## 1. Lewisian metaphysics: an overview



On a traditional conception, metaphysics aims to answer, in a suitably
abstract and fully general manner, two questions:


1. What *is there*?
2. What is it (that is, whatever it is that there is)
*like*?



Lewis fully endorses this conception: for him, metaphysicians are not
in the business merely of analyzing our "conceptual
scheme" (except insofar as doing so is an effective
*method* for finding answers to metaphysical questions), nor
need they pay any heed to the perennial philosophical calls for the
abolition of their subject. They are, rather, engaged in an
unproblematically factual inquiry into the nature of reality--one
whose recognizable epistemological pitfalls provide no grounds for
doubting its legitimacy:



> 
> Once the menu of well-worked-out theories is before us, philosophy is
> a matter of opinion. Is that to say that there is no truth to be had?
> Or that the truth is of our own making, and different ones of us can
> make it differently? Not at all! If you say flatly that there is no
> god, and I say that there are countless gods but none of them are our
> worldmates, then it may be that neither of us is making any mistake of
> method. We may each be bringing our opinions to equilibrium in the
> most careful possible way, taking account of all the arguments,
> distinctions, and counterexamples. But one of us, at least, is making
> a mistake of fact. Which one is wrong depends on what there is. (Lewis
> 1983a, p. xi)
> 



We can begin to get a handle on Lewis's audacious and
comprehensive answers to our two overarching questions by
distinguishing three components to his metaphysical program:



First, he offers an account of what the *fundamental ontological
structure* of the world is. Is, and *must*
be--although as we'll see, that qualification turns out to be
in a certain sense trivial. This account of fundamental ontology of
course presupposes that the word "fundamental"
*means* something, and in particular manages to cleanly
distinguish a certain central core of one's ontological
commitments from the rest. Suppose these commitments take the form of
views about what entities (or "particulars") there are,
and what properties and relations they stand in. Then we can
distinguish two questions. Are some entities more fundamental than
others--with, perhaps, an elite group of entities being the most
fundamental? Are some properties/relations more fundamental than
others--again, with, perhaps, an elite group being the most
fundamental? You might find a "yes" answer to both
questions attractive. (E.g., chairs exist, but they are not
fundamental-level entities--though perhaps quarks are. Likewise,
some chairs have the property of *being made of oak*; but this
is not a fundamental-level property--though perhaps the property
of having such-and-such electric charge is.) As for Lewis's own
views, with respect to the second question they are fairly
unambiguous: He is quite clear that a proper ontology must include not
just particulars but also properties and relations (see especially
1983b); he is equally clear that it is a perfectly objective and
determinate matter which of these properties/relations are more
fundamental (or, in his terminology, more "natural") than
others (*ibid.*); he is officially agnostic about whether some
properties/relations are *most* fundamental, or
*perfectly* natural (1986f). His views on the first question
are, to my eyes at least, more difficult to discern--but for
reasons that, in the final analysis, probably do not matter. See
the



> 
> Supplement on
>  Fundamental Entities
> 



What's more, to a very great extent he takes it that the route to
a proper theory of fundamental ontology is by way of *a priori*
philosophical inquiry. (An important qualification will be noted
shortly.)



Second, he offers an account of *modality*, his famous
"realism" about possible worlds. Lewis, like many
philosophers, takes talk of possibility and necessity to be best
explicated as disguised quantification over possible worlds (and
possible inhabitants thereof), and he was endlessly ingenious at
showing how to use the resources provided by a theory of
"possibilia" to produce analyses of a host of modal
locutions. But his realism about possible worlds consists in much more
than inclusion of such entities into his ontology; indeed, it would
probably be better to call Lewis a "reductionist" about
modality--reductionist in a way that distinguishes him from
virtually every other philosopher of modality. For a typical believer
in possible worlds will, if asked to explain what they are, give an
account that *uses modal notions* at some crucial point.
Perhaps she will say that possible worlds are maximal
*consistent* sets of sentences (in some appropriate language);
or perhaps she will say that they are certain kinds of maximal
properties that reality as a whole *could have* instantiated.
Lewis says no such thing: he offers a characterization of possible
worlds--and thus of modality generally--in *explicitly
non-modal terms*. This complete subordination of the modal to the
non-modal gives his philosophy of modality a quite radical character,
and also sheds light on some of his seemingly independent views about
the modalities involved in such concepts as causation, law of nature,
and chance. (For example, Lewis rejects philosophical accounts of laws
of nature that rely on any primitive modal notions.)



Third, Lewis offers an account of how facts about everything else
*reduce* to the sorts of facts laid out in his accounts of
fundamental ontology and modality. (Note that given the remarks in the
last paragraph, these reductions ultimately rest on facts about
fundamental ontology *alone*; no unanalyzed modal notions are
involved in them.) Better: he offers an assortment of distinctive
approaches for constructing such reductions, of which there are many
examples but no single, canonical exposition. At this point I wish to
make just three observations about these strategies. First, they can
be seen to be directed at providing answers to a distinctively
metaphysical kind of question, of the form, "What is it for
such-and-such a fact to obtain?" Examples will pin down the
idea:


* Question: What is it for an object to persist through time?
Lewis's answer: It is for that object to be constituted by
three-dimensional, instantaneous time-slices that exist at different
times. (Lewis 1988)
* Question: What is it for an object to have a certain property
essentially? Lewis's answer: It is for every one of that
object's counterparts in other possible worlds to have that
property. (Lewis 1968)
* Question: What is an event? Lewis's answer: It is a certain
kind of property of spacetime regions. (Lewis 1986d)
* Question: What is it for one event to be a cause of another?
Lewis's (preliminary) answer: It is for the second event to
counterfactually depend on the first, in the sense that had the first
not occurred, the second would not
 have.[1]
 (Lewis 1973a, 1986b)
* Question: What is an explanation of some event? Lewis's
answer: It is a quantity of information about that event's causes.
(Lewis 1986c)



And so on. It is this kind of question--albeit not always phrased
in this way, and accompanied by definite views about what constitutes
a philosophically appropriate answer--that animate what we might
call Lewis's "applied metaphysics": the application of
his basic positions in ontology and modality to a range of perennial
metaphysical topics. Note that the *reductionist* character of
his approach comes out when we pursue the obvious follow-up questions:
For example, what is it for one event to counterfactually depend on
another? Roughly, it is for the closest possible world in which the
second does not occur to be a world in which the first does not occur.
What is it for one world to be closer to actuality than another?
We'll skip the answer for now--but rest assured that it and
the answers to subsequent follow-up questions are designed to hang
together in such a way as to collectively display how facts about what
causes what ultimately reduce to facts about fundamental ontology. And
so it goes, for personal identity, free will, the mind, knowledge,
ethics, laws of nature, you name it.



The second observation is that it remains far from clear whether we
can dispense with the notion of "reduce to" (or
"determined by", "fixed by", etc.) in favor of
some philosophically more sanitized alternative; see the



> 
> Supplement on
>  Reduction
> 



The third observation I wish to make at this point is that Lewis is
strongly motivated by a desire for theoretical economy--both with
respect to ontology and with respect to ideology. His quest for
ontological economy shows up in the austerity of the *kinds* of
fundamental entities he admits into his ontology (he neither shows,
nor cares to show, any economy with respect to their *number*).
His quest for ideological economy shows up in several places, but
perhaps most notably in his utter rejection of *any* unanalyzed
modal notions, and--something that hasn't been mentioned
yet--in his attempted reduction of set theory to mereology and
plural quantification.



Let's take a somewhat closer look, now, at Lewis's account of
fundamental ontology.


## 2. Fundamental ontology: A simplified version



It will be useful to start with a view that is almost
Lewis's--almost, but not quite, as it is more opinionated
than he would be comfortable with. Stating the view takes but a few
lines; providing the needed commentary will take longer. Thus,
Almost-Lewis says the following:



The only *fundamental entities* that are *particulars*
are spacetime points.



What these particulars *are like* is given by what perfectly
natural monadic properties they instantiate, and what perfectly
natural relations they stand in to one another.



And that's it. That is, the facts about what fundamental
particulars there are, and what perfectly natural properties and
relations they instantiate, determine all other facts, even modal
facts. Almost-Lewis
(and Lewis) believes, of course, in *other* particulars besides
spacetime points; it's just that these particulars are not
fundamental: what it is for them to exist is to be explained, somehow,
in terms of facts about the fundamental entities. (See the



> 
> Supplement on
>  Fundamental Entities
> 



for some qualifications about Lewis's position.)



Notice one consequence: If the facts about what fundamental
particulars there are, and what perfectly natural properties and
relations they instantiate, determine *all* other facts, then
there is no reason to suppose that composite
particulars--particulars that have other particulars as proper
parts--ever instantiate perfectly natural monadic properties. (Of
course, they can perfectly well instantiate the very-but-not-perfectly
natural property of having *parts* that instantiate
such-and-such a perfectly natural *relation*.) Thus, if, for
example, my laptop has a mass of 3 kg, that is so only in a slightly
derivative sense: the laptop is composed of parts whose masses add up
to 3 kg.

 As noted, the position of Almost-Lewis is not that of
Lewis, and shortly we will need to review the key respect in which, by
Lewis's lights, it overreaches. But first we need to elaborate and
clarify the content of Almost-Lewis's position, by means of some
commentary.


Four questions demand attention: What are "perfectly
natural" properties and relations? What does it come to to say
that the fundamental particulars are *spacetime* points? What
does it come to to say that they are spacetime *points*?
Finally, what is the relationship between the fundamental ontology
posited by Almost-Lewis and Lewis's own celebrated thesis of
Humean Supervenience? Let's consider these topics in turn.



## 3. Perfectly natural properties and relations



Remember that laying out the foundations of one's ontology
requires two things: to say what, fundamentally, *there is*;
and to say what it *is like,* presumably by stating some
*facts about* the fundamental entities. But not just any facts
matter. For example, it may be true of some of the fundamental
entities that they coexist with at least one pig; but saying so does
nothing to help articulate the *fundamental structure* of
reality. To do that, Lewis thinks, one needs a *distinction*
among the properties and relations: some are special, in that it is
*their* pattern of instantiation among the fundamental entities
that constitutes the fundamental structure of reality--the
"joints" along which nature is to be ultimately carved.
These special properties and relations are the "perfectly
natural" ones.



(There are a variety of other uses to which Lewis puts the notion of
"natural" properties, some of which show that what he
needs is a distinction that admits of gradations, with the
*perfectly* natural properties at one extreme. See the



> 
> Supplement on
>  The Natural/Non-natural Distinction
> 



for an overview.)



It is not enough merely to *appeal* to such a distinction; for
metaphysics to do its job properly, it must also provide an
*account*. Now, one way to proceed would be to provide a theory
of what properties and relations are, in which it is stipulated that
all such things are to count as "perfectly natural". On
such an approach, while there may well be a property corresponding to
the predicate "has mass 5 kg" (for example), there will
almost certainly be no property corresponding to the predicate
"is green" (let alone that familiar gerrymander, "is
grue"). Lewis favors a different approach. Given his commitment
to set theory, he *already* believes in things that, by his
lights, deserve to be called the property of being green, and indeed
the property of being grue: these are merely certain
*sets*--sets of actual and possible objects. (See the
section on Lewis's modal metaphysics in the entry on
 David Lewis,
 and the supplement on
 The Natural/Non-natural Distinction.)
 The question for him, then, is how to *distinguish* among
these sets those that are perfectly natural. Here I will present
Almost-Lewis as being, almost like Lewis, agnostic as between four
broad alternatives. (Almost, because Lewis eventually decided that the
first alternative, according to which natural properties and relations
are Aristotelian universals, is unworkable; see his 1986f for the
reasons.)


* One could adopt a *theory of universals* of the kind
developed by David Armstrong (1978a,1978b): "...we could
call a property [viz., set of actual and possible objects]
*perfectly* natural if its members are all and only those
things that share some one universal." (1999 p. 13)
* One could treat "natural" as a *primitive*
predicate of sets of actual and possible objects: "...a
Nominalist could take it as a primitive fact that some classes of
things are perfectly natural properties; others are
less-than-perfectly natural to various degrees; and most are not at
all natural. Such a Nominalist uses 'natural' as a
primitive predicate, and offers no analysis of what he means in
predicating it of classes." (1999, p.
 14)[2]
* One could define "natural" in terms of a suitably
complex, and primitive, notion of *resemblance*:
"Alternatively, a Nominalist in pursuit of adequacy might prefer
to rest with primitive objective resemblance among things.
...Then he could undertake to define natural properties in terms
of the mutual resemblance of their members and the failure of
resemblance between their members and their non-members." (1999,
p. 14)
* One could adopt an ontology of *tropes*--roughly,
property-instances, entities that occupy a sort of ontological halfway
house between particulars and properties. (See Lewis 1986f, Williams
1953, Campbell 1990.)



Returning now to Almost-Lewis's fundamental ontology, the options
seem to be these: It might be that a spacetime point (or sequence of
points) instantiates a perfectly natural property (respectively,
relation) by instantiating a universal, in the sense of Armstrong. It
might be that it has it by having as one part a certain kind of trope,
in roughly the sense of Williams. (Whence we must amend slightly, and
take these *tropes* to be the fundamental entities.) It might
be that it has it by belonging to a special sort of set of (actual and
merely possible) spacetime points--special either on account of
the resemblances that unite its members and distinguish them from
non-members, or on account of simply *being perfectly natural*.
Regardless of which one chooses, Lewis thinks, one's theory of
natural properties and relations ought to respect four
philosophically-motivated constraints:



First, an adequate theory should be *minimal*, in the sense
that it posits just enough perfectly natural properties and relations
for their distribution among the fundamental particulars to fully and
determinately fix the nature of all of reality: "The guiding
idea, roughly, is that the world's universals should comprise a
minimal basis for characterizing the world completely. Universals that
do not contribute at all to this end are unwelcome, and so are
universals that contribute only redundantly." (1999, p. 12) It
is clear from the surrounding text that Lewis takes this constraint to
govern the various alternatives to a universals account of
 naturalness.[3]



Second, perfectly natural properties and relations are, Lewis thinks,
*non-modal*. What, exactly, this means will need to come in for
more discussion. For the moment, we can take it to
mean roughly this (though trouble for this characterization quickly
arises): the instantiation of a perfectly natural property by one
(fundamental) particular, or of a relation by several, places
absolutely no constraints of a logical or metaphysical kind on the
instantiation of any other perfectly natural property or relation by
that or any other particular or particulars.



Third, they are *intrinsic* to the particulars that instantiate
them--which, all too roughly, means that they characterize what
those particulars are like, independently of what any other distinct
particular is like. More: The intrinsic nature of any particular is
exhausted by what perfectly natural properties it
 instantiates.[4]
 This assumption also allows a theory of natural properties and
relations to yield, in a fairly simple way, a definition of
"perfect duplicate" applicable to any possible objects
*x* and *y* (not necessarily inhabiting the same
possible world): *x* and *y* are perfect duplicates iff
they share exactly the same perfectly natural
 properties.[5]
 A definition of "intrinsic" follows: a property
*P* is intrinsic iff any two duplicates *x* and
*y* (taken from any possible worlds) either both have
*P* or both fail to have *P*. Of course, what we really
have here is a tight circle that puts on display how the expressions
"intrinsic", "perfect duplicate", and
"perfectly natural" can be interdefined, with the help of
the modal notion of metaphysical possibility. (See Lewis 1983c and
Langton & Lewis 1998 for discussion of various strategies for
breaking out of this circle.)



The fourth constraint is purely negative: it is that it should be left
to the empirical sciences to fill in the details about *which*
perfectly natural *monadic properties* there are (at least, in
actuality: *philosophy* might teach us, or at least give us
some reason to believe, that there are, in other possible worlds,
so-called "alien" properties, perfectly natural properties
not instantiated in the actual world). Not just any empirical science
will do: given, in particular, the first of the four theses, it is
really the job of fundamental physics to fill in these details. The
special sciences get no say.



What about perfectly natural relations? Here matters are less clear.
Lewis certainly thinks that spatiotemporal relations are perfectly
natural; what is less obvious is whether, by his lights, physics could
rationally lead us to *reject* this claim. For now I will
simplify, and have Almost-Lewis add a *fifth*
constraint--one that is in tension at least with the spirit of
the fourth, and that the real Lewis certainly rejects. It is this: not
only are spatiotemporal relations perfectly natural, they are the
*only* perfectly natural relations. (The only *possible*
ones--though remember that given Lewis's reductionism about
modality, that is an idle addition.)



The picture that emerges is this: Reality consists of a multitude of
spacetime points. Each of these stands in spatiotemporal relations to
some others (though not to *all* others). Each instantiates various
perfectly natural, non-modal monadic properties. That is all there is;
anything putatively "extra"--facts about laws of
nature, or about persisting macro-objects, or about causation, or
about mentality, or about ethics, or about sets, etc.--must
somehow reduce to that stuff. For Almost-Lewis, this picture captures
a fundamental truth about the nature of existence. It is roughly right
that it is also a *necessary* truth--a status that would
seem to fall out automatically, given Lewis's reductionist account
of modality. The only unfinished *philosophical* business is to
work out the right theory of natural properties and relations, and to
work out the details of the reduction for particular cases.


## 4. Spatiotemporal relations and spacetime points



The foregoing Almost-Lewisian thesis about spatiotemporal relations is
too strong to be tenable: we now have reasonably good reasons, drawn
from quantum physics, for holding that even in the actual world, there
are perfectly natural relations other than the purely spatiotemporal
ones. (Roughly: the relations--whatever exactly they amount
to--coded up in the quantum mechanical wave-function.) Two points
in its defense are, however, worth brief mention: First, seemingly
obvious counterexamples--involving such basic physical relations
as *being more massive than*--in fact *aren't*
counterexamples, since Lewis can deny that they are genuinely
fundamental or *perfectly* natural, on the basis that facts
about their obtaining reduce to facts about the distribution of
monadic perfectly natural properties. (Still, they will certainly turn
out to be *very* natural.) Second, if we could at least
maintain, as a *contingent* thesis, that the only perfectly
natural relations are spatiotemporal ones, then we could plausibly
settle an unresolved and deeply vexed question about the content of
physicalism (the doctrine, to put it rather too crudely, that all
there is to the actual world is physical stuff), as explained in
the supplementary document



> 
> Physicalism
> 



At any rate, the thesis that spatiotemporal relations are at least
*among* the perfectly natural relations allows us to clarify
and simplify Almost-Lewis's position. Specifically, we can say
that all that it comes to to say that the fundamental entities are
*spacetime* points is that they stand in perfectly natural
spatiotemporal relations to one another. For more, see the
supplementary document on



> 
> Spacetime Points
> 



To say that they are spacetime *points*, finally, is to say
that they have no proper parts.



One upshot is that my original statement of Almost-Lewis's
ontology needs an amendment: for it was misleading to say that
according to him, the fundamental particulars are *spacetime*
points. That's true, but it wrongly suggests that he is making a
choice of one fundamental *kind* of particular, distinguished
from other possible choices by the essential nature of its members.
Not so. It is more accurate to describe his fundamental ontology
thus:


* There are particulars.
* They are, or are wholly composed of,
*simples*--particulars have no other particulars as proper
parts.
* These simples have various perfectly natural monadic
properties.
* They stand in various spatiotemporal relations to one
another.
* And that is all.


## 5. Humean Supervenience



Almost-Lewis's theses about what fundamental ontology comprises,
and how all other facts reduce to facts about it, bears a very close
relationship to Lewis's celebrated thesis of Humean Supervenience
(hereafter: "HS"). But they are not the same, and the
differences are worth keeping track of. Here is a typical statement of
HS (slightly stronger, as we'll see, than the version Lewis
officially endorses): No two possible worlds differ with respect to
what is true at them, without differing with respect to the
geometrical arrangement of their spacetime points, or with respect to
which perfectly natural properties are instantiated at those
 points.[6]
 (Note that so stated, HS is automatically metaphysically
*necessary*.) Thus, HS is a *supervenience* claim,
logically weaker than Almost-Lewis's claim of reduction. It is
also a claim that--for some good reasons and some bad
reasons--Lewis accepts only in a weaker form that is
metaphysically *contingent*. More significantly, it is no part
of HS that facts about possible worlds *themselves* reduce to
anything else; whereas both Almost-Lewis and Lewis are explicit in
their commitment to this further claim. Having said all this, it will
be worth remembering in what follows that Almost-Lewis's position
(which, remember, incorporates Lewis's modal realism)
*entails* HS. So, any doubts about HS will carry over to
Almost-Lewis's fundamental ontology.


## 6. Lewis v. Almost-Lewis



Let's consider now the most salient ways in which Lewis's own
positions about fundamental ontology diverge from those of
Almost-Lewis.



First, Lewis takes the lessons quantum physics teaches seriously
enough to withhold endorsement of Almost-Lewis's fifth thesis,
that the only perfectly natural relations are spatiotemporal
relations.



Second, Lewis is agnostic as to whether, in addition to spacetime
points, there might be (in this, or other possible worlds) fundamental
entities that are *occupants* of such points. But agnosticism
on this score is probably a bad idea: the proposed possibility is not
clearly intelligible, nor it is clear what its motivation could be.
For more, see the supplementary document on



> 
> Spacetime Points
> 



Third, on a plausible story about what *non-*fundamental
entities there are, it will turn out that on Almost-Lewis's view,
everything that exists is composed of simples (parts, that themselves
have no proper parts). Lewis is also agnostic on this score: he takes
it to be at least an epistemic possibility that there is
"gunk": something, every proper part of which itself has a
proper part (see for example Lewis 1991). Lewis says relatively little
either about the status of this possibility (in particular, is it more
than merely epistemic?), or about its potential ramifications for his
various positions in metaphysics. To keep things simple, I will
discount it for the remainder of this main essay.



Fourth, Lewis holds that his thesis of Humean Supervenience is, at
best, only *contingently* true. Of course, given that he
recognizes the (metaphysical) possibility of perfectly natural,
non-spatiotemporal relations, he *should* treat HS as at best
contingent. But he advances reasons for doing so of a quite different
sort. They are not particularly good reasons, and so we will pass them
by; but see the supplementary document on



> 
> The Contingency of Humean Supervenience
> 



for discussion.


## 7. Some criticisms



What, finally, should we make of Lewis's conception of fundamental
ontology? A complicated question; I will limit discussion to just two
important worries. Let's begin by noting the obvious influence of
a certain scientifically-informed conception of the world in shaping
Lewis's picture of reality. Lewis himself is quite explicit about
this influence:



> 
> The picture is inspired by classical physics. Humean Supervenience
> doesn't actually say that physics is right about what local
> qualities there are, but that's the case to keep in mind. But if
> we keep physics in mind, we'd better remember that physics
> isn't really classical. ...The point of defending Humean
> Supervenience is not to support reactionary physics, but rather to
> resist philosophical arguments that there are more things in heaven
> and earth than physics has dreamt of. (1994, p. 474)
> 



But there is a less acknowledged influence of first-order predicate
logic--an influence that is not entirely salutary. It is
undoubtedly tempting, for philosophers steeped in the use of
first-order logic as a clarifying tool, to assume that the proper
representation of the ultimate structure of reality must be by means
of some (interpreted) first-order language--a language whose
various *predicates* could be taken to express the various
*fundamental properties and relations* that characterize
reality at its most basic level. But if we look to physics
instead--as we surely ought to--we find that the basic
representational tools are *variables*, that correspond to
*physical magnitudes.* Taking seriously the picture of
fundamental ontology suggested by these representations turns out to
matter quite a bit: in particular, there are reasons to think that
none of the first three theses about natural properties and
relations--that they are minimal, non-modal, and
intrinsic--is tenable without some modification. This
issue--which we will mostly pass over in what follows, except
where it matters--is explored in more detail in the
supplementary document on



> 
> Physical Magnitudes
> 



The second significant source of concern about Lewis's conception
of fundamental ontology is the role--or rather lack
thereof--that *modal* notions have in it. This concern has
two aspects. First, one might hold that some, at least, of the
fundamental properties and relations that characterize reality have
modal aspects that are ontologically basic. Consider *mass*:
one might hold that it is metaphysically impossible for there to be a
world containing just two massive particles, accelerating away from
each other--and that this impossibility somehow flows from the
nature of mass itself.



Second, one might hold that it is one thing to state a thesis
concerning what the fundamental structure of reality *in fact*
happens to be; but that it is *another*, separate matter to
state how reality *could* be. Indeed, most metaphysicians, I
suspect, take it to be just blindingly obvious that these are
conceptually distinct tasks. Granted that one's views on what
there is, and what it is like, will have ramifications for one's
views on what there *could* be and what it *could* be
like (most obviously, because things *could* be the way they
*are*; but there may be more interesting and subtle connections
as well); still, the project of laying out the former views does not
*automatically* complete the project of laying out the
latter.



Of course there is a sense in which Lewis agrees: he takes it as
obvious, after all, that he must supply an account of modality. But
the strikingly reductionist character of that account shows that such
agreement as there is is mighty thin.


## 8. Counterpart theory


So far, this entry focused on the general shape of Lewis's
metaphysics. In this section, we turn to one of Lewis's more specific
metaphysical doctrines: counterpart theory. We will begin with an
overview of the theory, review its motivation, some applications, and
then turn to criticism and objections. In a supplementary document, we
review 
 Counterpart-theoretic Semantics for Quantified Modal Logic.



Counterpart theory was originally introduced in Lewis (1968) as a
formal theory about possible worlds and their inhabitants, but the
label quickly came to be used for a loosely defined approach towards
trans-world identity and the interpretation of modal or temporal
discourse. For the modal case, the central idea is that a *de
re* modal claim like 'Joe Biden might be immaterial'
is understood as saying that at some possible world, someone who
sufficiently resembles Joe Biden in certain
respects--a *counterpart* of Biden--is
immaterial. Accordingly, 'Biden is necessarily material'
is understood as saying that all of Biden's counterparts at all
worlds are material.



While counterpart theory was Lewis's invention, it was also a
product of its time, the 1960's, when quantified modal logic was
"in the air" and questions about essentialism and the
so-called problem of
 transworld identity
 gained attention (e.g. Quine 1953, Hintikka
 1962).[7]
 The counterpart-theoretic approach does, however, bear a striking
similarity to certain views of Leibniz concerning essential and
accidental
 properties.[8]
 (See Mondadori (1973) for a counterpart-theoretic interpretation of
Leibniz.)


### 8.1 Motivation and applications



Lewis assumed that modal statements in ordinary language can be
analysed in terms of quantification over possible worlds and
individuals: 'it is possible that there are talking
donkeys' is analysed as 'there are possible worlds at
which there are talking donkeys'. Lewis's modal realism
promises to turn this into a reductive analysis of
modality. Statements about what is or is not the case at some possible
world are ultimately reduced to non-modal statements. That there is a
world with talking donkeys, for example, means that there is a
spatiotemporally isolated aggregate of things that has a genuine
talking donkey as part.



A problem now arises for *de re* statements about what is
possible or necessary for a particular individual. Since Joe Biden
might have had three arms, there should be a world where he has three
arms. But if Biden were *part of* another world, then that
world and our world would have a part in common. Lewis denies that
different worlds can share parts. Even if we allowed for overlapping
worlds, it is hard to see how the other world could have a three-armed
Biden as a part, given that this part is also part of our world, where
Biden has two arms (compare Lewis 1986e, 199ff.). So 'Biden
might have had three arms' is not analysed as stating that there
is a world which contains a three-armed Biden as part. Rather, it says
that there is a world that contains a three-armed
*representative* or *counterpart* of Joe Biden.



What makes one object a counterpart of another? According to Lewis,
this is a matter of qualitative similarity in certain respects: a
counterpart of Biden is an individual that sufficiently resembles the
actual Biden in relevant respects, and more so than any other object
in its world. Sometimes Lewis drops the second clause (e.g. Lewis
1968, 114-115), and at one point he merely suggests (without
explanation) that counterparthood "usually involve[s]
similarity" (Lewis 1986e, 8).



The relevant respects of resemblance need not match intuitive
judgements of overall similarity (Lewis 1986e, 254f.), and they can be
highly extrinsic. For example, they might give high weight to
someone's origin (Lewis 1986e, 244f. and 252), or to a causal
connection to an epistemic subject (Lewis 1983e).



Lewis does not think there is a once-and-for all correct answer to
which respects of resemblance are relevant. He rejects the idea that
things have primitive essences which somehow determine or constrain
what is possible for them. Rather, it is ultimately up to us which
respects of similarity we prioritize. Often this choice is not fully
settled, leaving the counterpart relation vague and context-sensitive
(see Lewis 1971, 209-11; Lewis 1973b, 41; Lewis 1983d, 42-3;
Lewis 1986e, 251-5; Lewis 2003, 27-8).



For example, Lewis suggests that when we talk about people, we often
require the counterparts to have a very similar origin, while allowing
for different careers and lifespan. In such a context, we might judge
that Biden might have died as an infant. In other contexts, however,
we may wonder what would have been the case if Biden had been born to
different parents, relaxing the requirement of similar origin (see
e.g., Lewis 1973b, 41).



Since the counterpart relation is determined by similarity, everything
is its own counterpart. Other than that, Lewis imposes few general
restrictions on the counterpart relation. In Lewis (1968), he suggests
that nothing can be a counterpart of a different object in the same
world, but this assumption is later dropped (Lewis 1986e, 232,
fn.22). Unlike strict trans-world identity, counterparthood is not
assumed to be transitive, symmetric, or functional. An object can have
multiple counterparts at a world, and different objects can have a
common counterpart.



Lewis delivered counterpart theory as part of his modal realism, but
the core idea can be separated from this background, and even from the
assumption of world-bound agents. (See e.g. Stalnaker 1986,
Heller 1998, Sider 2002, Wang 2015, Woodward 2017; also Lewis 1986e,
237f. and 259f.) Many philosophers who reject modal realism have come
to appreciate counterpart theory because of its explanatory and
puzzle-solving power. We will give a few examples.



First, counterpart theory promises to explain the widespread
elusiveness and context-dependence of essentialist judgements (Lewis
1983d, 42-3; Lewis 1986e, SS4.5; cf. Kaplan 1979,
100-2). Could London have been located in Scotland? Could it have
been founded in the 16th century? The answers aren't obvious.
For Lewis, this is not because we have imperfect access to
London's true essence, but simply because we have not settled
exactly which other-worldly cities should qualify as counterparts of
London. A London counterpart is a city that sufficiently
*resembles* London, but the relevant resemblance criteria are
often vague and context-dependent.



Second, counterpart theory might offer an answer to certain puzzles
about identity. Is a statue identical to the piece of clay from which
it is formed? Is a person identical to their body? Lewis says yes.
However, intuitively persons and bodies, or statues and pieces of
clay, seem to differ in their modal properties: the piece of clay
could survive squashing, the statue could not. According to Lewis, the
explanation is that when we refer to an object as a statue, we give
more weight to similarity with respect to shape than when we think of
the same object as a piece of clay (see Lewis 1971; Lewis 1986e,
252ff.; Lewis 2003, 27-8; and Robinson 1982).



In this context, Lewis suggests that different ways of referring to an
object can "evoke" different similarity standards, and
thereby different counterpart relations, even within a single context
(cf. Lewis 1986e, 258ff; Lewis 2003). This idea also helps to
make sense of conditionals like 'If I were you, I'd hate
me', where 'I' and 'me' in the
consequent intuitively pick out different individuals in the imagined
person-swapping scenario, both of which somehow represent the speaker
(see Lewis 1973b, 43; Kocurek 2018).



Third, counterpart theory might offer an answer to puzzles involving
possible fission or time-travel. These might be analysed as scenarios
in which an individual has multiple counterparts relative to the same
counterpart relation. Lewis intuits that he "might have been
twins" (Lewis 1973, 40-1), because the fetus from which he
developed might have undergone fission. (See also Schwarz 2014, Karmo
1983.)



Fourth, counterpart theory offers a way to accept haecceitistic
intuitions without subscribing to haecceitism. Lewis assumes that all
truths supervene on qualitative truths: no two worlds agree in all
qualitative respects, while disagreeing with respect to who plays
which roles. A world of two-way eternal recurrence might look like a
counterexample: intuitively, we could have lived in any of the epochs
of such a world; these seem to be distinct possibilities, but they do
not correspond to any qualitative difference. Lewis suggests that the
different epochs really do represent different possibilities for us,
because we have counterparts in each of them. Here, too, Lewis invokes
multiple counterparthood relative to the same counterpart relation.
(See Lewis 1986e, 230-5.)



Fifth, the fact that the counterpart relation need not be an
equivalence relation has also been used to diffuse certain puzzles.
Consider an ordinary bicycle. Intuitively, such an object does not
have all its parts essentially: the bike could have had a different
chain, for instance. On the other hand, arguably it could not have
been composed of *entirely* different parts. A bike made of
entirely different parts would have been a different bike. But this
creates a puzzle (see Chisholm 1967, Chandler 1976): if the bike had a
different chain, it would still be an ordinary bike that does not have
all its parts essentially; it could have had a different saddle (say).
Iterating this line of thought, we can create a sequence that leads
from the actual bike to a bike composed of entirely different parts.
Each possible bike in the sequence could have been the next bike in
the sequence. But didn't we say that it is impossible that my
bike is composed of entirely different parts? Counterpart theory
offers a natural answer: a counterpart of a counterpart need not be a
counterpart (see Lewis 1968, 28f; Lewis 1986e, 243-6;
Ramachandran 2020).


So far, we have focused on metaphysical modality. The central idea
of counterpart theory has also been applied to other intensional
constructions.


For example, Sider (1996, 2001) and Hawley (2001) argue for a
temporal application, on which 'Joe Biden won the election' is
analysed as saying that a past counterpart of Biden won the
election. The temporal counterpart relation is here assumed to relate
short-lived temporal "stages" (see Schwarz 2014 for an alternative),
and is not a matter of qualitative similarity. As in the modal case,
this approach has been advertised as solving a range of puzzles, as
well as explaining the vagueness and context-dependence of temporal
judgements. Williams (2008) considers adopting the same perspective on
identity across space.


Several authors (including Lewis) have suggested applying
counterpart theory to epistemic modality. Consider the following
puzzle from Ninan (2018). A lottery has only two tickets, one blue and
one red. The tickets are numbered 1 and 2, but we don't know
which color goes with which number. We know that the blue ticket won,
but we don't know the number of the winning ticket. So, ticket
1 might be the winner, and likewise for ticket 2. Yet even though
one of them is the red ticket it doesn't seem to follow that the
red ticket is such that it might be the winner. Ninan suggests that
differing counterpart relations are evoked by different ways of
picking out the tickets. (See also Lewis 1983e, Stalnaker 1986, Shaw
2015, Rabern 2018, among others.)


An alternative to counterpart theory is to interpret terms for
ordinary individuals as denoting (partial) functions from worlds to
individuals, or aggregates of individuals at different worlds; a modal
predication like 'Biden might have been three-armed' is then
interpreted as saying that the function assigned to 'Biden' has a
three-armed value for some world, or that the aggregate assigned to
 'Biden' has a three-armed part located in some world. A temporal
analogue of this view (with aggregates) is Lewis's preferred account
of temporal discourse. In Lewis (1986f, 244) he endorses a similar
view for events--but see Bernstein (2014), Kaiserman (2017), and
McDonnell (2016) for reasons to prefer a counterpart-theoretic
treatment.[9]



The counterpart-theoretic analysis is more flexible insofar as it
easily allows for cases of multiple counterparts or asymmetric
counterparthood, which are difficult to model with trans-world
aggregates. Lewis argues that this added flexibility is more important
in the modal case than in the temporal case (Lewis 1971, 209; Lewis
1983d, 40-2; Lewis 1986e, 217-20).


### 8.2 Reactions and criticisms



Some early criticisms of counterpart theory seem to rest on
misunderstandings. For example, Plantinga (1974, 115f.) and Salmon
(1981, 232-38) complain that according to Lewis, all things have all
their properties essentially, since there is no world in which these
very things exist and have different properties. Kripke (1980)
similarly complains that according to Lewis, when we say that Humphrey
might have won the election, we are "not talking about something
that might have happened to *Humphrey*, but to someone
*else*" (45, fn. 13). In response, Lewis points out that
on his view it is indeed Humphrey himself who might have won, in
virtue of having a winning counterpart, cf. Lewis (1986e, 194-6,
246; 1983d, 41-2). Properly understood, Lewis's analysis
does not even deny that there are worlds at which Humphrey (himself)
won the election. What the analysis denies is only that there are
other worlds that contain Humphrey himself as a (winning) part. (See
also Hazen 1979, 320-4.)



One possible limitation of Lewis's account is that his
counterpart relation is assumed to be a qualitative similarity
relation. Feldman (1971) points out that "I could have been
quite unlike what I in fact am" is intuitively true, which seems
to require having non-similar counterparts. Lewis responds that there
are two similarity relations involved: at the relevant worlds, there
is someone who is similar to me in some respect (perhaps with respect
to origin), but dissimilar from me in other respects (see Lewis
1983d, 43; Lewis 1986e, 230-235). Heller (2005), Stalnaker
(1986), and Fara (2009) offer further considerations against analysing
counterparthood in terms of qualitative similarity.



Another limitation of Lewis's original presentation is that it
does not allow imposing constraints on the choice of counterparts for
different individuals. Suppose we hold that Elizabeth II is
essentially the daughter of George VI, and suppose there are worlds at
which both Elizabeth and George have multiple counterparts, perhaps in
different epochs of eternally recurring time. Some Elizabeth
counterpart will then not be the daughter of some George counterpart.
By Lewis's 1968 account, 'Elizabeth might not have been
George's daughter' then comes out true, contrary to our
supposition. This problem was raised in Hazen (1979). Hazen suggests a
revised analysis in terms of sets of counterpart functions. Lewis
(1983d, 44-5) instead suggests that we should understand
'Elizabeth might not have been George's daughter' as
a de re claim about the pair (or fusion) of Elizabeth and George.
Hazen (2012) argues that neither proposal fully avoids the
problem.



The most common objection to counterpart theory is probably that it
gives rise to a deviant and counter-intuitive modal logic. Lewis
(1968) gives translation rules from the language of quantified modal
logic to an extensional first-order language. These translation rules
invalidate the "necessity of identity" and the
"necessity of distinctness", as well as familiar
principles of modal logic such #(A [?] B) - #A.
Conversely, they do validate controversial principles such as the
Converse Barcan Formula and the "necessity of existence".
It has also been argued that the rules can't be extended to a
language with a well-behaved 'actually' operator.



Whether these are genuine problems is a matter of debate. In fact,
authors in mathematical logic have come to appreciate
counterpart-theoretic interpretations of modal logic as overcoming
certain problems of standard Kripke semantics. There are also ways of
reformulating Lewis's translation rules so as to avoid the
supposedly problematic features. See the supplementary document:



> 
> Counterpart-theoretic Semantics for Quantified Modal Logic
> 



for further details.



Lewis himself had no interest in preserving traditional principles of
modal logic. He always remained skeptical about the prospects of
formalising modal discourse with boxes and diamonds. (see
e.g. Lewis 1983d, 45; Lewis 1986e, 12-13). He thus rejected
the presupposition in these objections, that "the language of
boxes and diamonds affords a good regimentation of ordinary modal
thought" (Lewis 1986e, 12).