all emeralds are green and the proposition that all
emeralds are grue (where 'grue' was defined to mean
'either green and observed before \(t\) or blue and not observed
before \(t\)', for some fixed time \(t\)). Although Goodman
himself does not speak in these terms, one apparent contrast relates
to *rationality*: if one gathered many emeralds before \(t\)
and observed all to be green (and therefore also grue), it would be
rational to become confident that all emeralds are green, whereas it
would not be rational in the same circumstances to become confident
that all emeralds are
 grue.[56]
 Philosophers who talk about naturalness have generally seen some
connection between this contrast and the fact that greenness is more
natural than grueness. However, there has been little consensus about
how this connection should be worked out.



Following Goodman, the contrast is often summarised by saying that
'green' or greenness is "projectible", and
'grue' or grueness is not. Projectibility is also thought
of as coming in degrees, the idea perhaps being that a generalization
about a more projectible property requires less evidence to make a
given level of confidence rational than the corresponding
generalization about a less projectible property. The difficulty of
making this adequately precise, let alone of working backwards from an
assignment of degrees of projectibility to a description of the
rational response(s) to some given evidence, should not be
underestimated (see Earman
 1985
 for some distinctions). But insofar as we are willing to theorise in
terms of both projectibility and naturalness, we can consider the
simple proposal that a property's degree of projectibility is
equal to its degree of naturalness. One worry about this (see
Weatherson
 2013)
 is that one might want to say for other reasons that *being
green* is much less natural than, say, *being electrically
charged*, whereas there is no obvious sense in which it would take
less evidence to be rationally convinced that all emeralds are
electrically charged than that all emeralds are green.



Another approach links Goodman's contrast to the venerable idea
that simpler hypotheses are preferable to more complex hypotheses.
Induction can be understood (Harman
 1973;
 White
 2005),
 as a special case of "inference to the best explanation",
where being "best" is partly a matter of simplicity: the
suggestion is that the hypothesis (i.e. the proposition) that all
emeralds are green is simpler than the hypothesis that all emeralds
are grue. The degree of simplicity of a proposition might simply be
identified with its degree of naturalness (see
 SS3.3
 above).[57]



On a third approach, inductive inferences are reconstructed as
episodes of Bayesian conditionalisation (see the entry on
 Bayesian epistemology),
 so that the grue-green contrast is traced back to the fact that some
*prior credence functions* are more rational than others. In
this setting, the most straightforward way to bring in naturalness
would be to simplify identify the ideally rational prior credence
functions with the maximally natural functions (within the class of
functions meeting certain formal constraints, such as the axioms of
probability
 theory).[58]
 One might also take naturalness to be merely one of several factors
contributing to rationality. Or one could appeal to it in some other
way: for example, Dorr and Hawthorne
 (2013)
 suggest that reasonable prior credence functions might be weighted
averages of all sufficiently natural probability functions, in which
more natural probability functions receive higher weights.



One difficulty for all these ways of invoking naturalness in the
theory of rationality is that, given the popular doctrine of
externalism about the content of belief, according to which duplicate
thinkers in different environments can believe different propositions
(see the entry on
 externalism about mental content),
 they lead to the much less popular doctrine of externalism about
rationality, according to which duplicate thinkers can differ as
regards the rationality of the inductive inferences they are engaging
in. (For some arguments against externalism about rationality, see
Wedgwood
 2002).
 For example, the discovery (by nineteenth-century mineralologist
Alexis Damour) that the term 'jade' applies to two
different minerals, jadeite and nephrite, suggests that *being
jade* is a less natural property than, say, *being
turquoise*. The idea that naturalness is connected to the
rationality of induction threatens to imply that even before
Damour's discovery, rationality required less confidence in
universal generalizations about jade (e.g. that all jade can be
polished with bamboo) than it would have required for duplicate
believers in worlds where all the nephrite in the actual world had
been replaced by jadeite, so that the word 'jade'
expressed the property *being jadeite*. Indeed, even those who
embrace externalism about rationality on other grounds might not be
happy with this particular instance of it: there seems to be something
irrational about someone who, without the benefit of any special
observations like those of Damour, is simply disposed to be more
cautious in endorsing generalizations about jade than in endorsing
corresponding generalizations about turquoise.



To avoid this result without giving up on externalism about mental
content, one might say that what matters in the theory of rationality
the naturalness of *concepts* rather than *properties*,
and say that concepts *jade* and *turquoise* are equally
natural despite the fact that the properties they correspond to are
not. But for this to be illuminating, one will need to provide some
picture of what concepts are, and how naturalness for them relates to
naturalness for properties (see
 SS3.3).



Weatherson
 (2013;
 see
 Other Internet Resources)
 has a related objection to the claim that naturalness, understood as
playing certain other roles we have already discussed (e.g. in
the analysis of duplication) also makes for projectibility: he argues
that *being green* is highly projectible whereas *being an
electron* is not highly projectible, since merely finding that all
observed electrons have a certain mass does not strongly support the
conclusion that all electrons have that mass, unless one acquires
further evidence which supports the hypothesis that being an electron
is very natural. Opponents of externalism about mental content, such
as Lewis
 (1981),
 might respond that their theory of induction is in the first instance
about rationality in the formation of *narrow* beliefs and
credences, and propositions like *all electrons have mass
\(m\)* are not narrowly believed (see the entry on
 narrow mental content).
 But externalists about content will also need to be cautious about
embracing Weatherson's argument, since if they want to speak of
"degrees of projectibility" at all, there is pressure for
them to say that a single property can be denoted by different
concepts with different degrees of projectibility.


### 4.9 Epistemic value



Knowledge tends to have good effects, at least when combined with
benevolence. Similarly, true belief tends to have good effects, while
false belief tends to have bad effects. But some also hold that
knowledge or belief or high confidence in true propositions is
*non-instrumentally* good (good for its own sake), and perhaps
also that belief or high confidence in false propositions is
non-instrumentally bad (see the entry on
 the value of knowledge).
 These evaluative claims sometimes go hand in hand with slogans like
'belief aims at the truth' and 'belief aims at
knowledge', the thought being that the goodness of things that
have an aim or function consists in their achieving those aims or
fulfilling those functions. 



Once this starting point is granted, the question arises whether all
truths are equally non-instrumentally good to believe, and all
falsehoods equally bad. There is reason to think not: it would be a
warped kind of concern for the truth that led someone to memorise
phone books rather than doing science, on the grounds that this is
more efficiently generates epistemic value. The concept of naturalness
seems potentially useful in explaining which truths are most valuable
to believe. Indeed, if we are willing to apply 'natural'
to propositions as well as properties, the following simple
generalization looks tenable: the more natural a proposition is, the
better it is to know it or believe it or have high confidence in it if
it is true, and (one might add) the worse it is to believe it or have
high confidence in it if it is false.



Sider
 (2011)
 gives pride of place to evaluative considerations in his discussion
of naturalness: according to him, the fact that "it's
*better* to think and speak in joint-carving terms"
unifies many of the other roles for naturalness. He characterises
naturalness as a second "aim of belief", on a par with
truth. He furthermore holds that merely using words or concepts that
stand for unnatural properties is (non-instruementally) bad, even when
there is no difference in the propositions believed. His thought is
that simply using such words or concepts, no matter what beliefs one
forms, involves 'carving the world up
 incorrectly'.[59]
 Dorr and Hawthorne
 (2013)
 argue against Sider's picture of naturalness as a value
independent of truth, on the grounds that believing a false natural
proposition is *worse* than believing a false unnatural one,
whereas on Sider's account it should be better. For example, a
pill that induces many false beliefs about phone numbers should be
preferred to one that will induce a single false belief about whether
neutrinos have mass. They also claim that the value or disvalue in
using certain words or concepts is merely instrumental, flowing
entirely from their tendencies to promote or hinder belief.



The idea that naturalness matters to epistemic value strikes some as
mysterious. Dasgupta
 (2018)
 doubts that there is any more non-instrumental value in learning about the distribution
of natural properties than in learning about the distribution of 
unnatural ones, on
the grounds that there would need to be some deeper explanation for
such value-theoretic facts, and no adequate explanation is available.
His strategy is to consider some other feature of properties he calls
"graturalness", which grueness is stipulated to have to a
higher degree than greenness. When the opponent attempts to explain
the evaluative importance of naturalness by appealing to some feature
which naturalness necessarily has and graturalness necessarily lacks
(e.g. making for similarity), Dasgupta will respond by
introducing a corresponding "grueified" feature which
graturalness necessarily has and naturalness necessarily lacks
(e.g. making for "grimilarity"), and complain that
the explanatory challenge has merely been pushed back rather than met,
since we now need an explanation for the claim that the feature the
opponent pointed to has an evaluative significance that its grueified
counterpart lacks. This strategy is extremely general: indeed, it is
unclear whether *any* claim of non-instrumental value could be
explained in a way that would satisfy Dasgupta's criteria for
adequate explanation.



Those who take naturalness to correlate with epistemic value might
consider trying to *define* naturalness in value-theoretic
terms. McDaniel
 (2017b)
 attempts something like this, although he appeals to the concept of
*obligation* rather than value. He proposes that
for a property to be "fundamental" (i.e. natural) is for there to be a prima facie
obligation to form true beliefs about it. One challenge facing views
along these lines is that belief is an attitude to propositions,
whereas naturalness applies also to properties and relations. McDaniel
addresses this challenge using a theory of propositions as structured
entities: for *being F* to be more natural (to a certain
degree) than *being G* is for it to be the case that whenever
\(p\) and \(q\) are two true propositions differing only in that
*being F* appears in a position in the structure of \(p\) where
*being G* appears in the structure of \(q\), there is a
stronger (by a certain degree) prima facie obligation to believe \(p\)
than to believe \(q\). Note that if this scheme worked, it could also
used to recover naturalness comparisons for properties and relations
from naturalness comparisons for propositions, independent of the
connection to obligation. But it is certainly not uncontroversial than
this can be done, even setting aside concerns about the consistency of
the required fine-grained theory of propositions. One might think that
the naturalness of a proposition turns on factors like the
*diversity* of its "constituents" as well as the
sum of their individual degrees of naturalness, so that substituting a
less natural constituent for a more natural one might sometimes
increase the naturalness of a proposition by decreasing its diversity.
For example, the proposition that every grue gemstone is surrounded by
grue rock is not obviously less natural, or less valuable to believe if true, than the proposition
that every grue gemstone is surrounded by green rock.


### 4.10 Definability



There are various logical operations which map some properties to
others. One property can be the negation of another, or the
conjunction or disjunction of two or more others. A monadic property
can be the reflexivization of a binary relation; its existential or
universal quantification in one or other of its argument places, or
the result of saturating one or other of its argument places with an
object: given the relation of *loving*, reflexivization yields
*loving oneself*; quantification yields *loving
something*, *loving everything*, *being loved by
something*, and *being loved by everything*; while
saturation yields, for example, *loving Socrates* and *being
loved by Socrates*. In each of these cases, let us say that the
output property is "definable from" the input properties
and objects (taken collectively), and "partly definable
from" each of the input properties (individually). Let us also
assume that partial definability is transitive.



These examples do not amount to definitions of 'definable'
and 'partly definable'. Nevertheless, the notions seem
clear enough for our present purpose, which is to consider how
definability might relate to naturalness. Here are several
generalizations one might find tempting:


1. If \(p\) is partly definable from \(q\), \(p\) is less natural
than \(q\).
2. If \(p\) is partly definable from \(q\), \(p\) is not perfectly
natural.
3. If \(p\) is not partly definable from any \(q\), \(p\) is
perfectly natural.



A worry about these proposals is that on some accounts, every property
will count as partly definable from *itself*. Then (1) will be
inconsistent, (2) will entail that no property is perfectly natural,
and (3) will be vacuously true. The claim that partial definability is
reflexive might, for example, be motivated by the claim that every
property is the negation of its negation. It might also be motivated,
at least as far as relations are concerned, by the especially
plausible claim that every binary relation is the converse of its
converse, together with the claim that the the converse of any
relation is partially definable from
 it.[60]




These worries do not apply to the corresponding generalizations about
*strict* partial definability, where \(p\) is strictly
partially definable from \(q\) just in case \(p\) is partially
definable from \(q\) and \(q\) is not partially definable from \(p\):



4. If \(p\) is strictly partly definable from \(q\), \(p\) is less
natural than \(q\).
5. If \(p\) is strictly partly definable from \(q\), \(p\) is not
perfectly natural.
6. If \(p\) is not strictly partly definable from any \(q\), \(p\) is
perfectly natural.



And here are two other weakenings of (1) and (2) which are also not
trivialised even if partial definability is reflexive:


7. If \(p\) is partly definable from \(q\), \(p\) is not more natural
than \(q\).
8. If \(p\) is partly definable from \(q\) and \(p\) is perfectly
natural, \(q\) is perfectly
 natural.[61]



Generalizations like 1-8 are interesting only against the
background of a sufficiently fine-grained theory of properties. All
are trivialised by intensionalism, the thesis that necessarily
coextensive properties are always identical. Consider the following
consequence of intensionalism:



> 
> 
> (D) *being \(F\)* is identical to
> *being either \(F\) and \(G\) or \(F\) and not
>  \(G\)*.[62]
> 
> 
> 
> 



Since the conjunction and conjunction and disjunction of any two
properties count as partially definable out of each of those
properties and partial definability is transitive,
 (D)
 implies that every \(n\)-ary property is partially definable from
every other: *being \(F\)* is partly definable from *being
\(F\) and \(G\)*, which is in turn partly definable from *being
\(G\)*. This makes (1) inconsistent; makes (2) imply that no
property is perfectly natural; makes (3) and (6) imply that every
property is perfectly natural; makes (4) and (5) vacuously true; makes
(7) imply that all properties are equally natural; and makes (8) imply
that every property is perfectly natural if any are.



Intensionalism is controversial, so the fact that it conflicts with
our generalizations linking naturalness to definability is not by
itself a big problem for the generalizations. But once intensionalism
is denied, there is an urgent need for a systematic theory of
properties to replace it. For example: given that
 (D)
 is false for some \(F\) and \(G\), it would be good to know whether
it is false for all of them. If so, are there more general principles
which explain this? If not, what marks the difference between the
\(F\), \(G\) pairs for which
 (D)
 is true and those for which it is false? Without a background theory
to provide some guidance with these questions, generalizations linking
partial definability to naturalness will be of little use: we can
still know in many cases that \(p\) is partially definable from \(q\),
by knowing of some particular operations that can be applied to \(q\)
to get \(p\), but it will be unclear how to argue for any claim of
*strict* partial definability. The task of coming up with a
consistent non-intensionalist theory strong enough to answer answer
such questions is by no means straightforward. The most popular
fine-grained picture holds that properties (and propositions) are
"structured", where this is taken to imply that, for
example, *bearing \(R\) to \(a\)* is identical to *bearing
\(S\) to \(b\)* only if \(R\) and \(S\) are identical, and \(a\)
and \(b\) are identical. Unfortunately, when principles of this sort
are spelled out in the higher-order logic mentioned in SS3.4
above, and generalized in the obvious way, they lead to an
inconsistency: the "Russell-Myhill paradox", first
identified in Russell
 1903
 (appendix B), and subsequently rediscovered by Myhill
 (1958):
 see Goodman
 2017
 and Dorr
 2018
 (SS6). But this certainly is not the end of the story, even
within the higher order framework. For example, Dorr
 (2016)
 develops a consistent non-intensionalist theory that entails that
 (D)
 is false for every \(F\) and
 \(G\).[63]




Of course, even if one has a fine-grained view that makes partial
definability discriminating enough to be interesting, one may have
other reasons for rejecting generalizations like (1-8). Lewis
 (1970a;
 1986b, 56-9), Creswell
 (1985),
 and Soames
 (2002,
 ch. 11) propose theories on which there are both
"unstructured" and "structured" properties and
propositions, with the former conforming to intensionalism, and the
latter being in some sense "constructed"
(e.g. set-theoretically) out of the former. The notion of
"construction", or its transitive closure, can provide an
non-trivial interpretation of partial definability in these theories.
But a hallmark of this kind of theory is that every structured
property is necessarily coextensive with an unstructured one. Given
this, generalisation 6 would imply that every property is necessarily
coextensive with a perfectly natural property, a claim that is in
obvious tension with the proposals we have considered linking
naturalness to similarity, duplication, qualitativeness, modality,
lawhood, and mental and linguistic content.



The concept of *full* definability also has some notable
possible applications in the theory of naturalness. We could appeal to
it to provide an alternative articulation of the intuitive thought
about the "completeness" of the perfectly natural
properties, which Lewis captured using
 TSB:



*Definability*
  

Every property is fully definable out of some perfectly natural
properties and some
 objects.[64]




Setting aside doubts about the intelligibility of
 *Definability*,
 it would seem to be at least as strong as
 TSB.[65]
 At least in the paradigm cases where a proposition is fully definable
out of some properties and objects, the complete truth about which
objects there are and which ones instantiate each of those properties
will either necessitate that proposition or necessitate its negation.
By contrast, the inference from
 TSB
 to
 *Definability*
 is quite controversial. There are arguments which purport to show
that whenever a property supervenes on some properties, it is
definable in terms of them (Jackson
 1998;
 Stalnaker
 1996);
 but these arguments rely on an intensionalist theory of properties,
and raise difficult issues even in that setting (Leuenberger
 2018).
 



Assuming that
 TSB
 is weaker than
 *Definability*,
 it is arguably too weak to capture the intuitive idea that a
description of the world in terms of perfectly natural properties
should be a "complete" description. Indeed, the claim that
properties that not perfectly natural are definable in terms of
perfectly natural properties seems to be required for the
simplicity-of-definition criterion to yield sensible results.
Moreover, many of those who see a gulf between definability and mere
supervenience have also felt that there is something metaphysically
suspect about infinite descending chains of definability. If we can
find a way to spell out a ban on such chains, it will plausibly imply
that every property is fully definable out of *indefinable*
properties (and objects); and this claim will in turn
 *Definability*
 if we take indefinability to imply perfect naturalness, as in (3) or
(6) above. 



On the other hand,
 *Definability*
 also faces objections that
 TSB
 avoids. For example, many philosophers follow Moore
 (1903)
 in denying that moral properties are definable from physical
properties while still affirming that they supervene on physical
properties; some such philosophers might be reluctant to say that any
moral properties are perfectly natural, perhaps because they are
attracted to modal independence principles. 


### 4.11 Grounding



Most of the present section has focused on generalisations linking naturalness with relatively familiar, easy-to-grasp notions. We have not yet discussed possible links between naturalness and many other "heavy duty" metaphysical notions -- notions whose good standing and theoretical utility is as controversial as that of naturalness itself. Among the notions for which such connections might be worth investigating are: essence (Fine 1994); ontological dependence (see the entry on ontological dependence); truthmaking (see the entry on truthmakers); substance (see the entry on substance); reality and factuality (Fine 2001); modes of being (explained in terms of naturalness in McDaniel 2017b, ch. 1); and degrees of being (McDaniel 2017b, ch. 7). Space precludes a discussion of these topics. But to convey a sense of the possibilities, let us briefly consider some possible connections to *grounding*, a notion that has been especially prominent in the recent metaphysics literature (for background, see the
entry on metaphysical grounding).




Especially given the widespread use of 'fundamental' as
equivalent to 'ungrounded', it seems plausible that there
might be some intimate link between naturalness and
grounding. So far, however, most discussions of naturalness and
grounding tend to treat the notions as being in some sense rivals
(see, e.g., Sider
 2011,
 ch. 8), so there have been few attempts at systematic theorising
using both together. Here are some generalizations that seem worth
considering:


1. If the fact that \(F(x\_1,\ldots,x\_n)\) is partly grounded by the
fact that \(G(y\_1,\ldots,y\_m)\), then \(F\) is less natural than
\(G\).
2. If the fact that \(F(x\_1,\ldots,x\_n)\) is partly grounded by the
fact that \(G(y\_1,\ldots,y\_m)\), then \(F\) is not more natural than
\(G\).
3. If the fact that \(F(x\_1,\ldots,x\_n)\) is partly grounded by some
fact, then \(F\) is not perfectly natural.
4. If the fact that \(F(x\_1,\ldots,x\_n)\) is partly grounded by the
fact that \(G(y\_1,\ldots,y\_m)\), and \(F\) is perfectly natural, then
\(G\) is perfectly natural.
5. If the fact that \(F(x\_1,\ldots,x\_n)\) is not partly grounded by
any fact, then \(F\) is perfectly natural.



Note that (1) implies (2) and (3), which each imply (4); (5) is independent of (1-4). The notion of
partial ground might in turn be defined in terms of *full*
ground, which relates a plurality of facts (e.g. the fact that snow is
white and the fact that grass is green) to a single fact (e.g. the
fact that snow is white and grass is green): one fact partially
grounds another just in case it belongs to some plurality of facts
that fully grounds
 it.[66]




These suggestions are all quite tendentious. (1) is especially
problematic since it rules out cases where the fact that
\(F(x\_1,\ldots,x\_n)\) is grounded by the fact that
\(F(y\_1,\ldots,y\_n)\): for example, one might want to say that the
fact that Central Park is in New York City is partially grounded by
the fact that Central Park is in Manhattan. While (2) avoids this
problem, it is still hard to square with a thesis endorsed by most grounding
theorists: that an existential generalization is partially grounded by
each of its instances (Fine
 2012,
 59; Rosen
 2010,
 117). Consider a blob of clay \(x\) with a certain very specific
shape-property \(S\). The thesis about existential generalizations
entails that the fact that \(x\) *has some shape or other* is
partially grounded by the fact that \(x\) has \(S\). But the idea that
\(S\) is more natural than *having some shape or other* fits
poorly with many other parts of the naturalness role, such as the
connections to similarity, content, laws, and ampliative reasoning. A
worry about (3) is that it might make it too hard for a property to
count as perfectly natural. One might think that whenever the fact
that \(F(x\_1,\ldots,x\_n)\) is ungrounded, there are also some other
objects \(y\_1,\ldots,y\_n\) such that the fact that
\(F(y\_1,\ldots,y\_n)\) is grounded: perhaps, for example, the
ungrounded facts are all predications of mereologically simple objects
of properties which are also instantiated by certain mereologically
complex objects. Several of the generalizations also run into trouble
if we assume a moderately coarse-grained conception of facts, on which
(for example) the fact that Cleopatra loves Cleopatra is identical to
the fact that Cleopatra is a self-lover. For example, under this
assumption, (5) will have the surprising consequence that if the fact
that \(R(x,x)\) is ungrounded, not only the binary relation \(R\), but
also the monadic property *bearing \(R\) to oneself* is
perfectly
 natural.[67]
 (Grounding-theorists often assume a fine-grained account of facts
which would avoid this kind of problem; but there are worries about
the consistency of such accounts similar to the worries about
fine-grained theories of properties discussed in
 SS4.10.)
 



More ambitiously, one might attempt to provide necessary and
sufficient conditions for perfect naturalness in grounding-theoretic
terms, by strengthening (5) to (6), or (3) to (7):


6. \(F\) is perfectly natural if and only if it is possible for there
to be some \(x\_1,\ldots, x\_n\) such that \(F(x\_1,\ldots,x\_n)\) and the
fact that \(F(x\_1,\ldots,x\_n)\) is ungrounded.
7. \(F\) is perfectly natural if and only if it is possible for there
to be some \(x\_1,\ldots,x\_n\) such that \(F(x\_1,\ldots,x\_n)\), and it
is necessary that for any \(x\_1,\ldots,x\_n\) such that
\(F(x\_1,\ldots,x\_n)\), the fact that \(F(x\_1,\ldots,x\_n)\) is
ungrounded.



(6) and (7) are compatible, but their conjunction has a controversial
purely grounding-theoretic consequence, namely that whenever
\(F(x\_1,\ldots,x\_n)\) and the fact that \(F(y\_1,\ldots,y\_n)\) is
ungrounded, the fact that \(F(x\_1,\ldots,x\_n)\) is also ungrounded.




One could also consider trying to define grounding in terms of
naturalness. Here is a simple proposal, using the notion of
naturalness for propositions (see SS3.3),
and assuming that facts are equated with true propositions:


8. \(p\) is fully grounded by \(qq\) if and only if (a)
each of \(qq\) is true; (b) it is necessary that if all of \(qq\)
are true then \(p\) is true, and (c) each of \(qq\) is more natural 
than \(p\).[68]



Grounding theorists (who have generally been pessimistic about
attempts to define grounding in other terms) will likely object that
(8) wrongly entails that whenever \(p\) is fully grounded by \(qq\),
and \(r\) is both true and more natural than \(p\) obtains, \(p\) is also fully grounded by \(qq\)
together with \(r\) (even if \(r\) is intuitively completely
irrelevant to \(p\)). One might try to refine (8) to avoid this
objection by adding some further "relevance" condition,
also explained in terms of naturalness; but it is not obvious how this
should go. 



One further possible application of the ideology of grounding worth
considering is in making precise the idea about the
"completeness" of the description of the world in terms of
perfectly natural properties which Lewis cashed out as
 TSB.
 An obvious idea is simply to replace the notion of metaphysical
necessitation in
 TSB
 with that of grounding: 



*Grounding Completeness*
  

Necessarily, every fact is either fully grounded by, or a member of,
some collection of facts each of which is either (a) the fact that
\(R(x\_1,\ldots ,x\_n)\) or \(\neg R(x\_1,\ldots ,x\_n)\), for some
perfectly natural \(R\) and objects \(x\_1,\ldots ,x\_n\); (b) the fact
that \(\exists y(y=x)\), for some object \(x\), or (c) the fact that
every object is one of \(xx\), where \(xx\) are all the objects there
are. 



Given the standard assumption that full grounding entails
necessitation, this will be at least as strong as
 TSB,
 while arguably not being as strong as
 *Definability*
 (see
 SS4.10).


## 5. Doubts about naturalness



The question whether to admit distinctions of naturalness among
properties is often treated as a major choice point for theorising in
metaphysics. Lewis presents his various proposals involving
naturalness as part of an overarching argument for the good standing
of the notion of naturalness, based on its theoretical utility.
Meanwhile, the rejection of distinctions of naturalness is often (e.g.
by Taylor
 1993)
 thought of as bound up with the denial of "metaphysical"
or "full-blooded" realism, and associated with
controversial claims about mind-dependence or
perspective-relativity.



But it is not obvious what is at issue in such disagreements. If
"rejecting" naturalness means holding that
'natural' is literally *unintelligible* (on the
relevant use), it seems misguided. So long as we take understanding a
word to be a matter of induction into a practice rather than some
dubious "magical grasp" (Sider
 2011,
 SS2.1), it is hard to imagine some plausibly attainable standard
for intelligibility that the practice of using 'natural'
in the relevant way fails to meet. A more promising idea is simply to
equate "rejection" of naturalness with the claim that no
property is more natural than any other. (Similarly, atheists will do
better claiming that nothing is more holy than anything else than
claiming that the word 'holy' is unintelligible.)
Unfortunately, if we try to structure a debate around the question
'Are some properties more natural than others?', we will
run a serious risk of equivocation. One participant may treat it as
definitional that if there are any distinctions of naturalness,
naturalness plays a certain wide array of the roles that Lewis and
others have claimed for it in the theories of similarity, lawhood,
reference, epistemic value, etc.; such a philosopher will treat any
argument that no single ranking of properties plays all of these roles
as an argument that no property is more natural than any other.
Another participant might give pride of place to (say) the connection
between naturalness and similarity, treating as obvious that some
properties are more natural than others (since some things are more
similar to one another than others), while treating it as an open
question whether naturalness has any interesting connection to
lawhood, reference, etc. Such philosophers need not disagree over
anything important.



Dorr and Hawthorne
 (2013)
 argue that 'Are some properties more natural than
others?' is too sensitive to unimportant disagreements about how
philosophical terms of art get their meanings to be a useful
 question.[69]
 Inspired by Lewis's work on theoretical terms (Lewis
 1970b),
 they suggest that the central debates in this vicinity should focus,
instead, on the question which collections of the roles that
naturalness has been held to play in the literature are such that
there is *some* ranking of properties that plays all the roles
in the collection. "Naturalness enthusiasts" will hold
that there is a single ranking of properties playing all or most of
the roles; "naturalness sceptics", meanwhile, will hold
that different roles impose incompatible constraints, so that no
ranking satisfies more than a few of them. (For example, one might
think that there is no family of properties on which all others
supervene and which are also such that properties with short, simple
definitions terms of this family tend to be easier to refer to than
properties with only long, complex definitions.) Dorr and Hawthorne
give a long list of collections of roles whose co-satisfiability is
non-obvious. They also discuss one separate axis of disagreement: some
will think that 'natural', or at least 'perfectly
natural', is *precise*, or can readily be made so, while
others will regard it as vague, perhaps to a high
 degree.[70]




On this picture, the impression of a single choice point between
"admitting" and "rejecting" naturalness is
misleading: the true theoretical landscape is quite intricate.
Moreover, even the most "sceptical" positions do not seem
to have much of an *anti-realist* flavour. There is no obvious
way to argue from the claim that there are extensive failures of
co-satisfiability between the various alleged roles for naturalness,
or from the claim that 'natural' and 'perfectly
natural' are highly vague, to some controversial claim of
mind-dependence or relativity. Theories about naturalness provide ways
of sharpening and fleshing out what Lewis
 (1984, 228)
 calls "the traditional realism that recognises objective
sameness and difference, joints in the world, discriminatory
classifications not of our own making". But the ideas discussed
in this article do not seem particularly helpful for making any
nontrivial sense of the competing vision, on which questions of
similarity and classification are in some systematic way
"subjective" or "up to us".