## 1. Introduction



The standard way to introduce the distinction between intrinsic and
extrinsic properties is by the use of a few orientating
characterisations. David Lewis provides the following list:



> 
> 
> A sentence or statement or proposition that ascribes intrinsic
> properties to something is entirely about that thing; whereas an
> ascription of extrinsic properties to something is not entirely about
> that thing, though it may well be about some larger whole which
> includes that thing as part. (Lewis 1983a: 197)
> 
> 
> 
> A thing has its intrinsic properties in virtue of the way that thing
> itself, and nothing else, is. Not so for extrinsic properties, though
> a thing may well have these in virtue of the way some larger whole is
> ... (Lewis 1983a: 197)
> 
> 
> 
> If something has an intrinsic property, then so does any perfect
> duplicate of that thing; whereas duplicates situated in different
> surroundings will differ in their extrinsic properties. (Lewis 1983a:
> 197)
> 
> 
> 



The other way to introduce the subject matter is by providing examples
of paradigmatic intrinsic and extrinsic properties. One half of this
task is easy: everyone agrees that being an uncle is extrinsic, as is
being six metres from a rhododendron. The other half is more
challenging, since there is much less agreement about which properties
are intrinsic. Lewis (1983a, 1986a, 1988), for example, has insisted
that shape properties are intrinsic, but one could hold that an
object's shape depends on the curvature of the space in which it
is located at, and this might not even be intrinsic to that space
(Nerlich 1979), let alone the object (Bricker 1993, McDaniel 2007, and
Skow 2007). Lewis also mentions charge and internal structure as being
examples of intrinsic properties. (For ease of exposition, we will
assume below that shape properties are intrinsic. We will also assume
that properties like being made of tin, and having a mass of 500kg are
intrinsic.)


### 1.1 Philosophical Importance



The distinction between intrinsic and extrinsic properties plays an
essential role in stating several interesting philosophical problems.
Historically, the most prominent of these has to do with notions of
intrinsic value. G. E. Moore (1903: SS18) noted that we can make a
distinction between things that are good in themselves, or possess
intrinsic value, and those that are good as a means to other things.
To this day there is still much debate over whether this distinction
can be sustained (Feldman 1998, Kagan 1998), and, if it can, which
kinds of things possess intrinsic value (Krebs 1999). In particular,
one of the central topics in contemporary environmental ethics is the
question of which kinds of things (intelligent beings, conscious
beings, living things, species, etc) might have intrinsic value. While
this is the oldest (and still most common) use of the
intrinsic/extrinsic distinction in philosophy, it has not played much
role in the discussions of the distinction in metaphysics, to which we
now turn.



As P. T. Geach (1969) noted, the fact that some object \(a\) is not
\(F\) before an event occurs but is \(F\) after that event occurs does
not mean that the event constitutes, in any deep sense, a change in
\(a\). To use a well-worn example, at the time of Socrates's
death Xanthippe became a widow; that is, she was not a widow before
the event of her husband's death, but she was a widow when it
ended. Still, though that event constituted (or perhaps was
constituted by) a change in Socrates, it did not in itself constitute
a change in Xanthippe. Geach noted that we can distinguish between
real changes, such as what occurs in Socrates when he dies, from mere
changes in which predicates one satisfies, such as what occurs in
Xanthippe when Socrates dies. The latter he termed 'mere
Cambridge' change. There is something of a consensus that an
object undergoes real change in an event iff there is some
**intrinsic** property it instantiated before the event
but not afterwards.



David Lewis (1986a, 1988) built on this point of Geach's to
mount an attack on **endurantism**, the theory that
objects persist by being wholly located at different times, and that
there can be strict identity between an object existing at one time
and one existing at another time. Lewis argues that this is
inconsistent with the idea that objects undergo real change. If the
very same object can be both \(F\) (at one time) and not \(F\) (at
another), this means that \(F\)-ness must be a relation to a time, but
this means that it is not an intrinsic property. So any property that
an object can change must be extrinsic, so nothing undergoes real
change. Lewis says that this argument supports the rival theory of
**perdurantism**, which says that objects persist by
having different temporal parts at different times. While this
argument is controversial (see Haslanger (1989), Johnston (1987) and
Lowe (1988) for some responses), it does show how considerations about
intrinsicality can resonate within quite different areas of
metaphysics.



A third major area where the concept of intrinsicality has been put to
work is in stating various supervenience theses. Frank Jackson (1998)
defines physicalism in terms of duplication and physical duplication,
which are in turn defined in terms of intrinsic properties. Similarly,
Jaegwon Kim (1982) defines a mind/body supervenience thesis in terms
of intrinsic properties. As Theodore Sider (1993) notes, the simplest
way to define the **individualist** theory of mental
content that Tyler Burge (1979) attacks is as the claim that the
content of a thinker's propositional attitudes supervenes on the
intrinsic properties of the thinker. And many
**internalist** theories in epistemology are based around
the intuition that whether a thinker is justified in believing some
proposition supervenes on the intrinsic properties of the thinker.



A fourth area where the concept of intrinsically has been employed is
to state recombination principles intended to describe what possible
states of the world there are. (See Lewis 1986a.) An initial attempt
to state such a principle might be (R1), where \(x\) is wholly
distinct from \(y\) iff \(x\) and \(y\) have no parts in common. (As
is standard in philosophical usage, each thing is counted as a part of
itself. Hence, things are not wholly distinct from themselves.)



(R1)
For any properties \(p\) and \(q\) , it is possible that there are
two wholly distinct things, \(x\) and \(y\), such that \(x\) has \(p\)
and \(y\) has \(q\). 


Given an abundant theory of properties according to which each
(non-defective) predicate expresses a property, however, (R1) clearly
fails, since it isn't possible for there to be an \(x\) and a
\(y\) such that \(x\) has the property of being made of gold, and
\(y\) has the property of being such that nothing is made of gold.
This defect with (R1) can be fixed by replacing (R1) with (R2).



(R2)
For any *intrinsic* properties \(p\) and \(q\), it is
possible that there are two wholly distinct things, \(x\) and \(y\),
such that \(x\) has \(p\) and \(y\) has \(q\). 


While being made of gold and being such that nothing is made of gold
refute (R1), they don't refute (R2), since being such that
nothing is made of gold is clearly non-intrinsic.



In addition to the above four applications of the distinction between
intrinsic and extrinsic properties, the distinction has also been
employed for many other purposes. For example, it has been used to
formulate principles of universalizability for moral principles and
moral laws (Rabinowicz 1979), formulate a requirement on an
explanation being a good explanation, namely that it be an intrinsic
explanation (Field 1980, Maya 2014), and formulate a non-trivial
version of Mill's principle of the uniformity of nature
(Schlesinger 1990). The distinction (or distinctions definable in
terms of it) has also been used to argue against the magical ersatz
theory of possible worlds (Lewis 1986a, Sect. 3.4), defend the
transitivity of causation (Hall 2000, Hall 2004), defend the modal
theory of essence (Denby 2014), argue for the immanent theory of
properties (Audi 2019), and argue against the thesis that determinable
facts are always soley grounded by determinate facts (Shumener
2022).



Most of the philosophical applications of intrinsicality are
independent of its precise analysis. Work on its analysis, however,
has helped clarify these applications by allowing us to distinguish
between different notions of intrinsicality and different notions in
the vicinity of intrinsicality, and by giving us a greater
understanding of these various notions and of what properties satisfy
them. A good example of the latter kind of advance is Theodore
Sider's (2003) observation that most of the properties in folk
theory are 'maximal' and hence not intrinsic properties.
This observation provides a strong argument against various theories
that appeal to the intuitive intrinsicality of some everyday
property.


### 1.2 Global and Local



In addition to the "global" concept of a property being
intrinsic, there is a "local" concept of something having
a property in an intrinsic fashion (see Humberstone 1996, p. 206). For
each of Lewis's characterisations concerning the concept of an
intrinsic property quoted in section
 1.1,
 there is a corresponding characterisation for the concept of a thing
having a property in an intrinsic fashion. These are:


1. \(x\) has \(F\) in an intrinsic fashion iff a sentence ascribing
\(F\) to \(x\) is entirely about that thing.
2. \(x\) has \(F\) in an intrinsic fashion iff \(x\) has \(F\) in
virtue of how \(x\) itself, and nothing else, is.
3. \(x\) has \(F\) in an intrinsic fashion iff every duplicate of
\(x\) has \(F\).



It follows from at least the third of these characterisations that
something can have an extrinsic property in an intrinsic fashion. For
example, since every duplicate of a piece of tin has the property of
being made of tin, every duplicate of a piece of tin also has the
extrinsic property of being either made of tin or married. As a
result, it follows from the third characterisation that each piece of
tin has the property of being either made of tin or married in an
intrinsic fashion, despite this property being extrinsic.



Two plausible principles linking the local and global concepts of
intrinsicality are (GTL) and (LTG) (see Humberstone 1996, p. 228).



(GTL)
 If \(F\) is an intrinsic property, then it is necessary that
every \(x\) that has \(F\) has \(F\) in an intrinsic fashion 
(LTG)
 If it is necessary that every \(x\) that has \(F\) has \(F\) in
an intrinsic fashion, then \(F\) is an intrinsic property 


In addition to the two-place predicate 'has ... in an
intrinsic fashion', we can introduce a modifier
'intrinsically', where, adapting one of the
characterisations above, \(x\) is intrinsically \(F\) iff \(x\) is
\(F\) in virtue of how \(x\) itself, and nothing else, is. (cf.
Humberstone 1996, p. 228). This modifier takes predicates as arguments
rather than names of properties or variables ranging over properties.
Nominalists, who don't believe in properties, might attempt to
use this modifier to capture the intuitive distinctions associated
with intrinsicality without committing themselves to properties. For
example, a nominalist might claim that a table is intrinsically
rectangular, while claiming to consistently hold that there is no
property of being rectangular. (For simplicity, we will assume in the
following that nominalism is false and that there are abundantly many
properties and relations, some of which are intrinsic and some of
which are extrinsic.)



There might be further reasons to employ either the two place
predicate 'has ... in an intrinsic fashion', or the
modifier 'intrinsically', rather than the one-place
predicate 'is intrinsic'. For example, one might think
that at least one of the former two notions is more fundamental than
the latter notion, or that the latter notion is misguided in some way
that at least one of the former notions isn't. (See, for
example, Figdor 2008.)


### 1.3 Relations



While this article focuses on the distinction between intrinsic and
extrinsic one-place properties, it is important to recognize that the
distinction between intrinsic and extrinsic also applies to multiple
place relations. As in the case of one-place properties, the
distinction between intrinsic and extrinsic multiple place relations
can be introduced by characterisations and examples. One
characterisation is the following:



> 
> An \(n\)-place intrinsic relation is an \(n\)-place relation that
> \(n\) things stand in in virtue of how they are and how they are
> related to each other, and not in virtue of anything else; whereas,
> this is not the case for extrinsic \(n\)-place relations.
> 



Examples of extrinsic two-place relations include *being as popular
as*, and *having the same cousin as*. Possible examples of
intrinsic two-place relations include *being 1 m away from* and
*being made of the same type of metal as*. (The claim that
distance relations like *being 1 m away from* are intrinsic
will be denied by philosophers who deny that shape properties are
intrinsic.)



As well as the global notion of an \(n\)-place relation being
intrinsic, there is the local notion of \(n\) things standing in a
relation in an intrinsic fashion. A characterisation of this local
notion is:



> 
> \(n\) things stand in a relation \(R\) in an intrinsic fashion iff the
> \(n\) things stand in \(R\) in virtue of how they are and how they are
> related to each other, and not in virtue of anything else.
> 


## 2. Notions of Intrinsicality



A number of different distinctions have been called the
intrinsic/extrinsic distinction. As J. Michael Dunn (1990) notes, some
authors have used 'intrinsic' and 'extrinsic'
to mean 'essential' and 'accidental'. Dunn is
surely right in saying that this is a misuse of the terms. A more
interesting distinction is noted by Brian Ellis (1991; discussed in
Humberstone 1996: 206). (See also Figdor 2008 and Figdor 2014 for a
discussion of a related notion of intrinsicality.) Ellis suggests we
should distinguish between properties that objects have independently
of any outside forces acting on them (what we will call the
Ellis-intrinsic properties), and those that they have in virtue of
those outside forces (the Ellis-extrinsic properties). For many
objects (such as, say, a stretched rubber band) their shape will be
dependent on the outside forces acting on them, so their shape will be
Ellis-extrinsic. If one is committed to the idea that shapes are
intrinsic, one should think this means that the distinction between
the Ellis-intrinsic and Ellis-extrinsic properties is not the same as
the intrinsic/extrinsic distinction. Such a judgement may seem a
little hasty, but in any case we will turn now to distinctions that
have received more attention in the philosophical literature.


### 2.1 Relational vs. Non-Relational Properties



Many writers, especially in the literature on intrinsic value, use
'relational' for the opposite of intrinsic. This seems to
be a mistake since many properties seem to be both be relational and
intrinsic. For example, most people have the property *having
longer legs than arms*, and indeed seem to have this property in
an intrinsic fashion, even though the property consists in a certain
relation being satisfied. Maybe the property is not intrinsic if
whether or not something is an arm or a leg is extrinsic, so perhaps
this isn't a conclusive example, but it seems troubling. And, in
any case, there are other examples that can't be responded to in
this way. For example, the property of having a proper part is surely
intrinsic, but it also appears to be a relational property.



As Humberstone 1996 notes, some might respond by suggesting that a
relational property is one such that if an object has it, then it
bears some relation to a non-part of it. But this won't do
either. *Not being within a mile of a rhododendron* is clearly
relational, but does not consist in bearing a relation to any
non-part, as we can see by the fact that a non-rhododendron all alone
in a world can satisfy it.



A further response might be to say that a relational property should
be a property that involves a relation that can only relate wholly
distinct things, and that, given this construal of a relational
property, non-intrinsic properties are relational properties. However,
this response faces difficulties also. For example, this account would
presumably classify the property of being such that there is a cube as
being intrinsic. This property, however, is not intrinsic.



While the notion of a relational property should be distinguished from
the notion of a non-intrinsic property, it might be that the notion of
intrinsic properties can be given an account in terms of
relationality. One such account due to Francesscotti is discussed in
the supplement:



The Relational Account of Intrinsicality.
 


### 2.2 Local vs. Non-Local Properties



We now turn to the notion of intrinsicality characterised by the
characterisations listed in section
 1.
 Or perhaps better, we now turn to the *notions* of
intrinsicality characterised by these characterisations, since, while
Lewis thought that each of these characterisations characterise a
single notion of intrinsically, it is plausible that they characterise
different notions. In this subsection, we will discuss the notion
characterised by the first characterisation which involves aboutness.
In the next two subsections, we will discuss the notions characterised
by the other two characterisations.



Lewis's first characterisation is:



> 
> A sentence or statement or proposition that ascribes intrinsic
> properties to something is entirely about that thing; whereas an
> ascription of extrinsic properties to something is not entirely about
> that thing, though it may well be about some larger whole which
> includes that thing as part. (Lewis 1983a: 111-2)
> 



The notion of aboutness employed in this characterisation is plausibly
that of *intrinsic aboutness*, where intrinsic aboutness can be
intuitively characterised by (1).



(1)
 A state of affairs
\(s\) is intrinsically about a thing \(x\) iff \(s\) (either truly or
falsely) describes how \(x\) and its parts are and how they are
related to each other, as opposed to how \(x\) and its parts are
related to other things and how other things are. 


Using the notion of intrinsic aboutness, Lewis's first
characterisation can be more precisely stated by the schema (2).



(2)
The property of being \(F\) is intrinsic iff, necessarily, for
any \(x\), the state of affairs of \(x\) being \(F\) is intrinsically
about \(x\).


For any predicate \(F\) expressing a property \(p\), and any name
\(n\) referring to an \(x\), define the ascription of \(p\) to \(x\)
to be the state of affairs expressed by \(\ulcorner Fn\urcorner\).
Using this definition, we can replace the schema (2) with the sentence
(3).



(3)
 For any property
\(p, p\) is intrinsic iff, necessarily, for any \(x\), the ascription
of \(p\) to \(x\) is intrinsically about \(x\).


(3) plausibly classifies *being a cube* as intrinsic, since,
for any \(x\), the state of affairs of \(x\) being a cube plausibly
describes how \(x\) and its parts are and how they are related to each
other, as opposed to how they are related to other things and how
other things are. In contrast, (3) plausibly classifies *being an
uncle* as non-intrinsic, since its ascription to a thing is not
only about how it and its parts are and how they are related to each
other, but also about how it is related to things outside of it. (3)
also plausibly classifies being Obama as non-intrinsic since the
ascription of being Obama to Clinton, namely the state of affairs of
Obama being identical to Clinton, is not wholly about Clinton, but is
also about Obama. Finally, (3) also appears to classify being such
that there is a number as non-intrinsic, even if it is necessary that
there are numbers. This is because the ascription of this property to
Clinton, namely the state of affairs that Clinton is such that there
is a number, appears to be not wholly about how Clinton is, but is
also about how things wholly distinct from Clinton are. (We will
assume that the intuition underlying this judgement is correct.
However, this is not uncontroversial. See Sider 1996 for a rejection
of a closely related intuition and Eddon 2011 for a reply to this
rejection.)



In the following, we will call the properties classified as intrinsic
by (3) *local* properties, and call the properties classified
by (3) as not intrinsic *non-local* properties. (This notion of
a local property needs to be distinguished from the local conception
of something having a property in an intrinsic fashion discussed in
section
 1.2.)


### 2.3 Interior vs. Exterior Properties



The second characterisation listed in the quote by Lewis in section
 1
 was "A thing has its intrinsic properties in virtue of the way
that thing itself, and nothing else, is".

 This
characterisation can be stated more precisely by schema (4), where
'how \(x\) is intrinsically' abbreviates 'how \(x\)
and its parts are and how they are related to each other, as opposed
to how \(x\) and its parts are related to other things and how other
things are'.


(4)
 Being \(F\) is
intrinsic iff, necessarily, for any \(x\), if \(x\) is \(F\) then
\(x\) is \(F\) in virtue of how \(x\) is intrinsically. 


Call the properties that are classified as intrinsic by (4)
*interior* properties, and call the properties that are
classified as non-intrinsic by (4) *exterior* properties. An
important question is what meaning 'in virtue' is meant to
have in (4). One possible answer is that it is meant to have a reading
corresponding to metaphysical grounding, where metaphysical grounding
is the relation that stands to metaphysical explanation as causation
stands to causal explanation. Under such a reading of 'in
virtue', (5) is necessarily true.



(5)
 '\(\varphi\)
in virtue of it being the case that \(\phi\)' is true iff the
state of affairs expressed by \(\phi\) metaphysically grounds the
state of affairs expressed by \(\varphi\). 


While it might be initially appealing to think that 'in
virtue' in (4) expresses metaphysical grounding, Marshall 2015
has given an argument that (4) fails to characterise a notion of
intrinsicality when 'in virtue' has this reading. The
argument relies on (6) being necessarily true and (7) being true on
any notion of intrinsicality.



(6)
 '\(\varphi\)
in virtue of how \(x\) is intrinsically' is true iff there is a
sentence \(\phi\) that expresses an obtaining state of affairs that is
intrinsically about \(x\) such that '\(\varphi\) in virtue of it
being the case that \(\phi\)' is true. 


(7)
 It is possible for
there to be a sentence '\(a\) is \(F\)', where \(F\)
expresses an intrinsic property and '\(a\) is \(F\)'
expresses a foundational fact in the sense of being a fact that is not
metaphysically grounded by any other fact. 


The argument is the following. Suppose, for reductio, that (4)
characterises a notion of intrinsicality when 'in virtue'
expresses metaphysical grounding. It follows from this, and the
necessity of (5) and (6), that (8) is necessarily true.



(8)
 Being \(F\) is
intrinsic iff, necessarily, for any \(x\), if \(x\) is \(F\) then
there is an obtaining state of affairs \(s\) that is intrinsically
about \(x\) such that \(s\) metaphysically grounds \(x\) being \(F\).



By (7), it is possible for a sentence \(\ulcorner a\) is
\(F\urcorner\) to express a foundational fact, where \(F\) expresses
an intrinsic property. Hence, it follows from (7) and the necessity of
(8) that (9) is possibly true.



(9)
 \(a\) being \(F\)
is a foundational fact and there is an obtaining state of affairs
\(s\) that is intrinsically about \(a\) such that \(s\) metaphysically
grounds \(a\) being \(F\). 


However, (9) is necessarily false, since foundational facts cannot be
metaphysically grounded by any facts. Hence, the assumption that (4)
characterises a notion of intrinsicality when 'in virtue'
expresses metaphysical grounding is false.



Given 'in virtue' doesn't express metaphysical
grounding in (4) on its intended use, what does it express? One
attractive option is to think that 'in virtue' in (4) is
meant to be understood as having the same effect as 'is a matter
of' and that, on this reading, it expresses the identity
relation. On this reading, '\(\varphi\) in virtue of it being
the case that \(\psi\)' is true iff the state of affairs
expressed by \(\phi\) is identical to the state of affairs expressed
by \(\varphi\). Since it follows from (6) that, on this reading
'\(\varphi\) in virtue of how \(x\) is intrinsically' is
true iff there is a sentence \(\phi\) that expresses an obtaining
state of affairs that is intrinsically about \(x\), (10) follows from
(4) on this reading of 'in virtue'.



(10)
 Being \(F\) is
intrinsic iff, necessarily, for any \(x\), if \(x\) is \(F\) then the
state of affairs of \(x\) being \(F\) is intrinsically about \(x\).



An alternative option, which is investigated in Marshall 2016a, is to
hold that 'in virtue' may express a number of different
reflexive determination, or entailment, relations in (4), and that
each of these relations determines a different notion of
intrinsicality. In addition to identity, such entailment relations
might include metaphysical necessitation, nomic necessitation, and
weak metaphysical grounding, where \(p\) weakly metaphysically grounds
\(q\) iff either \(p=q\) or \(p\) metaphysically grounds \(q\). Given
these relations determine different notions of intrinsicality for each
relation \(R\), we may call the notion of interiority corresponding to
\(R\) '\(R\) interiority'. So, for example, the notion of
intrinsicality obtained from (4) by interpreting 'in
virtue' as expressing the identity relation might be called
identity interiority, while the notion of interiority obtained from
metaphysical necessitation might be called metaphysical necessitation
interiority, and the notion of interiority obtained from nomic
necessitation might be called nomic necessitation interiority. 



If 'in virtue' expresses the relation \(R\), then
'\(\varphi\) in virtue of it being the case that \(\phi\)'
is true iff the state of affairs of it being the case that \(\phi\)
stands in \(R\) to the state of affairs of it being the case the
\(\varphi\). It follows from this, (4) and (6) that the \(R\) interior
properties satisfy (11).



(11)
 Being \(F\) is
\(R\) interior iff, necessarily, for any \(x\), if \(x\) is \(F\) then
there is an obtaining state of affairs \(s\) that is intrinsically
about \(x\) such that \(s\) stands in \(R\) to the state of affairs of
it being the case that \(x\) is \(F\). 


Just as being cubical is intuitively a local property, while being an
uncle and being such that there is a number are intuitively non-local
properties, the former property is intuitively an identity interiority
property, while the latter properties are both intuitively not
identity interior. (Those that claim that being such that there is a
number is a local property, however, are likely to also claim that
this property is identity interior. See, for example, Sider (1993a),
Eddon (2011), Vallentyne (1997), Hoffman-Kolls (2014) and Francescotti
(2014) for the relevant debate.) The identity interior properties,
however, plausibly do not coincide with the local properties, since,
while being identical to Obama is not a local property, it is
plausibly an identity interior property. This is because, necessarily,
for any \(x\) that is identical to Obama, the state of affairs of
\(x\) being identical to Obama is plausibly about how \(x\) and its
parts are and how they are related to each other, as opposed to how
they are related to other things and how other things are.



Since each identity interior property is also an \(R\) interior
property, for each reflexive relation \(R\), the \(R\) interior
properties also do not coincide with the local properties for each
\(R\). The \(R\) interior properties also differ from the identity
interior properties for at least some \(R\). For example, the
metaphysical necessitation interior properties plausibly differ from
the identity interior properties since, while being such that there is
a number is plausibly not identity interior, it is metaphysical
necessitation interior. This is because, given the assumption that
numbers necessarily exist, any \(x\) is such that how it is
intrinsically necessitates that \(x\) is such that there is a
number.


### 2.4 Duplication Preserving vs. Duplication Non-Preserving Properties



The third characterisation Lewis lists when isolating the concept of
intrinsicality is that duplicates never differ with respect to their
intrinsic properties. Lewis holds a further principle that may not be
obvious from the above quote: that any property with respect to which
duplicates never differ is intrinsic. Adding this further principle to
the characterisation gives us (12).



(12)
 \(F\) is
intrinsic iff \(F\) never differs between duplicates. 


(12), however, has the following problem. Assume that no man has a
mass of 500 kg (although it is possible for a man to have a mass of
500 kg). Since nothing has the property of being a popular man who is
500 kg, given this assumption, no two duplicates differ in whether
they have this property. (12) therefore falsely classifies *being a
popular man who is 500 kg* as intrinsic.



David Lewis doesn't face this problem due to his concretist
theory of possible worlds. Lewis holds that, in addition to the world
we live in, there are many other concrete worlds of the same kind as
the world we live in. Moreover, he holds that contained in some of
these worlds are 500 kg duplicate men, some of whom are popular, and
some of whom are not. Given Lewis's concretist theory of
possible worlds, then, being a popular man who is 500 kg is classified
as extrinsic by (12).



Philosophers who instead endorse standard abstractionist theories of
possible worlds, however, can't accept (12). According to
standard abstractionist theories of possible worlds, possible worlds
do not contain concrete entities such as 500 kg men as parts.
According to standard abstractionists, then, there are no 500 kg men,
there are only possible worlds that represent that there are 500 kg
men. Philosophers who endorse standard abstractionist theories of
possible worlds, however, can replace (12) with (13) (see Moore 1922
and Francescotti 1999). (Or at least, they can replace (12) with (13)
given they hold that there are things that exist according to
non-actual possible worlds that don't exist according to the
actual world. Abstractionists who deny this will have to further
modify (13).)



(13)
 \(F\) is
intrinsic iff, for any \(x\) and \(y\), and for any possible worlds
\(u\) and \(v\) such that \(x\) at \(u\) is a duplicate of \(y\) at
\(v, x\) has \(F\) at \(u\) iff \(y\) has \(F\) at \(v\) 


Given a standard abstractionist theory of possible worlds, there is a
possible world \(u\) that represents there being a popular 500 kg man
\(x\), and there is another possible world \(v\) that represents there
being an unpopular 500 kg man \(y\), where \(x\) at \(u\) has the same
intrinsic properties as \(y\) at \(v\), and hence \(x\) at \(u\) is a
duplicate of \(y\) at \(v\). Given this, (13) classifies *being a
popular man who is 500 kg* as extrinsic.



Putting aside for the moment the question of whether (13) correctly
characterises a genuine notion of intrinsicality, let us call the
notion it does characterise duplication preservation. (13), understood
as a characterisation of duplication preservation has two important
consequences. First, it has the consequence that identity properties,
such as being Obama, aren't duplication preserving. The reason
for this is that, since Obama could have had a duplicate distinct from
himself, there is a possible world at which Obama has a duplicate who
is not Obama, and hence there is a possible world at which something
having the property of being Obama has a duplicate which lacks this
property.



A more general consequence can be stated using the distinction between
qualitative and non-qualitative properties, where a qualitative
property is intuitively a property that doesn't concern any
particular entities, while a non-qualitative property is a property
that does concern one or more particular entities. Examples of
qualitative properties include the property of being cubical and the
property of being next to a tin, while examples of non-qualitative
properties include the property of being Obama and the property of
being next to Clinton. It plausibly follows from (13), understood as a
characterisation of duplication preservation, that any non-qualitative
property that is possibly instantiated is not a duplication preserving
property.



Say that a property is indiscriminately necessary iff it is
necessarily had by everything. The second consequence of (13),
understood as a characterisation of duplication preservation, is that
any two properties that are indiscriminately necessary are duplication
preserving. Consider, for example, the indiscriminately necessary
property of being such that there is a number (where we are assuming
that numbers necessarily exist). Since all things at all possible
worlds have this property, it is a duplication preserving
property.



Since being such that there is a number is neither a local property or
an identity interior property, but is a duplication preserving
property (given numbers necessarily exist), the duplication preserving
properties do not coincide with either the local properties or the
identity interior properties. Since being Obama is a metaphysical
necessitation interior property, but is not a duplication preserving
property, the duplication preserving properties don't coincide
with the metaphysical necessitation interior properties either. (Moore
(1922) and Dunn (1990), in effect, both distinguish duplication
preservation from interiority on these grounds.) Hence, we have at
least four distinct kinds of properties that are characterised by
Lewis's three characterisations: local properties, identity
interior properties, metaphysical necessitation interior properties,
and duplication preserving properties.



If Lewis's three characterisations concerning intrinsicality
characterise different notions, the three characterisations described
in section
 1.2
 concerning having a property in an intrinsic fashion also
characterise different notions. In order to distinguish these notions,
we may employ these characterisations to define having a property in a
duplication preserving fashion, having a property in an interior
fashion (relative to different entailment relations), and having a
property in a local fashion.


### 2.5 Which is the real distinction?



In the last three subsections, we distinguished a number of distinct
notions that are characterised by Lewis's three
characterisation: locality, duplication preservation, and several
different kinds of interiority. We now need to address the question of
whether there is a single notion of intrinsicality, and, if there is,
which of these notions is it? More precisely, we need to determine
whether there is a single notion that philosophers typically use
'intrinsic' to express, and, if so, which notion is
it.



One might attempt to answer this question by first noting that it is
much more common for philosophers to use Lewis's second
"in virtue" characterisation (or some variant of it), than
either of the other two characterisations when explaining what they
mean by 'intrinsic'. (For example, as far as we know,
Lewis uses the "in virtue" characterisation whenever he
explains what he means by 'intrinsic', but only uses the
other characterisations once.) This suggests that philosophers
typically use 'intrinsic' to express a notion of
interiority rather than either of the notions of locality or
duplication preservation.



Given philosophers typically use 'intrinsic' to express
some notion of interiority, there are several reasons to think that
the notion of interiority they typically express is that of identity
interiority. First, identity interiority is more fundamental, or
simpler, than the other notions of interiority, and philosophers are
often interested in employing more fundamental notions than less
fundamental notions. Second, the intrinsic properties of a thing
should be the properties that thing has "in itself",
rather than merely the properties that are metaphysically
necessitated, or nomically necessitated, or metaphysically explained,
by the properties the thing has "in itself". But this
suggests that the intrinsic properties should be the identity interior
properties, rather than the metaphysical necessitation interior
properties, or the nomic necessitation interior properties, or the
weak grounding interior properties. Finally, while the identity
interiority of necessarily coextensive properties can intuitively
differ, this is not true for at least some of the other notions of
interiority, such as metaphysical necessitation interiority and nomic
necessitation interiority. (Properties \(p\) and \(q\) are necessarily
coextensive iff, necessarily, for any \(x\), \(x\) has \(p\) iff \(x\)
has \(q\).) Since it is a widespread intuition that necessarily
coextensive properties can differ in whether they are intrinsic, this
suggests these other notions of interiority aren't what
philosophers typically express by 'intrinsic'. (This last
reason also provides a further reason to think that duplication
preservation isn't the notion of intrinsicality philosophers
typically employ, since it also does not differ among necessarily
equivalent properties.)



There are therefore reasons to think that the notion philosophers
typically use 'intrinsic' to express is identity
interiority. These reasons, however, are not decisive. First,
philosophers often take intrinsic properties to be qualitative, and
use 'intrinsic' to distinguish only among qualitative
properties. (See Sider 1996.) Since each local property and each
duplication preserving property is qualitative, but this is not true
for interior properties, this possibly suggests that at least some
philosophers at least some of the time use 'intrinsic' to
express either locality or duplication preservation. Secondly, while
identity interiority can do much of the theoretical work
intrinsicality is meant to do, it arguably cannot do all of it, with
some of this work being better done by some of the other notions of
intrinsicality described above. For example, Sider (1996) has argued
that duplication preservation is better able to be used to define
individualism about mental states than identity interiority. Finally,
the failure of many philosophers to clearly distinguish which notion
of intrinsicality they are employing provides at least some reason to
suspect that those philosophers have often used
'intrinsic' indeterminately to express a multiplicity of
notions of intrinsicality.



An alternative view has been put forward by Plate 2018, who has argued
that there is a single notion of intrinsicality and that none of the
notions discussed above correspond to it. Plate points out that, on
each of the alleged notions of intrinsicality discussed above, either
the property of having Paris as a part is classified as non-intrinsic
or the uninstantiable property of being such that Socrates is
non-self-identical is classified as intrinsic. Plate then argues that,
since having Paris as a part is intrinsic while being such that
Socrates is non-self-identical is non-intrinsic, it follows that none
of the above notions is intrinsicality. A response to Plate's
argument is given by Marshall 2021, who argues that there is no notion
of intrinsicality on which these properties are respectively intrinsic
and non-intrinsic. Instead, according to this response, there are only
notions of intrinsicality on which either both are intrinsic or both
are non-intrinsic.



In the next section, we will consider a number of attempts to analyse
'intrinsic'. Unless otherwise made clear, we will use
'intrinsic' to express identity interiority, and we will
assume that identity interiority is the most dominant notion of
intrinsicality employed by philosophers. As we will see, however, a
number of these attempts are most charitably seen as attempts to
analyse notions of intrinsicality other than identity interiority.


## 3. Attempts at Analysis



One reason to attempt an analysis of intrinsicality is that the
orientating characterisations of intrinsicality discussed in section
 1
 and sections
 2.2
 -
 2.4
 are all to some degree circular. (Cf Lewis 1983a and Yablo 1999.) For
example, a property \(p\) whose ascription to a thing \(x\) is about
\(x\) is intrinsic only if it is about how \(x\) is
*intrinsically*; \(p\) is not intrinsic if the ascription of
\(p\) to \(x\) is about how \(x\) is *extrinsically*.
Similarly, a property \(p\) that is had by \(x\) in virtue of how
\(x\) is is intrinsic only if \(x\) has \(p\) in virtue of how \(x\)
is *intrinsically*; \(p\) need not be *intrinsic* if
\(x\) has \(p\) in virtue of how \(x\) is *extrinsically*.



Some of the following attempts to analyse intrinsicality are best seen
as attempts to analyse the distinction between duplication preserving
properties and properties that aren't duplication preserving,
while other attempts are best seen as attempts to analyse the
distinction between interior and exterior properties, or between local
and non-local properties. For a discussion of whether we can give, or
need to give, a reductive analysis of intrinsicality, see Skiles 2014
and Hoffman-Kolss 2018.)



In addition to the metaphysical project of determining what it is for
a property to be intrinsic, there is an epistemological project of
determining how we can come to know with respect to particular
properties whether they are intrinsic, and whether such knowledge is
possible. To some extent, the metaphysical project of determining what
it is to be intrinsic can help with this epistemological project. For
example, if it can be shown that being intrinsic can be analysed as
the possession of such and such features, and we have knowledge of
which properties have such and such features, then we can use this
analysis to explain how we can know of particular properties that they
are intrinsic. In addition to employing such analyses of
intrinsicality, however, we might hope to develop and describe further
methods for obtaining justified beliefs regarding what properties are
intrinsic. For further discussion that bears on this issue, see
McQueen and van Woudenberg 2016.


### 3.1 Broadly Logical Theories



It would be good if we could analyse intrinsicality using only broadly
logical notions, where broadly logical notions are exhausted by the
narrowly logical notions of conjunction, negation and existential
quantification, the modal notion of metaphysical possibility, the
mereological notion of parthood, and the basic notions associated with
property theory and set theory, such as the notions expressed by
'state of affairs', 'property',
'relation', 'possible world',
'instantiates', 'is a member of' and
'set'. (The locution 'state of affairs' is
used differently by different philosophers. Here it is being used to
refer to the zero-place analogues of one-place properties and multiple
place relations. Just as a property is a way of a thing is or fails to
be, a state of affairs, under our usage, is a way things are or fail
to be.)



It is at least initially appealing to think that, if an object has a
property in an intrinsic fashion, then it has it independently of the
way the rest of the world is. The rest of the world could disappear,
and the object might still have that property. Hence a *lonely*
object, an object that doesn't coexist with any contingent
objects wholly distinct from it, could have the property. Many
extrinsic properties, on the other hand, cannot be possessed by lonely
objects - no lonely object has the property of being six metres
from a rhododendron, for example. This suggests an analysis of
intrinsicality: \(F\) is an intrinsic property iff it is possible for
a lonely object to be \(F\). This analysis is usually attributed to
Kim (1982) (e.g. in Lewis 1983a and Sider 1993), though Humberstone
(1996) dissents from this interpretation. If this analysis is
successful then it would constitute a broadly logical analysis, since
the expressions on the right hand side of the account can each be
defined using only broadly logical vocabulary.



The major problem with this analysis is that the 'if'
direction of the biconditional is clearly false. As Lewis (1983)
pointed out, it is possible for a lonely object to have the property
of being lonely, but the property of being lonely is not
intrinsic.



One might try to deal with this problem by adding extra modal
conditions to the above account. Say that a property \(F\) is
*independent of accompaniment* iff the following four
conditions (taken from Langton and Lewis 1998) are met:


1. Possibly, there exists a lonely \(F\)
2. Possibly, there exists a lonely non-\(F\)
3. Possibly, there exists an accompanied (i.e. not lonely) \(F\)
4. Possibly, there exists an accompanied non-\(F\)



At first glance, if \(F\) is intrinsic, then whether or not an object
has \(F\) should not depend on whether it is lonely. If this is right,
then all four of these conditions should be satisfied. We might
therefore try to give the following broadly logical analysis: \(F\) is
intrinsic iff \(F\) is independent of accompaniment.



As Langton and Lewis (1998) point out, however, this strengthened
analysis still fails (see also Lewis 1983a). Given intrinsic
properties, such as being a cube, are independent of accompaniment,
the extrinsic property of being either a lonely cube or an accompanied
non-cube is also independent of accompaniment. Hence, provided the
strengthened analysis correctly classifies *being a cube* as
intrinsic, it will falsely classify *being either a lonely cube or
an accompanied non-cube* as intrinsic.



In the face of this failure, we might still hope that a yet more
sophisticated broadly logical analysis might succeed. An argument due
to Marshall 2009 and Parsons 2001, however, shows that, given standard
views about what is possible, and given the absence of special
assumptions about the abundance and structure of properties, no such
analysis can be given. This argument considers the intrinsic property
of being an electron and the extrinsic property of being either an
accompanied electron or a lonely positron. It then shows that, given
certain common views about properties and possibility, these
properties have the same pattern of instantiation through logical,
mereological and set-theoretic space. Since formulas containing only
broadly logical expressions cannot distinguish between properties
having the same such pattern, it then follows that no broadly logical
analysis of intrinsically is possible.



To give more of an idea of how this argument works, suppose for
simplicity that there are only three possible worlds, \(w\_1\), \(w\_2\)
and \(w\_3\), and only four possible individuals \(x\_1\), \(x\_2\),
\(x\_3\) and \(x\_4\). Suppose further that: 



At \(w\_1\), \(x\_1\) and \(x\_2\) are the only things that exist,
\(x\_1\) is an electron and \(x\_2\) is a positron 



At \(w\_2\), \(x\_3\) is the only thing that exists and \(x\_3\) is an
electron 



At \(w\_3\), \(x\_4\) is the only thing that exists and \(x\_4\) is a
positron. 



Now assume a possible worlds theory of properties, according to which
the property of being \(F = \{\langle x, w\rangle \mid x \text{ is } F
\text{ at } w\}\). Given this theory of properties:



\(q\_1\) = Being an electron = \(\{\langle x\_1, w\_1 \rangle, \langle
x\_3, w\_2\rangle\}.\) 



\(q\_2\) = Being either an accompanied electron or a lonely positron =
\(\{\langle x\_1, w\_1\rangle, \langle x\_4, w\_3\rangle\}.\) 



Suppose, for reductio, that the notion of an intrinsic property can be
analysed in terms of broadly logical notions. Then, since analyses
must be necessary, there is a formula \(\varphi(p)\) such that (A) is
necessarily true, where \(\varphi\) contains only broadly logical
vocabulary, such as variables, brackets, '&',
'\(\vee\)', '\(\exists\)', 'is a
possible world', 'at' and '\(\in\)'.



(A)
 \(p\) is intrinsic iff \(\varphi(p)\). 


Since \(q\_1\) is necessarily intrinsic, while \(q\_2\) is necessarily
not intrinsic, it follows from the necessity of (A) that
\(\varphi(q\_1)\) is necessarily true, while \(\varphi(q\_2)\) is
necessarily false. In other words, if there is a broadly logical
analysis of intrinsicality, then there is a formula \(\varphi\)
containing only broadly logical vocabulary that necessarily applies to
\(q\_1\), and necessarily does not apply to \(q\_2\). However, it can be
proved that there is no such formula. For example, while



\[\text{'}\exists w\_1 \exists w\_2 \exists x\_1 \exists x\_2 \exists x\_3(p =
\{\langle w\_1, x\_1\rangle , \langle w\_2, x\_3\rangle \})\text{'}\]


applies to \(q\_1\), it also applies to \(q\_2\). Hence, the reductio
assumption is false, and there is no broadly logical analysis of
intrinsicality. This argument can easily be adapted to show that there
is no broadly logical analysis of intrinsicality given more realistic
assumptions about what possible worlds there are.



It is important to note that, despite the existence of the above
argument, one might still be able to analyse intrinsicality in terms
of broadly logical notions if one is prepared to make special (and
controversial) assumptions about properties. As an illustration of
this, suppose one assumes that all intrinsic properties have no proper
parts while all extrinsic properties do have proper parts. Then one
might give a broadly logical analysis of intrinsicality by giving
account (14).



(14)
 \(p\) is an intrinsic property iff \(p\) is a property that has
no proper parts.

 
Even if an analysis like this one is true, however, its utility will
be limited. For example, unless we already know somehow which
properties have proper parts and which properties do not have proper
parts, (14) will be useless in helping us to determine which
properties are intrinsic and which properties are non-intrinsic.



Two more sophisticated attempts to give a broadly logical analysis of
intrinsicality are Yablo's contractionist account discussed in
section
 3.4,
 and Francescotti's relational account, which is discussed in
the supplementary document:



The Relational Account of Intrinsicality.
 



Both these accounts rely on controversial assumptions about properties
or related entities such as possible worlds. The most sophisticated
attempt to give a purely broadly logical analysis of intrinsicality
that has yet been proposed is that of Plate 2018. A response to
Plate's account is given in Marshall forthcoming.



Given that we cannot give a satisfactory account of intrinsicality
using only broadly logical notions, it is natural to investigate
whether we can give a satisfactory account using a larger set of
notions. In the next four subsections, we will consider whether
intrinsicality can be analysed in terms of the notions of perfect
naturalness, non-disjunctivity, possible world contractions, or
metaphysical grounding.


### 3.2 Perfect Naturalness Theories



The most influential attempt to analyse intrinsicality over the last
30 years has arguably been Lewis's 1986a account of
intrinsicality in terms of perfect naturalness. 



The perfectly natural properties and relations are the fundamental
properties and relations. Lewis wrote:



> 
> 
> Sharing of [the perfectly natural properties] makes for qualitative
> similarity, they carve at the joints, they are intrinsic, they are
> highly specific, the set of their instances are ipso facto not
> entirely miscellaneous, there are only just enough of them to
> characterise things completely and without redundancy. Physics has its
> short list of 'fundamental physical properties': the
> charges and masses of particles, also their is so-called
> 'spins' and 'colours' and
> 'flavours', and maybe a few more that have yet to be
> discovered. (Lewis 1986b, p. 60, author's emphasis)]
> 
> 
> 



The notion of a perfectly natural, or fundamental, property or
relation suggests an appealing approach to analysing intrinsicality.
Consider the non-intrinsic property of being an uncle. It is appealing
to think that the reason this property fails to be intrinsic is that
it can be analysed in terms of more fundamental relations, such as
being a sibling of and being a child of, in a manner that reveals how
something being an uncle is at least partially a matter of how it is
related to things that aren't part of it. In particular, it is
plausible to think that the reason being an uncle is non-intrinsic is
that for \(x\) to be an uncle is for \(x\) to have a sibling that has
a child, where this child fails to be part of \(x\). More generally,
it is plausible that, if being \(F\) is a non-intrinsic property, then
the fact that it is non-intrinsic can be revealed by analysing being
\(F\) in terms of more fundamental properties and relations in a
manner that reveals how something being \(F\) may partly be a matter
of being related to a non-part or may be partly a matter of how
non-parts are. If this is true, then it has the following two
consequences. First, it follows that every fundamental property is
intrinsic, since it being non-intrinsic cannot be revealed by analysis
in terms of more fundamental properties and relations. Secondly, and
even more importantly, it follows that whether a property is intrinsic
should be analysable in terms of how that property can be analysed in
terms of the fundamental properties and relations.



Lewis in effect adopts the above approach to analysing intrinsicality
by analysing intrinsicality in terms of a particular kind of
supervenience on the fundamental properties and relations.
Lewis's 1986 account (formulated so as to be compatible with
standard abstractionist theories of possible worlds) consists of (15)
and (16).



(15)
 \(p\) is an
intrinsic property iff (i) \(p\) is a property, and (ii) for any \(x\)
and \(y\), and for any worlds \(w\_1\) and \(w\_2\) such that \(x\) at
\(w\_1\) is a duplicate of \(y\) at \(w\_2\), \(x\) has \(p\) at \(w\_1\)
iff \(y\) has \(p\) at \(w\_2\).


(16)
 \(x\) at \(w\_1\)
is a duplicate of \(y\) at \(w\_2\) iff there is a one-to-one
correspondence \(f\) between \(x\)'s parts at \(w\_1\) and
\(y\)'s parts at \(w\_2\) such that, for any perfectly natural
property or relation \(R\), for any \(x\_1, x\_2,\ldots\) that are part
of \(x\) at \(w\_1\): \(x\_1, x\_2\ldots\) stand in \(R\) at \(w\_1\) iff
\(f(x\_1), f(x\_2)\ldots\) stand in \(R\) at \(w\_2\). 


Lewis's account, which I will call perfect naturalness account
1, is best seen as an attempt at analysing duplication preservation,
rather than as analysing identity interiority which we argued in
section
 2.5
 is the dominant notion of intrinsicality in philosophy. This account
rests heavily on the claim that all perfectly natural properties are
intrinsic, and, implicitly, that the perfectly natural properties and
relations are sufficient to characterise the world completely. The
last assumption is needed because the theory rules out the possibility
that there are two things that share all their perfectly natural
properties and lack proper parts, but differ with respect to some of
their intrinsic properties.



The first assumption that all perfectly natural properties are
intrinsic has been rejected by Weatherson (2006). (See also Yablo 1999
who holds that, even if every perfectly natural property is intrinsic,
this fact is at best a lucky accident and that this renders
Lewis's account unsuccessful as a philosophical account of
intrinsicality.) Weatherson claims that it is metaphysically possible
for the instantiated perfectly natural properties to be vector
properties. He then argues that, since vector properties must be
extrinsic, it follows that at least some perfectly natural properties
fail to be intrinsic, namely the perfectly natural properties
instantiated at such worlds. The claim that vector properties need to
be extrinsic, however, is highly contentious. It is natural to
understand a vector property to be a property that is usefully
represented for theoretical purposes by a certain kind of mathematical
object, namely a vector. Given this construal, however, there is no
reason to think that vector properties cannot be intrinsic. For
example, due to the similarity relationships between colours
represented by the colour solid it might be useful to represent
colours by vectors. However, doing this is compatible with
metaphysical theories of colour on which colours are intrinsic. There
might be other conceptions of being a vector property on which vector
properties cannot be intrinsic. However, it is unclear whether there
an conception of vector properties on which they cannot be intrinsic
but can be fundamental. (See Busse (2009) and Marshall (2016b) for
more discussion.)



The second assumption, that the perfectly natural properties and
relations are sufficient to characterise each possible world, has been
rejected by Sider (1995) and Schaffer (2004). Both claim that it is
metaphysically possible for there to be endless sequences of more and
more natural properties, without any set of perfectly natural
properties out of which all the other properties can be defined. In
response to this problem, Sider (1993b) has proposed a modification of
Lewis's account which he claims is consistent with there being
such endless sequences of more and more natural properties. Sider
(2011) and Marshall (2016b) have recently argued in favour of the
second assumption, with Sider's argument appealing to his Humean
theory of metaphysical possibility.



As noted above, Lewis's account fails to provide an account of
identity interiority, which is the most standard notion of
intrinsicality, and at best provides an analysis of duplication
preservation. Marshall 2016b has recently provided an account in terms
of perfect naturalness that, if successful, does provide an analysis
of identity interiority. If this account works, it can also be
modified to provide analyses of the other notions of intrinsicality
described in section
 2.4.



The basic idea behind this alternative account of intrinsicality,
which I will call perfect naturalness account 2, is that the
ascription of a property to a thing is intrinsically about that thing
iff the ascription can be expressed in fundamental terms while only
referring to, and quantifying over, the parts of \(x\). (A similar
account is briefly sketched in Skow 2007, p. 112.) In more detail,
perfect naturalness account 2 first analyses intrinsicality in terms
of intrinsic aboutness by (3).



(3)
 \(p\) is intrinsic iff, necessarily, for any \(x\), the
ascription of \(p\) to \(x\) is intrinsically about \(x\). 


It then provides an account of intrinsic aboutness in terms of perfect
naturalness and qualitivity by (17), where qualitivity is analysed by
(Q). (In (17) and (Q), perfectly natural operators and quantifiers are
fundamental operators and quantifiers.)



(17)
 \(s\) is intrinsically about \(x\) iff \(s\) is a non-qualitative
state of affairs that can be expressed by a sentence \(\phi\)
containing at most brackets, commas, lambda abstractors, names
referring to parts of \(x\), predicates expressing perfectly natural
properties and relations, operator expressions expressing perfectly
natural operators, and quantifier expressions that express perfectly
natural quantifiers and whose occurrences in \(\phi\) quantify only
over parts of \(x\). 


(Q)
 \(p\) is a
qualitative property iff \(p\) is a property, and \(p\) can be
expressed by a predicate containing at most brackets, commas,
variables, lambda abstractors, predicates expressing perfectly natural
relations, quantifier expressions expressing perfectly natural
quantifiers, and operator expressions expressing perfectly natural
operators. 


(17) entails that any state of affairs that is intrinsically about
something is non-qualitative. This consequence of (17) can be
justified by noting that, if \(s\) is intrinsically about \(x\), then
there must be at least one particular thing that it is about, and
hence it cannot be a qualitative state of affairs.



(17), as it stands, fails to provide a completely specified account of
intrinsic aboutness, since it contains the locution 'quantifies
only over parts of \(x\)' which needs to be defined, and giving
the needed definition is not straightforward. The account of
'quantifying over parts of \(x\)' given by perfect
naturalness account 2 can be simplified if we assume that
'some' expresses the only perfectly natural quantifier.
Say that a property \(p\) necessitates a property \(q\) iff,
necessarily, for any \(x\), if \(x\) has \(p\) then \(x\) has \(q\).
If \(\Phi\) is a set of formulas, let \(\bigvee\Phi\) be the
disjunction of the formulas in \(\Phi\). (So \(\bigvee\{\phi\_1 ,
\phi\_2,\ldots\}\) = \(\ulcorner\phi\_1 \vee \phi\_2
\vee\ldots\urcorner\).) Given these definitions and the above
simplifying assumption, perfect naturalness account 2 endorses (18),
where \(F\) and \(G\) are predicates expressing properties and
'\(\leq\)' symbolises 'is part of'.



(18)
 The initial occurrence of 'Some' in \(\ulcorner
[\text{Some } v|Fv]Gv\urcorner\) quantifies only over parts of \(x\)
iff, for any set \(\Pi \) of names of all the parts of \(x\), the
property expressed by \(F\) necessitates the property expressed by
\(\ulcorner\lambda v\bigvee\{\ulcorner v\leq a\urcorner \mid
a\in\Pi\}\urcorner\). 


For example, according to this account, if \(d\) has exactly three
proper parts (which are \(a\), \(b\) and \(c\)), then
'some' in '[Some \(v \mid Fv](Gv)\)'
quantifies only over parts of \(d\) iff the property of being \(F\)
necessitates the property of being part of either \(a\), \(b\), \(c\)
or \(d\) (that is, iff, necessarily, any thing that is \(F\) is part
of either \(a\), \(b\), \(c\) or \(d\)). More generally, this account
endorses (19).



(19)
 For any formulas
\(\phi\) and \(\psi\), the initial occurrence of 'Some' in
\(\ulcorner [\text{Some }v\mid \phi]\psi\urcorner\) quantifies only
over parts of \(x\) iff, for any set \(\Pi\) of names of all the parts
of \(x\), under any assignment \(g\) to the free variables in \(\phi\)
that are not \(v\), the property expressed by \(\ulcorner\lambda
v\phi\urcorner\) necessitates the property expressed by

\[\begin{align}
\ulcorner\lambda v\big( \bigvee\{\ulcorner v\leq u \urcorner \mid u&
\text{ is a free variable in } \phi \text{ that is not } v\} \\
 &\vee \bigvee\{\ulcorner v\leq a\urcorner \mid a\in\Pi\}\urcorner.
\end{align}\]



Given the simplifying assumption that 'some' is the only
perfectly natural quantifier expression, (3), (17), (Q) and (19)
provide the complete account of what it is for a property to be
intrinsic according to perfect naturalness account 2.



As with perfect naturalness account 1, perfect naturalness account 2
assumes that each perfectly natural property is intrinsic and that the
perfectly natural properties, relations, operators and quantifiers are
able to completely describe every possible world. If someone rejects
these assumptions, they will therefore be unable to endorse this
account.


### 3.3 Non-Disjunctivity Theories



Instead of analysing intrinsicality in terms of perfect naturalness,
one might try to analyse it in terms of non-disjunctivity, where
non-disjunctivity is regarded as an objective metaphysical feature of
properties rather than merely a syntactical feature of predicates. The
most prominent attempt to do this is the account of Langton and Lewis
1998. (A similar theory is in an appendix to Zimmerman 1997.)



Langton and Lewis's account involves the following three steps.
First, they analyse the non-disjunctive properties as follows:



> 
> [T]he *disjunctive* properties [are] those properties that can
> be expressed by a disjunction of (conjunctions of) natural properties;
> but that are not themselves natural properties. (Or, if naturalness
> admits of degrees, they are much less natural than the disjuncts in
> terms of which they can be expressed.) (Langton and Lewis 1998: 61,
> author's emphasis)
> 



Secondly, they say a property is *basic intrinsic* iff it is
non-disjunctive, is not the negation of a disjunctive property, and is
independent of accompaniment. Finally, they say that (i) two objects
are duplicates iff they share the same basic intrinsic properties, and
(ii) \(F\) is an intrinsic property iff two duplicates never differ
with respect to it.



The last step of their account is in a form that requires
Lewis's concretist theory of possible worlds. The last step in a
form suitable for a typical abstractionist theory of possible worlds
is instead:


1. \(x\) at \(u\) is a duplicate of \(y\) at \(v\) iff, for any basic
intrinsic property \(P, x\) has \(P\) at \(u\) iff \(y\) has \(P\) at
\(v\); and
2. \(F\) is intrinsic iff, for any \(x\) and \(y\), and for any
possible worlds \(u\) and \(v\) such that \(x\) at \(u\) is a
duplicate of \(y\) at \(v, x\) has \(F\) at \(u\) iff \(y\) has \(F\)
at \(v\).



Note that (ii) is just the abstractionist version of the duplication
characterisation for intrinsicality given by (13) in section
 2.4.



Langton and Lewis claim their account works given a variety of ways of
understanding 'natural' and 'more natural
than' in their analysis of 'non-disjunctive'. In
particular, they hold that the account will work if
'natural' is understood to express the notion of perfect
naturalness, or fundamentality, in the Lewis (1986a) account, and
'more natural than' is understood to mean 'is more
fundamental than'. They also claim that the account will work if
the natural properties are those taken to correspond to Armstrongian
universals, or the properties expressed by the predicates in canonical
formulations of physics, or the properties expressed by the predicates
in canonical formulations of common-sense. Langton and Lewis say that
it should not matter how we draw the distinction between natural and
non-natural properties for present purposes, as long as we have it,
and properties like *being lonely and round or accompanied and
cubical* are not natural. As a result of this, Langton and Lewis
think that their account is more neutral than Lewis's earlier
account and should therefore should appeal to a larger range of
philosophers than the earlier account.



Langton and Lewis put forward their account as only an analysis of
qualitative intrinsic properties, rather than intrinsic properties in
general. Their account, however, arguably fails as an account of
qualitative identity interior properties since, given numbers
necessarily exist, it classifies being such that there is a number as
intrinsic. Therefore, like the Lewis 1986a account, the Langton and
Lewis account is arguably most charitably interpreted as an account of
duplication preservation, or perhaps of qualitative metaphysical
necessitation interiority.



In addition to failing to give an account of the standard account of
intrinsicality, there are reasons to think that Langton and Lewis
cannot be as neutral as they claim to be regarding what account of
naturalness is combined with their account. As noted above, Langton
and Lewis claim that their account of intrinsicality is compatible
with any construal of naturalness on which the natural properties are
those that play a special role in our thinking. On one such conception
of naturalness, the natural properties include those properties for
which we have simple words or concepts, which do not strike us as
strange or odd, and on which we have relied in past inductions (Sider
2001: 362). Given this conception of naturalness, the property of
being a rock is a natural property, and hence is classified as
intrinsic on the Langton and Lewis account. However, as Sider 2001
argues, being a rock is plausibly not an intrinsic property. Just as
an object that is qualitatively intrinsically the same as a house and
is a large part of a house is not itself a house, an object that is
qualitatively intrinsically the same as a rock and is a large part of
a rock is not itself a rock. Hence, whether something is a rock is
partly a matter of what is the case outside of it. Hence, contra the
combination of the Langton and Lewis account with the above conception
of naturalness, being a rock is not intrinsic.



Another objection to the Langton and Lewis account is that there
appear to be counterexamples to the account, even given Lewis's
favoured conception of naturalness. On this conception, the perfectly
natural properties and relations are the fundamental properties and
relations, and a property \(p\) is more natural than a property \(q\)
iff the shortest definition of \(p\) in terms of the perfectly natural
properties and relations is shorter than the shortest definition of
\(q\) in terms of the perfectly natural properties and relations. (See
(Lewis 1986: 61).) As Marshall and Parsons 2001 argue, given this
construal of naturalness, the property of being such that there is a
cube appears to be neither a disjunctive property nor the negation of a
disjunctive property according to Langton and Lewis's account of
disjunctivity. If this is the case, however, then, since being such
that there is a cube is independent of accompaniment, the Langton and
Lewis account classifies this property as a basic intrinsic property,
and hence classifies it as an intrinsic property. This property,
however, is clearly not intrinsic.



Langton and Lewis 2001 have responded to this objection by claiming
that being such that there is a cube is less natural than both being a
cube and being accompanied by a cube, since (I) being such that there
is a cube can be expressed as the disjunction 'being either a
cube or accompanied by a cube', (II) disjoining unrelated
properties always reduces naturalness, and (III) being a cube and
being accompanied by a cube are unrelated properties. One problem with
this response is that it is not obvious why (II) is meant to be true
given Lewis's favoured conception of naturalness above, or what
other notion of naturalness it might be true of. A second problem with
the response is that, even granting (II) is true, it is not clear what
it is for two properties to be unrelated in the sense that is at issue
in (II), and hence it is not clear why being a cube and being
accompanied by a cube are meant to be unrelated. For further
discussion of attempts to analyse intrinsicality in terms of
non-disjunctivity, see the supplementary document:



More on Non-Disjunctive Theories.
 


### 3.4 Contractionist Theories



One attractive idea is that a thing's intrinsic properties are
those properties it would have if it were alone in the world. An
elegant theory of intrinsicality that develops this idea is that of
Peter Vallentyne (1997).



Vallentyne defines a *contraction* of a world as "a world
'obtainable' from the original one solely by
'removing' objects from it." (211) As a special case
of this, an \(x-t\) contraction, where \(x\) is an object and \(t\) a
time, is "a world 'obtainable' from the original one
by, to the greatest extent possible, 'removing' all
objects wholly distinct from \(x\), all spatial locations not occupied
by \(x\), and all times (temporal states of the world) except \(t\),
from the world", as well as all laws of nature from the world.
(211) Vallentyne claims that \(F\) is intrinsic iff, for any thing
\(x\), for any time \(t\), and for any worlds \(u\) and \(w\) such
that \(u\) is an \(x-t\) contraction of \(w, x\) has \(F\) at \(t\) at
\(u\) iff \(x\) has \(F\) at \(t\) at \(w\). In short, a property is
intrinsic to an object iff removing the rest of the world
doesn't change whether the object has the property.
Vallentyne's account classifies identity properties, like being
Obama, as intrinsic. It is therefore best seen as an account of
interiority. The account also classifies all indiscriminately
necessary properties as intrinsic. It is therefore arguably most
charitably interpreted as an account of metaphysical necessitation
interiority



As Vallentyne notes, this definition will not be very enlightening
unless we understand the idea of a contraction. This seems related to
the objection Langton and Lewis (1998) urge against Vallentyne. They
say that Vallentyne's account reduces to the claim that a
property is intrinsic iff possession of it never differs between an
object and its lonely duplicates, a claim they think is true but too
trivial to count as an analysis. Their position is that we cannot
understand contractions without understanding duplication, but if we
understand duplication then intrinsicality can be easily defined, so
Vallentyne's theory is no advance.



This criticism might be too quick. Stephen Yablo (1999) has argued
that Vallentyne should best be understood as working within a very
different metaphysical framework than Lewis. For Lewis, no (ordinary)
object exists at more than one world, so Vallentyne's
contractions, being separate worlds, must contain separate objects.
Hence \(x-t\) contractions can be nothing other than lonely
duplicates, and the theory is trivial. Yablo suggests that the theory
becomes substantive relative to a metaphysical background in which the
very same object can appear in different worlds. If this is the case
then we can get a grip on contractions without thinking about
duplications -- the \(x-t\) contraction of a world is the world
that contains \(x\) itself, and as few other things as possible.



There are, however, problems with Yablo's defense of
Vallentyne's contractionist approach to intrinsicality. One
problem, discussed in Parsons 2007 and Marshall 2014, is that, on
Yablo's version of contractionism, we need to be able to make
sense of sentences like 'In \(w\), Prince Philip is not a
husband', where \(w\) is the world consisting of just Prince
Philip. Making sense of such sentences, however, is difficult without
appealing to the distinction between intrinsic and non-intrinsic
properties. Appealing to this distinction when interpreting
Yablo's account, however, would render it circular. 


### 3.5 Grounding Theories



Over the last few years, there have been a number of attempts to
analyse intrinsicality in terms of metaphysical grounding, a prominent
instance of which is that of Rosen (2010). Other accounts of
intrinsicality in terms of grounding include those of Witmer, Butchard
and Trogdon (2005), Trogdon (2009), Bader (2013), and Witmer (2014).
For a discussion of the first two of these accounts, see the
supplementary document:



More on Grounding Theories of Intrinsicality.
 



The state of affairs of it being the case that \(A\) is grounded by
the state of affairs of it being the case that \(B\) iff \(A\) in
virtue of it being the case that \(B\), where the relevant notion of
"in virtue" is meant to be a metaphysically explanatory
one. Grounding is supposed to be a metaphysical analogue of causation:
just as causation is what connects the explanans of an explanation
with its explanandum in causal explanation, grounding is meant to be
what connects the explanans of an explanation with its explanandum in
metaphysical explanation. An example of a metaphysical explanation
would be an explanation of why a certain type of act is wrong in terms
of more fundamental moral facts.



Rosen's account presupposes Russellianism about facts, according
to which facts are structured out of things, properties and operators
in the same kind of way sentences are built up out of names,
predicates and operator expressions. Given this background,
Rosen's analysis can be stated as follows:



> 
> \(F\) is an intrinsic property iff, necessarily, for any \(x\): (i) if
> the ascription of \(F\) to \(x\) is grounded by a fact with
> constituent \(y\), then \(y\) is part of \(x\); and (ii) if the
> negation of the ascription of \(F\) to \(x\) is grounded by a fact
> with constituent \(y\), then \(y\) is part of \(x\)
> 



Given the negation of the ascription of being Obama to Clinton is
grounded by the negation of the ascription of identity to Obama and
Clinton (a claim plausibly required by Rosen's account) the
account classifies being Obama as non-intrinsic since Obama is not
part of Clinton. The analysis also classifies intuitively
non-intrinsic indiscriminately necessary properties, such as the
property of being such that there is a number, as non-intrinsic. Being
such that there is a number is classified as non-intrinsic by the
account since, for example, its ascription to Obama is grounded by the
fact that 2 is a number, which has a constituent, namely the number 2,
that is not part of Obama. These consequences of the account suggest
that it is best thought of as an account of the distinction between
local and non-local properties.



Marshall (2015) has given the following objection to Rosen's
account. (The objection is meant to generalise to all attempts to
analyse intrinsically in terms of grounding, including the attempts of
Witmer, Butchard and Trogdon (2005), Trogdon (2009), Bader (2013) and
Witmer (2014). For an objection that is meant to apply to
Rosen's account, but not Bader's account, see Bader
2013:fn. 53.) According to the objection: a) there is nothing
incoherent in supposing that a state of affairs that is intrinsically
about \(x\) might ground a state of affairs that is intrinsically
about something that is wholly distinct from \(x\), and b) on a number
of popular metaphysical theories, there are such grounding
relationships. If there are such grounding relationships, however,
Rosen's account fails.



To consider an example of such a grounding relationship, let us assume
that standard set theory is necessarily true, and that existence is a
local property. (If one thinks that existence is not a local property,
perhaps because one thinks that existence is the non-local
quantificational property of being such that, among all the things
there are, there is something that is identical to one, replace
existence with the property of being such that, among all the parts of
one, there is something identical to one, which is intuitively local.)
Let us also assume the common (though not universal) view that members
of sets are not literally parts of those sets. (See Lewis (1986a) for
an example of a prominent theory of sets that denies members of sets
are parts of those sets.) Given these assumptions, it is plausible
that the fact that \(\{Obama\}\) exists is grounded by the fact that
Obama exists. However, if this is correct, then, since the fact that
Obama exists has a constituent that is not part of \(\{Obama\}\), it
follows that Rosen's account falsely classifies existence as a
non-local property.



Borrowing from the recent account of Witmer (2014) (which is similar
to Rosen's account), we might attempt to fix Rosen's
account by replacing parthood in the account with ontological
dependence, where the relation of ontological dependence is defined by
(20).



(20)
 \(x\) ontologically depends on \(y\) iff the fact that \(y\)
exists at least partly grounds the fact that \(x\) exists.


Applying this modification to Rosen's account results in
(21).



(21)
 \(F\) is an intrinsic property iff, necessarily, for any \(x\):
(i) if the ascription of \(F\) to \(x\) is grounded by a fact with
constituent \(y\), then \(x\) ontologically depends on \(y\); and (ii)
if the negation of the ascription of \(F\) to \(x\) is grounded by a
fact with constituent \(y\), then \(x\) ontologically depends on
\(y\).


This modified version of Rosen's account appears to correctly
classify existence as a local property given the above metaphysical
theses. Given these metaphysical theses, however, it arguably ends up
classifying too many properties as local. For example, given the above
theses, it appears to classify the property of having a member as a
local property. It does this, since, given the above metaphysical
theses, necessarily, if \(x\) has a member, then the fact that \(x\)
has a member is plausibly grounded by the fact that \(y\) is a member
of \(x\), where \(y\) is some member of \(x\), and where \(x\) is
ontologically dependent on \(y\). Provided members of sets
aren't parts of those sets, however, the property of having a
member appears to be just as non-local as the property of having
something be two meters away from one.