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\title{No Free Lunch}
%\author{Louis deRosset}
\date{\today}


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\begin{document}
  \maketitle 
  
  Some philosophers (see \citep{Armstrong:1997}, \citep{Cameron:2008},
  \citep{Melia:2005}, and \citep{Schaffer:2007, Schaffer:2009,
    Schaffer:2010}) have recently suggested that explanations of a
  certain sort can mitigate our ontological commitments.  The
  explanations in question, \emph{grounding explanations}, are those
  that tell us what it is in virtue of which an entity exists and has
  the features it does.  These philosophers claim that the existence
  and qualitative properties (call these \emph{features}) of some
  entities can be given a grounding explanation solely by reference to
  the existence and properties of other things.\linespread{1}\footnote
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  {Here and generally throughout I use ``properties'' to stand in for
    both properties and relations.}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \linespread{1.3} Call an entity \emph{derivative} iff its existence
  and other features can all be explained solely by reference to the
  existence and properties of other things.  These philosophers argue
  that derivative entities are ``no addition to being,'' in the sense
  that an ontology is no less sparse for containing them than it is
  for containing the entities which ground them; derivative entities
  are an ``ontological free lunch.''\linespread{1}\footnote
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  {One striking feature of these authors' views that is not
    represented my exposition is that they have been developed in the
    pursuit of Armstrong-style truthmaker metaphysics.  Here I assume
    that truth-making incurs an explanatory commitment, following,
    \emph{e.g.}, \citep[p. 257]{Cameron:2010} and
    \citep{Schnieder:2006}.  I also put the view in the material mode,
    as a view about what explains certain facts, rather than a view
    about what makes certain truths concerning those facts true.  In
    this I follow \citep{Lewis:2001}.}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \linespread{1.3}
  
  Grounding explanations are exemplified by certain familiar
  scientific explanations, and are often marked by the locution ``in
  virtue of''.  (In what follows, my use of ``explain,'' ``because,''
  and cognate notions is artificially restricted to this kind of
  explanation.)  A given isotope of gold has a certain atomic mass in
  virtue of containing a certain number of protons and neutrons.
  Ethanol is miscible in water in virtue of containing a hydroxide
  group.  Diamond is hard because each carbon atom in its crystalline
  structure is bonded to each of its neighbors.
  
  Theorists who claim that grounding explanations provide an
  ``ontological free lunch'' have concentrated their efforts on
  defending a novel view of ontological commitment.  They argue that
  we should reject a familiar version of Ockham's Razor
  \begin{quote}
    Do not multiply entities beyond necessity!
  \end{quote}
  in favor of a less familiar variant
  \begin{quote}
    Do not multiply fundamental entities beyond
    necessity!\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {See esp. \citep[p. 361]{Schaffer:2009}, from which the
      formulation in the text is adapted. Schaffer's view differs from
      Cameron's \citeyearpar{Cameron:2008}.  Cameron argues that the
      ontological commitments of a theory are given by an inventory of
      fundamental entities; Schaffer, that ontological commitments are
      given by an inventory of the existent entities, but that only
      commitments to fundamental entities count when applying Ockham's
      Razor.  See \citep{Schaffer:2008} for a defense of Schaffer's
      side of the dispute.  Though the exposition in the main text
      fits Schaffer's view better than Cameron's, this dispute about
      the proper characterization of the technical notion of an
      \emph{ontological commitment} is irrelevant for present
      purposes.  Schaffer and Cameron agree that (\emph{i}) existence
      claims concerning derivative entities are an ``ontological free
      lunch,'' and (\emph{ii}) the class of derivative entities is
      large and diverse.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
  \end{quote}
  where an entity is \emph{fundamental} iff it is not derivative:
  either its existence or one of its other features is explanatorily
  basic.  If they are right, then grounding explanations are not (or
  not just) causal: the availability of a complete causal explanation
  of an explosion in terms of causal laws, initial conditions, and the
  ignition of a fuse does not show that the explosion is ``no addition
  to being,'' given the laws, conditions, and ignition.
  
  The dispute over the proper application of Ockham's Razor has
  interest only if the fundamental entities are a subclass of all of
  the entities.  If every entity is fundamental, then the application
  of the familiar version of Ockham's Razor yields the same results as
  the application of its restriction to the fundamental.  As one might
  expect, the theorists in question claim that the fundamental
  entities are an elite class.\linespread{1}\footnote
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  {Schaffer \citeyearpar{Schaffer:2010}, for instance, argues that
    every concrete entity other than the entirety of the concrete
    cosmos is derivative.  Another example is Armstrong's
    \citeyearpar{Armstrong:1997} contention that that most universals
    are derivative.  Melia \citeyearpar{Melia:2005} goes further,
    outlining a defense of the idea that all universals are
    derivative.}
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  \linespread{1.3}
  
  Here I argue that they are wrong: barring reduction, every entity is
  fundamental, in the sense that either its existence or its
  possession of at least one other feature is explanatorily basic.  Thus,
  the claim\hypertarget{EXPL}{}
  \begin{description}
    \item[(EXPLANATION)] Many entities are
      derivative: their existence and other features can be explained
      solely by reference to the existence and properties of other
      things.
  \end{description}
  should be rejected.  An upshot is that, whatever form Ockham's Razor
  should take, grounding explanations on their own do not provide an
  ``ontological free lunch.''\linespread{1}\footnote
  %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
  {Schaffer \citep{Schaffer:forthcoming-b, Schaffer:2009} has
    suggested that the basic idea for priority theorists is not
    explanation but \emph{grounding}.  We explain facts by reference
    to other facts, but, by Schaffer's lights, grounding may obtain
    between items of any category, including objects, facts, and
    properties.  Schaffer, however, agrees that grounding requires
    explanation: the existence and other features of the entities that
    ground an entity completely explain its existence and other
    non-relational features (private correspondence).  See, for
    instance, the claim at \cite[pp. 364-5]{Schaffer:2009} that
    grounding ``... is the notion the physicalist needs to explicate
    such plausible claims as `the fundamental properties and facts are
    physical and everything else obtains \emph{in virtue of} them'
    \citep[p. 39]{Loewer:2001})'' [citation and emphasis in original].
    \cite[p. 257]{Cameron:2010} also endorses this link between
    grounding and explanation.  The argument in this paper against
    (\hyperlink{EXPL}{EXPLANATION}) thus applies to views which
    rely on grounding rather than explanation to distinguish
    fundamental and derivative entities.\label{fn:grounding}}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
    
   Here's the plan.  \S\ref{sec:prems} sets out and defends the
   premises of the argument against (\hyperlink{EXPL}{EXPLANATION}).  The argument
   itself is outlined in \S\ref{sec:arg}, and its import clarified in
   \S\ref{sec:clarify}.  \S\S\ref{sec:dc_defense}-\ref{sec:existence}
   discuss responses that might be made to the argument.  I close with
   some remarks on the significance of the result in
   \S\ref{sec:conclusion}.
  
  \begin{section}{Determination, Diversity, Irreducibility}\label{sec:prems}
    Consideration of a standard way proposals for grounding
    explanations may fail motivates a constraint on this sort of
    explanation: the explanation of the existence and other features
    of a given entity must trace how those facts are determined.  A
    complete explanation of a derivative entity $d$'s having a certain
    feature $F$ in terms of facts $g_1, g_2, \dots$ should show why
    $d$ had to be $F$, given $g_1, g_2, \dots$.  The \emph{explanans}
    thereby provides means for distinguishing $d$ from entities which
    are not $F$, and thus showing why $d$, unlike those other things,
    is $F$.  There's something wrong with or missing from a proposed
    explanation if there is something just like $d$ so far as the
    \emph{explanans} goes, but which lacks $F$.  Call any entity which
    lacks $F$ but is just like $d$ so far as the \emph{explanans}
    goes, a \emph{confounding case} for the proposed explanation.  For
    instance, a confounding case for the explanatory claim
    \begin{example}
       This diamond is hard because it has such-and-such crystalline
       structure
    \end{example}
    would be an entity which has the crystalline structure in
    question, but is not hard.  
    
    An explanatory proposal is \emph{at best incomplete} if there is a
    confounding case for it: the proposal is either off on the wrong
    foot entirely, or requires supplementation.  ``At best
    incomplete'' is awkward to negate.  So, for reasons of style I'll
    call an explanation \emph{good} if it is not at best incomplete;
    good explanations are not off on the wrong foot entirely, and do
    not require supplementation.  Note that explanations that are
    ``good'' in this sense may fail to qualify as what we would
    ordinarily call ``a good explanation'': they may be complicated,
    long-winded, unenlightening, difficult to understand, \etc
      
    Here is an intuitive way of appreciating the point that
    confounding cases indicate that an explanatory proposal is at best
    incomplete.  A standard strategy for criticizing a proposed
    explanation of the form ``$x$ is $F$ in virtue of being $G$'' is
    to identify a confounding case: another object that's $G$ but not
    $F$.  Thus
    \begin{example}
       $x$ is a stable nucleus in virtue of being an oxygen
      nucleus\label{sent2}
    \end{example}
    is a transparently inadequate explanation, given the existence of
    short-lived radioactive oxygen isotopes.  Even the mere
    possibility of a confounding case shows that a proposed
    explanation is at best incomplete.  Suppose now that there happen
    not to be any short-lived radioactive oxygen isotopes, but it is
    possible to make some in a certain kind of research reactor.  The
    proposed explanation is still at best incomplete.  Explanatory
    proposals are thus subject to what I will call the
    \emph{determination constraint}:
    \begin{description}
    \item[(Determination Constraint)] An explanatory proposal of the
      form, ``$d$ has feature $F$ because $\phi(d, x_1, \dots, x_n)$''
      is at best incomplete if there is or might have been a
      confounding case for it: an entity $e$ and some entities $a_1,
      \dots, a_n$ such that $e$ (together with $a_1, \dots, a_n$)
      satisfies $\phi(y, x_1, \dots, x_n)$ but lacks
      $F$.\linespread{1}\footnote
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      {Two technical notes.  First, here and throughout, I am sloppy
        about use and mention whenever it is convenient and does not
        materially affect the argument.  Second, I am not assuming
        that any formula of the form $\phi(d, x_1, \dots, x_n)$
        contains $d$ (or, for that matter, any of the
        $x$'s).\label{fn:syntax}}
      %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
      \linespread{1.3}
    \end{description}    
    
    (\hyperlink{EXPL}{EXPLANATION}) requires that there be many derivative entities.
    Suppose that $d$ is one of them, and that $F$ is one of $d$'s
    features.  The argument against (\hyperlink{EXPL}{EXPLANATION}) will also use two
    further assumptions.  The first is:\hypertarget{DD}{}
    \begin{description}
    \item[(DERIVATIVE DIVERSITY)] There is a derivative entity $e$
      that lacks $F$.
    \end{description}
    (\hyperlink{DD}{DERIVATIVE DIVERSITY}) is innocuous: it's a
    natural concomitant of any view which proposes to use
    (\hyperlink{EXPL}{EXPLANATION}) to mitigate its ontological
    commitments.  Without (\hyperlink{DD}{DERIVATIVE DIVERSITY}), such a view has a
    severely limited scope: it can only claim an ``ontological free
    lunch'' with respect to a class of qualitatively indiscernible
    entities.
    
    The second assumption is:\hypertarget{IRRED}{}
    \begin{description}
    \item[(IRREDUCIBILITY)] $d$'s being $F$ and $e$'s not being $F$
      are irreducible.
    \end{description}
    The explication of the notion of reduction is highly contested.
    For present purposes I resort to stipulation: what I am calling a
    \emph{reduction} of one fact to another involves the
    identification of those facts.  No particular epistemological
    status is indicated, since an identity that undergirds a
    reduction, like many other identities, may not be \emph{a priori}.
    A hallmark of reduction is necessary equivalence: if one claim is
    reducible to another, then, as a matter of necessity, they have
    the same truth value; and if one fact is reducible to another,
    then they obtain at all the same possible worlds.  By way of
    contrast, explanation does not require necessary equivalence; a
    good explanation implies (by the determination constraint) that
    the \emph{explanans} be modally sufficient for the
    \emph{explanandum}, but it does not imply the converse.  So
    (\hyperlink{IRRED}{IRREDUCIBILITY}), on this stipulation, requires that $d$'s being
    $F$ is distinct from any fact involving only entities other than
    $d$ (and similarly for the fact that $e$ is not $F$).  Suppose,
    for instance, that $d$ is a diamond and $F$ is the property
    \emph{being hard}.  Then (\hyperlink{IRRED}{IRREDUCIBILITY}) implies that the fact
    that $d$ is hard is distinct from the fact that certain carbon
    atoms $c_1, c_2, \dots, c_n$ are bonded together in such-and-such
    a way.
    
    Assuming (\hyperlink{IRRED}{IRREDUCIBILITY}) is mostly a matter of methodology.
    Reductions are a traditionally recognized way to mitigate one's
    ontological commitments \citep{Quine:1948}.  But grounding
    explanations are weaker than reduction.  It is plausible to
    maintain that the average of my wife's and my heights is 5'6''
    solely in virtue of the fact that she is 5'4'' and I am 5'8''.
    But it is implausible to think that the fact that the average of
    our heights is 5'6'' reduces to our having those particular
    heights, since it is possible for a different combination of
    heights to yield the same average.  There is a reduction of
    average height ready to hand, but that should not distract us from
    the conceptual point: one fact can be explicable in terms of
    another without being reducible to it.  Thus, (\hyperlink{IRRED}{IRREDUCIBILITY})
    leaves open the hypothesis that, even though the diamond $d$'s
    being hard doesn't reduce to the bonding together of certain
    carbon atoms, $d$'s hardness is explained by that bonding.  What's
    distinctive about the idea that grounding explanations mitigate
    our ontological commitments is that they do so even in the absence
    of reduction.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {\cite[p. 76]{Melia:2005} explicitly draws a contrast between
      explanation and reduction.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Assuming (\hyperlink{IRRED}{IRREDUCIBILITY}) is designed to test this
    distinctive claim.
  \end{section}
  
  \begin{section}{Against (EXPLANATION)} \label{sec:arg}
    We are now in a position to state the argument against
    (\hyperlink{EXPL}{EXPLANATION}).  Here is a quick gloss.  On our assumptions, $d$ is
    derivative, so $d$'s being $F$ ultimately obtains in virtue of
    facts which \emph{do not involve $d$ at all}.  We get some
    explanation of the form, ``$d$ is $F$ because $\phi(t_1, \dots,
    t_n)$'', where $\phi$ may mention certain entities and
    features, but doesn't mention $d$.  That means that $e$ meets
    exactly the same conditions: it will be equally true of $e$ that
    (together with $t_1, \dots, t_n$) it satisfies $\phi$.  Since $e$
    is not $F$, it presents a confounding case for the proposed
    explanation.  Application of the determination constraint implies
    that the proposed explanation is at best incomplete.
    
    Some technical notions are required to state the more formal and
    more general version of the argument.  For present purposes, let's
    think of a fact as a distribution of certain properties and
    relations over certain entities, which I will term the
    entities \emph{involved} in the fact.  For instance,
    \emph{being male} is possessed by George Bush; this is a very
    simple way in which this property is distributed over the
    individual in question.  Likewise, Bush bears \emph{being the
      husband of} toward his spouse Laura; this is a somewhat less
    simple way in which this relation is distributed over those two
    individuals.  The latter fact involves both spouses; the former
    involves only George.  An explanatory proposal for the fact that
    $d$ has some feature $F$ says that this fact obtains in virtue of
    certain further facts, each of which is to be identified with a
    distribution of certain properties and relations over certain
    entities.  Thus, a proposal to explain $d$'s having $F$ can be
    expressed by a claim of the form,
    \begin{description}
      \item[(Prop)] $d$ is $F$ because $\phi(d, t_1, \dots, t_n)$
    \end{description}
    where all of the entities involved in the \emph{explanans} are
    denoted by exactly one term among $d, t_1, \dots, t_n$, and $\phi$
    says how the properties and relations in question are distributed
    over those entities.  When an explanatory proposal is expressed
    by a claim of this sort, I will say that the claim
    \emph{perspicuously articulates} the proposal.  A perspicuous
    articulation of an explanatory proposal names names: it specifies
    exactly which entities are involved in the proposed
    \emph{explanans} and \emph{explanandum}.  Thus, the claim, ``the
    diamond $d$ is hard in virtue of the arrangement and bonding of
    certain carbon atoms'' fails to be a perspicuous articulation of
    any explanatory proposal; on the other hand, if $c_1$, $\dots$,
    $c_n$ name the carbon atoms in question, then ``$d$ is hard in
    virtue of the arrangement and bonding of $c_1, \dots, c_n$'' is a
    perspicuous articulation of an explanatory proposal.  I will
    assume that every explanatory proposal that meets the needs of
    (\hyperlink{EXPL}{EXPLANATION}) has a perspicuous
    articulation.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {This assumption might be resisted on a variety of grounds.  (For
      instance, it might be held that some facts concerning the
      existence and other features of derivative entities can be
      adequately explained only by facts involving infinitely many
      entities, and that no infinitary perspicuous articulation
      exists.)  If the assumption fails, then the argument of this
      paper will have to be made at the level of facts.  This can be
      done by representing a fact (in the actual world) by a pair
      containing the set of entities $I$ it involves and the set of
      properties and relations $P$ it involves.  The fact represented
      by $\langle I, P \rangle$ is the distribution of the properties
      and relations in $P$ over the entities in $I$.  These
      representations can, in effect, play the role of perspicuous
      articulations of explanatory proposals.  This is not the place
      to work out the details of this alternative approach, so for
      present purposes I will rely on the assumption.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
    
    As a final bit of stage-setting, let's note a consequence of the
    determination constraint.  The determination constraint implies
    that good explanations don't have confounding cases.  Consider
    again a perspicuously articulated explanatory proposal of the form
    \begin{description}
      \item[(Prop)] $d$ is $F$ because $\phi(d, t_1, \dots, t_n)$
    \end{description}
    A confounding case for this explanatory proposal will be a
    situation in which some object $d^*$, along with some other
    objects $a_1, \dots, a_n$, has the properties required to satisfy
    the \emph{explanans} clause $\phi$, but in which $d^*$ lacks $F$.
    Thus, $d^*$, together with $a_1, \dots, a_n$, witnesses the truth
    of
    \begin{example}
      $(\exists y_1, \dots, y_n) (\exists x) (\phi(x, y_1, \dots, y_n)
      \wedge \neg Fx)$.
    \end{example}
    So, the determination constraint implies that any perspicuously
    articulated explanatory proposal of the form (Prop) is associated
    with a universal generalization
    \begin{example}
      $(\forall y_1, \dots, y_n)(\forall x)(\phi(x, y_1, \dots, y_n)
      \Rightarrow Fx)$.
    \end{example}
    According to the determination constraint, if the explanatory
    proposal is good, then its associated universal generalization is
    true.
    
    Here, then, is the argument.  Assume (\hyperlink{EXPL}{EXPLANATION}) for
    \emph{reductio}.  Recall our ancillary assumptions: $d$ is a
    derivative entity, $F$ is one of its features, and $d$ and $F$
    satisfy (\hyperlink{IRRED}{IRREDUCIBILITY}) and (\hyperlink{DD}{DERIVATIVE DIVERSITY}). Since $d$ is
    derivative, its possession of $F$ has a good explanation
    perspicuously articulated by a claim of the form,
    \begin{example}
      $d$ is $F$ because $R(t_1,\dots, t_n)$,\label{sent8}
    \end{example}
    where $R$ stands in for some (possibly very complex) relation, and
    none of the terms $t_1,\dots, t_n$ denote $d$.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {This is where (\hyperlink{IRRED}{IRREDUCIBILITY}) comes in.  If $d$'s being $F$ is
      reducible, then, for all we've said, it just comes to the
      obtaining of some relation $S$ among $t_1,\dots, t_n$.  Then,
      one might object that (\ref{sent8}) misrepresents the form of
      the explanatory proposal at hand, which is really
      \begin{example}
        $S(t_1,\dots, t_n)$ because $R(t_1,\dots, t_n)$.
      \end{example}
      This explanatory proposal does not succumb to the
      \emph{reductio}.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} But the \emph{explanans} clause $R(t_1,\dots,
    t_n)$ is also of the form, $\phi(d, t_1,\dots, t_n)$ (see the
    second remark of n.\ref{fn:syntax}).  So the explanation in
    question is also of the form
    \begin{example}
      $d$ is $F$ because $\phi(d, t_1,\dots, t_n)$.
    \end{example}
    The determination constraint implies that this explanation is good
    only if its associated universal generalization
    \begin{example}
      $(\forall y_1, \dots, y_n)(\forall x)(R(y_1, \dots, y_n)
      \Rightarrow Fx)$.\label{sent3}
    \end{example}
    is true.  Since $R(y_1,\dots, y_n)$ is $x$-free, standard
    quantificational logic yields
    \begin{example}
      $((\exists y_1, \dots, y_n)R(y_1, \dots, y_n)
      \Rightarrow (\forall x)Fx)$.\label{sent5}
    \end{example}
    An explanatory proposal is good only if the \emph{explanans}
    clause is true.  In this case, that requires that $R(t_1, \dots,
    t_n)$ be true.  But then the antecedent in (\ref{sent5}) is
    satisfied, and so $e$ is also $F$.  (\hyperlink{DD}{DERIVATIVE DIVERSITY}) implies
    that $e$ is not $F$.  Contradiction.
  \end{section}
  
  \begin{section}{Clarifications}\label{sec:clarify}
    Once this argument is stated, it is not difficult to see what's
    missing from the proposal to explain $d$'s being $F$ by
    $R(t_1,\dots, t_n)$.  We need only add explanatorily basic facts
    which indicate what the $t$'s have to do with $d$.  Suppose, for
    instance, that $d$ is a diamond, and we are trying to explain its
    hardness in terms of the arrangement and bonding of certain carbon
    atoms.  Then we could add the claim that $d$ is composed of those
    carbon atoms; $d$ is hard not just because some carbon atoms are
    arranged and bonded in such-and-such a way, but because \emph{its}
    carbon atoms are arranged and bonded in that way.  This proposal
    does not succumb to our argument; it is crucial for the argument
    that the \emph{explanans} not involve $d$.  If the
    \emph{explanans} clause mentions $d$, then the argument is
    invalid.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {To be precise, the step from the analogue of (\ref{sent3}) to the
      analogue of (\ref{sent5}) is blocked.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}  
    
    But our choice of derivative entities $d$ and $e$ and of feature
    $F$ was entirely arbitrary.  Thus, admitting explanatorily basic
    facts involving $d$ means giving up on (\hyperlink{EXPL}{EXPLANATION}) entirely.
    Our argument shows that there is a gap in any proposal for
    explaining any derivative entity's features that meets the demands
    of (\hyperlink{EXPL}{EXPLANATION}), so long as (\hyperlink{DD}{DERIVATIVE DIVERSITY}) and
    (\hyperlink{IRRED}{IRREDUCIBILITY}) are satisfied.  But filling this explanatory gap
    in the most obvious way, by admitting explanatorily basic facts
    involving $d$, means giving up (\hyperlink{EXPL}{EXPLANATION}).  The moral of our
    argument is: \emph{barring reduction, there do not exist two
      derivative entities $x$ and $y$ that differ on some feature
      $F$.}
  
    Our argument might be thought to show too much, on the grounds
    that it implausibly requires that every fact be explanatorily
    basic.  But the argument allows non-basic facts involving $d$.
    For instance, the argument does not show that we must accept that
    \begin{example}
       $d$ is a diamond. \label{sent1}
    \end{example}            
    is explanatorily basic.  (\ref{sent1}) imputes a sortal,
    ``diamond'' to our friend $d$.  It is plausible to think that
    $d$'s diamondhood is explicable in terms of the arrangement of its
    parts.  But the argument does not establish that (\ref{sent1})
    appears among the explanatorily basic facts.  The argument only
    shows that, among facts involving $d$, at least one is
    explanatorily basic; it does not show that any particular fact
    involving $d$ is explanatorily basic.  For this reason, the
    argument cannot show that there is no metaphysical utility in
    pursuing the explanation of such facts as (\ref{sent1}) in, say,
    partly microphysical terms.  For the same reason, there may even
    be a reasonable sense in which $d$ might be said to be ``less
    fundamental'' than the particles of which it is made: certain
    features of $d$ are, perhaps, explicable in part by reference to
    the particles, but not vice versa.  In sum, our argument does not
    provide any reason to doubt that claims like (\ref{sent1}) have
    explanations at all.  It only shows that such explanations cannot
    be used to show that asserting the existence of $d$ is
    ontologically innocent.
    
    Further, no claim about the epistemological status of the
    explanatorily basic $d$-involving facts is required by the
    argument.  For all the argument says, the missing facts might be
    knowable \emph{a priori}.  If so, when explaining $d$'s
    hardness to a suitably sharp audience, the crucial facts might
    go without saying.  Our argument shows that they still have to be
    among the explanatorily basic facts, no matter what their
    epistemological or conversational status.  The alleged \emph{a
      priority} of basic $d$-involving facts does not buy an
    ontological free lunch.  We cannot evade the argument, then, by
    suggesting that the needed facts involving $d$ are \emph{a
      priori}, or have some other epistemological or semantic status
    that makes them easy to overlook.
  \end{section}
  
  \begin{section}{The Determination Constraint Reconsidered}\label{sec:dc_defense}
    The best response to the argument is to deny the determination
    constraint.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Denying the determination constraint would not require us to
      abandon the claim that there is some sense in which the
      explanatorily basic facts fix everything else; there are lots of
      supervenience relations in the literature which would cause no
      trouble.  For instance, we could still affirm
      \emph{coincidents-friendly} supervenience (see
      \citep[p. 88]{Zimmerman:1995} and \citep{Rea:1997}), or weaker
      forms of global supervenience, including \emph{weak} (see
      \citep[p. 227]{Stalnaker:1996}, \citep[p. 214]{McLaughlin:1997},
      and \citep[p. 915]{Sider:1999}) and \emph{intermediate} global
      supervenience (see \citep[p. 503]{Bennett:2004}).}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Is rejecting the determination constraint
    plausible?  It might be argued that it sets too stringent a
    standard.  But note that lots of plausible explanatory proposals
    satisfy the determination constraint.  For instance, the
    explanation
    \begin{example}
       $A$ and $B$'s heights average 5'6'' because $A$ is
        5'4'' and $B$ is 5'8''
    \end{example}
    passes easily.  Any possible pair of individuals who have $A$'s
    and $B$'s respective heights will also share their average height.
    So the determination constraint does not set a standard that
    couldn't be met in principle.
    
    The determination constraint is motivated by general reflections
    on what a complete explanation must provide.  The idea that
    animates the determination constraint is that a complete
    explanation of an entity $x$'s having some feature $F$ must show
    why that entity, unlike, say, some non-$F$ entity $y$, is $F$.  It
    must therefore mention what distinguishes $x$ from $y$ in respect
    of $F$-ness; more colloquially, together with the explanation for
    $y$'s lacking $F$, it must provide the means for saying what
    \emph{makes them different} in this respect.
    
    The determination constraint is also motivated as a generalization
    from cases.  Consider again the explanatory proposal, concerning a
    particular oxygen nucleus $x$,
    \begin{list}{(\ref{sent2})}
      \item \setlength{\listparindent}{0pt} \raggedright $x$ is a
        stable nucleus in virtue of being an oxygen nucleus.
    \end{list}
    \noindent This proposal is transparently inadequate, given that
    there are radioactive oxygen nuclei.  The proposal does not
    suffice to explain why this nucleus, unlike those radioactive
    nuclei, is stable; it doesn't tell us what makes this nucleus
    different from them with respect to radioactivity.  It's
    inadequacy is a direct result, then, of its failure to meet the
    determination constraint.
    
    This is no isolated phenomenon.  Read any substantial swath of
    philosophy, and you will encounter explanatory proposals of the
    form 
    \begin{description}
      \item[(Exp)] $x$ is $F$ because $\phi(x)$.
    \end{description}
    You will also encounter arguments against such proposals of the
    form:
    \begin{itemize}
      \item[] $y$ is not $F$, but $y$ is such that $\phi(y)$\vspace{-0.3cm}
        \\\underline{\hspace*{2.5in}}\vspace{-0.2cm}
      \item[] (Exp) is at best incomplete.
    \end{itemize}
    The determination constraint says, in effect, that these arguments
    are valid.  Consider a crude example.  A utilitarian ethicist
    might propose that certain courses of action, like giving a
    substantial proportion of your income to OXFAM, are obligatory in
    virtue of the fact that they maximize utility.  Others have
    objected that there are confounding cases for this explanatory
    proposal: courses of action, like framing and punishing an
    innocent person to stop a crime wave, that maximize utility but
    are not obligatory.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {See \citep{Carritt:1950}.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} The utilitarian may dispute the premise, arguing,
    for instance, that framing the innocent in such circumstances is
    obligatory.  The utilitarian may amend the original explanatory
    proposal, arguing, for instance, that giving to OXFAM is
    obligatory in virtue of being enjoined by a rule the adoption of
    which maximizes utility.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {See the discussion in \citep{Smart:1973}.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} The utilitarian may \emph{not} blithely accept
    the premise and stick with her explanatory proposal.  The argument
    is valid, just as the determination constraint requires.
    
    To take a less crude example, consider the state of the debate
    over material constitution.  In the standard test case, a lump of
    clay Lumpl and a statue Goliath are coincident throughout the
    entirety of their careers.  They are made of the same particles,
    subject to the same physical pushes and pulls, \emph{etc.}
    \emph{Pluralists} about material constitution hold that Lumpl and
    Goliath are nevertheless distinct, typically on the grounds that
    they differ in their sortal and modal properties.  For instance, a
    pluralist typically argues that Lumpl differs from Goliath in that
    Lumpl, unlike Goliath, can survive being squashed.  A common
    objection\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {In fact, Wasserman \citeyearpar{Wasserman:2002} dubs it ``the
      standard objection.''}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} to pluralism is that the pluralist cannot explain
    the sortal and modal differences he alleges between Lumpl and
    Goliath.  This argument, called \emph{the grounding problem}, uses
    an instance of the determination constraint.  The idea is that
    Lumpl and Goliath are indiscernible with respect to all of the
    salient \emph{explanans}.  Lumpl, like Goliath, is composed of
    certain particles, in certain arrangements, and bearing certain
    causal and spatiotemporal relations to other things.  Thus, on the
    pluralist's view, any perspicuously articulated explanatory
    proposal of the form
    \begin{example}
      Goliath cannot survive squashing because it is composed of
      particles $p_1, \dots, p_n$ in such-and-such arrangement\label{sent6}
    \end{example}
    will find a confounding case presented by Lumpl, which can survive
    squashing.  This is a reason, as the determination constraint
    says, to think that any such explanation is at best
    incomplete.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Thus, \citep[p. 107]{Fine:2008}:
      \begin{quote}
        For if I use the fact that a given object $\phi$'s, for
        example, to explain why it has the modal profile that it does,
        then I had better be sure that a coincident object with a
        different modal profile does not also satisfy $\phi$.
      \end{quote}
    }
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
    
    Faced with this problem, pluralists either supplement the
    explanatory proposal (see, for instance, \citep{Fine:2008}),
    suggest that the salient sortal and modal features of Lumpl and
    Goliath are fundamental (see, for instance,
    \citep{Bennett:2004b}), or deny that Lumpl and Goliath are
    discernible in the relevant ways (see \citep{Sider:2008b}).  But
    if the determination constraint is rejected, they needn't bother:
    they can just blithely accept that the \emph{explanans} clause
    fits Lumpl as well as Goliath, even though Lumpl lacks the modal
    and sortal features in question.  This response is evidently
    unreasonable.  The determination constraint should not be
    rejected.
    
    Perhaps, however, there is an alternative constraint on good
    explanations that captures the inadequacies of the explanations we
    have been discussing, but cannot be used to establish the failure
    of (\hyperlink{EXPL}{EXPLANATION}).  Consider once again the transparently
    inadequate explanatory proposal
    \begin{list}{(\ref{sent2})}
      \item \setlength{\listparindent}{0pt} \raggedright $x$ is a
        stable nucleus in virtue of being an oxygen nucleus.
    \end{list}
    \noindent I have diagnosed the failure of this explanatory
    proposal as a failure to meet the determination constraint.  But
    it might be suggested that a better diagnosis appeals instead to a
    failure of entailment: The suggested diagnosis of the inadequacy
    of (\ref{sent2}) is that its \emph{explanans} does not entail its
    \emph{explanandum}.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {The sense of entailment in question is strict modal entailment:
      $P$ entails $Q$ in this sense iff $\square(P\Rightarrow{}Q)$.
      Thanks to Chad Carmichael and an anonymous referee for
      emphasizing the need to discuss this suggestion.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} The idea is that the explanation fails on the
    grounds that it is possible that $x$ be an oxygen nucleus that's
    unstable, and so the fact that $x$ is an oxygen nucleus does not
    entail that $x$ is stable.  In general, the suggestion is that
    explanatory proposals are subject to what I will call the
    \emph{entailment constraint}:
    \begin{description}
    \item[(Entailment Constraint)] An explanatory proposal of the
      form, ``$d$ has feature $F$ because $\phi(d, x_1, \dots, x_n)$''
      is at best incomplete if it is possible that $(\phi(d, x_1,
      \dots, x_n) \wedge \neg{}Fd)$.
    \end{description}
    
    Notice that the entailment constraint is strictly weaker that the
    determination constraint.  If the possibility that \emph{something
    or other} is a confounding case suffices for the inadequacy of a
    proposal to explain some feature of $d$, then the possibility that
    $d$ \emph{itself} is a confounding case will also suffice.  But
    an explanatory proposal can meet the entailment constraint without
    meeting the determination constraint.  Consider, yet another
    proposal to explain the fact that diamond $d$ is hard:
    \begin{example}
      $d$ is hard because $R(p_1, p_2, \dots)$\label{sent4}
    \end{example}
    where $R(p_1, p_2, \dots)$ exhaustively specifies the arrangement
    and other physical features of the totality of fundamental
    particles in the universe.  Nothing in the argument of
    \S\ref{sec:arg} rules out the idea that $R(p_1, p_2, \dots)$
    entails all of the facts, including the fact that $d$ is hard.
    So, nothing in that argument tells against the contention that
    (\ref{sent4}) passes the entailment constraint.  But the argument
    did establish that (\ref{sent4}) failed the determination
    constraint.  Since it is so weak, the entailment constraint cannot
    be used to show the falsity of (\hyperlink{EXPL}{EXPLANATION}).
    
    But the entailment constraint is also too weak to cover the cases
    plausibly diagnosed by the determination constraint.  It is
    plausible to think that the entailment constraint explains why
    (\ref{sent2}) is not a good explanation only if it is also
    plausible to think that $x$'s stability is merely contingent: if
    $x$ is necessarily stable (if it exists), then it is impossible
    for $x$ to be radioactive, and so also impossible for $x$ to be
    radioactive and an oxygen nucleus.  Truth be told, I suspect that
    any oxygen nucleus that is in fact stable is necessarily so,
    though, of course, the idiosyncrasies of my modal opinions are no
    sound basis for the determination constraint.  Still, it is easy
    to come up with explanatory proposals with the same kind of
    inadequacy as (\ref{sent2}) which pass the entailment constraint.
    Even skeptics about the necessary stability of $x$ may accept the
    necessity of $x$'s being an oxygen nucleus.  If so, then
    \begin{example}
      $x$ is an oxygen nucleus in virtue of the fact that $x$ contains
      at least two protons
    \end{example}
    will pass the entailment constraint.  Still, given the existence
    of, \emph{e.g.}, lithium nuclei, this explanatory proposal is
    obviously inadequate.  Similarly, notice that the entailment
    constraint won't help us pose the grounding problem for the
    pluralist.  Consider again the explanatory proposal
    \begin{list}{(\ref{sent6})}
      \item \setlength{\listparindent}{0pt} \raggedright Goliath
        cannot survive squashing because it is composed of particles
        $p_1, \dots, p_n$ in such-and-such arrangement.
    \end{list}
    \noindent The feature of Goliath whose explanation is in question
    is typically taken to be a feature that Goliath had to have.
    Being unable to survive squashing is no merely contingent feature
    of Goliath; Goliath not only cannot survive squashing, it
    necessarily cannot survive squashing.  If this plausible view is
    correct, the entailment constraint will not correctly diagnose the
    inadequacy of this explanatory proposal.
    
    In general, explanatory proposals that are, intuitively, at best
    incomplete but meet the entailment constraint can be generated
    using this recipe:
    \begin{enumerate}
    \item Find a case in which some necessary feature $F$ of an
      individual $x$ is plausibly explained by facts $G_1x, G_2x,
      \dots$.
    \item Make sure the case is also one in which there might have
      been some individual $y$ which has $G_1$ but lacks $F$ and any
      of the other $G$'s.
    \item The explanatory proposal that meets the entailment
      constraint by is at best incomplete will, then, be:
      \begin{example}
        $x$ is $F$ in virtue of the fact that it is $G_1$.
      \end{example}
    \end{enumerate}
    Here's one last example, derived using this recipe.  Consider the
    paperweight sitting on the desk in front of me.  This paperweight
    has, as a matter of necessity, both a rest mass and a location.
    Now consider the explanatory proposal
    \begin{example}
      This paperweight has both a rest mass and a location in virtue
      of having a location.\label{sent12}
    \end{example}
    This proposal is obviously at best incomplete, since the
    paperweight's rest mass is not accounted for at all.  The
    determination constraint says why, given the possible existence,
    \emph{e.g.}, of particles that have locations, but no rest mass
    (as photons actually do, according to some physical theories).
    The entailment constraint will not deliver this result, because
    the paperweight's necessary features are trivially entailed by any
    feature that suffices for its existence.  By virtue of its
    weakness, the entailment constraint is not an acceptable
    replacement for the determination constraint.
    
    In summary, abandoning the determination constraint presents two
    challenges.  First, we must state a plausible alternative
    constraint on adequate explanation with two features: (\emph{i})
    like the determination constraint, it correctly diagnoses the
    inadequacy of the explanatory proposals discussed in this section;
    but (\emph{ii}) unlike the determination constraint, it cannot be
    used to establish the failure of (\hyperlink{EXPL}{EXPLANATION}).  Second, we must
    deny the plausible view that the complete grounds for one entity's
    being $F$, together with the complete grounds for another entity's
    lacking $F$, provides the means for saying what makes these two
    particular entities different in this particular way.  
  \end{section}
  
  \begin{section}{Explaining Existence}\label{sec:existence}
    I have, from the very beginning, been calling an entity
    \emph{derivative} iff neither its existence nor any of its other
    features is explanatorily basic.  The argument so far has been
    carried out in terms of this notion of derivativeness.  An
    objector might contend that this notion of derivativeness is not
    the notion relevant to the application of Ockham's Razor in
    ontology.  The relevant notion of ``derivative'' is weaker: where
    the notion I have been working with requires that both the
    existence \emph{and other features} of a derivative entity be
    explicable in other terms, the objector's notion requires only
    that its existence be so explicable.  To avoid terminological
    confusion, I will keep using \emph{derivative} (and
    \emph{fundamental}) as I have been, and call an entity
    \emph{existentially derivative} iff its existence can be
    completely explained solely by reference to the existence and
    properties of other things.  Call an entity \emph{existentially
      fundamental} iff it is not existentially derivative.  The
    objector may then contend that we should reject (\hyperlink{EXPL}{EXPLANATION}) in
    favor of\hypertarget{EXPL_E}{}
    \begin{description}
    \item[(EXPLANATION)${}_\exists$] Many entities are
      \emph{existentially} derivative: their existence can be
      explained solely by reference to the existence and properties of
      other things.
    \end{description}
    Further, existentially derivative entities are an ``ontological
    free lunch,'' so we should endorse an even weaker form of Ockham's
    Razor
    \begin{quote}
      Do not multiply existentially fundamental entities beyond necessity!
    \end{quote}
    
    When this objection is stated in this raw, unvarnished way, it
    might appear \emph{ad hoc}.  But this appearance can be dispelled
    by attending to a particular case.  Consider a group of marbles
    $g$, whose constituents are exactly the marbles $m_1, \dots, m_n$.
    It is plausible to think that
    \begin{example}
      $g$ exists because $m_1, \dots, m_n$ are marbles grouped
      together\label{sent11}
    \end{example}
    is a good explanation.  Nothing more, it might seem, is required
    for $g$ to exist than for its constituents to be grouped
    together.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {This particular case is due to Mark Moyer, but a similar case has
      been suggested independently by Fabrice Correia and Benjamin
      Schnieder.  Thanks to Correia, Moyer, and Schnieder for
      suggesting the line of response explored in this section.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} But, if this explanation is good, then $g$ is
    existentially derivative.  The objector is suggesting that there
    is a large and diverse array of entities that are like $g$ in this
    respect, and that commitment to them is ``no addition to being'',
    given commitment to the existentially fundamental entities.
    
    Let's turn our attention to \hyperlink{EXPL_E}{(EXPLANATION)${}_\exists$}.  Is it
    true?  There are two ways of defending \hyperlink{EXPL_E}{(EXPLANATION)${}_\exists$}
    against the sort of argument I have given against (\hyperlink{EXPL}{EXPLANATION}).
    The first defense is to notice that, as it stands, the argument of
    \S\ref{sec:arg} does not generalize to show that
    \hyperlink{EXPL_E}{(EXPLANATION)${}_\exists$} fails.  Consider again
    \begin{list}{(\ref{sent11})}
      \item \setlength{\listparindent}{0pt} \raggedright $g$ exists
        because $m_1, \dots, m_n$ are marbles grouped together.
    \end{list}
    \noindent Notice that no analogue of (\hyperlink{DD}{DERIVATIVE DIVERSITY}) for
    this sort of explanation is true: there is no entity $e$ that
    lacks existence.  For the same reason, there is no (actual)
    confounding case for this explanatory proposal.  There is nothing
    that does not exist, and so \emph{a fortiori} nothing that is just
    like $g$ so far as the arrangement of $m_1, \dots, m_n$ goes, but
    lacking existence.  In the case of explanations of existence, the
    argument of \S\ref{sec:arg} gets no purchase.
    
    It turns out that a variant of the old argument can be mounted
    against \hyperlink{EXPL_E}{(EXPLANATION)${}_\exists$}.  The new argument relies on two
    new assumptions.  The first new assumption is that the explanation
    of the existence of an existentially derivative entity like $g$
    should also provide fodder for the explanation of the
    \emph{actual} existence of $g$.  Consider again the proposed
    explanation of $g$'s existence in terms of the grouping of $m_1,
    \dots, m_n$.  This explanatory proposal can be perspicuously
    articulated by a claim of the form
    \begin{example}
       $g$ exists because $R(m_1, \dots, m_n)$\label{sent9}
    \end{example}
    This explanation concerns the existence of $g$ in the actual
    world.  We can therefore generate an explanation for the actual
    existence of $g$ from the fact that it is actually the case that
    $R(m_1, \dots, m_n)$.  In brief, if the explanatory proposal
    (\ref{sent9}) is good, then so is
    \begin{example}
       $g$ \emph{actually} exists because \emph{actually} $R(m_1,
      \dots, m_n)$\label{sent10}
    \end{example}
    Call this assumption \emph{actualization}.
    
    The second new assumption constrains the range of candidates for
    explaining $g$'s existence.  Let $g$, $e$, \etc, exhaust the
    actual existent entities.  The second assumption, which I'll call
    \emph{permissiveness}, is that the facts (if any) that explain
    $g$'s existence do not entail that the only entities are $g$,
    $e$, \etc{}\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Strictly speaking, the assumption needed for the argument is that
      there is at least one existentially derivative entity whose
      existence does not have an explanatory basis which entails that
      there are no entities other than $g$, $e$, \etc{} But it is
      plausible that $g$ fits the bill if anything does.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} The assumption of permissiveness implies that no
    explanation for $g$'s existence rules out a world in which all
    facts in the \emph{explanans} obtain and yet there are some
    \emph{aliens}: some objects which don't actually
    exist.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Permissiveness will be rejected by anyone who rejects the
      possibility of aliens (\eg, \citep{LinskyZalta:1994}). It will
      also be rejected by anyone who thinks that the explanation for
      $g$'s existence includes a ``that's all'' fact, to the effect
      that there are no entities other than the actual entities;
      see \citep{ChalmersJackson:2001}.  So the explanatory basis for
      $g$'s existence, according to permissiveness, does not entail
      the non-existence of aliens.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} For instance, if (\ref{sent9}) is good, then
    permissiveness says that there is a possible world in which a
    non-actual entity exists and $m_1, \dots, m_n$ are grouped in the
    way specified by $R$.  Permissiveness is very plausible in this
    case.  The grouping of $m_1, \dots, m_n$ does not necessitate the
    nonexistence, for instance, of alien concrete objects elsewhere.
    
    Here, then, is a sketch of the variant of the argument.  Suppose
    for \emph{reductio} that
    \hyperlink{EXPL_E}{(EXPLANATION)${}_\exists$} is true.  Then some
    claim of the form of (\ref{sent9}) perspicuously articulates a
    good explanation of $g$'s existence.  Apply actualization: an
    explanatory proposal of the form
    \begin{list}{(\ref{sent10})}
      \item \setlength{\listparindent}{0pt} \raggedright $g$
        \emph{actually} exists because \emph{actually} $R(m_1, \dots,
        m_n)$
    \end{list}
    \noindent is also good.  Now apply permissiveness.  There is a
    world $w$ containing an alien $a$, and in which $R(m_1, \dots,
    m_n)$.  The alien $a$, together with $m_1, \dots, m_n$ present a
    possible confounding case for (\ref{sent10}): at $w$, $m_1, \dots
    m_n$ actually stand in $R$, but $a$ does not actually exist.  The
    determination constraint says an explanation is at best incomplete
    if it has possible confounding cases, so (\ref{sent10}) is not
    good.  Contradiction.
    
    Given actualization and permissiveness, the determination
    constraint implies that \hyperlink{EXPL_E}{(EXPLANATION)${}_\exists$} faces the same
    sort of problem as (\hyperlink{EXPL}{EXPLANATION}).  The argument can be avoided by
    denying either actualization or permissiveness.  But each of the
    new assumptions has considerable plausibility in the present case.
    We have, however, another defense of \hyperlink{EXPL_E}{(EXPLANATION)${}_\exists$}
    close to hand: deny the relevant applications of the determination
    constraint.  This is the second way to defend
    \hyperlink{EXPL_E}{(EXPLANATION)${}_\exists$} against the argument of \S\ref{sec:arg}.
    I have already argued that the determination constraint is
    supported as a generalization from cases.  But none of the cases
    in question concerned explanations of existence.  Thus, those
    cases can be accommodated if we deny applications of the
    determination constraint to explanations of the \emph{existence}
    of entities (and to their \emph{actual} existence), but accept its
    applications to explanations of other features.  The argument
    against \hyperlink{EXPL_E}{(EXPLANATION)${}_\exists$} crucially depends on the
    application of the determination constraint to the explanation of
    $g$'s (actual) existence.  If that application fails, then the
    argument is unsound.
    
    % Fabrice/Benjamin's Resp: DC doesn't apply to existence expl's.
    
    My worry about this way of defending \hyperlink{EXPL_E}{(EXPLANATION)${}_\exists$} is
    that denying the application of the determination constraint to
    explanations of existence threatens to trivialize its application
    to explanations of other features; call this the
    \emph{trivialization worry}.  Suppose we are confronted with an
    explanatory proposal of the form
    \begin{example}
       $x$ has $F$ because $P$
    \end{example}
    that fails the determination constraint.  If the determination
    constraint's applications to explanations of existence are
    rejected, then this explanatory proposal can be rehabilitated by
    claiming that what's being explained is instead an existential
    claim.  For instance, the proposal to explain the obligatoriness
    of giving a substantial proportion of your income to OXFAM can be
    recast as an explanation of the existence of a \emph{duty} to
    give:
    \begin{example}
      There exists a duty for you to donate a substantial portion of
      your income to OXFAM in virtue of the fact that doing so would
      be utility-maximizing.\label{sent14}
    \end{example}
    Similarly, a proposal to explain Goliath's modal or sortal
    features can be recast as a proposal to explain the existence of a
    certain \emph{essence} for Goliath.  More generally, a proposal to
    explain any entity's having some feature $F$ can be recast as a
    proposal to explain the existence of a certain $F$-ness trope
    possessed by it.  Such an explanatory proposal, on the view under
    consideration, can be good despite there being a confounding case.
    But it's obvious that the new explanation has just the same
    problem as the old one: recasting the explanation in existential
    terms makes the proposal no better.  The applications of the
    determination constraint to explanations of (actual) existence say
    why.
    
    A defender of \hyperlink{EXPL_E}{(EXPLANATION)${}_\exists$} should agree that these
    explanations of existence facts are at best incomplete, even
    though the restricted version of the determination constraint
    doesn't, by itself, deliver this verdict.  But, the defender might
    urge, there is reason to deny that the problematic explanatory
    proposal can avoid the determination constraint by being recast as
    explanations of existence facts.  Consider again the proposal to
    ground the existence of a duty to donate a substantial portion of
    your income to OXFAM in the fact that such a donation would be
    utility maximizing.  The defender may argue that the existence of
    a duty is to be explained by reference to the obligatoriness of
    courses of action: if you have a duty to donate, then that duty
    exists in virtue of it being obligatory for you to donate.  The
    proposed explanation of the existence of the duty in terms of
    utility maximization must, then, be understood as resulting from
    two explanatory claims:
    \begin{example}
      It is obligatory for you to donate to OXFAM in virtue of the
      fact that it would maximize utility \label{sent13}
    \end{example}
    and
    \begin{example}
      There exists a duty for you to donate in virtue of the fact that
      donation is obligatory for you.
    \end{example}
    The restricted version of the determination constraint still
    applies to (\ref{sent13}), which is inadequate given the
    possibility of confounding cases.  Similarly, the defender might
    argue, the existence of an essence for Goliath must be explained
    by appeal to Goliath's modal or sortal properties, and the
    existence of $F$-ness tropes must be explained by the relevant
    individual's being $F$.  The determination constraint, restricted
    so as not to apply to explanations of existence, will still apply
    to the explanations of features other than existence that, the
    defender urges, are more explanatorily basic than the existence
    facts in question.  So the trivialization worry is misplaced: the
    exception for explanations of existence facts cannot generally be
    used to skirt the determination constraint.
    
    This response to the trivialization worry commits the defender of
    \hyperlink{EXPL_E}{(EXPLANATION)${}_\exists$} to some pretty controversial theses
    concerning how the existence of duties, tropes, and essences are
    to be explained.  But let's suppose that the response is wholly
    convincing, and \hyperlink{EXPL_E}{(EXPLANATION)${}_\exists$} turns out to be true.
    There is now reason to deny that the existentially derivative
    entities are ``no addition to being,'' and so provide an
    ``ontological free lunch.''  Consider, for instance, a view on
    which the existence of mental states is completely explained by
    facts involving only microphysical entities, but those mental
    states had lots of explanatorily basic non-physical features,
    including irreducible phenomenological characters, primitive
    downward causal relations to microphysical entities, and the like.
    This is a view on which the only existentially fundamental
    entities are physical.  Is it also a view which vindicates
    physicalism, the idea that, given the physical entities and
    features, mental states are ``no addition to being''?  Evidently
    not.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Extant physicalists agree.  See, for instance,
      \citep{Wilson:2005} for discussion.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} This is a view on which mental states make a
    substantial ``addition to being.''  \emph{Existential
      derivativeness} is not the notion the physicalist needs to
    explicate the idea that there is nothing over and above the
    physical.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Similarly, if grounding everything in the physical allows that
      some features of mental states cannot be explained in purely
      physical terms, then grounding is not the notion needed to
      explicate physicalism.  This would contradict Schaffer's claim
      at \cite[pp. 364-5]{Schaffer:2009}.  (I believe that Schaffer is
      committed to the idea that grounding an entity requires that no
      facts involving it be explanatorily basic; see
      n. \ref{fn:grounding}.  If that's right, then Schaffer's actual
      view is not subject to this criticism, but also cannot rest on
      (EXPLANATION)${}\exists$.)}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} Existentially derivative entities are no
    ``ontological free lunch'' if some of their features cannot be
    explained in other terms.  
    
    But that is just what this defense of \hyperlink{EXPL_E}{(EXPLANATION)${}_\exists$}
    allows.  The defender is granting in effect that the determination
    constraint applies to the proposed explanations of some facts
    involving the possession of a property $F$ by $d$.  The fact that
    $d$ is $F$ is not an existence fact.  And the defense against the
    trivialization worry grants that we cannot evade the determination
    constraint, restricted so as not to apply to explanations of
    existence facts, by recasting a proposal to explain this fact as a
    proposal to explain the existence of something, \emph{e.g.}, an
    $F$-ness trope.  Thus, the argument of \S\ref{sec:arg} still
    applies, and so a fact to the effect that $d$ possesses some
    particular feature is explanatorily basic.  For this reason, the
    defense against the trivialization worry does not vindicate the
    contention that existentially derivative entities are an
    ``ontological free lunch.''
    
    The defender of \hyperlink{EXPL_E}{(EXPLANATION)${}_\exists$} who
    proposes to restrict the application of the determination
    constraint so that it does not apply to explanations of existence
    facts thus faces a dilemma.  Either the exception for explanations
    of existence facts allows us to skirt the determination constraint
    for the explanation of other kinds of facts, or it does not.  If
    it does, then the determination constraint is trivialized, and
    doesn't rule out explanations that are evidently at best
    incomplete.  If it doesn't, then
    \hyperlink{EXPL_E}{(EXPLANATION)${}_\exists$} cannot be used to
    show that the existentially derivative entities are an
    ``ontological free lunch''.
    
    In summary, there are two ways in which one might hope to use
    explanations of existence to evade the argument of
    \S\ref{sec:arg}: (\emph{i}) notice that the analogue of
    (\hyperlink{DD}{DERIVATIVE DIVERSITY}) fails in the case of explanations of
    existence; or (\emph{ii}) suggest that the determination
    constraint should be restricted so as not to apply to such
    explanations.  Response (\emph{i}) faces a version of the original
    argument.  Response (\emph{ii}) either falsely implies that
    recasting bad explanations as explanations of existence facts
    improves them, or won't secure the result that existentially
    derivative entities are an ``ontological free lunch.''
    
    This still leaves unresolved the question of what, if anything, is
    wrong with the proposal to explain the existence of the group of
    marbles $g$ in terms of the grouping of its constituent marbles
    $m_1, \dots, m_n$.  To tell the truth, I am unsure what to say
    about such a case.  In particular, I am unsure what is entailed by
    the claim that $g$ exists.  If it turns out that $g$'s existence
    does not tolerate replacement of its constituents over time, then
    I am tempted to think that its existence is reducible to $m_1,
    \dots, m_n$'s being grouped together.  If it turns out that $g$'s
    existence does tolerate replacement of constituents, so that there
    can be Ship of Theseus-style cases,\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {See \citep{Salmon:1979} and \citep{Noonan:1985a} for discussion.
      The Ship of Theseus is a ship $S_1$ that was originally
      constructed from some planks $p_1, p_2, \dots, p_n$.  These have
      all been replaced one-by-one with completely different planks
      $p_{n+1}, p_{n+2}, \dots, p_{2n}$.  It is plausible to think
      that in some such case, it is possible that the post-replacement
      and pre-replacement ships are identical.  If $g$ tolerates
      replacement of its constituent marbles in a similar manner, then
      it is plausible to think that $m_1, \dots, m_n$ can be traded
      out one-by-one, and then grouped to form a second group of
      marbles $g^*$.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} then matters are more complicated.  I suspect
    that Ship of Theseus-style cases show that the proposed
    explanation is inadequate,\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Argument: if a Ship of Theseus-style case is possible for $g$,
      then $m_1, \dots, m_n$ can be grouped together to form an alien
      group $g^*$.  It is plausible that, if those marbles explain
      $g$'s existence, then they also explain its existence at every
      time at which it in fact exists.  But, if they can be grouped to
      form $g^*$, then there can be a time at which the marbles are
      grouped as they actually are, and yet $g$ does not exist.  This
      strikes me as a compelling objection to the original explanatory
      proposal: the grouping of marbles at a given time fails to
      necessitate $g$'s existence at that time.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} but I am unsure how to supplement the proposal.
    
    Perhaps we need to add the claim that $g$ is composed of those
    marbles; $g$ exists not just because some marbles are grouped
    together, but because \emph{its} marbles are grouped together.  It
    is tempting to characterize this proposal by appeal to some notion
    of essence.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {Thanks to Benjamin Schnieder.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3} The idea would be that the existence of a
    material thing is explained by some congeries of facts involving
    other things, together with some specification of the essence of
    the thing.  In the case at hand, the claim is that the essence of
    $g$ is given by the fact that it is composed of $m_1, m_2, \dots,
    m_n$.  The existence of $g$ is explained by the fact that $g$ has
    this essential feature. together with the fact that those marbles
    are grouped together.  A notion of essence suited to play this
    role might be helpfully characterized in a variety of ways.  For
    instance, if the argument of \S\ref{sec:arg} succeeds, we can
    define a notion of essence that will do the trick: the
    \emph{essence} of a thing is the totality of its features that are
    explanatorily basic.  It is often held that the essential features
    of a thing play an important role in explaining its further
    features.  This suggestion enshrines this claim as definitive of
    essence.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {The exploration of this notion of essence is a task for another
      day.  For two further alternatives, see \citep{Almog:1991} and
      \citep{Fine:1994, Fine:2008}.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
    
    So far I have suggested that $g$'s existence be explained in part
    by appeal to its essence.  Perhaps, instead, we should accept that
    $g$'s existence is explanatorily basic.  After all, given
    analogues of (\hyperlink{IRRED}{IRREDUCIBILITY}) and
    (\hyperlink{DD}{DERIVATIVE DIVERSITY}), the application of the
    determination constraint to the explanation of any of $g$'s
    features other than existence implies that some of the facts
    involving $g$ are explanatorily basic.  Why not its existence?  It
    should be noted that the argument of \S\ref{sec:arg} can be
    adapted to show that $g$'s existence is not the only fact
    involving $g$ that is explanatorily basic.  Consider, for
    instance, a perspicuously articulated explanatory proposal of the
    form
    \begin{example}
      $d$ is $F$ because $d$ exists and $R(t_1,\dots,
        t_n)$,
    \end{example}
    where none of the terms $t_1,\dots, t_n$ denote $d$.  (DERIVATIVE
    DIVERSITY) ensures that $e$ is a confounding case for this
    explanatory proposal: $e$ exists, is such that $R(t_1,\dots,
    t_n)$, but is not $F$.  So, if the argument is sound, it shows not
    only that some fact involving $d$ is explanatorily basic, but also
    that some fact involving $d$ \emph{other than the fact that $d$
      exists} is explanatorily basic.
  \end{section}
  
  \begin{section}{Conclusion}\label{sec:conclusion}
    The argument of this paper has a general result: barring
    reduction, there do not exist two non-fundamental, qualitatively
    discernible entities.  I have focused on the consequences of this
    result for ontology.  In particular, I have argued that
    (\hyperlink{EXPL}{EXPLANATION}) cannot be used to claim that
    grounding explanations provide an ``ontological free lunch.''  But
    the general result bears on a more popular conception of reality.
    On this conception, reality comes in layers: at the bottom are
    (perhaps) the physical entities.  Higher up, we find chemical,
    biological, geological, psychological, sociological, economic,
    \emph{etc.}, entities: molecules, human beings, diamonds, mental
    states, nations, interest rates, and so on.  We might eliminate
    certain of these alleged higher-level entities from our ontology,
    denying their existence altogether; and we might hold that certain
    facts involving others are reducible.  But, on the layering
    conception, after we have eliminated and reduced as much as we
    can, the features of the remaining higher-level entities are to be
    explained entirely in terms of the existence and features of
    entities at lower levels.\linespread{1}\footnote
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    {See \citep{Kim:1993} for a seminal critical discussion of this
      conception; in Kim's discussion, the higher-level entities in
      question are mental properties and the lower-level entities are
      physical properties.}
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
    \linespread{1.3}
  
    If the argument of this paper is correct, this conception faces a
    problem.  Barring reduction and elimination, the features of
    molecules, human beings, diamonds, \emph{etc.} cannot be
    explained solely in terms of other things.  Grounding explanations
    still allow us to acknowledge that some \emph{facts} are more
    basic than others.  But if there is any such thing as a diamond,
    then, barring reduction, some facts involving diamonds are
    explanatorily basic.  Whether all of the explanatorily basic facts
    concerning erstwhile derivative entities are physical facts, or
    fit into any interesting category, is a matter for further
    investigation.
  \end{section}
  
  
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