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\title{Possible Worlds II: Nonreductive Theories of Possible Worlds}
\author{Louis deRosset}

\begin{document}
  \maketitle
  
  % \begin{abstract}
  %   It is difficult to wander far in contemporary metaphysics without
  %   bumping into talk of possible worlds.  But reference to possible
  %   worlds is not confined to metaphysics.  It can be found in
  %   contemporary epistemology and ethics, and has even found its way
  %   into linguistics and decision theory.  What are those possible
  %   worlds, the entities to which theorists in these disciplines all
  %   appeal?  Some have hoped that a theory of possible worlds can be
  %   used to reduce modality to non-modal terms.  This paper sets
  %   reductive theories aside, and articulates and applies a framework
  %   for evaluating nonreductive theories of possible worlds.  I argue
  %   that, if we abjure reduction, we should aim for a theory of
  %   possible worlds that is \emph{user-friendly}.  I then outline four
  %   leading contemporary theories and consider objections to each.  My
  %   conclusions are negative: every theory we discuss fails to be
  %   user-friendly in some significant respect.
  % \end{abstract}

  % It is difficult to wander far in contemporary metaphysics without
  % bumping into talk of possible worlds.  And reference to possible
  % worlds is not confined to metaphysics.  It can be found in
  % contemporary epistemology and ethics, and has even found its way
  % into linguistics and decision theory.  But what are those possible
  % worlds, the entities to which theorists in these disciplines all
  % appeal?  Call an attempt to answer this question a \emph{theory of
  %   possible worlds}.
  
  \bf In another review, I noted the spread of possible worlds as a
  tool for theorizing to fields far removed from the metaphysics of
  modality, including epistemology, ethics, linguistics, and decision
  theory.  I noted there that theorists in those other areas rely on:
  \begin{description}
    \item[(SC)]  it is possible that $\phi$ iff there is a $w$ such
      that $w$ is a possible world and `$\phi$' is true at $w$. 
  \end{description}    
  I briefly discussed a view (see \citep{Lewis:1986}) which attempts
  to use (SC) to provide a theory of possible worlds aimed at using
  (SC) to reduce modality.\linespread{1}\endnote
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  {\citep{deRosset:forthcoming}.}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \linespread{1.3}
  
  This review sets aside such reductive theories of possible worlds,
  and concentrates instead on four leading nonreductive theories. \rm
  Though some of these theories have reductive variants, the versions
  that figure most prominently in contemporary discussions disavow any
  ambition to reduce modality to non-modal terms.  I start by
  considering what theoretical purposes other than reduction possible
  worlds might serve (\S1).  I will then turn to the consideration of
  the four leading views (\S\S2-5).
  
  \begin{section}{What's a Theory of Possible Worlds Good For?}
    If possible worlds don't provide the means to reduce modality, why
    should we believe in them?  If (SC) does not provide a template
    for a reduction why think it's true?
    
    Providing a reductive theory of possible worlds is only one of the
    tasks to which possible worlds have been put.  Another ambition
    for a theory of possible worlds is to explain their nature in such
    a way as to illustrate and underwrite their fitness as tools for
    \emph{end-users} in fields far removed from the metaphysics of
    modality, the semantics of modal discourse, or the analysis of
    modal concepts.  \bf I have already noted that possible worlds are
    used as a tool in a wide variety of other philosophical (and
    not-so-philosophical) projects.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {\bf For instance, a conspicuous and seminal use of possible
      worlds is as a basis for a variety of semantic theories.  See
      the citations and very brief discussion at
      \citep[pp. XXXX]{deRosset:forthcoming} [CITE!] [PAGE NUMBERS
        NEEDED!].\rm}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
    
    \rm Let's call a theory of possible worlds suitable for satisfying
    this second ambition \emph{user-friendly}; and I will call a
    theory \emph{user-hostile} insofar as it fails to be
    user-friendly.  There are a number of ways in which a theory of
    possible worlds might be user-hostile.  Most obviously, a theory
    could be incoherent, or imply some contradiction or manifest
    absurdity.  But there are at least three subtler forms of
    user-hostility, and so three correlative constraints on
    user-friendly theories of possible worlds.  \bf First, it would be
    user-hostile to deny (SC); in particular, it would be user-hostile
    to deny the existence of those possible worlds required by
    instances of (SC).  End-users generally presume (SC) in their
    theorizing, so rejecting (SC) by denying the existence of the
    requisite possible worlds would count as a strike against
    user-friendliness.  Call this the \emph{constraint of
      existence}. \rm Second, it would be user-hostile to indulge in
    ontological extravagance, by requiring there to be entities whose
    existence we would ordinarily deny.  Securing the existence of
    possible worlds by claiming that there are flying pigs, talking
    donkeys, or purple penguins, for instance, would be user-hostile.
    The \emph{constraint of ontological modesty} is that theories of
    possible worlds should avoid ontological extravagance so far as
    possible.  Third, it would be user-hostile to require some
    substantial and controversial claim about what's possible, since
    it makes an end-user's theory hostage to the abstruse modal claims
    debated among metaphysicians of modality.  The \emph{constraint of
      modal modesty} is that theories of possible worlds should remain
    neutral on substantial, controversial modal theses so far as
    possible.
    
    User-friendliness does not hang on providing the means to reduce
    modality.  Appeal to modality is pervasive in end-users'
    theorizing.  They make liberal use of modal notions, like what
    good reasoning \emph{can} establish, or what actions it is
    \emph{possible} for an agent to perform.  If the condition
    \emph{$w$ is a possible world} turns out to be modal, then this
    would pose no additional threat to the cogency of their
    theorizing.  A theory of possible worlds can be user-friendly
    without being reductive.
  \end{section}
  
  \begin{section} {The Naive Theory}
    I begin with a theory of possible worlds on which the notion of a
    possible world is explained in terms similar to other notions
    conveyed by English expressions of the form ``possible $F$''.
    This theory explains \emph{being a possible world} on the model of
    such ordinary, and straightforwardly modal, notions as \emph{being
      a possible Republican nominee for president in 2008.}  Someone
    is a possible Republican nominee for president in 2008 just in
    case it is possible that he be a Republican nominee for president
    in 2008.  Mitt Romney, John McCain, and Rudolph Giuliani are all
    possible Republican nominees for president in 2008.  The gas
    station around the corner is not a possible Republican nominee for
    president in 2008.
      
    So suppose we explain the notion of \emph{being a possible world}
    along similar lines: $x$ is a possible world iff it is possible
    that $x$ be a world.  A world, we may assume, is a concrete cosmos
    of the same sort as the vast cosmos in which we all live, move,
    and have our being.  This view is defended by Kit
    Fine.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {See \citep{Fine:1985, Fine:2003a} for a defense.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} \bf I will call it \emph{the naive theory} to
    mark the fact that it is the theory of possible worlds we get by
    naively interpreting ``possible world'' as an ordinary English
    expression of the form ``possible $F$.''  \rm The naive theory
    clearly cannot be used to reduce possibilities to non-modal terms,
    since the notion of \emph{being a possible world} is explained in
    terms of possibility.  So the theory is not reductive.  There is
    reason to think that this view is not user-friendly either.
      
    First, apply the constraint of modal modesty.  `It is possible
    that Bush lost the 2000 electoral vote' is true and so part of the
    common-sense description of the modal facts.  \bf Next, apply the
    existence constraint, which enjoins the acceptance of (SC) and, in
    particular, the existence of those possible worlds required by
    instances of (SC). \rm (SC) requires that there be a possible
    world such that `Bush lost the 2000 electoral vote' is true at
    that world.  The concrete cosmos in which we live is a possible
    world, according to the conception we are considering, just as the
    Republican nominee in 2008, John McCain, is a possible Republican
    nominee in 2008.  But `Bush lost the 2000 electoral vote' is not
    true at the world in which we all live.  So the naive theory seems
    to require that some $w$ is a possible world distinct from the
    actual world.  Perhaps it is plausible to hold that there might
    have been a world $w$ distinct from our world.  But (SC) requires
    more.  The world $w$ in question must exist;\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {For the purposes of this paper, I am using `exist' as a shorter
    equivalent for `is among the things that there are.'  Hence, `$x$
    exists' in the sense of this paper is equivalent to `there is a
    $y$ such that $y = x$.'  Further, I am abstracting away from
    complications introduced by the fact that, for many things,
    existence is temporary.  Thus, in the parlance I am adopting here,
    `The Colossus of Rhodes exists' remains true, even though the
    Colossus of Rhodes is no longer among us.  I believe that this
    usage tracks at least one use of ``there is/are'' in English,
    which might sometimes get used, \emph{e.g.} in encyclopedia
    entries, as in `there are three great ancient Greek philosophers:
    Socrates, Plato, and Aristotle.'}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} we may not appeal to the truth of `Bush lost the
    2000 electoral vote' at $w$ if $w$ is something that doesn't
    exist, but might have.\bf So, to be clear, the naive theory and
    the existence and modal modesty constraints jointly require that
    there be a $w$ among the things that in fact exist that is a
    possible world, and is distinct from the actual world. \rm
      
    Now apply ontological modesty.  It is overwhelmingly plausible to
    claim that there are no worlds (no concrete cosmoses) other than
    the actual world.  Accommodating this evident ontological fact
    requires that our $w$ be a possible world, but not a world.  At
    first, this might seem to be perfectly fine.  Someone can be a
    possible Republican nominee for president in 2008 without being a
    Republican nominee for president in 2008.  Why should possible
    worlds be any different from possible nominees?  Why not claim
    that something can be a possible world without being a world?
      
    The problem is that we would also ordinarily deny that there is an
    individual that might have been a world but isn't.  Consider the
    non-worlds among us.  It is implausible to think that any among
    you, me, the antarctic ice sheet, the space shuttle
    \emph{Endeavour}, the First World War, the square root of two, the
    region between the tip of your nose and your chin, \emph{etc.},
    might have been the same sort of all-inclusive thing as our
    concrete cosmos.
    
    This problem can be avoided by violating any one of our three
    constraints.  Fine chooses to violate the existence constraint,
    denying that non-actual possible worlds exist.  The attendant cost
    in user-friendliness can be somewhat ameliorated by endorsing a
    variant of (SC), which replaces the quantifier ``there is a $w$''
    with ``\emph{it is possible that} there is a $w$'':
    \begin{description}
      \item[(SC)${}^-$] it is possible that $\phi$ iff it is possible
        that there is a $w$ such that $w$ is a possible world and
        `$\phi$' is true at $w$.
    \end{description}
    In summary, the naive theory abjures reduction and may be less
    user-friendly than we might initially have
    hoped.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {The main text is a simplification on two fronts.  First, Fine's
      view is that a claim is possible just in case it is possible
      that there be a world $w$ such that the claim is \emph{actually}
      true at $w$; see \citep{Fine:1977, Fine:1985, Fine:2003a} for
      discussion.  Second, Fine's view is that we need not reject (SC)
      outright.  Fine \citeyearpar{Fine:2003a} holds that the
      expression ``there is a $w$ such that $w$ is a possible worlds
      and \dots'' (or ``some possible world $w$ is such that \dots'')
      just means ``it is possible that there is a $w$ such that $w$ is
      a world and actually \dots.'' By Fine's lights, (SC)${}^-$ thus
      makes explicit what (SC) already means.  In the main text, I am
      assuming that end-users' quantifiers are unambiguous, and range
      only over existent things; more generally, I assume that the
      quantifiers in instances of (SC) on which an end-user relies are
      to be interpreted the same way as quantifiers in the rest of her
      theory.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
  \end{section}
  
  \begin{section}{Fictionalism}\label{sec:fictionalism}
    The naive theory appears to require an ontologically extravagant
    commitment to the existence of many worlds -- many concrete cosmoses
    -- if it is to accommodate some evident modal facts.  We have just
    seen that Fine suggests we avoid this problem by rejecting (SC) in
    favor of a close variant.  An alternative theory of possible
    worlds offers another way of pursuing this strategy.  According to
    this view, called \emph{fictionalism}, there are no possible
    worlds other than the actual world.  Taken literally and at face
    value, (SC) has false instances.  But there is a true surrogate
    for (SC).  For there is a story, a work of fiction, according to
    which there are lots of non-actual worlds.  Call this story $PW$.
    We obtain the surrogate for (SC) by replacing talk of what worlds
    $w$ there are in fact with talk about what worlds $w$ there are
    according to the story $PW$.  The fictionalist theory is
    introduced in \citep{Rosen:1990}.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Our discussion of fictionalism will focus on Rosen's development
    of the view.  Thus, views I attribute below to the fictionalist
    are often accurately attributable only to Rosen's version.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Fictionalists differ over the contents of $PW$,
    but by Rosen's lights that content is given by the theory of
    possible worlds offered by David Lewis \citeyearpar{Lewis:1986}.
    What's true concerning the existence and nature of possible worlds
    according to the fiction is just what Lewis's theory says on the
    matter.  In particular, the fiction $PW$ contains a number of
    postulates specifying an ontology borrowed from Lewis's theory:
    there are many cosmoses, and many inhabitants of those cosmoses.
    The inhabitants are a varied lot.  Our cosmos does not contain any
    animate headless bodies.  Other cosmoses do.  $PW$ guarantees this
    by containing both (\emph{i}) a complete account of all the
    non-modal facts concerning the actual world, and (\emph{ii}) a
    principle of recombination:
    \begin{description}
      \item[(RECOMBO)] If $x_1$ is an individual in a cosmos $w_1$,
        $x_2$ is an individual in a cosmos $w_2$, \dots, then there is
        a cosmos $w^\forall$ containing any number (including 0) of
        duplicates of $x_1$ and any number (including 0) of duplicates
        of $x_2$, and \dots, size and shape
        permitting.\linespread{1}\endnote
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        {\citep[pp. 87--90]{Lewis:1986}, \citep[p. 335]{Rosen:1990}.
          The qualification ``size and shape permitting'' is inserted
          to avoid problems concerning the maximum possible size for a
          cosmos.}
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        \linespread{1.3}
    \end{description}
    $PW$ asserts the existence in this cosmos of your head and also of
    the rest of your body, since that fact is reported by a complete
    account of the actual non-modal facts.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {I borrow the example from \citep[p. 88]{Lewis:1986}.  I am
      assuming for present purposes that the existence of both your
      head and the rest of your body is reported by a complete account
      of the actual non-modal facts.  This assumption requires that
      you have two undetached parts: your head and the rest of your
      body.  If this assumption fails, then it's not the case that,
      according to $PW$, there is a cosmos with an animate headless
      body.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} And (RECOMBO) thus requires that there be a
    cosmos containing some duplicates of your body but no duplicates
    of your head.
    
    With such a specification of $PW$ in hand, the fictionalist now
    proposes in effect to revise (SC) by inserting a reference to
    $PW$:
    \begin{description}
      \item[(SC)${}_F$] it is possible that $\phi$ iff \emph{according
        to $PW$} there is a $w$ such that $w$ is a possible world and
        `$\phi$' is true at $w$.\linespread{1}\endnote
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        {This exposition differs slightly from Rosen's.  Let $\phi$ be
          any possibility claim, and let $\phi^*$ be the analysis of
          $\phi$ according to Lewis's theory.  Rosen's fictionalist
          claims
          \begin{description}
            \item[(F)] $\phi$ iff according to $PW$, $\phi^*$.
          \end{description}
        }
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        \linespread{1.3}
    \end{description}    
    She then helps herself to Lewis's interpretation of the
    right-hand-side of (SC)${}_F$.  For instance, while Lewis's theory
    holds
    \begin{quote}
      There might have been some headless animate bodies iff there is
      a cosmos containing headless animate bodies
    \end{quote}
    the fictionalist claims instead
    \begin{quote}
      There might have been some headless animate bodies iff according
      to $PW$: there is a cosmos containing headless animate bodies.
    \end{quote}
    
    Asserting the existence of the fiction $PW$ does not appear to
    make any modal claims.  According to $PW$, being a possible world
    amounts to being a cosmos, and there doesn't seem to be anything
    modal about that.  So why think fictionalism is a nonreductive
    theory of possible worlds?  The answer is that the notion
    \emph{what's true according to $PW$} seems itself to be modal.  To
    see this, note that $PW$ leaves much about the plurality of worlds
    implicit.  For instance, $PW$ is inexplicit about the existence of
    a cosmos containing purple penguins.  The explicit claims of $PW$
    don't mention purple penguins at all (other than to say that there
    aren't any in our cosmos).  What the fictionalist needs is a way
    of extracting from the explicit claims of $PW$ the implicit
    detailed information about non-actual worlds and their inhabitants
    needed to satisfy the full panoply of instances of (SC)${}_F$.
    Modal means of extraction are ready to hand: Divers
    \citeyearpar[p. 335]{Divers:1999}, for instance, suggests an
    explicitly modal definition:
    \begin{description}
      \item[(DEF)] $\square(($According to $PW$, $\phi)$ iff
        $\square(PW\Rightarrow\phi))$.\linespread{1}\endnote
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        {Endorsed at \citep[p. 81]{Brogaard:2006}.}
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        \linespread{1.3}
    \end{description}
    But no such modal characterization will do if we are to use
    (SC)${}_F$ in the service of reducing necessity to non-modal
    terms.  It is far from obvious that such a non-modal means of
    extraction is available.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {This point is made by \citep[\S8]{Rosen:1990},
      \citep[p. 139]{Melia:2008} and \citep{Fine:2003a}.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} 
    
    This problem leads Rosen to suggest giving up on reduction, and
    insisting that (SC)${}_F$ can be used instead to license the use
    of the possible worlds idiom without committing us to ontological
    extravagance.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Brogaard \citeyearpar{Brogaard:2006} agrees that nonreductive
      fictionalism is the better sort.  Cf. Fine's development of an
      alternative form of fictionalism in \citep{Fine:2003a}, which,
      he claims, avoids the problems afflicting the version we are
      considering.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} That is, Rosen suggests we give up on reduction,
    and settle for user-friendliness.

    How user-friendly is fictionalism?  The replacement of (SC) by
    (SC)${}_F$ avoids ontological extravagance.  Fictionalism does not
    require that there be other cosmoses containing headless animate
    bodies.  It only requires that there be other such cosmoses
    \emph{according to a particular fiction}.  It is ontologically
    extravagant to assert that hobbits exist.  It is plain sense to
    assert that hobbits exist according to a certain fiction.
    Fictionalism is not perfectly user-friendly, however.  It does not
    endorse (SC), and in particular violates the constraint of
    existence.  \bf Also, as noted in my review of Lewis's
    theory,\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {\bf\citep[pp. XXXX]{deRosset:forthcoming}. [CITE!]  [PAGE NUMBERS
        NEEDED!]\rm}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} (RECOMBO) makes a substantial and controversial
    modal commitment against the necessity of laws of nature.  Modal
    modesty, which enjoins neutrality on such controversies, is
    thereby violated.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {\bf This violation of modal modesty applies only to versions of
      fictionalism, like Rosen's, that include (RECOMBO) in $PW$.  A
      fictionalist can evade the problem if she finds some other means
      for filling in the details concerning the contents of non-actual
      worlds.  Thanks to BW.\label{fn:RosenSpecific}\rm}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} \rm
    
    Less obvious but more serious forms of user-hostility loom.  The
    fictionalist's replacement (SC)${}_F$ for (SC) requires the
    impossibility of any claim $\phi$ if it's not the case that,
    according to $PW$, $\phi$ is true.  So, if it turns out that $PW$
    does not require there to be a cosmos containing purple penguins,
    then the fictionalist is committed to the impossibility of purple
    penguins.  But $PW$ does not require there to be a cosmos
    containing purple penguins.  The claims about there being lots of
    cosmoses with lots of inhabitants in them, imposes no requirement
    that some of those inhabitants be purple penguins.  (RECOMBO) is
    supposed to do some of this work.  But duplicate the penguins of
    this cosmos as often or as seldom as you like, and you will not
    come up with any purple ones.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {\bf See \citep{deRosset:forthcoming} for a more thorough
      discussion of this issue. [CITE!] [PAGE NUMBERS NEEDED!] \rm}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} And $PW$ doesn't say anything else about what
    inhabits the non-actual cosmoses.  So it's not the case that,
    according to $PW$, there is a cosmos containing purple penguins.
    Thus, the fictionalist is committed to the impossibility of purple
    penguins.  This is a violation of modal modesty, and implausible
    to boot.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Rosen \citeyearpar[\S7]{Rosen:1990} acknowledges the problem,
      posed with respect to the series of claims, for each cardinal
      number $\kappa$, that there might have been $\kappa$
      non-overlapping physical objects.  His suggested fix is to
      stipulate that, where the fiction $PW$ is silent, there are
      truth-value gaps.  This does not help with the charge of
      implausibility.  For in both the case of the upper limit on the
      possible number of non-overlapping physical objects, and the
      case of the possibility of purple penguins, the verdict ``no
      truth value'' is implausible; in the latter case, it is more
      implausible that the verdict ``false.''  \bf Like the prior
      charge of modal extravagance (see n. \ref{fn:RosenSpecific}),
      this charge depends on Rosen's incorporation of (RECOMBO) in
      $PW$. \rm}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
    
    The sort of fictionalism we have been discussing violates the
    constraints of existence and modal modesty, and has some
    implausible consequences.  On Fine's development, the naive theory
    avoids the implausibility in question and does better on modal
    modesty.  So we already have one theory of possible worlds that
    seems better off than the sort of fictionalism we have been
    considering.  There are others.
  \end{section}
  
  \begin{section} {Property-based conceptions}
    A variation on the naive theory's explanation of the nature of
    possible worlds yields a popular and influential family of
    theories.  This family of views, \emph{property-based
      conceptions}, results from replacing appeal to worlds with
    appeal to property- or state-like entities in the naive theory's
    explanation of the nature of possible
    worlds.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {I think this is a useful way to think of the view; its proponents
    do not present the view in this way, though the discussion in
    \citep{Stalnaker:1976} is suggestive.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} One version, for instance, holds that possible
    worlds are possible states of a world: $x$ is a possible world iff
    $x$ is a state such that it is possible that a world be in state
    $x$.  There are a wide array of variations on this theme.  Some
    theorists hold that possible worlds are maximal \emph{states of
    affairs} that either do obtain or could have obtained
    \citep{Plantinga:1973}; others that possible worlds are complete
    \emph{ways things might have been} \citep{Stalnaker:1976}; still
    others that possible worlds are complete possible \emph{histories}
    of the world \citep[p. 17]{Kripke:1980}.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Property-based conceptions also come with accounts of what it is
    for a sentence to be true at a possible world.  For instance,
    Plantinga \citeyearpar{Plantinga:1973} holds that a sentence is
    true at a state of affairs just in case, necessarily, the sentence
    is true if that state of affairs obtains.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
    
    The first thing to notice about property-based conceptions is that
    they will not underwrite a reduction of modality.  For instance,
    the notion of a possible world-state is clearly a modal notion.
    In fact, our explanation of the modal notion in question used the
    very same possibility operator that appears on the left-hand side
    of (SC).
    
    But property-based conceptions seem to do well on
    user-friendliness.  A conspicuous virtue of property-based
    conceptions is their apparent ontological modesty.  It is
    plausible to think that there are states the world could have had
    but doesn't.  Property-based conceptions seem also to be modally
    modest: they appear to accommodate the truth of all instances of
    (SC) without undertaking any implausible or even controversial
    modal commitments.  For instance, it is very plausible to hold
    that there is a world-state \emph{containing a loss by Bush of the
      2000 election}.  This is exactly the sort of thing required by
    (SC) according to property-based conceptions that identify
    possible worlds with possible world-states.  Similarly, we are
    antecedently inclined to affirm that there is no possible state of
    a world \emph{being such that 2 + 2 $\neq$ 4}.  So, though
    property-based conceptions will not serve the ambition of reducing
    modality, they do seem at first to be more user-friendly than
    either the naive theory or fictionalism.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {A caveat: if one is inclined to deny that there are property- or
    state-like entities, then one may be reluctant to accept a
    property-based conception, taken literally and at face value.  So
    the property-based conception fails on its face to be
    user-friendly for every sort of user.  Similarly, if one hoped
    (see \citep[\S1.5]{Lewis:1986}) to use possible worlds and their
    inhabitants to provide an analysis of properties, states, or
    histories, then the correlative property-based conception would be
    useless.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
    
    But there is reason to think that property-based conceptions
    ultimately must manifest one or another form of user-hostility.
    For the sake of exposition, I will assume that all property-based
    conceptions require that there is a property that a world might
    have had corresponding to each possible world-history,
    world-state, or way things might have been.  Consider, for
    instance, the property-based conception according to which
    possible worlds are \emph{ways things could be}.  Suppose,
    plausibly, that one of the ways things could be is that Bush lost
    the 2000 electoral vote.  Then, I am assuming, there is a
    corresponding property \emph{being a world containing a loss by
    Bush of the 2000 electoral vote}.  Likewise, I assume that the
    existence of a possible world-state $S$ implies the existence of
    the property \emph{being a world in state $S$}, and similarly for
    the existence of a world-history $H$.
    
    Let's call a property \emph{individual-involving} if it is
    specifiable only by reference to a particular individual.  It is
    possible that Charles Windsor, the Prince of Wales, have an
    eleventh granddaughter.  Let us consider, then, a situation in
    which Windsor had an eleventh granddaughter.  That eleventh
    granddaughter would be what metaphysicians call an \emph{alien
    individual}: an individual who does not exist at the actual world.
    In such a situation, there would be an individual-involving
    property of a world, call it \emph{elevenliness}, specifiable in
    terms of the particular alien individual who turns out to be
    Windsor's eleventh granddaughter: the property of \emph{containing
    her}.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Here I benefit from discussion of these issues with DC.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} An initial problem that the possibility of aliens
    poses for property-based conceptions is that, though it is
    possible that there have existed different individuals from the
    ones which actually exist, it is implausible to claim that there
    are properties involving such individuals.  It would violate
    ontological modesty, for instance, to affirm the existence of any
    property that, like \emph{elevenliness}, involves a particular
    possible eleventh granddaughter of Windsor.  Property-based
    conceptions appear to require such properties, given the
    possibility that Charles Windsor have an eleventh granddaughter.
      
    Faced with possibilities involving aliens, proponents of
    property-based conceptions propose to trade in commitment to
    properties involving alien individuals for commitment to
    properties involving only general conditions expressible without
    reference to any particular individual.  For instance, a proponent
    of a property-based conception may trade elevenliness in for the
    property \emph{being a world containing an eleventh granddaughter
      of Windsor}.  On this response, a property-based conception will
    not require the existence of a property, possession of which
    guarantees that a world contains a \emph{particular} possible
    eleventh granddaughter of Windsor; it will only require the
    existence of a property possession of which guarantees that a
    world contains an eleventh granddaughter of Windsor, no matter who
    that turns out to be.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {See, \emph{e.g.} Bennett's \citeyearpar[p. 317]{Bennett:2005}
    response to this problem.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
    
    But this response does not solve the underlying problem: trading
    individual-involving properties in for general properties leads to
    modal extravagance.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {This problem has been discussed by many commentators in one form
      or another, including \citep{McMichael:1983},
      \citep[pp. 158-65]{Lewis:1986}, and, most recently,
      \citep[p. 317]{Bennett:2005}.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} It seems possible that there be, in addition to
    an eleventh granddaughter of Windsor, a distinct child who, to all
    outward appearances, is just like Windsor's eleventh
    granddaughter.  The two girls might even be identical twins,
    genetically very similar, or even indiscernible, because they
    emerged from the same gametes.  The eleventh granddaughter came
    before the twelfth because she was born, say, a few minutes
    earlier -- another royal accident of birth.  It thus seems
    possible that Windsor have genetically identical eleventh and
    twelfth granddaughters, who might have switched places.  The birth
    order for the identical twins might have gone differently.
    
    So far, the modal claims are simple common sense, given the
    biological facts about identical twins.  But a more difficult
    question looms.  Take the situations in which Windsor has an
    eleventh and twelfth granddaughter who are identical twins.  In
    some such situation, the eleventh will have certain features,
    other than her mere distinctness from her sister, which allow us
    to discern her from the twelfth.  She might grow to be 5'11, while
    her sister only grows to 5'10.  She might reside in London, while
    her sister resides on a country farm.  She might dye her hair
    purple, while her sister sticks with her natural color.  Call this
    bunch of features the \emph{career} of the
    eleventh.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Note that the career of a twin may include such relational
    features as, \emph{e.g.} her being within two meters of the Duke
    of Cornwall at some particular time, and her being born 20,738,274
    seconds after the death of Louis XVI.  Thus, the career of an
    individual may indirectly embrace the entire history of the
    universe.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} The twelfth will also have a career, different in
    many ways.  Consider now whether the twins could have switched
    careers.  Is it possible that Windsor have had an eleventh
    granddaughter with a certain career, which might have been enjoyed
    in each exact detail by someone else?
      
    The question is vexed.  If the answer is `yes', then there is no
    interesting necessarily sufficient condition for being a
    particular possible 11th granddaughter of Charles Windsor.  This
    merits calling the `yes' view
    \emph{insufficientism}. Insufficientism is ripe for
    generalization.  Any one of us could have had a
    twin.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Forbes \citeyearpar[pp. 354--5]{Forbes:1980} dissents, arguing
    that no non-twin could be a twin.  So perhaps the claim that any
    one of us could have been a twin is a matter of controversy.  Even
    so, the underlying point still stands.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} And, if twins can switch careers, any one of us
    could have switched careers with our twin; there would be no
    interesting modally sufficient condition for being any particular
    person.  Insufficientism runs counter to a tradition which holds
    that identity facts must be determined by further intrinsic
    features.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Exemplars of this tradition include \citep{Forbes:1980},
    \citep[94--99]{Wiggins:1980} and \citep{Noonan:1985a}.}  %More
    %recently, Guy Rohrbaugh and I have %broached insufficientism
    %(under the label, the ``bare identities %view'') in
    %\citep{RohrAnddeR:2004} and \citep{RohrAnddeR:2006}}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Note that insufficientism is consistent with the
    view that there are interesting modally necessary conditions for
    being a particular person.  For instance, the idea that nothing
    coming from completely different gametes could be me is not ruled
    out; I could not switch with just anybody, but I could switch with
    my identical twin.  I confess that I suspect that insufficientism
    is true.  It is nevertheless a very controversial claim, a
    substantive, non-logical modal matter if ever there was one.  The
    modal modesty constraint requires that a theory of possible worlds
    remain neutral on this substantial and controverisal modal matter.
    
    The existence and ontological modesty constraints require
    property-based conceptions to deliver a `no' answer to our
    question: it is not possible for Charles Windsor to have had an
    eleventh granddaughter with a certain career, which might in turn
    have been enjoyed by someone else.  Suppose for the purposes of
    \emph{reductio} that there might have been such a granddaughter.
    There are two possibilities claimed here: the possibility for
    Windsor that he have had an eleventh granddaughter with a certain
    career, and the embedded possibility that that career have been
    enjoyed in every particular by someone else.  These possibilities
    are incompatible.  Meeting the constraint of existence will thus
    require the existence of two possible worlds (the first accessible
    from the actual world, and the other accessible from the first).
    These two worlds will be exactly alike, except with respect to
    which individual has the career in question.  In this sense, they
    will be ``qualitatively identical.''  On property-based
    conceptions, there must be two properties that both might have
    been properties of a world, but which differ only with respect to
    which individual has the relevant career.  Hence there is no
    specifying what it takes for something to possess either property
    except by reference to the particular individual who has that
    career.  The two properties are each individual-involving
    properties, differing only with respect to which individual is
    involved.
      
    Now apply the constraint of ontological modesty.  We are
    antecedently inclined to deny that there is any such thing as a
    property involving a certain individual if there is no such thing
    as the individual in question.  If Socrates had never existed,
    neither would the property of \emph{being him}.  The situation is
    no different for merely possible objects.  And we are antecedently
    inclined to deny that anyone who might have been the eleventh
    granddaughter of Charles Windsor actually
    exists.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {I picked the number 11 just for illustration.  If Windsor's sons
    turn out to be more fecund than is convenient for this argument,
    some higher number, \emph{e.g.} 111, may be required.  Similarly,
    if it turns out that some actual granddaughter of Windsor's might
    have been the eleventh, or by some miracle of modern medicine the
    $111^{th}$, then we may perhaps have to tweak the relation a
    little to exclude actual granddaughters.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} So affirming insufficientism while avoiding
    ontological extravagance requires that neither of these properties
    exist, contrary to the demands of the existence constraint.
      
    The fundamental problem with property-based conceptions is that,
    intuitively, the actual world does not provide the properties the
    view requires to back possibilities involving aliens.  Though
    there might have been a particular individual who was Windsor's
    eleventh granddaughter, there is no such thing as the property of
    \emph{being her}.  A symptom of this fundamental difficulty is
    that the conceptions require one of three forms of user-hostility:
    (i) \emph{modal extravagance}: incurring a substantial and
    controversial modal commitment by denying insufficientism; (ii)
    \emph{denying existence}: denying that all of the possible worlds
    required by the truth of all instances of (SC) exist; or (iii)
    \emph{ontological extravagance}: accepting the existence of
    properties involving merely possible eleventh granddaughters of
    Windsor.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Faced with this trilemma, some property-based theorists bite the
      bullet by violating the constraint of existence.  They propose
      to deny some instances of (SC) in favor of (SC)${}^-$.  See, for
      instance, \citep[pp. 17-8n.]{Salmon:1989}.  Other property-based
      theorists bite the bullet by accepting ontological extravagance.
      See, for instance, \citep[pp. 116--7]{Plantinga:2003b}.
      Bennett's \citeyearpar{Bennett:2005} discussion of this problem
      suggests that she proposes instead to pin a substantial modal
      commitment against insufficientism on end-users, violating modal
      modesty.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
    
    In the literature, this objection is often put in terms of
    requirements that property-based conceptions impose on the
    ontology of properties.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {See, for instance, \citep[pp. 159ff.]{Lewis:1986}.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Property-based conceptions appear to commit
    end-users to the existence of uninstantiated properties.  For
    instance, nothing in fact instantiates \emph{being a world
    containing a loss by Bush of the 2000 electoral vote}, but the
    property-based conception requires there to be such a property.
    
    A defender of property-based conceptions might hope to take refuge
    in the idea that the required uninstantiated properties can be
    constructed out of instantiated properties.  For instance, it
    might be held that \emph{being a world containing a loss by Bush
      of the 2000 electoral vote} is constructed out of a number of
    instantiated properties and relations, including \emph{being a
      world}, \emph{being an electoral vote}, the \emph{containing}
    relation, the \emph{losing} relation, \emph{etc.}  But this just
    puts off the problem.  If the defender of the property-based
    conception is to avoid the user-hostile reliance on controversial
    theses concerning modal facts, then she may not assume that it is
    impossible that there should be an instance of a \emph{simple}
    property $P$ that is not actually instantiated.  Call such a
    property an \emph{alien property}.  On the assumption that it is
    possible that there be such a property, the property-based
    conception seems to commit us to there being some such property as
    \emph{being a world containing an instance of $P$}, which seems in
    turn to require the existence of the alien property $P$.  At this
    point, the dialectic follows the script established by the dispute
    concerning alien individuals, with the causal role and pattern of
    instantiation of $P$ taking the place of the career of a possible
    eleventh granddaughter of Charles Windsor.
  \end{section}

  \begin{section}{Representational Theories}
    One final variation on the naive theory's explanation of the
    nature of possible worlds bears mention.  This sort of view, a
    \emph{representational theory}, results from replacing the naive
    theory's appeal to worlds with an appeal to representations as of
    worlds: $w$ is a possible world iff $w$ is a representation of
    what a world is like that might have been
    true.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Typically, representational theories will require that the
      representation be complete or maximal in some sense.  As with
      property-based conceptions, proponents of representational
      theories do not present their view as a variation of the naive
      theory.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} On this view, possible worlds are something like
    stories.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {I am assuming that stories are linguistic representations.  Thus,
    the view described in the text is only one variety of
    representational theory, on which the representation in question
    is linguistic.  Other varieties suggest instead that the
    representation is pictorial.  Still others decline to specify the
    nature of the representation.  See \citep[\S\S3.3,
    3.4]{Lewis:1986} for discussion of the latter two alternatives.
    The pictorial view is not really defended in the literature; see
    \citep{vanInwagen:1986} for a defense of a representational theory
    that declines to specify the nature of the representation.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Some stories describe situations that could not
    be true; according to the representational theory, possible worlds
    are stories which could have been true.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Historical developments of representational views may be found in
    \citep{Carnap:1947} and \citep{Adams:1974}.  More recent
    elaborations may be found in \citep{McGee:2006},
    \citep{Melia:2001} and \citep[ch. 5]{Nolan:2002}.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
    
    Explained in this way, representational theories seem useless for
    reducing modality.  The notion of a \emph{possibly true} story is
    evidently modal.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Perhaps a representational theorist might suggest instead a
      reductive theory of possible worlds on which a possible world is
      just the result of combinatorics on some set of basic
      representations.  For instance, the representationalist might
      hold that a story is a consistent set $w$ such that, for every
      atomic sentence $F\alpha_1, \alpha_2, \dots, \alpha_n$ in which
      $F$ stands for a basic property and each of the $\alpha$'s names
      a basic individual, $w$ contains either the sentence or its
      negation.  This is a proposal inspired by \citep{Carnap:1947}.
      Such an alternative avoids immediate appeal to modal notions in
      explaining what a possible world is.  But many authors have
      argued that some such appeal is still necessary; see, for
      instance, \citep[pp. 155-6]{Lewis:1986} and
      \citep[p. 139]{Melia:2008}.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} It appears, then, that representational theories
    of this sort will be nonreductive.
    
    How about user-friendliness?  Some have argued that
    representational theories fall prey to paradox.  It might be
    thought that the representationalist requires a one-one
    correspondence between possible individuals and their actual-world
    representations.  There are reasons to think that there is no
    upper limit on the number of possible individuals, in the sense
    that, for any possible world, there is a possible world with
    strictly more individuals.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {See Fine \citeyearpar[pp. 223-4]{Fine:2003a} for an argument, and
    \citep{Brogaard:2006} for discussion.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Since the actual cosmos in which we live must be
    accurately represented by some possible world on the
    representational theory, there is a possible world with strictly
    more individuals than there actually are.  The individuals of this
    bigger possible world cannot correspond one-one to the individuals
    that actually exist, contrary to the requirements of
    representational theories.  This leaves the representationalist a
    few options: (i) develop a representational theory that does not
    require a one-one correspondence between \emph{possibilia} and
    their actual-world representatives; (ii) violate the constraint of
    existence by denying (SC) in favor of something like (SC)${}^-$;
    or (iii) violate modal modesty by denying that there could be
    more individuals than there actually are.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {McGee \citeyearpar{McGee:2006} is committed to the third horn of
      the dilemma.  One might also argue that the question of how many
      individuals there actually are makes no sense, on the grounds
      that such a question would require absolutely unrestricted
      quantification, and there can't be absolutely unrestricted
      quantification.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} The second option is user-hostile because it
    requires us to deny that all instances of (SC), taken literally
    and at face value, are true.  The third option is user-hostile
    because it pins a substantial and implausible modal commitment on
    end-users.  So the first option seems indicated.
    
    Assuming representationalist theories can avoid the problem posed
    by needing a representation for every possible individual, they
    seem initially to be user-friendly.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Adopting a representational view may ruin the usefulness of
    possible worlds for theories of representation.  For instance, if
    one were to analyze propositions as sets of worlds, while claiming
    that worlds are sets of propositions, then the analysis of
    propositions would be circular.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} They aren't ontologically extravagant.  It is
    plausible to think that there are representations that aren't
    accurate but might have been.  The sentence ``Bush lost the 2000
    electoral vote'', for instance, might have been true.  And it's
    not too much of a stretch to suggest that there are richer
    representations as of entire worlds that aren't accurate but might
    have been, especially if these are somehow made up of
    representational elements in which we already believe.  
    
    But it turns out that, like the other nonreductive views we have
    canvassed, representational theories encounter difficulties
    accommodating the possibility of aliens.  The dialectic mirrors
    the one we encountered when considering the user-friendliness of
    property-based views: assuming that insufficientism is true drives
    a representationalist either to deny the existence of all of the
    worlds required by instances of (SC), or to some ontological
    extravagance.  The property-based conception appeared to require
    the existence of individual-involving properties that discern
    particular possible eleventh granddaughters of Charles Windsor.
    Representational theories appear instead to require the existence
    of representations that discern particular possible eleventh
    granddaughters of Charles Windsor, in the sense that those
    representations represent, refer to, or pick out one such
    individual but not another.
    
    But, given insufficientism, we are antecedently inclined to deny
    that there are any such representations.  There are no names,
    pictures, sculptures, photographs, \emph{etc.}, depicting one, but
    not another possible eleventh granddaughter.  Even if we allowed
    individuals to represent themselves, so that, \emph{e.g.}  unnamed
    real numbers represented themselves, it is implausible to claim
    that any such possible eleventh granddaughter exists.
    Insufficientism guarantees that there is no
    non-individual-involving condition which discerns them, and so no
    way of framing a description or picture of the career of such a
    possible eleventh granddaughter that would distinguish her from
    her twin.  Of course, if we were able to discern the two
    \emph{possible worlds} in question, then we could frame such a
    condition: \emph{being the individual who enjoys career $C$ at
      world $w$}.  But, given that possible worlds are themselves
    representations as of worlds, there seems to be nothing in virtue
    of which one of these representations could true while the other
    false, and so nothing to discern them so far as what's true at
    them.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Faced with this trilemma, some representational theorists bite
    the bullet by violating the constraint of existence.  See, for
    instance, \citep{Sider:2002}, and the discussion at
    \citep[pp. 120-1]{Nolan:2002}.  Nolan's
    \citeyearpar[pp. 113-4]{Nolan:2002} discussion of this problem
    suggests violating modal modesty by pinning a substantial modal
    commitment against insufficientism on end-users.  To the best of
    my knowledge, no representational theorist accepts ontological
    extravagance by affirming the existence of representations that
    discern distinct possible eleventh granddaughters of Charles
    Windsor, assuming insufficientism.  Lewis suggests that this last
    view is unacceptable because there is no good characterization of
    the sort of representation in question; for this reason he
    characterizes it as ``magical'' representation
    \citep[\S3.4]{Lewis:1986}.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
  \end{section}
  
  
  \begin{section}{Conclusion}
    The results of our brief, crude, and admittedly incomplete canvass
    have not been encouraging.  Every view we have discussed suffers
    from some user-hostility.  It should be possible to do
    better.\linespread{1}\endnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Thanks to TC, DP, GU, BW, and an anonymous referee for helpful
      comments on earlier drafts.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
  \end{section}
  
  
  
  \theendnotes
   \linespread{1}  
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\end{document}


