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\title{Possible Worlds for Modal Primitivists}
\author{Louis deRosset}

\begin{document}
  \maketitle
  
  \begin{abstract}
    Among the most remarkable developments in metaphysics since the
    1950's is the explosion of philosophical interest in possible
    worlds.  This paper proposes an explanation of what possible
    worlds are, and argues that this proposal, the \emph{interpreted
      models conception}, should be attractive to anyone who thinks
    that modal facts are primitive, and so not to be explained in
    terms of some non-modal notion of ``possible world.'' I articulate
    three constraints on any acceptable primitivist explanation of the
    nature of possible worlds, and show that the interpreted models
    conception meets the three constraints. 
  \end{abstract}
  
  \begin{section}{Possible Worlds and Modal Primitivism}
    Among the most remarkable developments in metaphysics since the
    1950's is the explosion of philosophical interest in possible
    worlds.  But what are those possible worlds?  Perhaps a clue is
    provided by the fact that every instance of the schema
    \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}    
    is widely assumed to be true.  (SC) reveals a systematic
    correspondence between certain modal facts and facts regarding the
    existence of possible worlds.  To some philosophers, (SC) and
    associated claims held out the hope of providing an explanation in
    non-modal terms of those modal facts which can be expressed by
    idioms representable in quantified modal logic.  Some even
    proposed that all modal facts could be explained in terms of facts
    about which possible worlds there were and what they were
    like.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {\citep{Lewis:1986}}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Providing a non-modal explanation of the notion
    of \emph{being a possible world} has occupied center stage in the
    thought of these philosophers.  It will not, however, occupy
    center stage here.  We are not going to require here that the
    notion of \emph{being a possible world} be explained in thoroughly
    non-modal terms.  Suppose, then, that \emph{modal primitivism} is
    true: some modal facts cannot be reduced to, or even explained in
    terms of, any congeries of exclusively non-modal facts.
    
    Even granting that (SC) does not provide fodder for an explanation
    of modal facts in non-modal terms, there remain a number of
    alternative views of (SC)'s significance.  One could suggest, for
    instance, that instances of (SC) are simply meaningless, since
    they all crucially employ the notion of \emph{being a possible
      world}, and that notion has not been explained.  Alternatively,
    one could simply deny that there are any possible worlds.  Then
    there will be instances of (SC) -- those involving true
    possibility claims -- that are false.
    
    If we take either of these skeptical lines, then much of
    contemporary modal metaphysics will turn out to be either
    meaningless or a pack of falsehoods.  And other disciplines that
    make liberal use of possible worlds, \emph{e.g.} some strands of
    linguistics, ethics, epistemology, and decision theory, will
    suffer from the same defect.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Examples of theorizing in these disparate fields that relies on
      possible worlds are abundant; a small sample of pieces includes
      \citep{Arntzenius:2003}, \citep{Blackburn:1985},
      \citep{Lewis:1979}, \citep{Meacham:2008, Meacham:2010}, and
      \citep{Stalnaker:1978}.  Perhaps a case can be made that, though
      modal metaphysics and other disciplines appear to share an
      ontology of possible worlds, the entities used in the other
      disciplines are radically different from those mentioned in
      (SC).  I will not be assessing the prospects for this line here.
      Instead, I will be assuming that appearances are not deceiving
      in this respect.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} This need not imply that talk of possible worlds
    in all of these disparate fields of inquiry is useless or
    unenlightening.  A theory incorporating (SC) might be a useful or
    edifying fiction, for instance.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {\citep{Rosen:1990}}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Alternatively, (SC) might be false, even though
    some close surrogate is true.  (SC) on this view would inherit its
    usefulness from the usefulness of its true surrogate.  Still, we
    might wonder whether a more charitable course is available.
    
    I will argue that there is a more charitable course available.
    Modal primitivists can provide an explicitly modal explanation of
    the notion of a possible world that, I will argue, renders
    instances of (SC) both meaningful and true.  Other writers have
    also advanced proposals for explaining the notion of a possible
    world in explicitly modal terms.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {See, for instance,\citep{Kripke:1980} \citep{Plantinga:1973},
      \citep{Salmon:1989}, \citep{Sider:2002} and
      \citep{Stalnaker:1976}.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Each such explanation, in conjunction with
    ancillary claims like (SC), generates a theory of possible worlds.
    Thus, many primitivist theories of possible worlds are available.
    The purpose of this paper is to propose one such theory.  Before I
    outline the proposed theory, however, I will articulate three
    constraints on theories of possible worlds.
  \end{section}
  
  \begin{section} {Constraints on an Acceptable Explanation}
    There are three constraints which a theory of possible worlds
    should meet.
    
    \begin{subsection} {Existence}    
      % Existence
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      First, the possible worlds should exist according to the
      theory.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {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} This is required if we are to take all
      instances of (SC) to be straightforwardly true, since some
      instances of its left-hand side are true.  Hence, this
      constraint is motivated by the very description of the project
      whose prospects we are exploring: providing an explicitly modal
      explanation of the notion of a possible world that does not
      require that we take a skeptical line on (SC).  Call this the
      \emph{constraint of existence}.
    \end{subsection}
    
    \begin{subsection} {Ontological Modesty}    
      The second constraint is that our theory should avoid, so far as
      possible, ontological extravagance.  That is, our theory should
      not, so far as possible, commit us to the view that there are
      things of a certain sort when, before we came to the theory of
      possible worlds, we would have thought that there are no such
      things.
            
      The constraint against ontological extravagance is motivated by
      the fact that those antecedent opinions may form the basis of
      objections to our theory of possible worlds.  If, for instance,
      our theory commits us to there being such things as talking
      donkeys, then the claim that there are no such things presents a
      \emph{prima facie} objection to the theory.  Perhaps every
      theory of possible worlds will have to indulge in extravagance
      at one point or another.  If so, then this constraint cannot be
      perfectly satisfied.  Even so, those theories that are less
      extravagant are less susceptible to such objections.  Call this
      second constraint the \emph{constraint of ontological modesty}.
    \end{subsection}
    
    \begin{subsection} {Modal Modesty}
      We can call the third constraint on an acceptable theory of
      possible worlds the \emph{constraint of modal modesty}: our
      theory of possible worlds should, as much as possible, commit us
      to no \emph{post hoc} conclusions about the non-logical modal
      facts.  Here, for instance, is an evident fact regarding George
      W. Bush: he might have lost the 2000 electoral vote.  We
      shouldn't endorse a theory of possible worlds that commits us to
      the claim that he could not have lost.  Of course, we might be
      wrong about what the modal facts are, so we should be open to an
      argument that, in this or that case, our common-sense
      description of modal reality is wrong.  But, in the absence of a
      compelling argument, \emph{e.g.} that Bush could not have lost,
      the claim that he might have lost provides the basis for an
      objection to any theory which contradicts that claim.
      
      But the constraint of modal modesty applies even in cases in
      which the modal facts are under serious dispute.  The constraint
      is motivated in these cases, not by the need for a materially
      adequate theory which avoids counter-example, but also by
      methodological considerations.  Our theory of possible worlds
      should provide a neutral framework within which to carry on
      non-logical disputes about what's possible.  These issues, then,
      should not be decided by the metaphysics and logic of the
      framework.  As with the case of ontological modesty, it may be
      impossible for our theory of possible worlds to be completely
      neutral on the non-logical modal facts.  But we can at least
      aspire to as much neutrality as may be had.
    \end{subsection}
    
    \begin{subsection}{The Significance of the Three Constraints}
      Our assumption of modal primitivism implies that possible worlds
      cannot be used to provide an analysis or reduction of modal
      claims or facts to non-modal terms.  But, as I've noted, the
      utility of possible worlds is not limited to this reductive
      project in the metaphysics of modality.  Possible worlds are
      useful in a wide variety of theoretical endeavors far removed
      from the metaphysics of modality.  Call the theorists who rely
      on (SC) and related claims in theorizing in areas other than the
      metaphysics of modality, the semantics of modal discourse, or
      the analysis of modal concepts \emph{end-users} for theories of
      possible worlds.
      
      Meeting the three constraints underwrites the usefulness of
      possible worlds for these end-users' theoretical endeavors.
      Conversely, violating the three constraints undermines the
      usefulness of possible worlds for end-users.  If we endorse a
      theory of possible worlds that violates the existence
      constraint, for instance, then we cannot interpret theorizing
      that relies on (SC) as literally true on the most straighforward
      reading.  If we endorse a theory of possible worlds that
      violates ontological modesty by endorsing implausible
      ontological claims, then we export those implausibilities to
      end-users' theories.  Likewise, violating modal modesty holds
      their theorizing hostage to abstruse disputes in modal
      metaphysics.  Each of these features makes possible worlds less
      serviceable for theoretical projects outside the metaphysics of
      modality.  It would be better to avoid them if we can.
    \end{subsection}
    
    \begin{subsection} {The Three Constraints in Action}
      \label{ssec:3ConstraintsIllustrated}
      The three constraints combine to make the problem of explaining
      the notion of \emph{being a possible world} rather more
      difficult than one might have hoped.  Suppose we explain
      ``possible world'' on the model of such ordinary, and
      straightforwardly modal, notions as ``possible Republican
      nominee for president in 2012.''  It is plausible to hold that
      someone is a possible Republican nominee for president in 2012
      just in case it is possible that she be a Republican nominee for
      president in 2012.  Sarah Palin, Mitt Romney, and Tim Pawlenty
      are all possible Republican nominees for president in 2012.  The
      gas station around the corner is not a possible Republican
      nominee for president in 2012.
      
      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.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {See \citep{Fine:1985, Fine:2003a} for a defense of a view along
      these lines.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3} Such a view will have difficulty simultaneously
      meeting all three constraints.
      
      First, apply the modal modesty constraint.  We take ``it is
      possible that Bush lost the 2000 electoral vote'' to be true and
      so part of our common-sense description of the modal facts.
      (SC) requires that there be a possible world such that ``Bush
      lost the 2000 electoral vote'' is true at that world.  The world
      in which we live is a possible world, according to the
      conception we are considering, but ``Bush lost the 2000
      electoral vote'' is not true at that world.  So modal modesty
      seems to require that some $w$ is a possible world distinct from
      the actual world.
      
      Now apply the existence constraint.  Perhaps it is plausible to
      hold that there might have been a world distinct from our world.
      But the analysis cannot appeal to the truth of ``Bush lost the
      2000 electoral vote'' in $w$ if $w$ is something that doesn't
      exist, but might have.  So the constraint requires that there be
      a $w$ among the things that \emph{exist} that is a possible
      world, and is distinct from the actual world.
      
      Finally, apply the ontological modesty constraint.  We are
      antecedently inclined to think that there are no worlds other
      than the actual world.  This constraint, then, 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 \emph{possible}
      Republican nominee for president in 2012 without being a
      Republican nominee.  Why, then, can't something be a possible
      world without being a world?  The problem is that we are
      antecedently inclined to deny that there is an individual that
      might have been a world, but isn't.  The view we are considering
      faces a \emph{prima facie} problem, then, satisfying all three
      constraints.
      
      As the example illustrates, violations of these constraints can
      often be traded off for one another.  Modal modesty can
      sometimes be secured with ontological extravagance.  We could
      secure the truth of ``Bush might have lost the 2000 electoral
      vote,'' for instance, by admitting entities that might have been
      worlds, but are not.  Ontological modesty can sometimes be
      bought by allowing ourselves reference to non-existent
      worlds.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {This is the response favored by \citep{Fine:1985, Fine:2003a}.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3} And we could embrace the theory of possible
      worlds we have been discussing without ontological extravagance,
      if we could swallow the extreme modal extravagance of
      Spinozistic necessitarianism.  Then there would be only one
      possible world, the world we all inhabit.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {The difficulty of meeting all three constraints is underscored
        by the fact that, arguably, extant primitivist theories do
        not. For instance, some theorists propose to violate the
        constraint of existence by denying some instances of (SC),
        interpreted literally and at face value.  See, for instance,
        \citep{Fine:1985, Fine:2003a},
        \citep[pp. 17-8n.]{Salmon:1989}.  Other primitivst theories
        have been accused of accepting ontological extravagance.
        Bennett \citeyearpar{Bennett:2005} and Fine
        \citeyearpar{Fine:1985} argue that this is the case for
        Plantinga's view; see
        esp. \citep[pp. 116--7]{Plantinga:2003b}.  Bennett's
        \citeyearpar[p. 317]{Bennett:2005} discussion of this problem
        suggests that she proposes instead to pin a substantial modal
        commitment on end-users, violating modal modesty.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3}
      
      Of course, failure to meet one of the constraints does not
      constitute a decisive objection against a theory.  But this
      concession should not obscure the fact that a proposal that
      meets one of the constraints is to that extent preferable to one
      that does not.  And we can always hope for perfection: an ideal
      proposal that meets all of these
      constraints.
    \end{subsection}

  \end{section}
  
  \begin{section}{Interpreted Models and Modal Facts}
    We have seen that a common sense explanation of the notion of
    \emph{being a possible world} has difficulty meeting our three
    constraints.  If we want a theory of possible worlds that does a
    better job at meeting the three constraints, we should stop
    treating ``possible world'' as a bit of ordinary English on the
    model of ``possible Republican nominee.''  We should, instead,
    treat it as a technical theoretical notion.
    
    What interpretation, then, of the technical vocabulary ``possible
    world'' should we give?  Well, undoubtedly the enthusiasm for (SC)
    and other applications of possible worlds theories is explained in
    part by the success of Kripkean model theories for modal logics.
    This is where I propose to start looking.  I will assume that the
    sort of model which we will be dealing with is a
    Kripke-model.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Throughout this section, I assume that our uninterpreted formal
      language is a first-order modal language with quantification,
      identity, and individual constants.  This extends the treatment
      at \citep[pp. 84, 90]{Kripke:1963}, which omits the treatment of
      individual constants.  Such a language may be inadequate to
      express the full panoply of modal facts; see \S\ref{sec:XPress}
      below.\label{fn:lang}}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} For a given language, each such model has a model
    structure $\langle w, W, R \rangle$, with $W$ a non-empty set
    (thought of as a set of ``possible worlds''), $w$ a member of that
    set (thought of as ``the actual world''), and $R$ an accessibility
    relation on the set.  I will say that $x$ is an \emph{element} of
    a model $M$ iff $M$ is a model with model structure $\langle w, W,
    R \rangle$, and $x$ is a member of $W$. The model assigns a domain
    of objects to every one of its elements, extensions to every
    predicate-element pair, and some object from some domain of some
    element to every individual constant.  The domains of different
    elements of $W$ may overlap, and truth in the model is defined in
    terms of satisfaction in some standard way.  The view I propose
    is, roughly, that being a possible world is being an element of
    some model which does not contravene the modal
    facts.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {If it turns out that Kripke-models are not suitable to represent
      the modal facts, because, say, identity turns out to be
      contingent, then we will have to use some other sort of model
      instead, \emph{e.g.}, the counterpart-theoretic models of
      \citep{Lewis:1968} or \citep{Hazen:1979}.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
    
    
    \begin{subsection}{Interpreted Models}
      What are we to make of the idea of such a model ``contravening
      the modal facts?''  It might be thought that the very idea that
      a model, a mathematical entity of a formal model theory,
      contravenes the modal facts is confused.  Models don't have
      anything to say about modal facts.  The ``predicates'',
      ``names'', and ``sentences'' which are assigned values in a
      model are sequences of uninterpreted formal strings, defined in
      abstraction from any natural language in which we express facts.
      
      It is plausible to argue that we get contact with the modal
      facts only if we suppose that the model theory somehow gives the
      semantics of some fragment of, say, English, or at least an
      analysis of the facts reported by truths expressible in that
      fragment of English.  But accepting that the model gives the
      semantics of some fragment of English would require that we
      accept that the real truth-conditions of such English sentences
      as ``it is possible that Bush lose the 2000 electoral vote''
      should be given in terms of truth in some model.  It is surely
      implausible to suggest that these English sentences have
      truth-conditions involving such model-theoretic entities as the
      members of the set $W$ of some model.  
      
      This objection relies on the idea that that the only way model
      theory could bear on the modal facts is by providing a semantics
      or analysis for the English vocabulary used to state those
      facts.  Our present aim, however, is a theory of possible
      worlds, not a semantics or analysis of ordinary English
      vocabulary.  Primitivism opens the way to denying that the
      possibility that Bush have lost the 2000 electoral vote needs
      interpretation.  The modal facts reported by English sentences
      like ``Bush might have lost the 2000 electoral vote'' can look
      after themselves, and stand in no need of analysis in other
      terms; likewise, there is no pressing need to specify such a
      sentence's truth-conditions in terms of quantification over
      possible worlds.  Theories invoking possible worlds can be
      useful and insightful, even when they take the modal vocabulary
      of natural language as basic.  This is the point of view we will
      be exploring.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {Other primitivists might accept the notion of \emph{being a
          possible world} as an unanalyzed bit of modal vocabulary, and
        insist that other sorts of modal facts have truth conditions
        stateable in terms of this more basic notion.  The point of the
        paragraph is that the demand that possible-worlds theory provide
        an analysis in terms of possible worlds of facts reported in more
        familiar modal vocabulary can be resisted if you do not already
        accept that modal claims in general stand in need of analysis in
        non-modal terms.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3} 
      
      The original problem still confronts us.  How are we to explain
      the notion of a model's ``contravening the modal facts?''  We
      need a way to interpret models, so that they say something about
      the modal facts.  To start with, we will need a \emph{key}, a
      way of matching the predicates and names of the uninterpreted
      formal language with properties and entities.  The key will not
      provide the means of interpreting English claims or modal facts
      in terms of the extensional model theory.  Rather, the key
      provides the means of interpreting the sentences of the hitherto
      uninterpreted object language of the model back into English (or
      some other natural language which we could understand).  We
      don't use the model to help us understand what the English
      claims; we use the English to help us understand what the model
      ``claims.''
      
      Call a key \emph{acceptable} iff it maps `=' to identity,
      assigns a formal predicate to every property expressible in
      English,\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {This formulation may verge on paradox, if, for instance, there is
        such a property as \emph{being heterological}, and this property
        is expressible in English.  On its own, however, the formulation
        is not paradoxical.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3} and assigns an individual constant to every
      designee of an English name.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {If the restriction to English is deemed too parochial, we may
        stipulate instead that the key maps formal predicates to those
        individuals and properties picked out by \emph{any natural
          language whatsoever}.  A particular language in the class of
        formal languages I am treating (see n. \ref{fn:lang} above)
        may be specified by its stock of non-logical predicates and
        constants.  Thus, the set of languages interpreted by a given
        key $K$ can be read off of the stock of predicates and
        constants in the domain of $K$.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3} An acceptable key provides us with everything
      we need to interpret the formal language of a Kripke-model.  We
      can ``read out'' any sentence of a formal language whose
      predicates and terms are all mapped by the key to properties and
      entities expressible in English.  For instance, if $t$ is mapped
      to George Bush, and $F$ is mapped to the property designated by
      ``lost the 2000 electoral vote'', then ``$\lozenge Ft$'' can be
      read as ``it is possible that Bush lost the 2000 electoral
      vote.''
      
      Where a key maps one or more of the predicates or terms of a
      formal sentence to properties or entities not designated by any
      English expression, there will be no reading in English.  But
      the key does give us the means to recursively define an
      interpretation for the sentence.  The general idea is that open
      sentences are interpreted as complex predicates expressing
      relations, and then their closures under existential
      quantification are interpreted as expressing the claim that
      there are things that stand in the relevant relation.  For
      instance, ``$(Fx \wedge Gy)$'' is interpreted as the relation
      that any pair of objects stands in just in case the first member
      of the pair is $F$ and the second member of the pair is $G$.
      ``$(Fx \wedge Gx)$'' is interpreted as the 1-place relation that
      any object stands in just in case it is both $F$ and $G$.
      ``$(\exists x) (Fx \wedge Gx)$'' is interpreted as the claim
      that there is something that is both $F$ and $G$.  The
      interpretations for negations, conjunctions, and
      necessity-claims are given in the obvious
      way.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {See Appendix A for details.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3}
      
      Call the pair of a model $M$ and an acceptable key $K$ an
      \emph{interpreted model}.  We can now define the notion of an
      interpreted model's contravening the modal facts.  An
      interpreted model $\langle M, K\rangle$ contravenes the modal
      facts iff there is some $\phi$ true in $M$, but whose
      interpretation according to the key $K$ is not true.  Some of
      the interpretations of sentences true in a model $M$ will make
      straightforwardly modal claims.  For instance, there is an
      acceptable key according to which ``$\lozenge Ft$'' can be
      interpreted as the claim that it is possible that Bush lose the
      2000 electoral vote.  The truth, or lack thereof, of such a
      modal claim is itself a modal matter.  Hence it will be a modal
      matter whether $\langle M, K\rangle$ contravenes the modal
      facts.
      
      I will say that something is an \emph{element} of the interpreted
      model $\langle M, K\rangle$ if it is an element of $M$.  A precise
      statement of the thesis, then, is that being a possible world is
      being an element of an interpreted model $\langle M, K\rangle$
      which does not contravene the modal facts.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {Here and for most of what follows I will ignore complications
        required to take account of the accessibility relation $R$.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3} There is a corresponding notion of what it is for
      there to be a possible world at which some claim is true: roughly,
      there is a world at which $S$ is true iff a formal sentence that
      $S$ interprets is true at some element of some interpreted model
      which does not contravene the modal facts.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {See Appendix B.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3} Call this the \emph{interpreted models
        conception} of possible worlds.
    \end{subsection}
    
    \begin{subsection}{The Three Constraints Satisfied}
      This explanation of the notion of \emph{possible world} appears
      to meet the three constraints.  Possible worlds exist if the
      model-theoretic entities of the requisite sort do.  The
      existence constraint seems thereby to be met.  At the very
      least, we pass the buck for meeting the existence constraint for
      possible worlds to meeting a similar existence constraint for
      the model-theoretic entities.  If one thinks that model theory
      is a useful fiction, for instance, then the present view may not
      meet the existence constraint.  Taking the model theory at face
      value, however, secures the existence of possible worlds on the
      interpreted models conception.
      
      The interpreted models conception is also designed to be modally
      modest.  Notice that, unlike some possible worlds theories, what
      it takes to be a possible world is explicitly explained in terms
      of the modal facts -- exactly the sorts of facts reportable by
      instances of (SC)'s left-hand side.  In essence, the present view
      reverses the interpretation of (SC) favored by those who propose
      to use it to analyze modal facts in terms of possible
      worlds.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {Here I principally have in mind the theories of
        \citep{Carnap:1947} and \citep{Lewis:1986}.  Thanks to an
        anonymous referee.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3} On the interpreted models conception, what it
      takes to be a possible world is explained in terms of what might
      have been the case.  Thus, the modal facts determine which
      possible worlds there are.
      
      Finally, consider the requirement of ontological modesty.
      Possible worlds on the interpreted models conception are just
      whatever sort of entities are appealed to in ordinary,
      mathematical model theories.  Their metaphysical status and
      nature are no more extravagant than any other entities appealed
      to in model theory, for instance, of first-order logic.
      Importantly, we make appeal to no such entities as, \emph{e.g.},
      Barack Obama's possible seventeenth granddaughter.
    \end{subsection}
    
    \begin{subsection}{Menzel's Theory}
      The interpreted models conception is close in spirit to one
      offered by \citep{Menzel:1990}.  Though the views differ on many
      details, they share a common focus on the interpretation of
      models.  Nevertheless, the view I have described enjoys two
      advantages over Menzel's proposal, one philosophical, one
      technical.  The philosophical advantage first: Menzel repudiates
      the constraint of existence, claiming that there are no
      non-actual possible worlds.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {\citep{Chihara:1998} also has a similar view, though his view
        may fail to meet modal modesty, by requiring the truth of all
        axioms of S5.  For an argument that the axioms of S5, even if
        true, are not logically true, see \citep{Salmon:1989}.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3} From the present point of view, this rejection
      of possible worlds is both regrettable and unnecessary.  It is
      regrettable because it requires rejecting most instances of
      (SC), implying that the uses made of possible worlds by
      end-users must be rejected if they are to be taken at face
      value.  Rejecting possible worlds is unnecessary because the
      interpreted models conception shows how they can be accommodated
      without sacrificing modal or ontological modesty.
      
      Menzel's rejection of possible worlds might be defended by
      claiming that the uses of apparent quantification over possible
      worlds in end-users' theorizing is a mere \emph{fa\c{c}on de
        parler}.  According to the defense, end-users aren't really
      committing themselves to the truth of instances (SC) and the
      corresponding existence of possible worlds; what they say can be
      unpacked in such a way that what it expresses are the very
      model-theoretic facts in virtue of which possible worlds exist
      and satisfy instances of (SC) according to the interpreted
      models conception.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {See (ISC) in Appendix B below for a concise statement of the
        relevant correspondence between the modal and model-theoretic
        facts.  Thanks to an anonymous referee for suggesting this
        avenue of response.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3} Despite appearances, for instance, end-users
      don't \emph{really} commit themselves to the claim that it is
      possible that Bush lost in 2000 iff there is a possible world at
      which `Bush lost the 2000 electoral vote' is true; they only
      commit themselves to the claim that it is possible that Bush
      lost in 2000 iff there is an interpreted model of the
      appropriate sort that fails to contravene the modal facts.
      
      We may differ about how serious a cost in plausibility is
      incurred by this proposed reinterpretation, but there is little
      doubt that there is some cost.  End-users certainly seem to be
      committing themselves to the truth of (SC).  It is better to
      interpret their theories at face value unless there is
      compelling reason to do otherwise.  But, I have argued, the
      interpreted models conception shows that neither modal nor
      ontological modesty provide any such reason.  Since there's no
      need to reinterpret end-users' apparent quantification over
      possible worlds, it's better to take it at face value.  In this
      respect, then, the interpreted models conception has an
      advantage over any view which, like Menzel's, denies the
      existence of possible worlds.
      
      Now for the technical advantage.  Associated with each possible
      world in a Kripke-model for a modal language is a Tarski-style
      model for a non-modal predicate calculus.  Each world has a
      domain over which quantifiers range, and corresponds to an
      assignment of extensions to all of the predicates and
      terms.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {I am ignoring for the purposes of the present discussion the fact
        that a Kripke-model may assign to a term a value outside the
        domain of a given world.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3} Menzel suggests, in effect, that such a
      Tarski-model represents a possibility in virtue of a certain
      modal fact: the model has an interpretation on which it could
      have been an accurate representation of the way things
      are.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {See esp. \citep[p. 371]{Menzel:1990}.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3} For instance, a Tarski-model which has an
      interpretation on which Gore is in the extension of ``won the
      2000 electoral vote'' could have been an accurate representation
      of how things are; if Gore had won, that model, so-interpreted,
      would have been accurate.  But now there is a problem: some
      modal claims themselves concern further possibilities.  Take,
      for example, the essentialist claim that \emph{humanity} is a
      necessary feature of anything that has it; and, further, that
      this fact is not just contingent, but necessary.  Imagine now
      two Tarksi-models.  One of those models (on a given
      interpretation) represents a possibility in which there are more
      humans than there actually are, by including an entity in the
      extension of ``human'' which does not represent any actual human
      being, though it could have represented some non-actual human
      being.  The other model (on the same interpretation) represents
      a possibility in which there are more turnips than there
      actually are by including the very same entity in the extension
      of ``turnip.''  Both models could have been accurate
      representations of how things are non-modally.  But they don't
      ``fit together'' on our essentialist assumption, since, taken
      together, they jointly represent the claim that there could have
      been something which is contingently
      human.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {For Menzel's full treatment of this and related problems, see
        \citep[pp. 373--80]{Menzel:1990}.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3} So, on pain of modal extravagance, Menzel needs
      a way to ``fit together'' the interpreted non-modal models so
      that these kinds of essentialist claims are accommodated.  The
      problem engenders quite a bit of complication in Menzel's
      proposal.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {See, for instance, the condition $(C_{nm})$, characterized as
        ``admittedly hairy'' by Menzel, on
        \citep[p. 379]{Menzel:1990}.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3}
      
      This problem is avoided entirely by the interpreted models
      conception.  The starting point for the interpreted models
      conception are interpreted Kripke-models whose embedded
      Tarski-models collectively are an accurate representation of
      modal reality.  Assuming that modal reality can be modeled using
      the apparatus of possible worlds, the ``fit'' among the embedded
      Tarski-models required to accommodate essentialist claims is
      thereby guaranteed without further complication.  If an
      interpreted model contravenes some essentialist truth, then its
      elements are not possible worlds on the interpreted models
      conception.  Thus, the interpreted models conception avoids the
      complications attending Menzel's proposal.
    \end{subsection}
  \end{section}
  
  \begin{section} {\emph{Being a Possible World} is a relational condition}
    \label{sec:relational}
    A couple of problems for the interpreted models conception suggest
    themselves immediately.  On the interpreted models conception,
    \emph{being a possible world} is a relational condition.
    Something is a possible world only if there is an interpreted
    model to which it bears a certain relation.  This gives rise to
    two problems.
    
    \begin{subsection}{Is Everything a Possible World?}
      First, since model theories are typically silent on the underlying
      nature of the entities they quantify over, there seems to be no
      bar to any individual's being a possible world.  If there is any
      interpreted model at all that fails to contravene the modal facts,
      then for any given individual there is an interpreted model that
      fails to contravene the modal facts and has that individual as an
      element.  So Bush is a possible world, as are you and I, the
      Antarctic ice sheet, the square root of 2, \emph{etc.}.  Some
      commentators have suggested that this presents a problem for, or
      even a refutation of, the interpreted models
      conception.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {The need to address this problem was pressed independently by
        Michael Fara and Harold Hodes.}%
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3}
      
      The fact that the interpreted models conception implies that
      everything is a possible world would indeed pose a problem, if it
      were intuitively evident that some things are not possible worlds.
      Some theories,\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {See \citep{Lewis:1986}, \citep{Plantinga:1973}, and
        \citep{Stalnaker:1976}.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3} together with evident facts, recommend this
      opinion.  And perhaps some theorists would be surprised to
      discover that absolutely everything is a possible world.  But no
      one should think it intuitively evident that some individuals are
      not possible worlds.  Once we have convinced ourselves that the
      notion of a "possible world" is a technical notion, we should be
      no more surprised to discover that everything is a possible world
      than we are surprised to discover, \emph{e.g.}, that everything is
      an element of some boolean algebra.
      
      Still, it is important to distinguish the technical claim that
      everything is a possible world from an apparently non-technical
      claim expressed by similar verbiage.  There are, we have seen,
      ordinary English constructions of a form apparently similar to
      "possible world" that make perfectly good sense, \emph{e.g.},
      "possible 2012 Republican presidential nominee".  Suppose we
      take "possible world" as a bit of ordinary English on this
      model.  So-interpreted, the claim, "everything is a possible
      world" is false, since, for instance, Bush is not such that it
      is possible that he be a world.  The interpreted models
      conception does not say otherwise.  I have only claimed that
      this is not the correct way to take possible worlds claims as
      they occur in theorizing about, \emph{e.g.}, the metaphysics of
      mind or morality, epistemology, linguistics, \emph{etc.}  I
      think it would pose a serious problem for a theory if it implied
      this ordinary English claim; but the interpreted models
      conception does not.
      
      Some theorists will still object to the deflationary upshot of
      the interpreted models conception.  According to the interpreted
      models conception, a possible world is just any old thing (or
      any old pure set) that plays a certain role in a certain kind of
      model.  Surely, the objection goes, we have an intuitive, if
      somewhat inchoate, conception of what possible worlds are: they
      are complete \emph{ways things could have been}.  This intuitive
      conception motivates the endorsement of (SC); it is what makes
      the further theorizing by end-users seem promising.  But the
      interpreted models conception does not underwrite this intuitive
      conception.  So, the objector concludes, the interpreted models
      conception renders (SC) unattractive and fails to capture what
      seemed so promising about theorizing that uses possible
      worlds.
      
      I myself have no very strong attachment to the conception of
      possible worlds as complete ways things could have been.  Does
      rejecting that admittedly popular conception in favor of the
      interpreted models conception rob possible worlds of their
      utility for theorizing by end-users in other fields?  To my
      mind, the answer is clearly `no': the interpreted models
      conception does not undermine the utility of possible worlds for
      theorizing by end-users.  In my view, what makes (SC) attractive
      is that it has been theoretically useful, if not for giving a
      reduction of modal facts or modal discourse, then for a host of
      other enterprises.  The usefulness of possible worlds for such
      theorizing does not depend on giving an account of their nature
      that verifies the particular conception to which the objector
      appeals.
      
      Still, it might be contended that if there are any worlds at
      all, there must be \emph{some} feature that demarcates them from
      non-worlds, even if there's no reason to cleave to any
      particular conception of their nature.  That is, if there are
      any possible worlds, then there must be some feature that's
      distinctive of them, and so possessed by all the worlds and
      lacked by at least some non-worlds.  Metaphysicians differ on
      the nature of possible worlds, but they have until now agreed
      that there is some feature shared by possible worlds and lacked
      by non-worlds.  As I have noted, however, there is no such
      feature on the interpreted models conception.  Thus, the
      objection goes, the interpreted models conception is a view on
      which there aren't \emph{really} any possible worlds; there is
      only a certain model-theoretic \emph{role} -- a role which
      anything can play.  For this reason, the objector concludes, the
      interpreted models conception does not satisfy the existence
      constraint after all.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {Thanks to an anonymous referee for suggesting the need to
        respond to these last two concerns.}%
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3}
      
      The objection relies on the contention that the existence of
      possible worlds requires there to be a feature which possible
      worlds have and which some non-worlds lack.  There is no reason
      to accept this contention.  On the interpreted models conception
      being a possible world is a relational condition.  It does not
      follow that there are no worlds.  Being not made of phlogiston
      is also a relational condition, and one that is satisfied by
      everything.  We should not conclude on this basis that there
      aren't \emph{really} any things not made of phlogiston, nor that
      there is only a certain scientific \emph{role} -- a role which
      anything can (and does) play.  Similarly, the fact that being a
      possible world is a relational condition does not imply that
      there aren't \emph{really} any worlds.  It does imply that there
      is no ``intrinsic nature'' that makes something a possible
      world.  It also implies that possible worlds are not an
      identifiable sub-category of entity.  Thus, there is less to say
      about the distinctive nature of possible worlds in the
      interpreted models conception than there is on other theories of
      possible worlds.  But any theory of possible worlds will have as
      an upshot that all competing theories of possible worlds are
      wrong in some respect, so there's no special grounds for
      rejecting the interpreted models conception on this score.
    \end{subsection}
    
    \begin{subsection}{Relations Among Worlds}
      The fact that being a possible world is a relational condition
      also introduces a second problem, encountered when attempting to
      discern the features of possible worlds.  We may want to know
      whether a certain relation obtains among worlds.  For instance,
      we might want to know whether there are at least two possible
      worlds.  There will be two ways to take this question on the
      interpreted models conception.  On one way of taking this
      question, a `yes' answer requires that there be a single
      interpreted model with at least two elements.  On another way of
      taking this question, a `yes' answer can also be given when
      there are two interpreted models each with only a single
      element.  This means that sometimes questions about the features
      of possible worlds lack a single, determinate answer, contrary
      to what we might have hoped or expected.
      
      It seems to me that, though we must acknowledge that this is a
      problem, we should not make too big a deal out of it.  We
      encounter a similar problem with respect to other relational
      conditions.  For instance, being a parent, like being a world, is
      a relational condition.  Someone is a parent only if there is an
      individual to whom she bears a certain relation.  There will
      sometimes be a problem when attempting to discern features of
      parents.
      
      Imagine, for instance, a village in which every pair of parents
      has only one child, and that each member of each pair always
      agrees with the other.  We can take the question, ``Do parents
      ever disagree?'' in two ways.  On one way of taking it, the
      \emph{coordinated interpretation}, the answer is `no', since
      parents of a given child always agree.  On the other way of
      taking the question, the \emph{uncoordinated interpretation},
      the answer may still be `yes', since parents of different
      children may disagree.  Also, sometimes on the coordinated
      interpretation of a question about the features of parents, the
      right first response to a yes-no question is, ``it depends.''
      Suppose, for instance, that a new family moves into the village,
      and that the parents of this family disagree on some matters.
      Consider the question, ``Do parents disagree?''  On the
      coordinated interpretation of this question, the right first
      response is, ``it depends on which parents are under
      discussion.''
      
      The problem of disentangling the coordinated and uncoordinated
      interpretations of these questions is handled without
      significant misunderstanding in most cases.  Where there is
      misunderstanding, it can usually be corrected by delineating
      which interpretation is salient.  A similar strategy should work
      for problems understanding claims regarding the features of
      possible worlds.  In the discussion that follows, I always
      intend the coordinated interpretation of questions about worlds;
      that is, I intend to address questions about whether there is a
      single interpreted model in which worlds have the requisite
      features.
      
      % Let's start with a question that was briefly raised in note
      % \ref{fn:BushIsAWorld} above.  Nothing I have said so far rules out
      % the claim that Bush is a possible world.  Though I have always
      % thought of the elements of models as pure sets, model theory is
      % typically silent on the particular nature of the objects it
      % quantifies over.  So model theory does not imply that Bush is not
      % an element of a model.  And, if he is the element of some model or
      % other, and there are any models which fail to contravene the modal
      % facts, then Bush will be an element of some model which does not
      % contravene the modal facts.  On the present proposal, this is all
      % it takes to be a possible world, so Bush is one.
      % 
      % So far, so good.  Once we realize that possible worlds aren't
      % worlds in the colloquial sense (\emph{i.e.} aren't planets, or
      % universes, or spacetimes) and also aren't things that could be
      % worlds in that sense, the only thing that stands in the way of
      % Bush's being one is explaining how (SC)'s truth could be secured
      % using him.  In this respect, the features of the models do most of
      % the explanatory work; the worlds of the model just serve as points
      % or indexes of evaluation, so the features they have independently
      % of the models are moot.
      % 
      % But there is a residual question.  Is Bush a possible world at
      % which ``Bush does not exist'' is true?  Suppose that I am correct to
      % suggest that at least one interpreted model does not contravene
      % the modal facts, and suppose with common sense that Bush's
      % possible nonexistence is among those facts.  Then there is a world
      % in this model at which ``Bush does not exist'' (roughly)
      % interprets some formal sentence true at some world of the model.
      % Then, assuming that Bush is a possible world, there will be a
      % model such that, ``Bush does not exist'' is true in the model at a
      % certain possible world, namely Bush.  There will also be a model
      % in which ``Bush does not exist'' is not true at Bush.  
      % 
      % Where does this leave our original question?  Well, it seems to me
      % that there are two natural interpretations of the question in the
      % present framework.  On one natural interpretation, broached in
      % note \ref{fn:BushIsAWorld}, the right answer is ``yes,'' because
      % there is a model of the requisite sort.  On another natural
      % interpretation, the right answer is ``it depends,'' since there
      % are models of the right sort, but also other models of the wrong
      % sort.  The second interpretation, according to which ``it
      % depends'' is the right answer to the question, is more natural
      % than the first by my lights.  The first interpretation might,
      % perhaps, be more natural if we asked instead, ``Is Bush
      % \emph{ever} a possible world in which ``Bush does not exist'' is
      % true?''  Whichever interpretation is best, I hope that the
      % underlying facts of the situation are clear.
      % 
      % Is it an objectionable feature of the interpreted models
      % conception that it allows that (at least in some models), Bush is
      % a possible world at which ``Bush does not exist'' is true?  I
      % believe that it is not objectionable, so long as we remember two
      % things.  First, a sentence's truth at an element of a model has
      % nothing to do with the features that element has independently of
      % the model.  Rather, the sentence's truth at that element is
      % determined by how the model represents things as being at that
      % element.  Second, once we acknowledge that \emph{being a possible
      % world} is a technical notion in need of explanation, we should no
      % longer be inclined to think we have much in the way of
      % pre-theoretic intuitions about them.  Thus, whatever features they
      % turn out to have on our explanation, so long as we secure the
      % truth of (SC) and meet the three constraints, we shouldn't worry
      % so much that the explanation produces results that would be
      % counter-intuitive according to some pre-theoretic beliefs about
      % what possible worlds are supposed to be like.  For my own part, I
      % simply have no such pre-theoretic beliefs, any more than I have a
      % pre-theoretic beliefs regarding the nature of ultrafilters.
      
      % How many worlds?
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      If there are any interpreted models which do not contravene the
      modal facts, then there will be non-isomorphic models which do not
      contravene the modal facts.  Some of these models will deliver
      different answers to the question, ``How many possible worlds are
      there?''\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {For a quick and dirty illustration, assume that $\langle w, W, R
        \rangle$ is the model structure for some interpreted model $M$
        which does not contravene the modal facts.  Just add to $W$ all
        the members of some strictly larger set, making sure that in the
        new model, each new element is a ``copy'' of some old element,
        along the lines suggested in the next paragraph.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3} The appropriate response to this difficulty is
      to acknowledge it, but not to overestimate its importance.
      Suppose that we are given two interpreted models, neither of
      which contravenes the modal facts, but which contain different
      numbers of possible worlds.  Whatever else there may be to say
      about such models, they agree in all respects on those aspects
      of modal reality which they model.  A difference on the question
      of how many worlds there are appears to make no difference to
      the correct description of what might have been the case.
      
      % indiscernible worlds
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      The view at hand makes short work of the question of whether there
      are distinct worlds that are indiscernible with respect to which
      claims are true in them.  Suppose we are given an interpreted
      model which does not contravene the modal facts.  Now just create
      a new model by adding an element $w^*$ to the set of worlds of the
      old model, so that $w^*$ is a ``copy'' of some world $w$ in the
      old model.  The extensions of terms and simple predicates in $w^*$
      are exactly the same as in $w$, and exactly the same worlds are
      accessible from, and to, $w^*$.  The new interpreted model will
      verify exactly the same claims as the old, and so will not
      contravene the modal facts.  So there is a model in which there
      are distinct worlds that are indiscernible in the relevant sense.
      Some may see this as a problem with the account.  I think of it
      instead as an interesting (and optional)\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {We can rule out distinct, indiscernible possible worlds by
        requiring that every pair of worlds differ with respect to
        domain or assignment of extensions to non-logical vocabulary.}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3} feature.  
      
      In summary, these questions involving the features of possible
      worlds pose no serious problems.
    \end{subsection}
  \end{section}
  
  \begin{section} {Expressibility Problems}
    
    \label{sec:XPress}
    One problem for the interpreted models conception involves
    limitations on which modal facts can be expressed using primitive
    modal vocabulary.  A language containing quantifiers over worlds
    is expressively more powerful than a language containing only
    sentential modal operators.  In particular, it is possible to
    express certain relations between worlds in a language with
    quantifiers over worlds that cannot be expressed in QML.  For
    instance, one cannot express in QML the claim that there is a
    world $w^\prime$ such that every world $w$ contains some
    individual also contained in $w^\prime$.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {\citep{Hazen:1976}}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Let's suppose, for simplicity, that all modal
    claims stateable in a primitive modal vocabulary are representable
    in QML.  Then the primitive modal vocabulary has less expressive
    power than a language containing quantifiers over worlds. I have
    been suggesting, in effect, that this extra expressive power is
    wasted.  Insofar as such questions are not settled by those modal
    facts expressible in a primitively modal language, we need not
    worry too much about which answers come out.  
    
    We have supposed that all modal claims stateable in a primitive
    modal vocabulary are representable in QML.  But some theorists
    have argued, independently of any theory of possible worlds, that
    there is more to modal reality than can be said using the
    sentential modal operators, the ``boxes and diamonds'',
    characteristic of QML.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {\citep[pp. 13-17]{Lewis:1986}.} % ALSO: WIGGINS
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Let's suppose now that these theorists are
    correct.  There will then be another failure of expressive power
    that cannot be so quickly dismissed.  Consider for illustration
    the argument of \citep[pp. 13-17]{Lewis:1986} that some
    supervenience claims cannot be expressed in the idioms
    representable in QML.  Let's focus in particular on the claim that
    psychological facts supervene on physical facts.  Lewis gives a
    vague but intuitive gloss of this claim in explicitly modal terms:
    there could be no psychological difference without a physical
    difference.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Let's ignore the problem of saying what counts as a physical
      difference.  Let's also ignore the interesting differences
      between alternative modal characterizations of supervenience
      relations; see \citep{Teller:1984}.  The comments in the main
      text should apply equally well to all of the alternatives.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} He argues that this ``no difference'' claim is
    not implied, and hence not adequately represented, by the claim we
    get using only a sentential modal operator: ``it is not possible
    that two things should differ with respect to their psychological
    properties but not with respect to their physical properties''.
    (The problem is that the ``no difference'' claim Lewis intends
    fails if there are two individuals of different worlds which are
    psychologically different but physically the same, while the
    proposed ``box and diamond'' gloss need not fail.)  Rather than
    focus on whether someone might offer an alternative ``box and
    diamond'' gloss of the ``no difference'' claim that more
    adequately captures its import, let's just suppose that Lewis is
    right: there are two interpreted models that do not contravene the
    modal facts in the sense under discussion, and so agree on all the
    ``box and diamond'' modal facts, but that differ with respect to
    other modal facts.  In particular, let's suppose that there are
    two such interpreted models which differ with respect to
    supervenience facts.
        
    This supposition gives rise to a serious objection.  I have been
    claiming that many questions regarding worlds, including how many
    there are, whether there are distinct but indiscernible worlds,
    and so on, are to be answered, ``it depends on which interpreted
    model the worlds are in.''  I have suggested that there is no
    pre-theoretical reason to believe that this is the wrong way to
    respond.  But now consider the question of whether psychological
    facts supervene on physical facts.  Surely, the objector urges,
    the question of whether the supervenience relation holds is a
    matter of modal fact.  We may not know the right answer, and there
    may be difficulties interpreting the claim, but at least we know
    that ``it depends on the model'' is wrong.  According to this
    objection, our proposed explanation of the notion of a possible
    world does not meet the constraint of modal modesty: it requires
    that we give an evidently incorrect response to questions
    regarding supervenience relations.
    
    The objection relies the assumption that there are two models
    which do not ``contravene the modal facts'' in the sense we have
    given to that expression, but which deliver different answers to
    supervenience questions.  This assumption requires that each of
    our two interpreted models \emph{delivers an answer} to
    supervenience questions, and that for some such questions, they
    deliver different answers.  An interpreted model can only deliver
    an answer to a supervenience question if there is some way of
    interpreting that question in terms of some feature of the
    interpreted model.  So the objection assumes what we might call
    the \emph{interpretability of supervenience}: that there is a way
    of interpreting supervenience questions in terms of some feature
    of the interpreted model.
    
    For instance, an interpreted model might be thought to deliver a
    ``no'' to the question of whether psychological facts supervene on
    physical facts in virtue of containing two worlds which assigned
    the same extensions to predicates mapped by the key to physical
    properties, but assigned different extensions to predicates mapped
    by the key to psychological properties.  Another interpreted model
    might be though to deliver a ``yes'' answer to the same question
    in virtue of containing no such pair of
    worlds.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {This would be a plausible interpretation only for a particularly
      weak brand of supervenience, which has never been explicitly
      advanced to the best of my knowledge.  This toy interpretation
      is a mere illustration.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} But if models are interpretable in this way, then
    we have evidently drawn the notion of ``contravening the modal
    facts'' too narrowly: we need only augment the explanation of that
    notion with a specification of how a model can contravene
    supervenience facts in addition to contravening ``box and
    diamond'' facts.  If supervenience facts can be interpreted in any
    possible worlds theory in terms of which individuals have which
    properties in which worlds, then the interpreted models conception
    can use that interpretation to restrict the class of models which
    do not contravene the modal facts.  And then the original
    explanation of the notion of a possible world as an element of an
    interpreted model which does not contravene the modal facts will
    not have the untoward consequence that there are two such models
    which differ on matters of supervenience.
    
    Suppose we reject the interpretability of supervenience:
    interpreted models just have nothing to say about whether there
    could be a psychological difference without a physical difference.
    It seems, then, that our theory of possible worlds is ill-suited
    to modeling supervenience claims.  But notice that any interpreted
    model provides the means of saying which individuals have which
    properties in which worlds.  For instance, an interpreted model,
    together with a specification of which properties are physical and
    which are psychological, provides the means of determining whether
    there is anything which has exactly the same physical properties,
    but different psychological properties in two worlds of the
    interpreted model.  In this respect, our interpreted model can say
    as much about what possible worlds there are and what their
    denizens are like as any other possible worlds theory.  So,
    rejecting the interpretability of supervenience implies that
    supervenience claims are not interpretable in terms of any theory
    of possible worlds.  If this is a strike against the interpreted
    models conception, it is equally a strike against any other
    theory of possible worlds.
    
    The point generalizes.  Any aspect of modal reality is either
    interpretable in terms of features of worlds and what goes on in
    them or it is not. If it is interpretable in such terms, then we
    can use the interpretation to express a constraint that a model
    must meet in order not to contravene the modal facts.  If it is
    not interpretable in such terms, then this is a weakness of
    possible-worlds approaches generally, and so provides no reason to
    favor some alternative theory of possible worlds over the one we
    have been discussing.
    
  % \end{section}
  % 
  % \begin{section} {Satisfiability Problems}    
  %   \label{sec:Sat} The objection regarding the interpretability of
  %   supervenience showed that the notion of contravening the modal
  %   facts that has been developed here is perhaps not restrictive
  %   enough, so too many interpreted models will fail to contravene the
  %   modal facts, in our sense.  But we might also have the opposite
  %   worry: that not even one model will meet the constraint.  If every
  %   interpreted model contravenes the modal facts, then there are no
  %   possible worlds, according to the interpreted models conception.
  %   This would render the view unfit for simultaneously meeting
  %   existence and modal modesty.
  %   
  %   If we were sure that the modal facts were reportable by a
  %   consistent set of sentences amenable to treatment in first-order
  %   QML, then a completeness proof would warrant the assumption that
  %   there is at least one interpreted model which does not contravene
  %   the modal facts.  But there are reasons for skepticism.
  %   
  %   % One reason stems from the worry that modal reality cannot be
  %   % modeled using a set of possible worlds.  On one version of this
  %   % worry, there are ``too many'' possible worlds to fit into a single
  %   % set.\linespread{1}\footnote
  %   % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   % {[CITE!] Pollock, Fine}
  %   % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   % \linespread{1.3} Since every model comes with a set of possible
  %   % worlds, every model and key would contravene the modal facts, if
  %   % this is among the facts which are not to be contravened.  From the
  %   % present point of view, however, it is not clear how compelling the
  %   % claim that there are too many possible worlds to form a set should
  %   % be.  Until we know how the technical notion of \emph{being a
  %   % possible world} is to be explained, we should be agnostic on the
  %   % question of whether there are too many to form a set.  On the view
  %   % of possible worlds here urged, there is no reason to suppose that
  %   % there are so many.
  %   
  %   One reason stems from the fact that some plausible theses about
  %   the relations between propositions and possible worlds lead to
  %   paradox.  David Kaplan,\linespread{1}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {\citep{Kaplan:1995}}
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   \linespread{1.3} for instance, notes that
  %   \begin{description}
  %     \item[(A)] $(\forall p) \lozenge (\forall s) (Qs \Leftrightarrow
  %       p = s)$ 
  %   \end{description}
  %   , for `$p$' and `$s$' propositional variables, and `$Q$' a
  %   non-logical sentential operator is unsatisfiable on standard
  %   semantic assumptions.  On those assumptions, possibility is
  %   interpreted in accordance with (SC) as requiring the existence of
  %   at least one world of a certain sort.  (A)'s truth would thus
  %   require that, for every proposition, there is at least one world
  %   where it and no other proposition has some property $Q$; it and it
  %   alone, for instance, is uttered.  This requires that there be at
  %   least as many worlds as propositions; if there were fewer worlds
  %   than propositions, then some proposition or other would be left
  %   without a world in which it and no other proposition has $Q$.
  %   Kaplan has argued that, when we do intensional logic, we have
  %   reason to believe that, to every set of possible worlds, there is
  %   a proposition which is true at exactly those possible
  %   worlds.\linespread{1}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {As Kaplan notes, this does not imply that it be possible to
  %   express all such propositions in English or any other single
  %   language.}
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   \linespread{1.3} Hence there are at least as many propositions as
  %   sets of worlds.  (A) is therefore unsatisfiable given the standard
  %   semantical treatment: its satisfaction requires that there be at
  %   least as many worlds as propositions; we have independent reason
  %   to believe there are at least as many propositions as sets of
  %   worlds; and there can't be at least as many worlds as sets of
  %   worlds.
  %   
  %   Thus, if the interpretation of (A) on some key is true, then,
  %   holding the standard semantic assumptions fixed, the modal facts
  %   cannot be modeled using a possible-worlds model theory for a
  %   language which includes (A).  There will be no possible worlds,
  %   according to the theory of possible worlds on offer
  %   here.\linespread{1}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {Technically, there will still be possible worlds according to the
  %   letter of the theory under discussion, since (i) QML does not
  %   allow the use of propositional quantifiers, nor such sentential
  %   operators as $Q$; and (ii) there is perhaps no property of
  %   propositions expressible by a sentential operator of English which
  %   would yield a true interpretation of (A).  Thus, an acceptable key
  %   could fail to map $Q$ so as to yield a true interpretation of (A).
  %   But these technicalities are small comfort, given that we can and
  %   should extend the notion of a model's contravening the modal facts
  %   to treat non-logical sentential operators, and, if there is any
  %   property of propositions which behaves as (A) requires, then we
  %   could easily extend English to have a sentential operator which
  %   picks that property out.}
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   \linespread{1.3} This does not show that there are no possible
  %   worlds, but it shows that there are conditions under which there
  %   would be no possible worlds, according to the interpreted models
  %   conception.  More seriously, it shows that the possible worlds
  %   theory put forward here is not modally modest: if we add those
  %   standard semantic assumptions to our theory, then (SC) can be used
  %   to yield the falsity of any interpretation of (A).
  %   
  %   This is not the place to attempt to say what the appropriate moral
  %   of Kaplan's argument is.  It is enough for now to note that the
  %   assumptions that are required to generate the unsatisfiability of
  %   (A) are really quite minimal.  The problem, then, stems not so
  %   much from some peculiarity of the explanation offered here, as
  %   from the general commitment to the claim that there are at least
  %   as many propositions as worlds, and the standard satisfaction
  %   conditions (including (SC)) for formal sentences like (A).  To the
  %   extent there is some reason to believe that the theory offered
  %   here cannot accommodate the truth of some interpretation of (A),
  %   there is similar reason to believe that there is no theory of
  %   possible worlds which can both secure the truth of every instance
  %   of (SC), and accommodate the truth of some interpretations of (A),
  %   if we hold those standard semantic assumptions fixed.  Kaplan,
  %   then, has posed a challenge generally to any possible worlds
  %   semantics for (A), according to which no such semantics is modally
  %   modest.  If the argument is correct, then of course the present
  %   view is not modally modest with respect to interpretations (A),
  %   but then neither is any other extant view.\linespread{1}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {Kaplan gives a sketch of the sort of view which might avoid the
  %   result: a hierarchical type-theory for worlds.  I will not pause
  %   here to develop or assess the suggestion.}
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   \linespread{1.3}
        
    % To summarize: the interpreted models conception holds that
    % possible worlds are elements of interpreted models which do not
    % contravene the modal facts.  There is a possible world at which
    % $\phi$ is true iff there is an interpreted model in which $\phi$
    % is the interpretation of some formal sentence which is true at
    % some world in the model.  This conception unabashedly reduces
    % facts about possible worlds to modal facts; on this score it is no
    % worse off than any other non-reductive possible worlds theory.  On
    % the interpreted models conception, possible worlds are among the
    % things that there are: they actually exist.  The view is
    % ontologically modest to the extent that talk of relations and
    % model theoretic entities is ontologically modest.  In particular,
    % the view requires no primitive quantification over things that
    % might have existed, but do not actually exist.  The view is as
    % modally modest as a possible worlds theory can be.  A complicating
    % feature of the view is that often the right first response to
    % questions regarding the features of possible worlds is, ``it
    % depends.''  This presents no difficulty in principle, since we
    % take ``possible world'' to be a technical term, and so should have
    % no pre-theoretic intuitions about the nature of possible worlds
    % which such a result would contradict.
    
    % These results suggest that the interpreted models conception is a
    % contender.  It's not the only contender; there are other
    % non-reductive explanations of what it takes to be a possible world
    % available.  If they do at least as well at meeting the constraints
    % as the view we have been discussing, then we have no reason
    % favoring one view over the other.  A consideration of all
    % competing views would be piecemeal, so I won't aspire to
    % completeness here.  Instead, I will consider in some detail one
    % very plausible primitivist explanation of the notion.  Even if the
    % argument is successful, then, I will not have established that the
    % view I am defending is the best of all possible primitivist
    % theories of possible worlds.  I will be content if I can show that
    % it is better than a very plausible competitor.
    
  \end{section}
  
  % \begin{section} {Another Primitivist Proposal}
  %   \label{sec:propBasedConception}
  %   According to the interpreted models conception, possible worlds
  %   are a certain kind of model-theoretic object.  A more typical
  %   primitivist strategy has been to claim that possible worlds are
  %   state- or property-like entities.  Possible worlds are maximal
  %   states of affairs that either do obtain or \emph{could have}
  %   obtained \citep{Plantinga:1973}; possible worlds are complete
  %   \emph{ways things might have been} \citep{Stalnaker:1976};
  %   possible worlds are complete possible \emph{states} or
  %   \emph{histories} of the world \citep[p. 17]{Kripke:1980}.  Call
  %   this family of explanations the \emph{property-based conception}
  %   of possible worlds.\linespread{1}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {I will make no attempt to further specify in detail the
  %   metaphysical similarities and differences among these views.  For
  %   present purposes, it would not matter if it turned out that the
  %   only things that all versions of the property-based conception
  %   have in common are modal primitivism and the identification of
  %   possible worlds with state- or property-like entities.}
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   \linespread{1.3}
  %   
  %   A conspicuous virtue of the property-based conception is its
  %   apparent ontological modesty.  As Lewis has
  %   argued,\linespread{1}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {\citep{Lewis:1973}.  \citep{Stalnaker:1976} has also noted the
  %   apparent ontological modesty of Lewis's presentation in this
  %   passage.}
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   \linespread{1.3} we already think that there are many ways things
  %   might have been, other than the way that they actually are.  Not
  %   only are we apparently not inclined to deny the ontology of one
  %   version of the property-based conception, we are even antecedently
  %   inclined to affirm it.  Neither are we antecedently inclined to
  %   reject the ontological claims of other versions.  We are not
  %   antecedently inclined to deny that there are possible states a
  %   world could be in, other than the state the world is actually in;
  %   likewise, we are not antecedently inclined to deny that there are
  %   possible histories for a world other than the actual history of
  %   the world; and similarly for possible states of affairs.
  %   
  %   The property-based conception seems to meet the existence and
  %   modal modesty constraints, too.  Since, by all appearances, it is
  %   ontologically modest, there is no pressure to mitigate ontological
  %   extravagance by denying that possible worlds exist.  So the
  %   proposal should meet the existence constraint.  What about meeting
  %   the modal modesty constraint?  We are antecedently inclined to
  %   affirm that, among the possible properties, histories, or states
  %   in question will be exactly the sorts of properties, histories, or
  %   states which witness the truth of paradigmatic instances of (SC).
  %   For instance, it is very plausible to hold that there is a
  %   (non-actual) possible history of a world according to which Bush
  %   lost the 2000 electoral vote.  This is exactly what would be
  %   required by (SC) according to the property-based conception, since
  %   ``it is possible that Bush lost the 2000 electoral vote'' is true.
  %   Similarly, we are antecedently inclined to affirm that there is no
  %   possible history of a world according to which 2 + 2 $\neq$ 4.
  %   
  %   What we appear to have, then, are two primitivist conceptions of
  %   possible worlds, the property-based conception and the interpreted
  %   models conception, which both meet our three constraints equally
  %   well.  Indeed, both views rely on an ontology of properties and
  %   relations (or at least such property-like entities as states and
  %   histories) for their explanation of the notion of a possible
  %   world.  There is, then, an underlying affinity between these two
  %   conceptions.
  %   
  %   There is also an important difference.  Though both conceptions
  %   assume an ontology of properties, the interpreted models
  %   conception is more conservative: it takes the modal facts as
  %   given, and uses the properties to interpret formal sentences and
  %   evaluate their truth in light of those facts.  The property-based
  %   conception, by contrast, takes an ontology of modal properties --
  %   ways things might have been -- as given, and identifies worlds
  %   with those properties.  Via (SC), that means that the ``box and
  %   diamond'' modal facts are directly sensitive to the vagaries of
  %   the ontology of properties.  For instance, if it turns out that
  %   there is no such property of worlds as \emph{containing a
  %   philosophizing cat}, then, on pain of violating the constraint of
  %   existence, (SC) will allow us to derive the falsity of ``it is
  %   possible that there is a philosophizing cat.''
  %   
  %   I will argue that this sensitivity of the property-based
  %   conception to the ontology of properties provides reason to think
  %   that it faces a serious difficulty: it cannot meet the three
  %   constraints.  The problem centers on how the property-based
  %   conception handles possibilities regarding individuals that might
  %   have existed but do not actually exist.  In the literature, such
  %   individuals are called \emph{mere possibilia}, or
  %   \emph{aliens}.\linespread{1}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {This problem has been discussed by many commentators in one form
  %   or another, including \citep{McMichael:1983}, \citep{Lewis:1986},
  %   and, most recently, \citep{Bennett:2005}.}
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   \linespread{1.3} For the purposes of exposition, it is useful to
  %   settle on a particular version of the property-based conception.
  %   So let's consider a version according to which a possible world is
  %   a maximal property that a world could have.\linespread{1}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {This is the version defended in \citep{Stalnaker:1976}.}
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   \linespread{1.3}
  %     
  %   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.  In such a situation,
  %   there would be an individual-involving property of a world, call
  %   it \emph{elevenliness}, specifiable in terms of the particular
  %   individual who turns out to be Windsor's eleventh granddaughter:
  %   the property of \emph{containing that very
  %   woman}.\linespread{1}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {Here I benefit from discussion of these issues with
  %   [XXXXXXXXXX].}
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   \linespread{1.3} Elevenliness would be an individual-involving
  %   property, distinct from other individual-involving properties,
  %   such as \emph{containing Charles Windsor}, or \emph{containing
  %   $x$} for any other individual $x$.  It is also distinct, and even
  %   independent, from the non-individual-involving property of
  %   \emph{containing some person or other who is the eleventh
  %   granddaughter of Charles Windsor.}  If there were such a property
  %   as elevenliness, it would be a property that a world could have.
  %   Ontological modesty, however, requires that we not claim that
  %   elevenliness actually exists: we are antecedently inclined to deny
  %   that there are individual-involving properties if the relevant
  %   individual does not exist.\linespread{1}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {Notice that our common-sense ontology does not imply that
  %   individual-involving properties cannot be instantiated if they do
  %   not exist.  Robert Stalnaker (in a paper in draft) has argued that
  %   individual-involving properties can be instantiated even when they
  %   do not exist.  Stalnaker has urged, for instance, that the
  %   property \emph{not containing me} is instantiated by any world in
  %   which I fail to exist, despite the fact that it does not exist in
  %   that world.}
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   \linespread{1.3}
  %     
  %   An initial problem that the possibility of aliens poses for the
  %   property-based conception is that, though it is possible that
  %   there have existed different individuals from the ones which
  %   actually exist, there are no properties involving such
  %   individuals.  Consider again the possibility that Charles Windsor
  %   have an eleventh granddaughter.  It might seem as if this
  %   possibility, according to the property-based conception, requires
  %   the existence of the individual-involving property, elevenliness,
  %   which is instantiated by a world only if it contains a certain
  %   particular individual, who does not actually exist.  But asserting
  %   the existence of such a property, I have claimed, would be
  %   metaphysically extravagant.
  %     
  %   Faced with possibilities involving aliens, proponents of the
  %   property-based conception propose to trade in commitment to the
  %   existence of alien individual-involving properties for properties
  %   involving only general conditions expressible without reference to
  %   any particular individual.  For instance, a proponent of the
  %   property-based conception will trade such individual-involving
  %   properties as elevenliness for the property \emph{being a world
  %   containing an eleventh granddaughter of Windsor}.  On this
  %   response, the 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.  I
  %   will argue that this response does not solve the underlying
  %   problem: trading individual-involving properties in for general
  %   properties leads to modal extravagance.
  %     
  %   Let's return to the possibility that Charles Windsor have an
  %   eleventh granddaughter.  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.  For
  %   instance, 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, 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}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {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.  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}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {Exemplars of this tradition include \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 non-logical modal matter.
  %   
  %   Existence and ontological modesty require the version of the
  %   property-based conception under discussion 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.  Standard possible worlds
  %   theories which meet the existence constraint 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 the version of the
  %   property-based conception we are considering, there must be two
  %   maximal properties that both might have been properties of a
  %   world, but which differ only with respect to which individual has
  %   the relevant career.\linespread{1}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {Other versions of the property-based conception will instead
  %   require that there be two individual-involving states, histories,
  %   \emph{etc.}  I believe that the argument could be adapted to apply
  %   to these alternatives, but I won't attempt that task here.}
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   \linespread{1.3} The difference between these two properties lies
  %   only in which individual has the career.  Hence there is no
  %   specifying what it takes for something to possess either property
  %   except by reference to the particular individual who has the
  %   relevant career.  The two properties are each individual-involving
  %   properties, differing only with respect to which individual is
  %   involved.
  %     
  %   To continue our \emph{reductio} argument, 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}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {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 ontological modesty requires that neither of
  %   these properties exist, contrary to the existence constraint.
  %     
  %   The fundamental problem with the property-based conception is
  %   that, intuitively, the actual world does not provide the
  %   properties the view requires to back possibilities involving
  %   merely possible things.  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 that very woman}.  A
  %   symptom of this fundamental difficulty is that the view will
  %   violate one or another of our three constraints.
  %         
  %   There are three avenues of response for a proponent of the
  %   property-based conception.  First, the proponent might deny the
  %   principle in the ontology of properties used in the application of
  %   ontological modesty.
  %   \begin{description}
  %   \item[(Property Dependence)] The existence of an
  %     individual-involving property necessarily depends on the
  %     existence of all of the individuals it involves.
  %   \end{description}
  %   The argument requires that we should be antecedently inclined to
  %   accept (Property Dependence).  But, the response goes, we should
  %   not be antecedently inclined to accept it; true, the
  %   property-based conception does require the existence of
  %   individual-involving properties in the absence of the relevant
  %   individual; but that should not count as any ontological
  %   extravagance.
  %     
  %   To my mind, if we think it possible for there to be any
  %   individual-involving properties at all, then (Property Dependence)
  %   has a great deal of plausibility.  Further, proponents of the
  %   property-based conception are often motivated by the thought that
  %   a theory of possible worlds requiring the existence of merely
  %   possible entities is objectionable on account of its ontological
  %   extravagance.  But denying (Property Dependence) threatens to blur
  %   the line between the property-based conception and such
  %   extravagant alternatives.  For the partisan of possibilia can
  %   simply identify one of the possible fat men in the doorway, for
  %   example, with the property of \emph{being that individual}.  Then
  %   we can make adjustments elsewhere in the theory, replacing what
  %   the old, extravagant theory said about him with suitably doctored
  %   claims about the property of \emph{being him}.  Where the old
  %   theory might have said that he occupies no space, but might have
  %   occupied space, the new theory says instead that \emph{being him}
  %   is instantiated by no space-occupying thing, but it might have
  %   been instantiated by some space-occupying thing.  Such a technical
  %   fix would render the view ontologically modest if we denied
  %   (Property Dependence).  The result nevertheless looks like a
  %   re-formulation of the original, possibilia-based view.  Nothing
  %   seems to be left of the property-based theorist's motivation for
  %   rejecting such a view.\linespread{1}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {Similar sentiments are expressed in \citep{Fine:1985}.}
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   \linespread{1.3} Perhaps it is a measure of the intuitive
  %   plausibility of (Property Dependence) that there seems little to
  %   distinguish the old, extravagant, theory from its replacement.
  %     
  %   % ADD: BACK OFF FROM FULLY GENERAL CLAIM, (PROPERTY DEPENDENCE),
  %   % TO A MORE MODEST CLAIM INVOLVING ONLY THE CASE AT HAND:
  %   % PROPERTIES INVOLVING THE RELEVANT POSSIBILIA DO NOT EXIST.
  %     
  %   Despite its initial appeal, we might perhaps chip away at
  %   (Property Dependence).  Perhaps, even if there were no water,
  %   there would still be such water-involving properties as
  %   \emph{being water} so long as there were hydrogen and
  %   oxygen.\linespread{1}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {This position is suggested by the discussion in \citep[pp. 116,
  %   120]{Burge:1982}.}
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   \linespread{1.3} Similarly, perhaps, even if the assembly of
  %   widgets at a certain plant were terminated just before the
  %   emergence of a certain widget, there would still be such
  %   individual-involving properties as \emph{containing that very
  %   widget}, so long as the parts out of which it would have been
  %   assembled exist.  \linespread{1}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {This position is suggested by the discussion in
  %   \citep[p. 608]{Kaplan:1989b}.}
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   \linespread{1.3} Such apparent counterexamples might move us to
  %   abandon (Property Dependence) in favor of
  %   \begin{description}
  %     \item[(Property Dependence)${}^C$] The existence of an
  %       individual-involving property necessarily depends on the
  %       existence of either of all of the individuals it involves, or
  %       of the components out of which those individuals could be
  %       made.
  %   \end{description}
  %   
  %   But we don't really need (Property Dependence) in its full
  %   generality to make the argument.  The argument only uses the claim
  %   that no property exists that involves any nonexistent individual
  %   who might have been an eleventh granddaughter of Charles Windsor.
  %   The components out of which she might have been made (whatever
  %   those are) don't seem to exist either, so rejecting (Property
  %   Dependence) in favor of (Property Dependence)${}^C$ won't help the
  %   the property-based conception meet the three constraints.
  %   Whatever may be the ultimate verdict on (Property Dependence), we
  %   are antecedently inclined to deny the existence of any property
  %   involving a particular merely possible eleventh granddaughter of
  %   Charles Windsor.
  %   
  %   The second avenue of response is to deny the possibility claim
  %   with which we started.  The argument only shows that, if
  %   insufficientism is true, then the property-based conception must
  %   either accept some ontological extravagance, or violate the
  %   constraint of existence.  So, goes the response, the argument is
  %   avoided if we deny insufficientism.
  %     
  %   We have already seen that insufficientism is a matter of
  %   significant controversy.  Though my sympathies lie with it, no one
  %   should think that its truth is obvious.  So my objection cannot be
  %   that the property-based conception must deny some obvious truth if
  %   it is to avoid ontological extravagance and accept the existence
  %   of possible worlds.  The objection I have been urging is somewhat
  %   different: our theory of possible worlds should not take sides in
  %   this interesting, controversial, non-logical dispute.  Our theory
  %   of possible worlds should, so far as it can, provide a neutral
  %   framework within which such disputes can be pursued.  Perhaps any
  %   framework has to take sides on certain logical questions,
  %   \emph{e.g.}, whether identity is necessary or not.  But a theory
  %   makes a better framework within which to situate end-users'
  %   theories if it does not force their answers.  Perhaps complete
  %   neutrality is a pipe dream, but we can at least hope for more
  %   neutrality than is delivered by the present view on this response.
  %     
  %   Perhaps for these reasons, property-based theorists who have
  %   considered the question tend to opt for the third avenue of
  %   response: accept that, if insufficientism turns out to be true,
  %   then the possible worlds needed to witness the relevant
  %   possibilities do not exist.  This response bites the bullet on the
  %   constraint of existence, rather than on ontological or modal
  %   modesty.  As a matter of fact, the actual world does not provide
  %   the properties the view requires to back possibilities involving
  %   merely possible things.  (SC), read flatly as requiring the actual
  %   existence of possible worlds, turns out to fail.  But this is a
  %   contingent failure.  Had Charles Windsor had an eleventh
  %   granddaughter, then the requisite properties involving her would
  %   have also existed. Hence, though the properties required by the
  %   property-based conception do not exist, they might have.  We need,
  %   then, only a fairly modest revision of (SC), so that `it is
  %   possible that $\phi$' is true iff there is a possible world $w$ of
  %   the right sort, or there might have been, or possibly there might
  %   have been, or possibly possibly there might have been,
  %   \emph{etc.}\linespread{1}\footnote
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   {See, for instance, \citep{Salmon:1989}, and \citep{Fine:2003a}.
  %   \citep{Sider:2002} also denies that possible worlds exist, though
  %   he does not take the view described in the main text.}
  %   %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  %   \linespread{1.3}
  %     
  %   Is this response ultimately satisfactory?  Suppose that there
  %   might have been modally incompatible entities: canonically, there
  %   might have been an individual $x$ and there might have been an
  %   individual $y$, such that, necessarily, if $x$ exists, then $y$
  %   does not.  The question of whether this is true is at least as
  %   vexed a question as insufficientism.  If events turn out to be
  %   individuals, then on plausible metaphysical assumptions, some
  %   pairs of possible events will be modally incompatible.  For
  %   instance, it is possible that Bush have lost the 2000 electoral
  %   vote.  And, of course, it is possible that Bush won that very
  %   vote.  Consider now two possible events, an electoral vote loss
  %   for Bush, and Bush's actual victory.  It is very plausible to
  %   think that, though it is possible that either of these events
  %   occur, it is not possible that both should occur.  If these events
  %   are modally incompatible individuals, then, on (Property
  %   Dependence), it is not possible that there be properties involving
  %   both.  For instance, it is not possible that there be such a
  %   property as \emph{being a world in which the one event occurs and
  %   not the other}.  We might worry, then, that some of the
  %   individual-involving properties that seem to be required by the
  %   property-based conception are not even possible.
  %     
  %   The view that particular events involving Bush and the 2000
  %   electoral vote are modally incompatible makes a big pile of
  %   substantial metaphysical claims.  Though these claims are
  %   plausible, they are by no means incontrovertible.  This view,
  %   however, was only an example designed to illustrate why one might
  %   think that there might have been modally incompatible entities.
  %   The assumptions are substantial, but the view is only invoked to
  %   help explain the import of the fairly abstract modal claim at
  %   issue.  The point for present purposes is not that this
  %   controversial and complicated view is true.  The point is rather
  %   that, so far as possible, a theory of possible worlds should be
  %   non-committal on such questions as whether there might have been
  %   modally incompatible entities.  What is not clear, then, is
  %   whether giving up on the constraint of existence really secures
  %   modal modesty for the property-based conception.
  %     
  %   The property-based conception faces problems meeting our three
  %   constraints when claims regarding aliens are at issue.  The
  %   interpreted models conception seems to meet the three constraints,
  %   and, in particular, is neutral regarding the possibilities for
  %   aliens.  Assume that insufficientism is true.  Then there will be
  %   an interpreted model that fails to contravene the modal facts, in
  %   which two worlds assign the same career to distinct elements.
  %   Since the elements themselves are not \emph{actual} individuals,
  %   we do not face a choice between denying that possible worlds
  %   witnessing the truth of insufficientism exist, or affirming the
  %   existence of individual-involving properties concerning non-actual
  %   granddaughters of Charles Windsor.  In general, by using actual
  %   model-theoretic entities to stand in for the aliens, the
  %   interpreted models conception stays neutral on what's possible
  %   concerning aliens, without violating the existence or ontological
  %   modesty constraints.  This provides a reason for favoring the
  %   interpreted models conception over the property-based conception.
  % \end{section}
  
  \begin{section} {The Point of Possible Worlds} 
    
    The interpreted models conception is frankly reductive: the notion
    of \emph{being a possible world} is explained in terms of what the
    possibilities are.  This stands the standard possible worlds
    reduction of necessity and possibility on its head: the facts
    regarding worlds are explained in terms of the possibilities,
    rather than \emph{vice versa}.
    
    Some will complain, then, that the point of adopting a theory of
    possible worlds has been lost: we adopt a theory of possible
    worlds to provide an analysis of either the modal facts or our
    modal concepts in terms of ordinary quantification over
    extraordinary entities.  Further, possible worlds, on the
    interpreted models conception, will not play any role in the
    explanation of modal principles.  For instance, we can't use
    quantifier negation laws to explain why the necessity of a claim
    is equivalent to the impossibility of its negation.  On the
    interpreted models conception, the order of explanation is
    reversed.  For instance, the fact that such modal principles hold
    explains the fact that ``Socrates is human'' is true at every
    possible world just in case ``Socrates is not human'' is true at
    no possible world.  On the interpreted models conception, possible
    worlds play no interesting role in any account of modality or of
    modal concepts.
    
    If the only point to adopting a theory of possible worlds were to
    provide such an account, then the objection would be warranted.
    But it would apply equally well to any other primitivist account
    of possible worlds.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Perhaps a weaker version of the complaint applies to the present
      view.  It is consistent with primitivism that a possible worlds
      theory that verifies instances of (SC) provides the makings of
      an analysis of the facts which we can report using the
      sentential operators ``it is necessary that\dots'' and ``it is
      possible that\dots'' in terms of more basic, and still
      \emph{modal}, facts regarding possible worlds.  The point of
      possible worlds theories, on this more moderate view, is to
      analyze these ``box and diamond'' modal facts.  Endorsing the
      interpreted models conception requires abandoning even this more
      modest analytical aim.  I believe the appropriate response to
      this complaint is to claim that this aspect of the theory is a
      feature, rather than a bug.  There is no reason to think that
      the ``box and diamond'' modal facts are any less fundamental or
      more problematic than the facts about possible worlds; they seem
      to stand less in need of analysis than the facts which the more
      modest view proposes as their ground.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Since the aim here is not to defend primitivism,
    but to suggest a theory of possible worlds that should be
    attractive to primitivists, the objection should be addressed in
    detail on another occasion.
    
    More significantly, nobody, primitivist or not, should accept that
    the only point to the invocation of possible worlds is to provide
    a reduction of modal facts to non-modal facts.  As I have
    emphasized several times already, (SC) and possible worlds have
    become embedded in rather a wide swath of theorizing by end-users
    about such (apparently) non-modal matters as the nature of mind,
    language, morality, and rationality.  Possible worlds are
    \emph{tools} in a wide variety of other philosophical (and
    not-so-philosophical) projects.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Thanks to Jessica Wilson for discussion.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} A theory of possible worlds should illustrate and
    underwrite their fitness as tools for inquiry in fields far
    removed from the metaphysics of modality, the analysis of modal
    concepts, or the semantics of modal discourse.  The fitness of
    possible worlds for theorizing in philosophy of mind, linguistics,
    ethics, epistemology, \emph{etc.}, does not hang on their
    providing the means to analyze modality or explain modal
    principles.
    
    Our three constraints were motivated by this conception of
    possible worlds as theoretical tools.  Neutrality on modal and
    ontological matters is desirable because it frees end-users from
    untoward commitments in ontology or the metaphysics of modality.
    End-users' theories don't appear to be hostage to the abstruse
    ontological or modal claims debated among metaphysicians of
    modality, and a theory of possible worlds shouldn't make them seem
    so.  Similarly, the constraint of existence turns out just to be a
    matter of interpreting literally and at face value theories that
    incorporate (SC).  The fact that the interpreted models conception
    meets all three constraints is therefore a conspicuous virtue of
    the account.  Possible worlds are here to stay.  I have argued
    that the interpreted models conception shows how they can be
    comfortably accommodated.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Thanks to Roberta Ballarin, Ben Caplan, David Christensen, Tyler
      Doggett, Michael Fara, Harold Hodes, Paul Hovda, Mark Moyer,
      Derk Pereboom, Guy Rohrbaugh, Jessica Wilson, an anonymous
      referee, and an audience at Dartmouth College for invaluable
      comments and criticism.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}

  \end{section}
  
  \section*{Appendix A:  How to Interpret a Model}
    
    The interpreted models conception presumes that formal
    Kripke-style models can be interpreted so that they say something
    about the modal facts, and that, so-interpreted, they can be
    assessed for accuracy, \emph{i.e.} for whether they ``contravene
    the modal facts.''  In Kripke-style model theories the formal
    sentences of QML are interpreted using a standard apparatus of
    assignments of values to variables.  The familiarity of this
    standard apparatus gives rise to a natural worry for the
    interpreted models conception.  If we use the standard apparatus
    of Kripke-style semantics in our attempt to interpret what a
    formal model says about the \emph{real} modal facts, then, one
    might worry, we will have to find ``real'' counterparts for the
    model-theoretic indices: real possible worlds in which sentences
    will be evaluated, and real \emph{possibilia} to assign as values
    to the variables.  It is difficult to see how such an
    interpretation can avoid ontological extravagance.
    
    Fortunately, we can set this worry aside.  There is another way to
    interpret the formal sentences of QML, which does not use the
    standard Kripke-style apparatus of assignments of values to
    variables.  Thus, we can interpret QML without any appeal to
    features of \emph{possibilia}.  Instead of giving an
    interpretation of the open formal sentences of QML in terms of
    satisfaction under assignments of values to variables, we can
    interpret those sentences as complex predicates, expressing
    properties and relations.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {The treatment given here bears a significant resemblance to the
    methods of \citep{Quine:1960} and \citep{Bealer:1983}.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Correlatively, the free variables of a formula
    are thought of as syntactic markers of argument-places for the
    relation expressed by the formula, rather than as terms which
    refer (on an assignment).  The technical trick which allows us to
    interpret the formal sentences of QML without ontological
    extravagance is the trick of recursively assigning relations to
    wffs of QML as their interpretation.  (We think of
    \emph{properties} as 1-place relations, and \emph{claims} as
    0-place relations.)
    
    The recursion, however, is not directly on the complexity of the
    formula.  We will need to assign different relations to $Fx
    \Leftrightarrow Fx$ and $Fx \Leftrightarrow Fy$, even though we
    assign the same relations to $Fx$ and $Fy$.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Fine's worry in \citep{Fine:2003} about the tenability of an
    ordinary satisfaction-based semantics is related to this point.  I
    originally came to the sort of semantics on offer here as a result
    of reflection on Fine's puzzle.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Without fancy footwork, recursion on the
    complexity of a formula will not allow this.  Instead, we will use
    some novel syntactic notions to carry out the idea.
    
    The use of individual constants introduces some complications.  I
    will therefore start by showing how to give interpretations for
    constant-free formulae.  The idea of our novel syntax is to think
    of argument-places for relations as successively ``coordinated'' by
    replacement of free variables.  For instance, $Fx \Leftrightarrow
    Fx$ is obtained from $Fx \Leftrightarrow Fy$ by ``coordinating''
    the two argument places marked by $x$ and $y$.  The result of a
    coordination of a formula is its ``coordinand'', and uncoordinated
    formulae are called ``clumsy.''  More canonically:
    \begin{flushleft}
      \begin{description} 
        \item A formula $\phi$ is \emph{the $x_n{}-x_m$ coordinand} of
          $\psi$ iff:\vspace{-0.1 in}
           \begin{enumerate}
             \item[i.] $x_n$ occurs free in $\psi$;\vspace{-0.1 in}
             \item[ii.] $x_m$ occurs free and exactly once in $\psi$;\vspace{-0.1 in}
             \item[iii.] $x_n$ does not occur after $x_m$ in $\psi$;\vspace{-0.1 in}
             \item[iv.] $x_n$ is free for $x_m$ in $\psi$; and \vspace{-0.1 in}
             \item[v.] $\phi$ is the result of replacing the only occurrence
               of $x_m$ in $\psi$ with $x_n$.
           \end{enumerate}
        \item A formula $\phi$ is \emph{clumsy atomic} iff it is $F^n
           \vars{1}{n}$, for $n$ distinct variables $\vars{1}{n}$.
        \item A formula $\phi$ is a \emph{clumsy conjunction} if it is
          a conjunction whose conjuncts share no free variables in
          common.
      \end{description}
    \end{flushleft}
    
    We can think of a formula as being built up by coordination,
    negation, \emph{etc.} from clumsy constituents.  We obtain formal
    specification of this construction by defining the notion of a
    \emph{coordination analysis} for a formula.
    
    \begin{flushleft}
      A \emph{coordination analysis} for a formula $\phi$ is a
      tree such that:
      \begin{enumerate}
        \item[i.] $\phi$ is the root node;\vspace{-0.1 in}
        \item[ii.] all leaves are clumsy atomic formulae;\vspace{-0.1
          in}
        \item[iii.] every node has at most two
          children;\vspace{-0.1 in}
        \item[iv.] if a node $\phi$ has two children $\psi_1$ and
          $\psi_2$, then \vspace{-0.1 in}\\ $\phi$ is the clumsy conjunction $(\psi_1
          \wedge \psi_2)$; \vspace{-0.1 in}
        \item[v.] if a node $\phi$ has exactly one child $\psi$,
          then either: \vspace{-0.1 in}
          \begin{enumerate}
            \item[a.] $\phi$ is $\neg\psi$;\vspace{-0.1 in}
            \item[b.] $\phi$ is $(\exists x)\psi$;\vspace{-0.1 in}
            \item[c.] $\phi$ is $\square\psi$; or \vspace{-0.1 in}
            \item[d.] $\phi$ is the $x_n - x_m$ coordinand of $\psi$;
          \end{enumerate}
      \end{enumerate}
    \end{flushleft}
    
    It can be shown that every formula $\phi$ has a coordination
    analysis by a somewhat tedious induction on the complexity of
    $\phi$.  The two interesting cases are when $\phi$ is atomic, and
    when $\phi$ is a conjunction.  For the atomic case, since leaves
    must be clumsy atomic formulae, we must show how an arbitrary
    coordinated atomic formula can be analyzed in terms of step-wise
    coordination of argument-places.  Similarly, for the case of
    conjunction, we must show how any conjunction can be analyzed in
    terms of step-wise coordination from some clumsy conjunction.  In
    both cases, we can prove that $\phi$ has a coordination analysis
    by induction on the number of distinct free variables in $\phi$,
    and then by induction on the number of occurrences of the last
    multiply-occurring free variable in $\phi$.
    
    A formula will generally have many coordination analyses.  More
    than one is more than is convenient, so we should define some nice
    well-ordering $\mathcal{R}$ on coordination analyses, and then let
    \emph{the} coordination analysis of $\phi$ be the
    $\mathcal{R}$-least coordination analysis of $\phi$.
    
    Recall that an acceptable key $K$ assigns an $n$-ary relation to
    every $n$-place predicate $F^n$, and the identity relation to
    `='. We will assume that the relations are closed under operations
    corresponding to the syntactic operations of clumsy conjunction,
    negation, necessitation, coordination of argument-places,
    \emph{etc.}  Now we can finally define the interpretation of
    $\phi$ by $K$ ($[\phi]^K$) by recursion on the depth of the
    coordination analysis of $\phi$:
    \begin{enumerate}
        \item[i.] if $\phi$ is clumsy atomic (and so a leaf), then \\
          $[\phi]^K$ is $K(F^n)$;\vspace{-0.1 in}
        \item[ii.] if $\phi$ is the $x_n - x_m$ coordinand of
          $\psi(\vars{1}{n}\dots\vars{m}{j})$, then \\ 
          $[\phi]^K$ is the relation $\vars[a]{1}{n},\dots ,a_{m-1},
           \vars[a]{n}{j}$ \emph{stand in} $[\psi]^K$;\vspace{-0.1 in}
        \item[iii.] if $\phi$ is a clumsy conjunction of its children
          $\psi_1$ and $\psi_2$ then \\ 
          $[\phi]^K$ is the relation \vspace{-0.1 in}\\
          \hspace*{0.1 in}$\vars[a]{1}{n}$ \emph{stand in}
          $[\psi_1]^K$ \emph{and} 
          \hspace*{0.1 in}$\vars[b]{1}{m}$ \emph{stand in}
          $[\psi_2]^K$;\vspace{-0.1 in}
        \item[iv.] if $\phi$ is the negation of its only child $\psi$, then \\
          $[\phi]^K$ is the relation $\vars[a]{1}{n}$ \emph{do not
            stand in} $[\psi]^K$;\vspace{-0.1 in} 
        \item[v.] if $\phi$ is an existential generalization over
          $x_i$ of its only child $\psi$, then \\
          $[\phi]^K$ is the relation\vspace{-0.1 in}\\
          \hspace*{0.1 in} \emph{there is an $a_i$ such
            that} $\vars[a]{1}{i},\dots,a_n$ \emph{stand in}
          $[\psi]^K$;\vspace{-0.1 in}   
        \item[vi.] if $\phi$ is the necessitation of its only child
          $\psi$, then \\ 
          $[\phi]^K$ is the relation\vspace{-0.1 in}\\
          \hspace*{0.1 in} \emph{it is necessary that}
          $\vars[a]{1}{n}$ \emph{stand in} $[\psi]^K$.          
    \end{enumerate}
    
    Adding individual constants is relatively straightforward.  Recall
    that a key $K$ assigns an individual to every individual constant
    $c$.  The individual constants must be ordered, \emph{e.g.},
    alphabetically, and we need both to define a way of introducing a
    constant into a formula, and to extend some of our definitions.
    \begin{flushleft} 
      \begin{description}
        \item A formula $\phi$ is \emph{the $c{}-x_m$ coordinand} of
          $\psi$ iff:\vspace{-0.1 in}
           \begin{enumerate}
             \item[i.] $c$ is an individual constant that does not
               occur in $\psi$; and \vspace{-0.1 in}
             \item[ii.] $\phi$ is the result of replacing all free
               occurrences of $x_m$ in $\psi$ with $c$.
           \end{enumerate}
         \item Add to clause (v.) of the definition of a coordination
           analysis:
           \begin{enumerate}
             \item[e.] $\phi$ is the $c - x_m$ coordinand of $\psi$;
           \end{enumerate}
         \item Add to the definition of $[\phi]^K$:
           \begin{enumerate}
             \item[vii.] if $\phi$ is the $c - x_m$ coordinand of its
               only child $\psi(\vars{1}{m},\dots , x_n)$, then \\
               $[\phi]^K$ is the relation $\vars[a]{1}{m-1}, K(c),
               \dots, a_n$ \emph{stand in} $[\psi]^K$
           \end{enumerate}
      \end{description}
    \end{flushleft}
    
    Notice that constants, unlike variables, are added ``all at
    once;'' this measure simplifies the proof that every formula has a
    coordination analysis (in the new sense).  The old proof
    establishes that every constant-free formula has a coordination
    analysis.  We complete the new proof by supplementing the old
    proof with inductions on the number of constants in $\phi$ in both
    the atomic and conjunction cases.
    
  \section*{Appendix B:  How to Interpret (SC)}
    
    I have been focused for the entirety of this paper on giving a
    theory of possible worlds that meets the constraints of existence,
    ontological modesty, and modal modesty.  I have not been directly
    concerned with providing an analysis or interpretation of (SC),
    beyond specifying what it takes for something to be a possible
    world.  There is, however, a natural interpretation available,
    which can be reached in one of two ways.  Both ways use the notion
    of an (English) sentence $\phi$'s \emph{interpreting} some formal
    sentence $\psi$ according to a key $K$.  I take the notion of an
    (English) sentence's \emph{expressing} a claim as
    primitive.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {If we think of the substituends for $\phi$ in (SC) as claims,
      rather than natural language sentences, then we should say that
      every claim expresses itself, and a claim $\phi$ interprets
      $\psi$ according to $K$ iff $K$ assigns $\phi$ to $\psi$.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Then, $\phi$ interprets $\psi$ according to $K$
    iff $\phi$ expresses the claim assigned to $\psi$ by $K$.  Call
    the formal sentence interpreted by $\phi$ according to $K$
    $\phi_K$.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Technically, $\phi$ can interpret many sentences according to
      $K$.  If so, let $\phi_K$ be the least sentence (according to
      some well-ordering) that $\phi$ interprets.  This wrinkle will
      not matter for our discussion.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Also, for the sake of brevity, I will call an
    interpreted model \emph{good} iff it fails to contravene the modal
    facts.
    
    The first way to our natural interpretation of (SC) is to define
    all at once the relation \emph{$w$ is a possible world at which
      $\phi$ is true}:
    \begin{description}
      \item[(PW+TA)] $w$ is a possible world at which `$\phi$' is true
        iff there is a good interpreted model $\langle M, K\rangle$,
        such that (\emph{i}) $w$ is an element of $\langle M,
        K\rangle$, and (\emph{ii}) $\phi_K$ is true at $w$ in
        $M$.\linespread{1}\footnote
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        {In what follows, let `$M\vDash_w\phi_K$' abbreviate (\emph{ii}).}
        %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
        \linespread{1.3}
    \end{description}
    Substituting the right-hand side of (PW+TA) in for its left-hand
    side in a colloquial variant of (SC):
    \begin{description}
      \item[(CSC)] it is possible that $\phi$ iff there is a $w$ such
        that $w$ is a possible world at which `$\phi$' is true.
    \end{description}
    yields the interpretation of (SC):
    \begin{description}
      \item[(ISC)] it is possible that $\phi$ iff there is a $w$ such
        that there is a good interpreted model $\langle M, K\rangle$,
        such that (\emph{i}) $w$ is an element of $\langle M,
        K\rangle$, and (\emph{ii}) $M\vDash_w\phi_K$.
    \end{description}
    Given (PW+TA), the interpreted models conception of \emph{being a
      possible world} is a natural upshot:
    \begin{description}
      \item[(PW)] $w$ is a possible world iff there is a good
        interpreted model $\langle M, K\rangle$ such that $w$ is an
        element of $\langle M, K\rangle$.
    \end{description}
    
    This way of arriving at an interpretation of (SC) presents (PW) as
    an upshot, rather than a necessary input, of the interpretation.
    This is not the most natural way to present it.  It is more
    natural to offer a definition of the relation \emph{$\phi$ is true
      at $w$} that can be used along with (PW) to yield an
    interpretation of (SC).  The second, less direct way of reaching
    (ISC) follows that more natural course.  Here's a definition of
    \emph{$\phi$ is true at $w$} consilient with the interpreted
    models conception:
    \begin{description}
      \item[(TA)] $\phi$ is true at $w$ iff there is a good
        interpreted model $\langle M, K\rangle$ such that
        $M\vDash_w\phi_K$.
    \end{description}
    Plugging the right-hand sides of (PW) and (TA) into the requisite
    positions in (SC) yields:
    \begin{description}
      \item[(ISC)${}^*$] it is possible that $\phi$ iff there is a $w$
        such that (\emph{i}) there is a good interpreted model
        $\langle M, K\rangle$ that has $w$ as an element, and
        (\emph{ii}) there is a good interpreted model $\langle
        M^\prime, K^\prime\rangle$ such that
        $M^\prime\vDash_w\phi_{K^\prime}$.
    \end{description}
    (ISC) and (ISC)${}^*$ are equivalent, since, for any $\phi$,
    $M^\prime\vDash_w\phi_{K^\prime}$ only if $w$ is an element of
    $\langle M^\prime, K^\prime\rangle$.  So our two ways of
    interpreting (SC) reach the same destination.
    
    Notice that, according to (TA), for $\phi$ some contingent truth,
    if $\phi$ is true at $w$, then $\neg\phi$ is also true at $w$.
    Further, if we adopt a natural definition of the correlative
    notion of $\phi$'s being false at $w$:
    \begin{description}
      \item[(FA)] $\phi$ is false at $w$ iff there is a good
        interpreted model $\langle M, K\rangle$ such that
        $M\vDash_w\neg\phi_K$
    \end{description}
    then it turns out that $\phi$ is both true and false at $w$.  Is
    this objectionable?  Not if we remember that \emph{being true at
      $w$}, like \emph{being a possible world}, is a condition that
    implicitly involves a relation to an interpreted model.  As we saw
    in section \ref{sec:relational} above, when we speak of such
    conditions, what we say is sometimes ambiguous between coordinated
    and uncoordinated interpretations.  As with possible worlds, most
    of the interesting issues arise with respect to the coordinated
    interpretation.  It's only on an uncoordinated interpretation that
    a question of the form, ``Is $\phi$ both true and false at $w$?''
    should get an affirmative answer.  On the more interesting,
    coordinated interpretation, the answer is clearly
    no.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Thanks to an anonymous referee for suggesting the need for this
      appendix, and offering some proposals for how the interpretation
      of (SC) might be carried out.}%
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
    
    
      
  \linespread{1} \bibliographystyle{plainnat}
  \bibliography{PossWorlds}
\end{document}



