## 1. Quantifier Accounts of Ontological Commitment


### 1.1 Quine's Criterion, Preliminaries


Quine's criterion of ontological commitment has dominated
ontological discussion in analytic philosophy since the middle of the
20th century; it deserves to be called the orthodox
view. For Quine, such a criterion played two distinct roles. First,
it allowed one to measure the ontological cost of theories, an
important component in deciding which theories to accept; it thus
provided a partial foundation for theory choice. Moreover, once one
had settled on a total theory, it allowed one to determine which
components of the theory were responsible for its ontological costs
(see Quine (1960: 270) for the purported benefits of such ontological
bookkeeping). Second, it played a polemical role. It could be used to
argue that opponents' theories were more costly than the theorists
admitted (see Church (1958), who wields the criterion polemically
against Ayer, Pap, and Ryle). And it could be used to advance a
traditional nominalist agenda because, as Quine saw it, ordinary
subject-predicate sentences carry no ontological commitment to
properties or universals.


The *locus classicus* for Quine's criterion is "On
What There Is". Quine writes: 



> 
> A theory is committed to those and only those entities to which the
> bound variables of the theory must be capable of referring in order
> that the affirmations made in the theory be true.
> [2] 
> (Quine 1948: 33)
> 
> 
> 


To illustrate: if a theory contains a quantified sentence
'[?]*x* Electron(*x*)', then the bound
variable '*x*' must range over electrons in order
for the theory to be true; and so the theory is ontologically
committed to electrons. (For an introduction to the logic of
quantifiers and bound variables, the basics of which are presupposed
in this article, see Shapiro (2013).)


Talk of "commitment" has an unfortunate connotation: it
applies more naturally to persons than to theories. But Quine's
criterion should be understood as applying to theories primarily, and
to persons derivatively by way of the theories they accept (Quine
1953: 103). It would be more perspicuous to speak simply of
the *existential implications*, or *ontological
presuppositions*, of a theory. But 'ontological
commitment' is well entrenched, and it would be pointless to try
to avoid it.


It is important to note at the start that Quine's criterion
is *descriptive*; it should not be confused with
the *prescriptive* account of ontological commitment that is
part of his general method of ontology. That method, roughly, is this:
first, regiment the competing theories in first-order predicate logic;
second, determine which of these theories is epistemically best (where
what counts as "epistemically best" depends in part on
pragmatic features such as simplicity and fruitfulness); third, choose
the epistemically best theory. We can then say: one
is *ontologically committed* to those entities that are needed
as values of the bound variables for this chosen epistemically best
theory to be true. Put like this, the account may seem circular:
ontological commitment depends on what theories are best, which
depends in part on the simplicity, and so the ontological commitments,
of those theories. But there is no circularity in Quine's ontological
method. The above account of ontological commitment is prescriptive,
and applies to persons, not to theories. What entities *we ought to
commit ourselves to* depends on a prior descriptive account of
what entities *theories are committed to*. The prescriptive
account of ontological commitment serves as a crucial premise in the
Quine-Putnam indispensability argument for the existence of abstract
entities (the *locus classicus* is Putnam 1971). This article,
however, will focus entirely on descriptive criteria of ontological
commitment.


It is also important to distinguish at the outset
the *ontological* commitments of a theory from
its *ideological* commitments (following Quine
1951a). The ontological
commitments of a theory are, roughly, what the theory says exists; a
theory is ontologically committed to electrons, for example, if the
truth of the theory requires that there be electrons. The ideological
commitments of a theory are, roughly, those concepts, logical or
non-logical, that are expressible within the theory. Ontological
commitment and ideological commitment are largely independent of one
another. For example, a theory may be ontologically committed to there
being *K*s, even if the concept of being a *K* is not
expressible in the theory. In the other direction, a theory may be
ideologically committed to relational concepts that
add *structure* to the entities posited by the theory without
adding new kinds of entity to the ontology. There has been a trend in
recent metaphysics to take ideological commitment to be as
important--if not more important--than ontological
commitment with respect to providing a criterion of theory choice, at
least for fundamental theories (see especially Sider 2011:
12-15). In this article, however, only ontological commitment
will be discussed.


### 1.2 *Relata* of Ontological Commitment


Ontological commitment, on its face, is a relation between theories
and entities, or kinds of entity. Before attempting to explicate and
evaluate Quine's criterion of ontological commitment, it is important
to specify more carefully the *relata*. A *theory*, for
Quine, is just a set of sentences (or a single sentence) of some
language. But what language? Strictly speaking, Quine's criterion is
only to be applied to theories couched within the following
especially austere form of first-order predicate logic: there are
first-order existential and universal quantifiers but no other
variable-binding operators; there are predicates, including identity,
but no function symbols or individual constants. Thus, all names and
definite descriptions have been eliminated from the language;
reference rests entirely with the variables. The elimination of
singular terms serves both to simplify the criterion and to avoid
age-old confusions. Unlike bound variables, singular terms sometimes
do and sometimes do not carry ontological commitment: 'Socrates
is wise' presumably does; but 'Pegasus does not
exist' presumably does not. Whenever singular terms do carry
ontological commitment, it will be reflected in the assignment to the
variables of the regimented theory. Thus, whenever a sentence
containing a definite description is ontologically committed to some
entity satisfying the description, so will be the regimented sentence
that results from applying Russell's theory of descriptions (which
was endorsed by Quine). Whenever a sentence containing a name is
ontologically committed to some entity denoted by the name, so will
be a regimented sentence that contains a predicate applying
exclusively to that entity. For example, if instead of using the name
'Socrates' we use a predicate 'Is-Socrates'
interpreted to mean *is-identical-with-Socrates* (sometimes
abbreviated 'Socratizes'), the ontological commitment
incurred by the name is transferred without loss to the bound
variable in '[?]*x*
Is-Socrates(*x*)' (or '[?]*x*
Socratizes(*x*)').[3] 


Unless noted otherwise, 'theory' will always mean
'interpreted theory'; uninterpreted theories say nothing
and have no ontological commitments. When the theory is couched within
first-order logic, the interpretation is given by standard Tarskian
semantics. In particular, each monadic predicate of the language of
the theory has been assigned a class as its interpretation;
each *n*-ary predicate has been assigned a class
of *n*-tuples; the quantifiers are interpreted objectually by
way of assignments to the variables. On the other hand, it is not
supposed that the domain of the quantifiers has been fixed once and
for all. Sometimes, in applying the criteria, alternative domains are
considered, and the theory is evaluated for truth under the various
alternatives.


The restriction to theories couched within first-order predicate
logic is severe. Philosophers have been no less interested in the
ontological commitments of theories couched within other logical
frameworks, such as higher-order or intensional logics
(see SS1.7.2
and SS1.7.3). Worse, if the criterion is to
have application to the actual ontological debates among philosophers,
it will have to apply to theories couched within ordinary language as
well. This led Quine--at least sometimes--to endorse
an *extended criterion* of ontological commitment: a theory is
ontologically committed to an entity or kind of entity if and only if
every acceptable paraphrase of the theory into (austere) first-order
predicate logic is ontologically committed to that entity or kind of
entity. This provides philosophers with a method for *avoiding*
ontological commitment: find an acceptable paraphrase of the theory
into first-order predicate logic that is *not* ontologically
committed to the questionable entities (Quine 1948: 32, 1953:
103-104). (What counts as an "acceptable
paraphrase", and whether any sort of "paraphrase"
should be required, is discussed
in SS4.2.)


What about the second *relatum*? What is it that theories
are committed to? Sometimes we say that a theory is committed to a
particular entity. For example, we say that the sentence
'Socrates exists' (or, in our austere language,
'[?]*x* Socratizes(*x*)') is committed
to Socrates. But more often we speak of ontological commitment to some
entities in the plural, or to some *kind* of entity. For
example, we speak of theories being committed to universals, or
numbers, or electrons. But ontological commitment to some entities
plurally, or some kind of entity, is clearly not the same as
ontological commitment to each of the entities singly. Consider a
theory that contains '[?]*x*
Elephant(*x*)'. It is ontologically committed to
elephants, or to the kind *elephant*, but not to any particular
elephant: no particular elephant need be the value of the variable
'*x*' in order for the sentence to be true. A
theory is committed to *K*s, or a kind *K*, just in case
the values of the variables must include *at least
one* *K* in order for the theory to be true, in which case
the existence of *K*s is *in some sense* entailed by the
theory. (More on different relevant senses of entailment
in SS1.5
and SS2 below.) 


But there is a problem (Cartwright 1954; Scheffler and Chomsky
1958; Jackson 1989; Michael 2008). Suppose first that ontological
commitment takes a plural argument in its second place, that a
commitment to *K*s is a relation to the *K*s
themselves. Then, on plausible assumptions, the meta-ontologist
cannot assert that a theory is ontologically committed to *K*s
without being committed to *K*s herself; for the
meta-ontologist's own domain of quantification must include at least
one *K* in order for '*T* is ontologically
committed to *K*s' to be true. And that is wrong.
Suppose instead, then, that ontological commitment is a relation, not
to the *K*s, but to the *kind K*. If a kind *K*
can exist without *K*s existing, this may help. But, on
plausible assumptions, it still has the untoward consequence that the
meta-ontologist cannot herself assert that a theory is ontologically
committed to a kind *K* without being committed
to *kinds*. And, whether kinds are construed as properties, or
sets of *possibilia*, or taken to be *sui generis*,
that still seems wrong. This suggests that the logical form of
statements attributing ontological commitment is misleading, and that
ontological commitment is not a genuine relation at all; only then
will meta-ontological claims about ontological commitment themselves
be ontologically neutral, or as ontologically neutral as
possible.[4] In
this article, the locutions 'ontological commitment
to *K*s' and 'ontological commitment to the
kind *K*' will be used without presupposing that
ontological commitment is a relation to the *K*s plurally, or
to the kind *K*, or, indeed, that it is a relation at all. The
different accounts of ontological commitment will treat the matter
differently.


Finally, it will be useful in what follows to interpret
'*K*' and 'kind' broadly. So, for
example, *not-being-an-electron* is a kind,
and *being-an-electron-or-a-chair* is a kind, and so is being
one of some random collection of entities. We may not be interested in
asking whether a theory is ontologically committed to negative, or
disjunctive, or gerrymandered kinds; but it is harmless to allow the
criterion to supply answers. Moreover, this allows us to assimilate
ontological commitment to single entities to ontological commitment to
kinds of entity: to be ontologically committed to a single
entity, *a*, is just to be ontologically committed to the
kind, *is-identical-with-a*.


### 1.3 Quine's Criterion: Extensional Versions


We turn now to the task of providing a precise formulation of
Quine's criterion. This task is none too easy: formulations that keep
within the strictures of Quine's overall philosophy are woefully
inadequate; the farther one is allowed to stray from Quinean
strictures, the better things look; but, by the time one gets to a
defensible criterion, one has left Quine far behind. I will consider
in turn extensional, metalinguistic, and modal formulations.


Quine's statement of his criterion (quoted above) appears to
be *modal*. It asks: what entities *must* be taken to
be values of the bound variables in order for the sentences of the
theory to be true? But given Quine's general hostility to modality
(e.g., in Quine 1953: 139-159), it is all but certain that he
thought the 'must' was eliminable. Moreover, Quine
explicitly claimed that his account of ontological commitment was
within the theory of reference, not the theory of meaning (Quine
1953: 130-131). This suggests that Quine had in mind a purely
extensional criterion, and an extensional individuation of kinds.
Indeed, Quine provides an extensional criterion for a special case:
the ontological commitments of an existential
sentence.[5] He
writes:



> 
> To say that a given existential quantification presupposes objects
> of a given kind is to say simply that the open sentence which follows
> the quantifier is true of some objects of that kind and none not of
> that kind. (Quine 1953: 131)
> 
> 
> 


This criterion seems to give reasonable results for *true*
existential sentences. Thus, '[?]x
Elephant(*x*)' is committed to
("presupposes") elephants because the open sentence
'Elephant(*x*)' applies to all and only elephants,
and there are some. It is likewise committed to any kind that contains
all the elephants: to mammals, to animals, to living things. (It has
these commitments whether or not these kinds are expressible in the
object language.) And it is not committed to any kind that leaves out
any elephants, such as *male elephant*. The
kind *elephant* is the smallest kind to which
'[?]*x* Elephant(*x*)' is committed. So
far, so good.


But when the criterion is applied to *false* existential
sentences, trouble immediately arises. For example, since there are no
centaurs, the open sentence 'Centaur(*x*)' is not
true of any objects of any kind, and so '[?]*x*
Centaur(*x*)' is not ontologically committed to any kind
of entity. The same goes for '[?]*x*
Unicorn(*x*)', and all other false existential
sentences. But, then, all false existential sentences have the same
ontological commitment, namely, none. And that's wrong on two counts:
'[?]*x* Centaur(*x*)' and
'[?]*x* Unicorn(*x*)' *are*
ontologically committing; and their ontological commitments are not
the same.


Let us return, then, to Quine's favored formulation of the
criterion involving the 'must' and explore whether it is
compatible with an extensionalist account. Indeed, ordinary uses of
'must' and other modals are often used to signal
generality without making the content of what is asserted modal.
Consider the following example: "A set of five numbers between 1
and 9, inclusive, must contain at least one odd number". The
content is simply that every set of five numbers between 1 and 9,
inclusive contains at least one odd number. (Perhaps the
'must' is also epistemic, but not in a way that affects
the content of what is asserted.) Similarly, Quine's criterion can be
interpreted as innocent quantification: 



> 
> *Quine's Criterion, Extensional Version*. A
> theory *T* is *ontologically committed* to *K*s
> iff every domain of quantification that makes the theory true contains
> at least one *K*. 
> 
> 
> 


(In considering whether a domain makes the theory true, the
interpretation of the predicates occurring in the theory is
restricted to the domain if need be.) This account is purely
extensional: if a theory is committed to a kind *K*,
and *K'* has exactly the same members as *K*,
then the theory is also committed to kind *K*'. And that
will be its downfall. For, as above, '[?]*x*
Centaur(*x*)' and '[?]*x*
Unicorn(*x*)' will have the same ontological
commitments. The only difference is that, because the condition is
universal rather than existential, the criterion will have both of
these sentences be (vacuously) committed to every kind of entity. No
improvement there.[6]


What to do? Of course, this result could be avoided by populating
domains of quantification with centaurs, unicorns, and all manner of
fictional being. That would allow one to charge a profligate opponent
with an ontological commitment to centaurs, and a distinct ontological
commitment to unicorns--but only by taking on these very same
ontological commitments in the metalanguage. For Quine, at least, the
cure is as bad as the disease.


Alternatively, one could restrict the criterion to *true*
theories. That would still allow the criterion to be used to allocate
ontological costs, to say which theories are responsible for which
costs. Polemical uses of the criterion would have to be foresworn; but
perhaps they are problematic in any case. The primary role for the
criterion, however, would have to be given up: its role in providing a
foundation for theory choice. For if the criterion of ontological
commitment is to be applied only to true theories, then one would have
to determine which theories are true prior to applying the
criterion. One might as well have as one's criterion of theory choice:
"Choose true theories!" So, the idea of restricting
ontological commitment to true theories is a non-starter. Could one
instead restrict the criterion to the theories *one takes to be
true*? That wouldn't so much solve the problem of false
existentials as merely hide it from view. But, in any case, one would
be no better off with respect to theory choice. One would have to
determine which theories one takes to be true prior to applying the
criterion. And so, again, ontological commitment could not serve as a
foundation for theory choice. One might as well have as one's
criterion of theory choice: "Choose theories you take to be
true!" Restricting the application of the criterion of
ontological commitment is not the solution to the problem of false
existentials.


### 1.4 Quine's Criterion: Metalinguistic Versions


If Quine is to avoid the problem of false existentials without
exiting the theory of reference, he will need to resort to semantic
ascent. Ontological commitment to a kind *K* must somehow
involve the predicate '*K*'. That will allow
ontological commitment to centaurs and ontological commitment to
unicorns to be distinguished, even though 'centaur' and
'unicorn' are co-extensional. Indeed, Quine does suggest
"withdrawing to a semantical plane" when the ontological
commitments of the theory one is addressing outrun the ontological
commitments of one's own theory. He writes, referring to the
ontologically profligate philosopher McX: 



> 
> So long as I adhere to my ontology, as opposed to McX's, I cannot
> allow my bound variables to refer to entities which belong to McX's
> ontology and not to mine. I can, however, consistently describe our
> disagreement by characterizing the statements which McX
> affirms. Provided merely that my ontology countenances linguistic
> forms, or at least concrete inscriptions and utterances, I can talk
> about McX's sentences. (Quine 1948: 35)
> 
> 
> 


One way of implementing this retreat to a semantical plane would be
to interpret Quine's criterion as the following schema (with
'*K*' now a schematic letter ranging over
predicates of the object language):



> 
> *Quine's Criterion, Metalinguistic Version.* A
> theory *T* is *ontologically committed* to *K*s
> iff *T* logically entails '[?]*x*
> K*x*' iff, for every interpretation that
> makes *T* true, there is some entity in the domain of the
> interpretation that is in the extension of
> '*K*'.[7]
> 
> 
> 


Indeed, this captures *some* sense in which the truth
of *T* requires that there be *K*s among the values of
the variables. For example, any interpretation that makes
'[?]*x* Unicorn(*x*)' true will have
some entity in its domain also be in the extension of
'Unicorn'. This entity needn't be a unicorn, of
course. But that's all to the good. It is the price one pays, the
non-believer in unicorns might say, if one wants to be able to say
that a theory is ontologically committed to unicorns. But note that
the notion of ontological commitment is now entirely formal. Because
interpretations are generalized over, the actual interpretation of
'*K*' plays no role. Ontological commitment is
essentially intra-linguistic, rather than a relation between language
and the world. 


Suppose, for the moment, that one could live with the world being
well lost. There are still objections to the adequacy of the
metalinguistic
 criterion.[8] 
First, there is the problem that, in
making entailment purely formal, all interconnections between kinds
have been lost: ontological commitment to bachelors won't carry with
it ontological commitment to males; ontological commitment to
elephants won't carry with it ontological commitment to
animals. There are no constraints on the joint interpretation of
'bachelor' and 'male', or of
'elephant' and 'animal'. One can get these
connections back, however, by adding additional postulates
to *T*, such as '[?]*x*(Bachelor(*x*)
[?] Male(*x*))', or
'[?]*x*(Elephant(*x*) [?]
Animal(*x*))'. (These might or might not be taken to be
meaning postulates, or metaphysically necessary postulates; for Quine
they would just be more theory.) This has the awkward result that the
original theory *T* is not ontologically committed to males if
it just contains '[?]*x*
Bachelor(*x*)', or to animals if it just contains
'[?]*x* Elephant(*x*)'; only *T*
with the postulates added is so committed. But perhaps one can live
with that.


A second, more serious problem is that ontological commitment is
entrapped in the object language: if '*K*' is not
a predicate of the object language, we have no way of assigning
ontological commitment to *K*s. Say that the *explicit*
ontological commitments of a theory are given by the existential
sentences that the theory logically
entails.[9] The
metalinguistic criterion above takes all ontological commitment to be
explicit ontological commitment. But ontological commitment can fail
to be explicit--or so it appears--in one of two ways. A
theory may have *implicit* ontological commitments by
entailing existential sentences or propositions according to some
entailment relation more generous than logical entailment
(see SS1.5
and SS2). Or a theory may
have *implicit* ontological commitments by entailing
existential propositions that are not expressible in the language of
the theory. For example, a theory containing '[?]*x*
Elephant(*x*)', one might hold, is ontologically
committed to animals, whether or not the language of the theory
contains the predicate 'Animal'. A defender of the
metalinguistic criterion, however, should simply deny that there are
implicit ontological 
commitments.[10] 
As with the previous objection, we can
enhance the theory by adding new predicates and new postulates
involving the new predicates. Indeed, we can suppose that the object
language is as rich as the metalanguage with respect to
non-semantical predicates. Again, it is awkward to have to say that
only the enhanced theory in the enhanced language, not the original
theory in the original language, carries ontological commitment to
animals. But, again, perhaps we can live with that. (For more on the
problem of implicit commitment, see 1.7
and SS2.1)


The real problem lies elsewhere. Although there is a sense in which
according to the criterion a theory that is ontologically committed
to *K*s must have *K*s among the values of its
variables, it appears to be *the wrong sense*. If the
interpretation of '*K*' is not held fixed, then the
elements of the domain that are in the extension of
'*K*' have nothing to do with whether *K*s
exist, as '*K*' is actually interpreted. It may
seem a good thing that ontological commitment to unicorns has nothing
to do with unicorns--since there aren't any. But now ontological
commitment to elephants also has nothing to do with elephants, and
that's wrong. The ontological commitments of a theory should depend on
the meanings of its terms, not just on its logical form. Could one
instead go for a hybrid criterion, metalinguistic for empty kinds,
extensional for non-empty kinds? (This is suggested by Quine's talk
of "retreating to a semantical plane".) That would only
compound the problem. For then we could not know which criterion to
apply to a theory prior to knowing what exists, and the criterion
could not play a foundational role in ontology. 


### 1.5 Modal Quantifier Criteria


It is high time to take the 'must' seriously when
considering what entities must be taken to be values of the bound
variables in order for the theory to be true. The resulting criteria
would not be endorsed by Quine. But, as long as they tie ontological
commitment to the values of the bound variables of theories couched
within first-order logic, they are still Quinean in spirit; they are
still versions of the quantifier account of ontological
commitment.


We begin by interpreting the 'must' as metaphysical
necessity, and interpreting metaphysical necessity as quantification
over all metaphysically possible worlds. According to the simplest
modal account, a theory is (perhaps implicitly) committed
to *K*s whenever the existence of *K*s is necessarily
implied by the theory. This account isn't even Quinean in
spirit--it makes no mention of the values of the
variables--but it serves as a useful foil to Quinean accounts,
and will come in for later discussion
(SS2.2). It is assumed that each possible
world has an associated domain of entities: roughly, all the entities
that exist in the world. (These entities, however, need not all be in
the range of the first-order quantifiers.) Moreover, it is assumed
that when sentences of first-order predicate logic are evaluated
relative to a possible world, the sentences' quantifiers range only
over entities in the domain of the world in question. This leads to
what will be called a *modal entailment criterion* (other
entailment criteria make use of entailment relations other than
metaphysical necessitation): 



> 
> *Modal Entailment Criterion.* A theory *T*
> is *ontologically committed* to *K*s iff, for every
> world *w*, *T* is true at *w* only if the domain
> of *w* contains at least one *K*.
> 
> 
> 


In other words, a theory is ontologically committed to *K*s
just when it necessarily implies that *K*s
exist.[11]


Unlike extensional criteria, the modal entailment criterion has no
problem with (contingently) false existentials. The sentence
'[?]*x* Unicorn(*x*)' is ontologically
committed to unicorns because, at every world at which it is true,
the domain of that world contains at least one unicorn. In saying
this, the meta-ontologist may or may not herself be ontologically
committed to unicorns depending on how the meta-ontologist construes
talk of possible worlds and possible individuals. A *modal
realist*--someone who takes *possibilia* to be
concrete--will be so committed. But an *ersatz modal
realist*--someone who takes *possibilia* to be in
some sense abstract--will interpret "the domain of that
world contains at least one unicorn" in a way that does not
commit the meta-ontologist to unicorns; abstract worlds
can *represent* unicorns as existing whether or not unicorns
exist.[12]


The modal entailment criterion, however, appears to be inadequate
with respect to the vast majority of philosophical applications, at
least as metaphysical necessity is standardly interpreted
(but see SS2.2 for possible lines of
defense). This is because many philosophically controversial kinds of
entity are taken to exist necessarily if they exist at all, for
example, numbers, or universals. But necessarily existing kinds of
entity make trouble for the criterion: for any theory *T*, any
world at which *T* is true is a world whose domain contains
numbers. So any theory *T* is ontologically committed to
numbers, whether or not the content of *T*, intuitively, has
anything to do with numbers. This suggests that we return to a Quinean
criterion that ties ontological commitment directly to the values of
the variables; numbers need not be values of the variables for a
theory *T* that has nothing to do with numbers. To make the
variable assignments relevant, we can quantify doubly over worlds and
sub-domains of worlds. This leads to the *modal quantifier
criterion*:



> 
> *Modal Quantifier Criterion.* A theory *T*
> is *ontologically committed* to *K*s iff, for every
> world *w* and every sub-domain of the domain
> at *w*, *T* is true at *w* with its variables
> ranging over that sub-domain only if that sub-domain contains at least
> one *K*.
> 
> 
> 


(This criterion requires, in effect, that whenever a theory is true
of some *part* of a world, considered in isolation, the theory
is ontologically committed to *K*s only if that part contains
at least one *K*.) The modal quantifier criterion takes care
of the problem of irrelevant necessary existents. In effect, it
introduces a narrower modal entailment relation that is more
sensitive to the content of *T*. Only if *T* entails
the existence of *K*s according to this narrower entailment
relation is *T* ontologically committed
to *K*s.[13]


But some less pressing objections remain that suggest that
metaphysical necessity is too blunt an instrument to capture the
notion of ontological commitment. Three will be mentioned here.
First, just as extensional criteria could not distinguish between
ontological commitments to co-extensional kinds, the modal criteria
cannot distinguish between ontological commitments to co-intensional
kinds. A theory that is committed to triangular polygons will also be
committed to trilateral polygons, and *vice versa*, because,
necessarily, a polygon is triangular if and only if it is
trilateral. A theory that is committed to round squares will also be
committed to even prime numbers greater than two, and *vice
versa*, because, necessarily, (it is vacuously true that)
something is a round square if and only if it is an even prime number
greater than two. Second, metaphysically impossible theories, theories
that are true at no world, will be (vacuously) committed to every kind
of entity. But that seems wrong: naive set theory is committed to
sets, but not to angels that dance on the head of a pin. Third, the
modal criteria may introduce spurious commitment if there are
metaphysically necessary connections between distinct kinds of
entity. For example, if whales are necessarily mammals, then a theory
committed to whales would automatically be committed to mammals, even
if the theory explicitly claims that there are no mammals and that
whales are fish. A theory committed to Hesperus would automatically be
committed to Phosphorus, assuming the necessity of identity, even if
the theory explicitly claims that Hesperus exists and that Phosphorus
doesn't.


All of these objections to modal criteria for ontological
commitment have a common source: ontological commitment appears to be
sensitive to distinctions that metaphysical necessity cannot
accommodate. One solution, then, would be to seek a more
discriminating entailment relation. Thus, the third objection
suggests that *a priori* entailment should replace
metaphysically modal entailment. In the modal quantifier criterion,
one could quantify over domains and sub-domains of worlds that
are *conceptually*, rather than *metaphysically*,
possible.[14]
That won't help with the first two objections, however, since the
existence of triangular polygons is *a priori* entailed by the
existence of trilateral polygons, and, assuming classical logic
is *a priori*, the existence of angels (and everything else)
is *a priori* entailed by naive set theory (or any
inconsistent theory). For these, we can retreat further to some sort
of relevant entailment relation within a paraconsistent logic. The
quantifier criterion could be expressed by quantifying over domains
and sub-domains of worlds that are both conceptually and
metaphysically
 impossible.[15]
 Alternatively, one could revert to an
entailment account, since relevant entailment by itself can handle
the problem of irrelevant necessary existents. It is not the case
that any theory *relevantly* entails that numbers exist. 


Perhaps the retreat to a sufficiently weak entailment relation will
result in a workable criterion. Another response, however, holds the
line at modal
 entailment.[16] 
It claims that all the above objections
are based on a confusion between the ontological commitments of a
theory and the ontological beliefs of someone who holds the theory.
Although there needs to be some hyper-intensional notion of belief
according to which someone could believe that trilateral polygons
exist without believing that triangular polygons exist, or believe
that naive set theory is true without believing that angels exist, or
believe that whales are fish, we should not identify the ontological
commitments of a theory with the ontological beliefs of those that
accept the theory. On this account, even an ideally rational person
might accept a theory without being able to determine its ontological
commitments. More worrisome, if the ontological commitments of
theories do not parallel the ontological beliefs of a person who
accepts the theory, then the foundational role of ontological
commitment in theory choice will be diminished. (For an extended
discussion of entailment accounts,
see SS2.)


### 1.6 Evaluating Quantifier Accounts: Sufficiency


This section and the next provide an evaluation of quantifier
accounts of ontological commitment. It is still assumed, following
Quine, that the target theory is couched within first-order predicate
logic. (For discussion of ontological commitment in ordinary language,
see SS4.) The critique is divided into
two parts: in this section, it is asked whether the quantifier
criterion is *sufficient* for ontological commitment; in the
next section, it is asked whether it is *necessary*.
Objections that were considered above in presenting specific versions
of the quantifier criterion will not be repeated.


A criterion fails to be *sufficient* for ontological
commitment if it *overgenerates*, if it sometimes holds that a
theory is committed to some kind of entity that the theory should not
be committed to. I suppose that any quantifier criterion introduces
some sort of entailment relation between theories and existence
claims, an entailment relation that depends upon the values of the
bound variables of the sentences of the theory. (That's what
distinguishes "quantifier accounts" from the broader
"entailment accounts" to be discussed
in SS2.) Now ask: How could a theory
entail that *K*s exist without being ontologically committed
to *K*s? How could the sufficiency of the quantifier criterion
fail to be entirely trivial? Is there room to deny that existential
consequences of a theory, as given by sentences of first-order logic
with the existential quantifier, do not amount to ontological
commitments of the theory? Four ways that one might challenge
sufficiency will be considered. The first two require only
clarifications or minor adjustments to a quantifier criterion. The
third and fourth, however, raise philosophically substantial
issues.


#### 1.6.1 Inclusive Logic/Free Logic


 The first objection to sufficiency is not so much a critique of
the criterion, but of the logic to which it is applied. For Quine
first-order predicate logic excludes the empty domain: its valid
formulas are those that come out true under all interpretations of the
non-logical symbols over all *non-empty* domains. It follows
from this that, on the quantifier account, universal sentences carry
the exact same ontological commitments as their existential
counterparts. It follows because, in classical first-order predicate
logic, universal sentences have existential import:
'[?]*x* ph(*x*)' logically entails
'[?]*x* ph(*x*)'. If we want to allow
'[?]*x* ph(*x*)' to be true even
when there are no phs (because there is nothing at all), then
we do not want it to carry any ontological commitment. Ontological
commitments should reside entirely with the existential
quantifier. Implementing this is easy. We simply do logic so as to
include interpretations with an empty
domain--so-called, *inclusive logic*. According to the
truth conditions for quantifiers in inclusive logic, all universal
sentences are true in an empty domain, and all existential sentences
are false. Once we have made the shift to inclusive logic, we can also
say, what seems right, that conditional existential
sentences--such as, '[?]*x* ph(*x*)
[?] [?]*x* *y*(*x*)'--carry no
ontological commitment.


Consider now Quine's insight, on which the quantifier account is
based, that it is bound variables rather than singular terms that
carry ontological commitment. To implement this insight, Quine simply
eliminated singular terms from the language. But suppose one wanted
to be able to apply the quantifier criterion to theories containing
singular terms. One could instead switch from standard first-order
predicate logic to *free logic*. In free logics, because one
cannot in general derive the existential generalization
'[?]*x* ph(*x*)' from
'ph(*t*)', for singular term
'*t*', singular terms fail to carry ontological
commitment. Arguably, then, an inclusive, free logic provides the
ideal logical environment for applying the quantifier
account.[17]


#### 1.6.2 Substitutional vs. Objectual Quantifiers


 A second objection to the sufficiency of the quantifier account
relates to the interpretation of the quantifiers of first-order
logic. If the quantifiers are interpreted *substitutionally*,
instead of *objectually*, the criterion may
overgenerate. According to the objectual interpretation (due to
Tarski), an existentially quantified sentence '[?]*x*
ph(*x*)' is true if and only if the open formula
'ph(*x*)' is satisfied by some object in the
domain. According to the substitutional interpretation,
'[?]*x* ph(*x*)' is true if and only
if 'ph(*t*)' is true, for some singular term
'*t*' of the language. When the quantifiers are
given a substitutional interpretation, Quine's criterion will
overgenerate if the language has empty singular terms. For example, a
theory that contains (a symbolic translation of) the sentence
'Sherlock Holmes is a fictional detective' would be
ontologically committed to fictional objects, and that seems
wrong.


This objection might have force against the application of a
quantifier criterion to ordinary language if it could be argued that
the quantifiers of ordinary language are sometimes (or always)
substitutional.[18] 
But it is not in any way an objection to
a quantifier criterion *as Quine conceived of
it*. Quine *stipulated* that the quantifiers of the
first-order language were to be interpreted objectually--indeed,
the substitutional interpretation cannot be applied to Quine's
austere language with no singular terms--and he explicitly
acknowledged that, if interpreted substitutionally, all questions of
ontological commitment are moot (Quine 1968: 106, 1974: 139). Thus,
this could only be a critique of a quantifier account of ontological
commitment if it could be argued that the objectual interpretation
were somehow incoherent, or somehow had to be reduced, ultimately, to
the substitutional interpretation. That would undermine the
quantifier criterion, to be sure; it would reverse the Quinean order
of things, requiring that names, not bound variables, be the primary
locus of reference. But justifying such a position would be a tall
order.


#### 1.6.3 Meinongian Quantifiers


 According to the third objection to sufficiency, the quantifiers
of first-order logic, properly understood, do not carry existential
commitment; they are not "existentially loaded". Indeed,
calling '[?]*x*' the "existential
quantifier" is a misnomer; it would be better to call it the
"particular quantifier" in contrast with the
"universal quantifier". Ordinary language, on its face,
supports the view that quantification need not be existentially loaded
(see SS4). For example, if we assert
"some fictional detective is smarter than Sherlock
Holmes", we do not ordinarily take ourselves to be committed to
the existence of fictional entities. Similarly, the objection claims,
the straightforward translation of this into first-order logic using
'[?]*x*' for 'some' should carry no
existential import. One only gets existential import when one
restricts the quantifier, explicitly or implicitly, with an existence
predicate. Thus, the correct translation into first-order logic of
"dragons exist" is not '[?]*x*
Dragon(*x*)', but '[?]*x*
(Exists(*x*) and Dragon(*x*))'. The quantifier
'[?]*x*', by itself, is neutral with respect to
existence. (Pronounce '[?]*x*' "for
some *x*", not "there exists
an *x*".)


Distinguishing a restricted quantifier that has existential import
from an unrestricted quantifier that does not only has a point if
existence is a substantial property that some things have and other
things lack. Barack Obama has it; Sherlock Holmes does not. The
planet Venus has it; the postulated planet Vulcan does not. This
objection to Quine's criterion, then, is based on the Meinongian view
that some things do not
 exist.[19] 
These things that do not exist typically
are taken to include fictional entities, postulated entities of
science, and all manner of intentional objects; sometimes
(mere) *possibilia*, *impossibilia*, and various
abstract entities are also included among the
non-existent. Non-existent entities can be quantified over, referred
to, and truly attributed properties. Existence is not a prerequisite
for being talked about.


Quine and his followers claim not to understand this substantial
property of existence, a property that is conceptually independent of
the quantifiers (Quine 1948; van Inwagen 1998). For Quine, the
existence predicate, 'Exists(*x*)', should be
defined simply as '[?]*y* *y*
= *x*'. Thus, for Quine, to say that some thing doesn't
exist is to say that some thing is such that no thing is identical
with it--a logical contradiction. But, Quine claimed, whether or
not to accept the Meinongian account of existence is a sideshow when
it comes to the question of ontological commitment. In responding to
the fictional Meinongian Wyman, Quine offers "to give Wyman the
word 'exist'. I'll try not to use it again; I still have
'is'" (Quine 1948: 23). That is to say, theories are
ontologically committed to *K*s if they entail that *there
are K*s, whether or not they entail that *Ks
exist*. Indeed, Quine could also have given the (radical)
Meinongian the word 'is' and any other predicate that
purports to impose an ontological restriction on the quantifier; for
what is central to Quine's criterion is that one
cannot *quantify* over entities without incurring ontological
commitment to those entities. To use quantifiers to refer to entities
while denying that one is ontologically committed is to fail to own up
to one's commitments, and thereby engage in a sort of intellectual
doublethink. Quantification is the basic mode of reference to objects,
and reference to objects is always ontologically committing.


A Meinongian *need* not quarrel with any of this; she could
accept that the unrestricted quantifier is ontologically committing,
even though it carries no existential import. In that case, the
Meinongian view would not be a challenge to the sufficiency of Quine's
criterion. But some Meinongians (e.g., Routley 1982) explicitly deny
that they are ontologically committed to what they quantify over when
their quantifiers are not existentially loaded; they explicitly reject
the orthodox Quinean doctrine of ontological commitment. In trying to
understand this claim, the Quinean faces a problem of
interpretation. Since the Quinean rejects the distinction between
existentially loaded and existentially unloaded quantifiers, he has to
decide which of these to translate as his one and only
quantifier. Routley assumes that his loaded quantifier should be
identified with Quine's one and only quantifier, because that
quantifier is likewise loaded for Quine. If we take him at his word,
then we say that, when using his unloaded quantifier, he quantifies
without incurring ontological commitment; but that is to make his
statements incoherent by Quinean lights, since for the Quinean it is
constitutive of the quantifier that it is bound up with ontology. (As
Lewis (1990: 27) says, he "quantifies without
quantifying".) It is better, then, to translate his
existentially unloaded quantifiers as unrestricted quantifiers, and
his existentially loaded quantifiers as quantifiers restricted by an
existence predicate. That does not eliminate the mystery in his view:
the Quinean still claims not to understand the property that the
existence predicate supposedly expresses. But this is preferable to
attributing the outright incoherence that comes with saying he has a
radically different understanding of the quantifier. On this second,
better interpretation, the Meinongian has a radically expansive
ontology filled with fictional entities, *possibilia*, and
perhaps *impossibilia* as well. Although the Meinongian may
claim to be a "noneist", the Quinean's best way of
understanding him makes him an "allist". On this way of
understanding the Meinongian, the quantifier account of ontological
commitment emerges unscathed.


#### 1.6.4 Realism and Reduction


 A fourth objection to sufficiency has to do with how the criterion
relates to issues of realism and reduction, to the distinction
between kinds of entity that are fundamental and kinds that are
derived. It might seem that in cases of ontological reduction the
quantifier criterion sometimes imputes ontological commitments where
there should be none. Consider a view that identifies chairs with
sums of particles, and takes only the particles, not their sums, to
be fundamental. Consider the sentence '[?]*x*
Chair(*x*)', and ask: is it ontologically committed to
chairs? The answer, on the quantifier criterion, is
"yes": chairs must be among the values of the variable
'*x*' in order for the sentence to be
true.[20] But,
one might object, since chairs are not *fundamental* on the
view in question, they should not be counted among the
sentence's *ontological* commitments. One can hold that there
are chairs without being *ontologically* committed to
chairs.


The objection is misguided if directed at the quantifier
criterion *as it has standardly been interpreted* (and was
interpreted by Quine). The quantifier criterion was not designed to be
sensitive to whether the entities assumed by a theory are fundamental
or derived (see SS3.2 below). If the
values of the bound variables must include *K*s, then the
theory is ontologically committed to *K*s, whether *K*s
are fundamental or derived. That is simply how the criterion is to be
understood. If we want a way to distinguish between ontological
commitments to fundamental kinds and ontological commitments to
derived kinds, we need to look elsewhere.


Fine has argued, however, that *qua* ontologists we do need
to look
 elsewhere.[21] 
Suppose, with Fine, that there is a
primitive distinction between those entities that are
"fundamentally real", and those that are not. Let
'Real(*x*)' be a predicate that marks this
distinction. (Unlike the Meinongian existence predicate considered
in SS1.7.3, this predicate is not derived
from a special quantifier.) The ontologist's task, then, is to
determine which entities and kinds of entity are fundamentally
real. But the quantifier account, being insensitive to this
distinction, will be of no use in carrying out this task and should
therefore be rejected. The quantifier account provides at best a
criterion for *ordinary* or *scientific* commitment,
but not the specifically *ontological* commitment of interest
to metaphysicians. Indeed, since ordinary and scientific theories do
not contain the predicate 'Real(*x*)', they do not
by themselves carry any ontological commitment. Only philosophical
theories that make claims about what entities or kinds of entity are
fundamentally real carry such commitment. Ontology is an autonomous
discipline; it cannot (as Quine would have it) be subsumed by
science.


Once one accepts that a "reality" predicate is needed
to characterize ontological commitment, a puzzle about the logic of
ontological commitment admits of a natural solution. The puzzle is
this (from Fine 2009). Because the natural numbers are included within
the integers, an ontological commitment to the integers, it seems,
should be a stronger commitment than a commitment to the natural
numbers. But quantifier accounts of ontological commitment get this
backwards: since commitment to *K*s is associated with
the *existential* claim,
'[?]*x* *K**x*', a commitment to
natural numbers entails, and is thus stronger than, a commitment to
integers. This can be set right, Fine holds, by associating
ontological commitment to *K*s with a *universal* claim:
[?]*x* (*K*(*x*) [?]
Real(*x*). Quantifier accounts, because they lack a
"reality" predicate, are unable to capture the logic of
ontological commitment.


A defender of the quantifier account, however, can respond by
noting that there is a distinction between ontological commitment
to *K*s and ontological commitment to *the K*s; the
quantifier account aims only to analyze the former. Indeed, it seems
plausible to say (as noted in SS1.3)
that an ontological commitment to elephants carries with it an
ontological commitment to mammals, just not an ontological commitment
to *the* mammals. An ontological commitment to *the K*s,
on the quantifier account, can be understood as an ontological
commitment to each and every particular *K*.


### 1.7 Evaluating Quantifier Accounts: Necessity


A criterion fails to be *necessary* for ontological
commitment if it *undergenerates*, if it sometimes fails to
hold that a theory is committed to some kind of entity that the theory
should be committed to. Quantifier criteria will undergenerate if they
fail to account for all the implicit ontological commitments of
theories. Whereas the problems canvassed
in SS1.6 for the sufficiency of a
quantifier account can all be addressed by a devoted Quinean, the
problems with necessity are more serious. Indeed, they suggest that
Quinean approaches that focus on the values of the variables of
quantification should be jettisoned for a more general entailment
approach to ontological commitment
(SS2).


#### 1.7.1 Commitment to Properties or Universals


 To play its appointed role, a criterion of ontological commitment
should be ontologically neutral: it lets the theory speak for itself,
as it were, without imposing ontological assumptions from the
outside. Quinean criteria, however, by focusing only on the values of
the (individual) variables, have been accused of being ontologically
biased against realism about properties or universals because they
fail to attach ontological significance to the
predicate.[22]
Consider '[?]*x* Swims(*x*)'. According
to Quinean criteria, this sentence is committed to entities that
swim, but not to a property of swimming; for no property of swimming
need be among the values of the bound variables in order for the
sentence to be true. One way to ensure ontological neutrality is to
switch to an entailment criterion. An entailment criterion--such
as the modal entailment criterion
in SS1.5--avoids bias by allowing
any component of a sentence to contribute to its ontological
commitments, not just the (first-order) quantifiers and bound
variables. If universals must exist in order for a theory to be true,
then the theory is ontologically committed to universals; if not,
not. And the same goes for particulars, or any other ontological
category. This neutrality, however, comes at a price. The ontological
commitments of a theory--indeed, of a simple subject-predicate
sentence--cannot be determined prior to resolving the debate
between nominalists and realists. Ontological commitments will not be
transparent. Even a professed nominalist may in fact be committed to
universals, willy-nilly. (For more on entailment accounts,
see SS2.)


In response to the charge of ontological bias, the Quinean might
try to turn the tables on the objector: to take predicates to carry
ontological commitment would be to be biased *against*
nominalism. If even '[?]*x* Swims(*x*)'
were committed to properties or universals, no theory would be
compatible with nominalism. In any case, the Quinean will reject the
charge of ontological bias. Ontological commitment to universals is
not excluded by the quantifier criteria; a theory couched in a
first-order language whose variables range over universals can be so
committed. For example, the sentence
'[?]*x*[?]*y* (Universal(*y*)
& *y*=*Swim* &
Instantiates(*x*,*y*)' is committed to universals,
as is, more simply, '[?]*y*
Universal(*y*)'. The Quinean, then, simply asks that the
commitment to universals be made explicit in the same way that the
commitment to particulars is made explicit: by quantifying.


#### 1.7.2 Second-Order Languages


Another reason for refusing to allow predicates to carry
ontological commitment parallels Quine's reason for refusing
ontological commitment to names: just as there can be empty names,
there can be empty predicates. (An empty predicate is not a predicate
with the null set for its extension; it is a predicate with no
extension.) If I introduce the name 'Herman' stipulating
that it refers to the one and only monster under my bed, I introduce
an empty name; and sentences containing 'Herman' should
not thereby be ontologically committed to monsters. Similarly, if I
introduce a predicate 'whish' stipulating that it
expresses a biological kind property that applies to whales and fish
but excludes rabbits and rhinoceri, I introduce an empty predicate;
and sentences containing 'whish' should not thereby be
ontologically committed to any biological kinds. Parallel problems
admit of parallel solutions: we can eliminate predicates in favor of
second-order variables, and take ontological commitment, if any, to be
carried by the second-order variables rather than the predicates. (But
we cannot make our second-order language too austere: we need to have
some non-logical descriptive vocabulary to express the (non-logical)
content of the theory.) The charge of ontological bias against Quinean
criteria is simply shifted: why allow *first-order* variables
to carry ontological commitment, but not *second-order*
variables?


For Quine, this question simply doesn't arise. Second-order logic
is illegitimate because it treats predicates as if they were
names:



> 
> Predicates have attributes as their "intensions" or
> meanings (or would if there were attributes), and they have sets as
> their extensions; but they are names of neither. Variables eligible
> for quantification therefore do not belong in predicate
> positions. (Quine 1970: 67)
> 
> 
> 


Second-order logic is "set theory in sheep's clothing":
the ontological commitments of set theory have been hidden under the
guise of logic. To make ontological commitment transparent, Quine
holds, second-order theories must be transformed into first-order
theories by replacing second-order variables with first-order
variables ranging over sets, and taking predication to be set
membership.


For the proponent of second-order theories, no such transformation
is needed: the quantifier criterion can be applied directly to the
values of the second-order variables. The ontological commitments of
second-order theories will then depend on how the second-order
variables are interpreted. If the second-order variables range over
properties ("attributes"), then second-order theories are
committed to properties; if instead they range over sets, then
second-order theories are committed to sets. On either interpretation,
any world at which a second-order theory is true will have either
properties or sets in its domain, and so the modal quantifier
criterion (from SS1.5) can be applied
without amendment. Extending the quantifier criterion to second-order
theories, then, would seem to be within the spirit, if not the letter,
of Quine's account of ontological commitment.


If the (monadic[23]) 
second-order quantifiers are
interpreted as plural quantifiers, as recommended by Boolos
(1984), then, arguably,
they carry no additional ontological commitments beyond those already
carried by the first-order quantifiers. Plural quantification
is *ontologically innocent* according to Boolos: to refer to
some things plurally--some books, for example--is not to
refer to a plural entity--a set, or set-like entity, with books
as members--but to refer to the things--the books
themselves--in an irreducibly plural way.
[24] 
Individual and plural quantifiers range over the same domain. Since
quantifier criteria attribute ontological commitment based on the
domain of quantification, they reveal no hidden ontological cost to
theories couched in an extension of first-order logic that includes
plural quantification. To get this result, we simply extend
quantifier criteria to apply to plural as well as individual
variables. In slogan form: "to be is to be the value of some
variable or the values of some variables" (Boolos 1984). A plural variable takes as
its semantic value a plurality of entities, not a plural entity (such
as a set or class). Plural entities are not needed to serve as
semantic values--not, in any case, if the language in which the
semantics is done is itself a plural language.


Within the framework of plural logic, however, there are two
distinctively different ways that a theory can be ontologically
committed to entities, or kinds of entity, corresponding to the
distinction between distributive and collective predication (see Rayo
(2007: 435-7), who calls the latter "plethological
commitment"). Suppose that '*K*' is a
predicate that applies to plural terms. If '*K*'
applies distributively, then a theory that affirms the plural
quantification "there are some things such that they
are *K*s" introduces no commitment beyond that of the
singular quantification "there is something such that it
is *K*". But if '*K*' is a predicate
that applies collectively, then the plural quantification "there
are some things such they are *K*s" is committed
to *K*s in a distinctively plural way that, in general, cannot
be captured in terms of its ontological commitments. For example, the
mereological nihilist who holds that "there are simples arranged
chairwise" is thereby ontologically committed only to simples,
not to any plural entity containing simples; but she is also
committed, in an irreducibly plural way, to there being simples
arranged chairwise.


#### 1.7.3 Intensional Languages


Quine's criterion is silent on the ontological commitments of
theories couched within intensional languages--languages with
intensional operators such as 'it is possible that' or
'it will be that' or 'it is believed that'. If
it were directly applied to such languages, it would run the risk of
undergenerating ontological commitments: perhaps there are commitments
hidden by the intensional operators, as would be the case if they are
disguised quantifiers. Quine's extended criterion involving
paraphrase can be applied; but now there is a danger of
overgenerating. For paraphrases of theories couched within an
intensional language into first-order predicate logic will almost
inevitably have to treat the intensional operators as disguised
quantifiers, and it may be that any adequate domain for those
quantifiers will require controversial entities that do not correctly
represent the ontological commitments of the original intensional
theory. For example, if modal theories are "translated"
into first-order logic in the standard way, where
'possibly' means 'at some possible world' and
'necessarily' means 'at all possible worlds',
they will be ontologically committed to possible worlds according to
the extended quantifier criterion. But this seems to just beg the
question against the modalist who does not think she is so
committed. (For problems with the extended
criterion, see SS4.2.)


Entailment accounts of ontological commitment
(SS2) get this right, but only by
dodging the hard questions. Thus, if modal operators *are*
disguised quantifiers over possible worlds, then modal theories will
be ontologically committed to possible worlds according to an
entailment account: possible worlds must exist in order for the
theories to be true. If modal operators are *not* disguised
quantifiers over possible worlds, then modal theories need not be
ontologically committed to possible worlds according to an entailment
account. An entailment account will accurately reflect the ontological
truth; but it does nothing to help us discover that truth. 


#### 1.7.4 Implicit Commitment: Extrinsic Properties


 Consider the sentence '[?]*x*
Parent(*x*)'. Any criterion that focuses on the
question--what entities need to be among the values of the
variable '*x*' in order for this sentence to be
true?--appears to get the ontological commitments of the sentence
wrong (see Rayo 2007: 431-2). For the sentence appears to be
ontologically committed to *children* no less than
to *parents*; but children do not need to be taken to be values
of the variable in order for the sentence to be true (assuming, that
is, that it is possible for a parent not also to be a child). We have
here, it seems, a stark division between two approaches to ontological
commitment: the quantifier approach, which looks only to the values of
the variables of the sentences of a theory, and the entailment
approach, which looks to the existential entailments of the theory,
and ignores whether or not the entities whose existence is entailed
are values of the theory's variables. In cases where predicates
express extrinsic properties, the quantifier approach undergenerates;
it misses ontological commitments.


Before addressing the objection from extrinsic properties, it is
important to note an ambiguity in the informal expression of the
quantifier account. When we ask, following Quine, "what entities
must be values of the variables" in order for a
theory *T* to be true, we might mean: what entities must
be *assigned to the variables of the sentences in T* in order
for *T* to be true. Or we might mean: what entities must
be *included in the domain of the interpretation of T* in order
for *T* to be true, whether or not those entities must be
assigned to the variables of the sentences in *T*. When
predicates of the theory express extrinsic properties, this difference
matters. In this article, Quine's criterion--and quantifier
criteria more generally--have always been understood in the
first, stronger way. But those philosophers who take Quine's criterion
to be compatible with an entailment approach to ontological commitment
are best understood as interpreting him in the second, weaker way.


There are two ways that one might try to defend the quantifier
account, neither very satisfactory. First, one might simply restrict
the predicates of the language of the theory to those that express
intrinsic properties (and relations). Perhaps one could argue that the
theories Quine had primarily in mind, theories of mathematics and
natural science, obey this restriction already with respect to their
primitive predicates (though this has been contested even for physics,
and is not plausible for the special sciences). But this would be a
rather severe restriction; and there is no evidence that Quine had
intended the criterion to be restricted in this way. Moreover, it
would needlessly inject controversies over which properties and
relations are intrinsic into the application of the criterion,
threatening both its foundational and polemical roles.


Second, one might dig in one's heels and deny that
'[?]*x* Parent(*x*)' is committed to
children. One familiar gloss on "the ontological commitments of
a theory" takes this to include only "what the theory
says there is". If "says" is interpreted narrowly,
a theory that contained only '[?]*x*
Parent(*x*)' does not *say* that children exist,
even though this follows analytically from the theory. In defense of
the view, one can point out that the intuitive judgment that
'[?]*x* Parent(*x*)' is ontologically
committed to children comes from including the definitions of
'Parent(*x*)' and 'Child(*x*)'
as an unspoken part of the theory. (Compare the defense of the
metalinguistic criterion in SS1.4.)
Thus, if we suppose the language contains, in addition to the unary
predicate 'Parent(*x*)', the unary predicate
'Child(*x*)' and the binary predicate
'Parent(*x,y*)'; and we suppose the theory
contains, in addition to '[?]*x*
Parent(*x*)', the sentences '[?]*x*
(Parent(*x*) if and only if [?]*y*
Parent(*x,y*))' and '[?]*x*
(Child(*x*) if and only if [?]*y*
Parent(*y,x*))', then the modal quantifier account gets
the ontological commitments of the beefed-up theory right: for every
world and sub-domain of the domain of that world such that the theory
is true at that world with its variables ranging over that
sub-domain, children are in the sub-domain. Perhaps the existence of
children is an *implicit* commitment of
'[?]*x* Parent(*x*)'; but, on the view
being considered, the quantifier account should not be required to
capture all the implicit commitments of the theory.


One problem with this view, at least as a defense of
the *modal* quantifier account, is that it requires an
invidious distinction between different sorts of analytic entailment;
for, although '[?]*x* Parent(*x*)' is
not committed to children according to the modal quantifier criterion,
'[?]*x* Bachelor(*x*)' *is*
committed to males. The chief problem, however, with accepting a
criterion that omits implicit ontological commitments is that it
undermines the philosophical use of the criterion. Surely, the
implicit ontological commitments of the theory must be included in
tabulating the ontological costs of a theory. Enjoining the theorist
to "make all implicit ontological commitments explicit"
does not solve the problem because the theorist may not know, and may
be in no position to know, the implicit commitments of her
theory. Such ignorance, however, does not make the theory less
costly. It seems, then, that the problem of extrinsic
properties--and, more generally, the problem of capturing
implicit ontological commitment--provides a genuine
counterexample to quantifier accounts of ontological commitment.


#### 1.7.5 Implicit Commitment: Sets and Wholes


 The problem of extrinsic properties arises because there are
analytic connections between one predicate holding of a thing and
another predicate holding of some other thing. Similar problems arise
for quantifier criteria if there are metaphysically necessary
connections between non-identical things. Thus, if a set cannot exist
without its members existing, then it appears that a theory that
contains '[?]*x* *x* =
{*a*, *b*}' is ontologically committed to the
existence of *a*. But *a* need not be assigned as a
value to the variable '*x*' in order for the theory
to be true. Similarly, if a whole cannot exist without one of its
parts existing, then there appears to be implicit commitment. Suppose,
for example, that a human cannot exist without his head. Then a theory
that contains '[?]*x* *x* = Obama' is
ontologically committed to Obama's head. But, again, Obama's head need
not be assigned as a value to the variable '*x*' in
order for the theory to be true.


Responses can be given that parallel the responses to the problem
of extrinsic properties. And a parallel conclusion applies: to capture
these cases of implicit ontological commitment, one must switch from a
quantifier account to an entailment account.


## 2. Entailment Accounts of Ontological Commitment


### 2.1 Overview


According to an entailment account of ontological commitment: A
theory *T* is *ontologically committed* to *K*s
if and only if *T* entails that *K*s exist. One must
speak of *an* entailment account because different entailment
relations lead to different accounts of ontological commitment; the
challenge is to find the *right* entailment relation. But in
any case, entailment accounts do not "look to the values of the
variables" for the ontological commitments of a theory.
Quantifiers and variables have no privileged role; they figure into
the ontological commitments of a theory only to the extent that they
figure into the entailments of the theory, just as with other
components of the sentences of the theory.


Entailment accounts, as understood here, have the feature that
ontological commitment depends only on the *content* of a
theory, not on its *logical form*. (Thus, the metalinguistic
account discussed in SS1.4 is not
included.) Two theories that have the same content, then, have the
same ontological commitments. If we say that theories have the same
content just when they mutually entail one another, then each
entailment account comes with its own notion of content; the problem
of finding the right entailment relation is just the problem of
finding the right notion of content, the notion that is relevant to
ontological commitment.


It is convenient, though not obligatory, to hypostasize content and
speak of the *propositions* expressed by the sentences of the
theory. That allows us to neatly separate the problem of determining
the ontological commitments of a theory (set of sentences) into two
components: What propositions are expressed by the sentences of the
theory? And, what are the ontological commitments, that is, the
existential entailments, of those propositions? Indeed, we could then
choose to construe theories as sets of propositions and treat all talk
of sentences as outside the bounds of an account of ontological
commitment proper. But propositions are controversial entities, not
accepted by all parties to the debate. To maintain neutrality, we will
continue to apply ontological commitment to theories construed as sets
of sentences, but with the understanding that, for the friend of
propositions, the ontological commitment of sentences is just the
ontological commitment of the propositions those sentences
express.


The principle advantage of entailment accounts over quantifier
accounts is that they have no problem with implicit ontological
commitment. (Or, more exactly, since different entailment accounts
will differ as to what counts as implicit content, entailment accounts
have no problem with what counts as implicit ontological commitment by
that account's own standards.) If '[?]*x*
Human(*x*)' entails not only that humans exist, but that
humanity exists, then it is ontologically committed to properties or
universals, even though no property or universal is the value of any
quantified variable. If '[?]*x*
Parent(*x*)' entails that children exist, then
'[?]*x* Parent(*x*)' is ontologically
committed to children, again, even though it need not "quantify
over" any children. An entailment account, unlike the quantifier
account, gets these cases right--supposing, that is, we want our
account to capture *implicit* ontological commitment.


For a stark illustration of the difference between entailment and
quantifier accounts, consider a theory *T* that contains the
single sentence '[?]*x* *K*(*x*)'
and the theory *T\** that contains the single sentence
'*True(T)*' (see Peacock (2011: 81-2), who
calls *T\** a "meta-theory"). *T*
and *T\** mutually entail one another (assuming
'*K*' is not a semantical predicate); they thus
have the same ontological commitments on an entailment account. And
this seems right. Surely, a philosopher cannot eliminate her
commitment to *K*s by switching from *T* to *T\**;
it matters not that *T\**'s commitment to *K*s is only
implicit. But this implicit ontological commitment is invisible to the
quantifier account because *T\** has no quantificational
structure. Once again, by missing implicit commitments, the quantifier
account fails to capture the ontological costs of a theory.


The principle disadvantage of an entailment account as compared
with a quantifier account is that it fails to provide a neutral
platform for assessing the ontological costs of theories:
disagreements over ontology will often reappear as disagreements over
the existential entailments of a theory. Here it is important to
distinguish two things that might be meant by "criterion of
ontological commitment". (This isn't always done; but Jackson
(1989) and Rayo (2007) are clear on the distinction.) One might mean
a *test* by which ontological commitments can be discovered. Or
one might mean an *analysis* that says what ontological
commitment consists in. An adequate *test* would be a procedure
that could be applied by all parties in a neutral way, independently
of those parties' prior ontological beliefs. Quine believed that the
quantifier criterion provided such a test. He was aware, of course,
that there would be disputes over how best to regiment theories within
first-order logic. But--or so he thought--once theories were
regimented, the quantifier criterion would supply unequivocal
verdicts.


A correct *analysis* of ontological commitment, however
theoretically valuable, might do nothing towards providing an adequate
test. Such an analysis, unaccompanied by a test, would be of little
help in determining the ontological costs of theories, or resolving
ontological disputes. Entailment accounts have good claim to providing
a correct analysis of ontological commitment; but they do not provide
a test. For one thing, the parties to an ontological dispute will
often disagree as to what propositions are expressed by the sentences
of a theory, that is, as to what the truth conditions of those
sentences are. Quine's criterion bypasses this problematic step by
applying the criterion only to already regimented theories. For a
second thing, parties to an ontological dispute will often disagree
over the existential entailments of a theory (see
SS2.2 below). Quine's criterion (as he understood it) avoids
this problem by focusing on the formal relation of logical entailment
between sentences, a relation that is well understood. The material
relations of entailment between the propositions, however--be
they modal, or conceptual, or *a priori*--are matters of
much dispute. If the goal is to find an adequate test for ontological
commitment, entailment accounts fall short.


Initially, one might have hoped to combine an entailment
account, *qua* analysis, with a quantifier
account, *qua* test. (Arguably, this is the best way to
understand Quine's project.) But the problems with implicit commitment
suggest that the entailment analysis and the quantifier test do not
match up, that the test will undergenerate with respect to the
analysis. One could dampen one's ambitions, and be content with a
one-sided test, a test that in general provides only a sufficient
condition. (This is suggested by Rayo (2007: 432) as a response to the
problem of extrinsic properties.) But a one-sided test won't be of
much help in comparing the ontological costs of theories. Such
comparisons require that *all* costs be put on the table.


### 2.2 Modal Entailment Criteria


The problem of applying an entailment account to ontological
disputes is especially acute if the entailment relation is taken to
be *necessary implication*. That choice results in the
following criterion (introduced above
in SS1.5):



> 
> *Modal Entailment Criterion.* A theory *T*
> is *ontologically committed* to *K*s iff *T*
> necessarily implies that *K*s exist iff, for every
> (metaphysically) possible world *w*, *T* is true
> at *w* only if the domain of *w* contains at least one
> *K*.[25]
> 
> 
> 


On this criterion, every theory is ontologically committed to every
necessarily existing kind of entity, what was called above
(SS1.5) the *problem of irrelevant
necessary existents*. The problem can be ignored when comparing
the ontological costs of theories that disagree only over contingently
existing kinds: the irrelevant commitments are the same for both
theories, and cancel one another out. But many traditional ontological
disputes are over the existence of necessarily existing kinds of
entity, such as mathematical entities or intensional
entities. Consider, for example two theories that disagree only over
whether numbers exist. On the modal entailment criterion, these
theories have the same ontological commitments; moreover, supposing
numbers do exist, the theory that explicitly denies the existence of
numbers is nonetheless committed to numbers. Clearly, the modal
entailment criterion fails to capture the intuitive notion of
ontological commitment. Applying the criterion in this case will do
nothing to further the ontological enterprise.


Jackson (1989) highlights the problem of irrelevant necessary
existents by comparing the following two arguments:


* B1.
 Materialism is true.
* B2. 
 Materialism is ontologically committed to non-actual possible worlds.
* BC. 
 Therefore, non-actual possible worlds exist.


* C1. 
 Modal Realism is true.
* C2. 
 Modal Realism is ontologically committed to non-actual possible worlds.
* CC. 
 Therefore, non-actual possible worlds exist.


Suppose, for the sake of argument, that non-actual possible worlds
necessarily exist. Then the second premises, B2 and C2, are both true
on the modal entailment account. Moreover, on the modal entailment
account, both of these arguments are necessarily truth preserving. It
seems to follow that the conclusion that non-actual possible worlds
exist can be supported *either* by arguing for B1, that modal
realism is true, or by arguing for C1, that materialism is true. But,
surely, an argument for materialism provides no support for the claim
that possible worlds exist. Or, to turn this around, an objection to
the claim that possible worlds exist does not cast doubt on
materialism.


Jackson defends the modal entailment account in the following
way. Although the two arguments are on a par with respect to validity,
and the truth of their premises,




>  there is an enormous epistemological difference
> between the arguments. ... doubts about the existence of
> non-actual possible worlds cannot cast doubt on B1, but can cast doubt
> on C1. (Jackson 1989: 197-8)
> 
> 
> 


Thus, there is an epistemological relation between propositions,
"casts doubt on", which holds between CC and C1 but not
between CC (=BC) and B1. Or, turning this around once again, there is
an epistemological relation--presumably some sort of *a
priori* entailment relation--according to which modal realism
entails that non-actual possible worlds exist, but materialism does
not entail that non-actual possible worlds exist. Jackson says nothing
to further characterize this relation, let alone provide a semantic
analysis. But, however the relation is precisely characterized, the
natural question to put to Jackson is: why not use *that*
relation, in place of modal entailment, to characterize ontological
commitment? For, clearly, this "epistemological"
entailment relation better tracks our judgments as to the ontological
commitments of theories; and it is the "epistemological"
relation, not modal entailment, that will have to be brought to bear
when comparing the ontological costs of theories positing necessary
existents.


Peacock (2011) also defends the modal entailment account from the
problem of irrelevant necessary existents. In particular, he argues
that the ontological costs of theories positing necessary existents
can be compared by considering, not the theories' *actual*
commitments (which cannot be known prior to determining the truth of
the theories), but by considering "what ontology
we *would* regard a theory as implicitly committed to, were it
true" (Peacock 2011: 93). This suggests that there is some
counterfactual relation between propositions according to
which--to return to Jackson's example--non-actual possible
worlds would have to exist if modal realism were true, but it is not
the case that non-actual possible worlds would have to exist if
materialism were true. This counterfactual relation, of course, could
not be given a standard possible worlds analysis lest it apply
vacuously to impossible antecedents. But even supposing a semantics
for such a counterfactual relation could be given, the natural
question to put to Peacock is: why not use *that* relation, in
place of modal entailment, to characterize ontological commitment?
For it is this counterfactual relation, not modal entailment, that
will have to be brought to bear when comparing the ontological costs
of theories positing necessary
existents.[26]
In sum: neither Jackson nor Peacock give any positive reason to hold
on to a modal entailment account, which is puzzling in light of the
fact that each is committed to there being an entailment
relation--be it epistemological or counterfactual--that
better fits the role that the notion of ontological commitment was
introduced to play.


### 2.3 *A Priori* Entailment Criteria


If the notion of ontological commitment is to play a role in theory
choice or clarifying ontological disputes, then the ontological
commitments of a theory must be *rationally accessible*;
failure to acknowledge the ontological commitments of one's theory
must be a rational failing (see Michael 2008). That puts the following
epistemic constraint on the entailment relation that is invoked by an
entailment account of ontological commitment: the entailments of a
theory must be knowable by any ideally rational agent, that is to say,
must be knowable *a priori*. Does modal entailment meet this
constraint? With respect to the examples considered thus
far--theories positing necessary existents--it may depend on
how "ideally rational" and "*a priori*"
are understood; some philosophers would argue that entities that
necessarily exist can be known *a priori* to exist. But if
(following Kripke 1980) we accept that there are necessary
propositions that cannot be known *a priori*, then modal
entailment will run afoul of the epistemic constraint. Suppose, for
example, that 'Water is H2O' is
(metaphysically) necessary but not knowable *a priori*. Then
the theory containing the single sentence, 'Water exists',
modally entails that hydrogen exists. But, according to the epistemic
constraint, the existence of hydrogen should not be an entailment on
an entailment account of ontological commitment. It appears that some
sort of *a priori* entailment relation must take the place of
modal entailment in the analysis of ontological commitment:



> 
> *A Priori Entailment Criterion.* A theory *T*
> is *ontologically committed* to *K*s
> iff *T* *a priori* entails that *K*s exist iff,
> for every conceptually possible world *w*, *T* is true
> at *w* only if the domain of *w* contains at least one
> *K*.[27]
> 
> 
> 


Switching to an *a priori* entailment relation goes partway
towards restoring the applicability of an entailment account to
ontological disputes. It allows that philosophers--if ideally
rational--will be able to agree on the ontological commitments
of *propositions*. But there is still the problem of
determining what propositions are expressed by the sentences of a
theory, that is, what the truth conditions of those sentences are. If
the sentences are couched within ordinary or philosophical English, so
that their truth conditions are not just a matter of stipulation,
there may be no resolving the dispute, even among ideally rational
philosophers (see SS4).


## 3. Truthmaker Accounts of Ontological Commitment


### 3.1 Overview


Quantifier accounts of ontological commitment, we have seen, are
liable to miss the *implicit* ontological commitments of
theories because they look exclusively to the values of the variables
of quantification. Entailment accounts succeed in capturing implicit
commitments, but are liable to overgenerate if the entailment
relation fails to be sensitive to relevant ontological
distinctions. According to truthmaker accounts of ontological
commitment, being a *truthmaker* for a theory is just such a
relevant ontological distinction: to find the ontological commitments
of a theory, we should look towards the truthmakers for that theory;
a theory is ontologically committed to *K*s only
if *K*s are needed in order to make the theory true. According
to truthmaker theorists, focusing on truthmakers will eliminate
spurious ontological commitments of an entailment theory. The problem
for a truthmaker account, however, is how to do this without also
eliminating genuine ontological
commitments.[28]


Truthmaker accounts, like entailment accounts, have the feature
that ontological commitment depends only on the *content* of a
theory, not on its *logical form*. That will be so if the
primary truthbearers are taken to be non-linguistic propositions
(individuated by content) rather than sentences. In that case,
ontological commitment applies directly to the propositions, or to
theories construed as sets of propositions, and only indirectly to
the sentences, or sets of sentences, that express those propositions.
That has the benefit, as with the entailment account, of freeing the
notion of ontological commitment from language (see
SS2.1). But taking propositions to be truthbearers will not
be to every truthmaker theorist's liking. Moreover, different views
on the nature of propositions--possible worlds accounts,
Russellian accounts, Fregean accounts, to name a few--lead to
somewhat different truthmaker accounts. To remain neutral in what
follows with respect to the existence and nature of propositions,
sentences will be taken to be the bearers of truth, and ontological
commitment will continue to be applied to theories construed as sets
of sentences.[29]


Say that an entity *makes true* (or is a *truthmaker
for*) a sentence or theory if the existence of that entity is, in
some sense, *sufficient* for the truth of the sentence or
theory. (It is left open for now just how to understand this notion of
sufficiency.) In some cases, truthmakers are easy to come by. For
example, the sentence 'Socrates exists' has Socrates as a
truthmaker (assuming, at any rate, that Socrates has *real*
existence; see SS3.2 below). The
sentence 'Elephants exist' has multiple truthmakers: each
and every particular elephant (assuming elephants are necessarily
elephants; see SS3.3 below). Other cases
are more problematic. Consider the conjunction: 'Socrates exists
and Plato exists'. Neither Socrates nor Plato, by himself, is a
truthmaker for the conjunction; we need the *pair* of Socrates
and Plato. But how should we understand 'pair' here? There
are three options. If one is a universalist about mereological
composition--for any things, there is a mereological sum composed
of just those things--then perhaps the sum of Socrates and Plato
can serve as a truthmaker for the conjunction. If one is a realist
about sets--and sets exist "in the world" when their
members do--then perhaps the set with Socrates and Plato as its
members can serve as a truthmaker. But if we reject mereological
universalism and set theoretic realism, then we will have to allow
that the truthmaker relation takes a plural argument: Socrates and
Plato together are truthmakers for the conjunction, but there is no
single entity that is a truthmaker. In what follows, it will be
simplest to speak as if, whenever a theory has truthmakers, it has a
single entity as a truthmaker; but only minor emendations would be
needed to allow for plural truthmaking.


Some sentences, it is natural to think, have neither singular nor
plural truthmakers (see Lewis 2001). For example, a negative
existential sentence such as 'There are no unicorns' is
made true, it seems, not by what exists, but by what fails to exist:
it is true for lack of any falsemakers. And whether a (contingent)
predication such as 'the ball is red" is made true by the
ball in question will depend on how the notion of sufficiency is
understood (see  Supplement to
SS3.3). *Truthmaker maximalists* hold
that *every* truth (or at least every contingent truth) has a
truthmaker. Typically, truthmaker maximalists will
introduce *states of affairs* or *facts* to serve as the
required truthmakers: *totality states of affairs* can serve as
truthmakers for negative existentials (see Armstrong 2004:
72-5); the state of affairs of the ball's being red, not the red
ball itself, is the truthmaker for 'the ball is red' (see
Armstrong 1997: 115-6). There is a danger here that truthmaking
could be trivialized: could any true sentence *S* be said to
have *the state of affairs that S* as its truthmaker?
Typically, truthmaker maximalists avoid such trivialization by holding
to a sparse theory of states of affairs. For example, a disjunctive
sentence '*S* or *T*' is not made true by a
disjunctive state of affairs: there are none. Rather, the truthmakers
for *S* (if any) and the truthmakers for *T* (if any)
are separately truthmakers for the disjunction '*S*
or *T*'. In any case, truthmaker maximalism is not
required for the truthmaker approach to ontological commitment. What
matters, rather, is that the truthmakers and the ontological
commitments of a theory are aligned.


How might a truthmaker criterion for ontological commitment be
formulated? It will be useful to separately consider two versions, one
for ontological commitment to particular entities, the other for
ontological commitment to kinds:



> 
> *Truthmaker Criterion for Particulars*: A theory *T*
> is *ontologically committed* to a particular *a* iff,
> necessarily, if *T* is true, then *a* is a truthmaker
> for *T*.
> 
> 
> *Truthmaker Criterion for Kinds:* A theory *T*
> is *ontologically committed* to *K*s iff, necessarily,
> if *T* is true, then *K*s are among the truthmakers
> for *T*.
> 
> 
> 


Note that in the criterion for particulars the necessity is *de
re*. The particular *a* itself must be a truthmaker
for *T* in any world where *T* is true. But in the
criterion for kinds the necessity must be taken to be *de
dicto* (with respect to '*K*') if the criterion
is to give the desired results. For example, 'there are
elephants' is ontologically committed to elephants according to
the criterion if *some elephant or other* is a truthmaker for
'there are elephants' in any world in which it is true,
whether or not any *actual* elephant is a truthmaker in those
worlds.


How should we understand the truthmaking relation, and the way in
which the existence of the truthmaker is *sufficient* for the
truth of the theory? On the orthodox necessitation approach to
truthmaking (Fox 1987; Bigelow 1988: 125; Armstrong 1997: 115; Lewis
2001), this sufficiency is analyzed in modal terms as
follows: *x* is a *truthmaker* for *T*
iff *T* is true and, necessarily, if *x* exists,
then *T* is true. (A plural version replaces the singular
variable '*x*' with a plural variable
'*xx*' (read "the *x*s").) This
leads to a familiar problem when truthmakers are sought for necessary
theories: any entity (or plurality of entities) is a truthmaker for
any necessary theory on the analysis. And there is the familiar fix:
replace modal entailment (in the analysis of truthmaking) with a more
discriminating entailment relation, perhaps some sort of relevant
entailment (Restall 1996). But for many truthmaker theorists, no
entailment relation will be able to capture the idea that truthmakers
must be *fundamental* entities, and must *ground* the
theories they make true. This drives these truthmaker theorists to
introduce a primitive grounding or in-virtue-of relation to
characterize the truthmaking relation: *x* is
a *truthmaker* for *T* iff *T* is true and,
necessarily, *T*s truth is *grounded* in *x*
(or *T* is true *in virtue of* the existence
of *x*) (see Schaffer 2010 and Cameron forthcoming). Call
this the grounding account of
truthmaking.[30]


According to *truthmaker necessitarianism*, the existence of
the truthmaker *necessitates* the truth of the sentence: any
possible world in which the truthmaker exists is a world in which the
sentence is true. Truthmaker necessitarianism is automatic on the
necessitation analysis of truthmaking. On a grounding account,
truthmaker necessitarianism can be upheld by endorsing an axiomatic
constraint on the notion of grounding: if an entity *grounds*
the truth of a sentence, then, necessarily, if the entity exists,
then the sentence is true. The problems to be discussed
(in SS3.3
and SS3.4) apply to any truthmaker
account of ontological commitment that endorses truthmaker
necessitarianism.[31]


### 3.2 Deep vs. Shallow Ontological Commitment


The truthmaker criterion for ontological commitment to kinds, as
formulated above, *strengthens* the modal entailment criterion
(SS2.2) since, necessarily,
if *K*s are among the truthmakers for *T*,
then *K*s exist. Thus, on a truthmaker account, for a theory
to be ontologically committed to *K*s, it is required not only
that there exist *K*s if the theory is true, but that there
exist *K*s *that are truthmakers for the theory*, if
the theory is true. The chief reason given for the strengthening
(Cameron 2008, 2010) is this: if the *K*s are
not *fundamental* entities, if they lack *real
existence*, then a theory can entail that *K*s exist
without thereby incurring ontological commitment to *K*s. For
example, suppose one holds that, although it can truly be said that
chairs exist, only (mereological) simples (or states of affairs
involving those simples) are fundamental, or have real
existence. Moreover, suppose that only entities that are fundamental,
or have real existence, can play the truthmaking role, in line with
the grounding account of truthmaking. Then, the (true) sentence
'chairs exist' is made true, not by chairs, but only by
simples arranged chairwise. It then follows on the truthmaker account
of ontological commitment that 'chairs exist' is
committed only to simples arranged chairwise, not to chairs. And,
arguably, this is the right result: chairs should not be counted
among the ontological commitments of 'chairs exist'
because, on the view being supposed, they do not add to the
ontological cost of accepting 'chairs exist'. In
Armstrong's memorable phrase: they are "an ontological free
lunch" (Armstrong 1997: 12-3). This result cannot be had
by the quantifier account or the entailment account because these
accounts are both insensitive to the crucial distinction between
existence and *real* existence. These accounts inevitably
overgenerate commitments, thus ruling out by fiat a minimal ontology
that accepts only simples.


Much more would have to be said in a defense of the truthmaker
account about the distinction between existence and real
existence.[32]
But let us suppose for what follows that the distinction is
legitimate, and has the ontological significance attributed to
it. Still, one might wonder whether we have just changed the
subject. On the traditional notion of ontological commitment, as
introduced by Quine, it is supposed to be trivially true that
'*K*s exist' is ontologically committed
to *K*s (see SS1.6.4). Instead of
providing an account of a theory's ontological commitment as
traditionally understood, the truthmaker account is an account of a
theory's *fundamental* ontological commitments. Looked at in
this way, the quantifier account and the truthmaker account are not
in competition, and can live side by side (see Schaffer 2008). 


Even if the dispute is partly terminological, however, sometimes
the choice of terminology matters. It is important to be clear on what
the different notions of ontological commitment are, and how they fit
in to the general project of ontology. Given a distinction between
existence and real existence, the notion of ontological commitment
bifurcates into two distinct notions, one *shallow* and
one *deep*. On the shallow notion, entities that exist but do
not really exist are to be counted among the ontological commitments
of a theory: it is trivial or analytic that 'Socrates
exists' is committed to Socrates, and that 'elephants
exist' is committed to elephants, whether or not Socrates or
elephants have real existence. On the deep notion, only entities that
really exist count towards the theory's ontological
commitments. Ontological commitment, arguably, has traditionally been
given the shallow interpretation; but that is because, within the
orthodox Quinean setting, there simply is no ontologically relevant
distinction between existence and real existence. Once the distinction
is accepted, however, and the project of ontology is construed to be
the project of discovering not what there is, but what
there *really* is, then it is only the deep notion of
ontological commitment that resonates with the ontological
 enterprise.[33] 
It is an analysis of this deep notion, as given by the truthmaker 
account, that matters for ontology.


Just as there are shallow and deep notions of ontological
commitment, so there are shallow and deep notions of truthmaking. On
the necessitation account of truthmaking, truthmaking is shallow:
Socrates is a truthmaker for 'Socrates exists' and
elephants are truthmakers for 'elephants exist', whether
or not Socrates or elephants have real existence. On the grounding
account of truthmaking, truthmaking is deep: only real existents can
serve as truthmakers. Clearly, if we pair a shallow interpretation of
ontological commitment with a deep interpretation of truthmaking, then
a truthmaker account of ontological commitment is off the table: the
ontological commitments and the truthmakers of a theory will not be
aligned. In evaluating a truthmaker account, then, it must be
supposed from the start that either both notions are taken to be
shallow or both notions are taken to be deep. In what follows, both
will be taken to be deep in accord with the truthmaker theorist's
intentions.


### 3.3 The Problem of Inessential Predication


Let us begin an evaluation of the truthmaker account with a problem
for the truthmaker criterion for ontological commitment to
particulars: the *problem of inessential predication*. Consider
the sentence 'Socrates is wise', and suppose that Socrates
really exists. (If not, switch to an inessential predication whose
subject is an entity that really exists.) Then it seems undeniable
that 'Socrates is wise' is ontologically committed to
Socrates. But, given truthmaker necessitarianism and the orthodox
understanding of essential properties, Socrates will not be a
truthmaker for 'Socrates is wise' on the truthmaker
account. Wisdom, presumably, is not an essential property of Socrates;
and so Socrates' mere existence does not necessitate his being
wise. But, then, the truthmaker account of ontological commitment for
particulars wrongly claims that 'Socrates is wise'
is *not* ontologically committed to Socrates. Note that this
problem doesn't arise for *essential* predications. Suppose,
for example, that Socrates is essentially human. Then Socrates' mere
existence necessitates his being human, and 'Socrates is
human' has Socrates both as a truthmaker and as an ontological
commitment. But pointing this out only seems to make matters worse for
the truthmaker account: why should 'Socrates is human' but
not 'Socrates is wise' be ontologically committed to
Socrates? Essential and inessential predications should be treated on
a par with respect to ontological commitment.


An analog of the problem of inessential predication applies to the
truthmaker criterion of ontological commitment to kinds. Consider the
sentence 'philosophers exist', and suppose that
philosophers *really* exist. Then it seems undeniable that
'philosophers exist' is ontologically committed to
philosophers. Presumably, however, philosophers are not essentially
philosophers. And so, given truthmaker necessitarianism, no
philosopher is a truthmaker for 'philosophers exist': any
philosopher is such that he or she could have existed in a
philosopher-less world. The truthmaker criterion for kinds, then,
wrongly claims that 'philosophers exist' is *not*
ontologically committed to philosophers. Call this the *problem of
inessential generalization*.


Different accounts of truthmaking--of which entities are
truthmakers, and what their essential properties are--will lead
to different responses. Since adequate responses appear to be
available to the truthmaker theorist, these responses are relegated to
the supplementary document.


### 3.4 The Problem of Sufficiency: Implicit Commitment Revisited


The truthmaker account of ontological commitment--like the
quantifier account, but for an entirely different reason--has a
problem with capturing implicit commitment. The problem arises
because the truthmakers for a theory, but not the ontological
commitments of a theory, are entities whose existence
is *sufficient* for the truth of the theory. The problem
afflicts both the truthmaker criterion for particulars, and for
kinds. Indeed, assuming truthmaker necessitarianism, it follows from
the former criterion that a theory is ontologically committed to a
particular only if the existence of the particular is *both*
necessary and sufficient for the truth of the theory; it follows from
the latter criterion that a theory is ontologically committed to a
kind only if the existence of some members (or other) of the kind
is *both* necessary and sufficient for the truth of the
theory. But the examples of implicit commitment considered
in SS1.7.4
and SS1.7.5 suggest that sufficiency
is not required for ontological commitment. The truthmaker account,
like the quantifier account, undergenerates
commitments.[34]


To illustrate the problem of sufficiency, consider the following
examples. Suppose, as is standard, that the existence of a set
necessitates the existence of any of its members. Then, intuitively,
the sentence '{*a*, *b*} exists' is
implicitly committed to *a*, a result that is captured by
entailment accounts of ontological commitment. But since the existence
of *a* is not sufficient for the existence of
{*a*, *b*}, *a* is not a truthmaker for
'{*a*, *b*} exists', and the truthmaker
criterion for particulars fails to capture this commitment. For a
second example, suppose that, necessarily, elephants have trunks, but
that any trunk could exist without there existing any elephants.
Then, intuitively, the sentence 'elephants exist' is
ontologically committed, not only to elephants, but to elephants'
trunks, a result that is captured by entailment accounts. But since
trunks are not among the truthmakers for 'elephants
exist', the truthmaker criterion for kinds fails to capture this
commitment. For a final example, consider the sentence
'bachelors exist'. Since it is analytic (suppose) that
males exist if bachelors exist, this sentence, intuitively, is
ontologically committed to males, a result that is captured by both
quantifier and entailment accounts. But no male is a truthmaker for
'bachelors exist', since any male could exist without
there existing any bachelors. So, again, the truthmaker criterion for
kinds fails to capture an implicit commitment.


How might a defender of the truthmaker account
respond?[35] It
won't do to claim that the notion of ontological commitment targeted
by the truthmaker account doesn't include the missing implicit
commitments. For one thing, the truthmaker criterion will let in
those implicit commitments that pass the sufficiency test, and there
seems to be no principled reason for admitting these implicit
commitments, but not the others. More importantly, excluding implicit
commitments does not sit well with the chief motivation for a
truthmaker account, viz., that ontological commitments should reflect
fundamental ontology; for commitments are no less fundamental for
being implicit. The mismatch between the notions of ontological
commitment and truthmaking--that the ontological commitments of
a theory are entities whose existence is necessary, given the truth
of the theory, whereas the truthmakers of a theory are entities whose
existence is sufficient for the truth of the theory--seems to be
a serious problem for a truthmaker account of ontological commitment,
at least as presented above.


Perhaps the best response to the problem of sufficiency for a
truthmaker theorist who wants to capture a deep notion of ontological
commitment is to omit the requirement that the ontological commitments
of a theory be truthmakers *for that theory*. It is enough that
they are truthmakers for some theory or other, that is, that they are
entities that really exist. This leads to the following criteria:



> 
> *Weakened Truthmaker Criterion for Particulars*: A
> theory *T* is *ontologically committed* to a
> particular *a* iff *T* entails that *a* really
> exists.
> 
> 
> *Weakened Truthmaker Criterion for Kinds:* A
> theory *T* is *ontologically committed* to *K*s
> iff *T* entails that *K*s are among the entities that
> really exist.[36]
> 
> 
> 


But it is doubtful that this is properly called a *truthmaker
account* of ontological commitment. The (deep) ontological
commitments of a theory are no longer being analyzed in terms of the
truthmaker commitments of the theory, but instead in terms of its
fundamental commitments, its commitments to entities that really
exist. And these, we have seen, are not the same.


## 4. Ontological Commitment in Ordinary Language


### 4.1 Overview


If a quantifier account of ontological commitment is to be applied
directly to ordinary language, we need to know which ordinary language
quantifiers or uses of quantifiers (if any) are ontologically
committing. The same goes for entailment and truthmaker accounts,
notwithstanding our relaxed attitude in applying the accounts to
ordinary language examples. For on both accounts ontological
commitment depends on the entailment of existentially quantified
sentences, and we need to know whether the quantifiers of the entailed
sentences carry ontological commitment.


Quine's considered view was that his criterion of ontological
commitment cannot be applied to theories couched in ordinary
language. Before applying the criterion, the theory must be
paraphrased into first-order predicate logic (without names); and
then, strictly speaking, it is only the first-order theory that
carries ontological commitment, not the ordinary language theory.
[37] 
(The proponent of the theory can be convicted of holding the
ontological commitment if--and only if--she accepts the
paraphrase.) For Quine, this view reflected his general approach to
ordinary language and ontology: the speakers of ordinary language,
typically, are not *ontologically serious*; it is the task of
philosophers, not ordinary folk, to determine which *prima
facie* commitments are genuine. There were also specific reasons
why a direct application of his criterion to ordinary language might
lead to erroneous results. For one thing, theories couched in ordinary
language containing non-extensional constructions might have
ontological commitments not revealed by quantificational structure (see SS1.7.3). For another thing, quantification
in ordinary language might fail to carry ontological commitment if it
is not *objectual*, not amenable to the standard Tarskian
model-theoretic semantics
 (see SS1.6.2). Only the quantifiers
 of first-order logic, Quine
held, are guaranteed to be objectual (because we stipulate their
semantics); and only objectual quantification carries ontological
commitment.[38]


On closer inspection, however, Quine is in no position to claim
that theories couched in ordinary language *never* carry
ontological commitment. In giving the truth conditions for the
quantifiers of first-order logic, Quine explicitly makes use of
ordinary language quantifiers (see, for example, Quine 1970). Thus,
his claim that the first-order quantifiers are objectual, and carry
ontological commitment, can't help but carry over to the ordinary
language quantifiers from which they derive their meaning. There
must, then, be some portion of ordinary language that can do all the
work, vis-a-vis ontological commitment, that the first-order language
was introduced to 
do.[39] 
This doesn't eliminate the need to seek
paraphrases of ordinary language theories
(see SS4.2 below). But such theories
need not be paraphrased into the language of first-order predicate
logic; it is enough if they are paraphrased into that regimented
portion of ordinary language that can be counted on, in appropriate
contexts, to be ontologically serious. The problem that needs to be
addressed by the Quinean is then: how to recognize which quantifiers,
or which uses of quantifiers, make up this ontologically serious
portion of ordinary language. Those who reject the Quinean approach
to ontological commitment, on the other hand, need to argue that
there is no principled way to separate out those quantifiers, or uses
of quantifiers, that carry ontological commitment in ordinary
language; or, more radically, that quantification in ordinary
language is never ontologically committing.


There is a wide spectrum of views on whether ordinary language
quantification is ontologically committing. On one end of the
spectrum is the view that *all* ordinary assertive uses of the
quantifiers 'there is' and 'some' are
ontologically committing, a view that may be dubbed
"allism" (see Lewis 1990,
and SS1.6.4 above). On this view,
discovering the ontological commitments of ordinary speakers is
easy. In asserting that "some prime numbers are greater than
100", we commit ourselves to the existence of numbers; in
asserting that "there are properties that all great leaders
have in common", we commit ourselves to the existence of
properties. Ontological commitment cannot be avoided by paraphrasing
these sentences into the ontologically serious portion of our
language because these ordinary assertions are already ontologically
serious.[40]


Hofweber (2005) agrees that quantification in ordinary language is
sometimes ontologically committing, but not always, and not when we
quantify over abstract entities, such as numbers and properties, in
ordinary (i.e., non-philosophical) contexts. Hofweber claims that
there are two distinct readings of ordinary language quantifiers,
readings that make different contributions to truth conditions and
perform different communicative functions. One reading, which he calls
the "domain conditions reading", is tied to ontology: it
places conditions on what exists in reality. If I say
"something fell on my head" (in a standard context), I am
thereby committed to the existence of an object meeting the condition
"fell on my head". But, Hofweber argues, there is another
reading of ordinary language quantifiers that is not tied to ontology,
what he calls the "inferential role reading". On this
reading, the inference from *F*(*t*)
to *F*(*something*) is trivially valid, even in cases
where we do not take the term *t* to refer to anything in
reality. On this reading, I can infer "there is someone whom I
admire" from "I admire Sherlock Holmes" without my
quantified assertion conveying any ontological commitment.


Yablo (1998) holds that the distinction between quantified
statements that are ontologically committing and those that are
not--if it can be made out at all--will rest not on
different readings of the quantifiers, but on the distinction between
literal and figurative uses of language. Indeed, all parties agree
that figurative language can fail to be ontologically committing; one
can assert, truly, that she has butterflies in her stomach without
thereby being committed to the existence of butterflies. But
figurative language is very widespread, even in the sciences. A case
can be made, for example, that ordinary talk of numbers is merely a
pretense we engage in to more easily communicate facts about the
concrete world; and similarly for talk of other abstract entities. The
project of applying ontological commitment to ordinary language
theories, then, is none too easy: it depends upon being able to
demarcate language that is literal from language that is partly
figurative; and it depends on there being a stripped-down core of
ordinary language that, when interpreted literally, has the
representational capacity for fully describing the world.


On the far end of the spectrum is the view that ordinary language
quantifiers, by themselves, are never ontologically committing. A
Carnapian might hold this view as a consequence of her general view
that ontological commitment, in a metaphysically loaded sense of the
term, is incoherent (see SS5.1). A Meinongian
might hold this view on the grounds that it is only the existence
predicate, not the quantifier, that conveys ontological
commitment. The Meinongian, however, is in a weak position when she
claims that quantification over non-existent objects fails to be
ontologically committing (see SS1.6.3); for
non-existent objects don't differ from existent objects in their
instantiation of properties, or their having natures independent of
our theorizing. Azzouni (2004), like the Meinongian, takes the
existence predicate to be what conveys ontological commitment; but,
unlike the Meinongian, he denies that our talk of non-existent objects
requires that they have *any* sort of independent being. The
tie between ontology and quantification is completely severed. All
quantification--like Hofweber's "inference role"
reading of the quantifier--is ontologically neutral.


How do the various views along this spectrum impact the Quinean
project of using paraphrase to provide substitutes for ordinary
language theories that can be evaluated for ontological commitment?
For Quine, as already noted, there is no presumption that ordinary
language quantification is ontologically serious. But it is crucial
to Quine's project that some portion of ordinary language can be used
for ontologically serious discourse, and that when so used,
quantification holds the key to ontological commitment. The allist
view, although compatible with the Quinean project, will lead to
attributions of ontological commitment wildly different from Quine's
own. It makes ontological commitment hard to avoid, leading to
overpopulated ontological slums; the desert landscape favored by Quine
will be hard to come by. Hofweber's view is more congenial to the
Quinean project, so long as the "domain conditions"
reading of the quantifiers can be stipulated to be in effect. That can
be done, Quine seems to presuppose, merely by taking an appropriately
serious attitude to the discourse in question. Inside the philosophy
room--or, better, the ontology room--that attitude is
pervasive. Outside the philosophy room, the philosopher is no less
free than other speakers to engage in ontologically frivolous
discourse (see Quine 1960: 210). Yablo's view would be congenial to
the Quinean project if it were possible to make out the line between
literal and figurative discourse: Quine's ontologically serious
discourse could then be identified with thoroughly literal discourse,
assertively uttered. But Yablo is skeptical that any such project can
succeed, and sees that as dooming the Quinean approach to
ontology. Azzouni's view, of course, is outright incompatible with the
Quinean project. It leads towards a substantially different account of
ontological commitment in which quantification plays no role (see
Azzouni 2004: ch. 4).


### 4.2 The Method of Paraphrase, and its Problems


If criteria of ontological commitment cannot be applied directly to
ordinary language, then attributions of ontological commitment to
philosophers--or ordinary folk, but we will focus on
philosophers--cannot be made just by examining the statements or
theories that they accept. Those statements or theories may
have *prima facie* ontological commitments, to be sure, in
virtue of their quantificational or referential apparatus. But some
method is needed to determine which *prima facie* commitments
count as genuine commitments. Enter here the *method of
paraphrase* endorsed by Quine and his modern day
followers. According to this method, we must first ask the philosopher
to provide paraphrases of her statements into the regimented portion
of our language, paraphrases that are acceptable by standards agreed
to in advance. We then apply our criterion of ontological commitment
to the regimented paraphrases. If the philosopher provides acceptable
paraphrases that are not ontologically committed to *K*s, then
she is acquitted of commitment to *K*s. If the paraphrases she
provides are ontologically committed to *K*s, then she is
guilty of such commitment. If she opts out, refusing to provide
paraphrases into the regimented portion of language, then attributions
of ontological commitment are moot.


The method of paraphrase dominated ontological discussion
throughout much of the twentieth century. For a detailed illustration
of the method from its heyday, one can do no better than to turn to
David and Stephanie Lewis's article "Holes" (Lewis and
Lewis 1970), a dialogue between two philosophers, a nominalist and a
realist about abstract objects. Suppose that the nominalist accepts
statements that are *prima facie* committed to
(immaterial) *holes*, such as "there are holes in this
piece of cheese". To be acquitted of this *prima facie*
commitment, the nominalist seeks a paraphrase that avoids
it--not just for this one sentence, but for ordinary discourse
about holes generally. To be acceptable, the paraphrases should
preserve the truth of those statements about holes, and the validity
of those arguments about holes, that are endorsed by ordinary
speakers. As a first stab, she might make use of the predicate
'perforated' and paraphrase "there are holes in
this piece of cheese" as "the cheese is
perforated". This allows the nominalist to continue to
"speak with the vulgar" without taking on a commitment to
holes. But the predicate 'perforated' won't help in
providing paraphrases for statements that count holes, such as
"there are seventeen holes in this piece of cheese".
Here the nominalist could introduce a separate predicate,
'*n*-perforated', for each *n*. That,
however, would violate strictures against infinitely many primitive
predicates in the regimented language; and it wouldn't be of much
help in paraphrasing numerical comparisons such as "there are
as many holes in this piece of cheese as there are crackers on this
plate". In the case of discourse about holes, the nominalist
does better to turn to a different strategy of paraphrase:
ontological reduction.[41] 
An ontological reduction maintains the
quantificational structure of the discourse, but restricts the
underlying domain to uncontroversial entities, in this case, material
objects. For example, "there are holes in this piece of
cheese" can be taken to quantify, not over immaterial holes,
but over material hole-linings; talk of holes is interpreted as talk
of the hole-linings that, intuitively, surround the holes. Problems
persist: identity conditions for holes and hole-linings do not agree,
making trouble for the paraphrasing of identity statements involving
holes. But let us end the story here: enough has been said to
illustrate how the method of paraphrase might used to avoid *prima
facie* ontological commitments.


Avoiding ontological commitment, however, is not the only role that
paraphrase plays in ontological disputes. The method of paraphrase has
also been used to bring hidden ontological commitments of ordinary
discourse to the fore. One prominent example is Davidson's (1967)
argument that action sentences are committed to an ontology of
events. For example, to explain the logical inference from "John
walked slowly" to "John walked", we should treat the
adverb 'slowly' as a description of an event over which
these sentences tacitly quantify. Paraphrases of action sentences into
the regimented portion of our language will thus be ontologically
committed to events. A second well-known example is Lewis's argument
from *Counterfactuals* that modal discourse, as he understands
it, is committed to possible worlds: 



> 
> It is uncontroversially true that things might be otherwise than
> they are. ... Ordinary language permits the paraphrase: there are
> many ways things could have been besides the way they actually
> are. ... I believe permissible paraphrases of what I believe;
> taking the paraphrase at its face value, I therefore believe in the
> existence of entities that might be called 'ways things could
> have been'. I prefer to call them 'possible worlds'.
> (Lewis 1973: 84)
> 
> 
> 


In both of these cases, a version of the method of paraphrase is
used to support, rather than avoid, commitment to controversial
ontology.


The method of paraphrase provides a clear, well-defined
strategy--some would say the only such
strategy[42]--for
 determining the ontological
commitments of the two parties to an ontological dispute. But there
have been objections to the method, both early and late. Early on,
Alston (1958) posed more or less the following dilemma (see also
Jackson 1980). Either the paraphrase and the sentence being
paraphrased have the same truth conditions, or they do not. If they
do, then they cannot differ in their ontological commitments; for the
role of truth conditions, in part, is to specify what must exist in
order for the sentence to be true. But then the method of paraphrase
cannot be used to avoid ontological commitment. Suppose then that the
paraphrase and the sentence being paraphrased do not have the same
truth conditions. Then the paraphrase does nothing to support the
claim that the original sentence can be used without incurring
ontological commitment. Indeed, whatever the precise relation
between the paraphrase and the sentence being paraphrased, nothing
justifies reading the ontological commitments of the original
sentence off the paraphrase rather than the other way around (compare
the neo-Fregean account in SS5.2).


The best response on behalf of the defender of the method of
paraphrase is to reject the assumption that the goal of paraphrase is
to capture the truth conditions, or the ontological commitments, of
the original. For Quine, an acceptable paraphrase is just a sentence
that "serves any purposes [of the original] that seem worth
serving" (Quine 1960: 214). A stronger notion of paraphrase is
ruled out because the truth conditions, and so the ontological
commitments, of sentences in ordinary language are, for the most
part, semantically indeterminate. In applying the method of
paraphrase, we ask the philosopher to make her ordinary language
theory precise through the process of regimentation, not to uncover
the hidden commitments of the original
theory.[43] If
she complies, then the ontological commitments of the regimented
theory can be attributed to her. But there is no going back to
assign ontological commitments to the original theory couched in
ordinary language.[44]


More recent critics of the method of paraphrase see the search for
paraphrase as fundamentally misguided. Sider (1999), for example,
suggests that the project of discovering the ontological commitments
of ordinary language should proceed by searching, not for paraphrases,
but for *underlying truths* (see also Melia 1995 and Yablo 1999
for related critiques). Suppose some sentence in ordinary language
that we use in describing the world is *prima facie* committed
to *K*s. Nonetheless, the attitude that we take to the sentence
may be something weaker than believing the sentence true. In that
case, our use of that sentence need not commit us to *K*s, even
if all acceptable paraphrases are committed to *K*s. To get at
our commitments, we should look, not to paraphrases, but to the truths
that underlie the descriptive role of the sentence; our commitment is
the commitment of those underlying truths. Roughly, we can say that
the underlying truths are true propositions that would still have been
true and would have been sufficient for the truth of the original
sentence, if there had been *K*s (see Sider 1999:
332-3). To illustrate with a simple example, consider the
sentence: "the number of moons of Jupiter is sixty-seven".
Suppose a philosopher uses this sentence to describe the world, but
does not want to accept its *prima facie* commitment to
numbers. If, as seems plausible, the underlying truths for the
sentence are just propositions stating what physical objects are in
the vicinity of Jupiter and what their motions are, then the sentence
is ontologically committed only to those physical objects, not to
numbers. We may not know what these underlying truths are, or be able
to express them in ordinary language. That matters not. For, unlike
the method of paraphrase, the philosopher is under no obligation to
provide the underlying truths in order to acquit herself of
the *prima facie* commitments. She can justify her use of
sentences that are *prima facie* committed to numbers as long
as she holds that there are underlying truths, and that the underlying
truths are not so committed.


## 5. Frege and Neo-Fregeanism


### 5.1 Frege


Although explicit discussion of ontological commitment postdates
Frege's writings, an account of ontological commitment can be garnered
from his defense of arithmetical platonism in *The Foundations of
Arithmetic* (Frege 1884). That account is embedded in the
following argument for the claim that numbers exist as
"self-subsistent objects" (see Wright (1990), who calls
the argument "deceptively simple"). The semantic role of
singular terms (Frege's *proper names*) is to refer to
objects. When singular terms occur in true sentences of an appropriate
kind--extensional sentences generally, and identity
sentences in particular--they cannot help but to refer; for that
is their contribution to the truth of the sentence. Appropriate
sentences involving numerical singular terms are sometimes true;
witness 'the number of planets is nine' and
'1+1=2'. Therefore, these singular terms refer to
objects. Moreover, the argument may be continued, since some of these
singular terms--those of the form 'the number
of *F*s'--figure in the identity condition
definitive of the concept of number (Hume's Principle, see below), the
objects referred to are properly called
"numbers". Therefore, numbers exist.


This argument might be challenged in two different ways. First,
one might deny that sentences containing numerical singular terms are
(literally) true; one might, for example, endorse a fictionalist
account of mathematics (Field 1980). For our purposes, we can sidestep
this objection and focus on the other premise of the argument, that
extensional sentences containing numerical singular terms, if true,
refer to numbers--which is to say, in other words, that such
sentences, whether true or not, are *ontologically committed*
to numbers. If this is to serve as a criterion of ontological
commitment, there must be some purely *syntactic* way of
determining which expressions are genuine singular terms; if genuine
singular terms are characterized *semantically*--for
example, as expressions that refer to objects when they occur in true
extensional sentences--then the account of ontological commitment
goes in a circle. Thus, Fregean accounts are committed to
a *syntactic priority thesis*: ontology depends on ontological
commitment plus truth; and ontological commitment in turn depends on
syntax (see Wright 1983 for extended discussion).


In various scattered remarks, Frege appears to endorse the
syntactic priority thesis. For example, he writes (1884: SS57):
"we say 'the [number] 1', and by means of the
definite article, set up 1 as an object". And he notes that
numerals occur, not only attributively, for example, in 'Jupiter
has four moons', but also in identity statements, such as
'the number of Jupiter's moons is four'; and the sense of
such an identity statement is just that "the expression
'the number of Jupiter's moons' denotes the same object as
the word 'four'". The difficulty of identifying the
appropriate use of the definite article, or when 'is'
signals a genuine identity statement, was, perhaps, not sufficiently
appreciated by Frege. But recent work (especially Dummett (1973) and
Hale (1996)) has given some reason to think that genuine singular
terms in English can be recognized syntactically in terms of their
inferential role.


Suppose that the task of providing syntactic criteria for
identifying genuine singular terms can be accomplished. Suppose,
moreover, that there are syntactic criteria for saying when a genuine
singular term purports to refer to objects of kind *K*. Then
the Fregean can put forth the following sufficient condition for
ontological commitment: 



> 
> *Fregean Account.* Suppose that *t* is a singular
> term purporting to refer to objects of kind *K*. Then, any
> extensional theory containing the term *t* is *ontologically
> committed* to *K*s.
> 
> 
> 


Note that, unlike the criteria for ontological commitment
considered heretofore, this is intended only as a *sufficient*
condition for ontological commitment. For one thing, just as Frege
took singular terms to be a vehicle for commitment
to *objects*, he took incomplete expressions to be a vehicle of
commitment to *concepts* and (more
generally) *functions*. Unlike Quine, Frege treated all
meaningful expressions on a par with respect to ontological
commitment. But even if we restrict our attention to ontological
commitment to objects, there is no reason to think that Frege's
criterion is necessary as well as sufficient. Frege could well have
allowed existentially quantified sentences to be ontologically
committed to objects, whether or not they were equivalent to
disjunctions of sentences containing singular terms.


In one important respect, Frege's approach to ontological
commitment is in agreement with Quine's. Both hold that the smallest
unit of ontological commitment is the complete sentence. For Frege,
this can be seen as an uncontroversial application of his
famous *Context Principle*: "it is only in the context of
a proposition that words have any meaning" (Frege 1884:
SS62). It follows that attribution of ontological commitment to
persons always goes by way of the theories (sentences) that they
accept. This approach to ontological commitment is common to all
mainstream views in contemporary analytic philosophy. It could be
denied, however, by a philosopher who holds that we stand in
intentional relations, not only to propositions, but also to
objects. For such a philosopher, plausibly, a person is ontologically
committed to the objects of her thought, and the kinds they exemplify,
independently of what propositions she takes to be true of those
objects, or even whether there are any such propositions.


The Fregean approach to ontological commitment differs from
orthodox Quinean approaches most obviously in its emphasis on singular
terms as opposed to quantifiers and variables. But this may reflect
more a difference in their accounts of singular terms than a
substantial difference in their accounts of ontological commitment;
both approaches, after all, focus on the *referential*
apparatus of the language. Quine eliminated all singular
terms *prior* to assessing ontological commitment: definite
descriptions were eliminated via Russell's theory of descriptions;
names were eliminated in favor of predicates satisfied by at most one
thing. This allowed Quine to sidestep issues arising from ostensibly
empty names (see SS1.2 above). Frege, on
the other hand, took singular terms to be the locus of commitment to
objects. In Frege 1892, he writes: "when we say 'the
moon' ... we presuppose a reference". Singular terms,
then, are ontologically committing in this sense: they presuppose
reference to objects. This applies no less to empty names that in fact
fail to refer. Frege writes:




>  anyone who seriously took the sentence
> ["Odysseus ..."] to be true or false would ascribe to
> the name "Odysseus" a reference.
> 
> 
> 


This has a couple of awkward consequences. For one, it follows that
denials of existence cannot be made with empty names: "Odysseus
never existed" is neither true nor false. But perhaps these
denials won't much be missed, since we can always say instead that the
name 'Odysseus' doesn't refer. For another, sentences with
fictional names are always ontologically committing. But, ultimately,
it is the ontological commitment of persons that we care about, not
sentences. A person may use a sentence containing a singular
term--for example, an actor on the stage--without inheriting
the sentence's ontological commitments. Only when a person judges the
sentence to be true is she committed to an object as referent of the
singular term, and to the kinds of entity to which the object
belongs.


Another difference between Frege and Quine (but not all Quineans)
is that Frege's account of ontological commitment applies directly to
sentences of ordinary language. Quine's account applies only
indirectly to sentences of ordinary language via the method of
paraphrase, or, in what seems to be his "official" view,
not at all. This difference leads to different, and more onerous,
demands being placed on the Fregean. As already noted, the Fregean
must discover syntactic criteria for identifying genuine singular
terms in ordinary language. Quine never engaged in the corresponding
project: finding syntactic criteria for identifying objectual
quantifiers in ordinary language. As we saw
in SS4.2, that task is none too easy, and
there is substantial disagreement even over basic
cases. But, *contra* Quine, it is a task that must be carried
out if quantifier approaches to ontological commitment are to have
philosophical application. Philosophers, after all, do not often
speak or write in Quine's target language: first-order predicate
logic. Quinean approaches--if they are to provide a
theory-neutral criterion of ontological commitment applicable to
ordinary language--are no less beholden to the syntactic priority
thesis than Fregean approaches.


### 5.2 Neo-Fregeanism


Thus far, we have canvassed Frege's argument that, if sentences
containing numerical singular terms are true, then those terms refer
to numbers. The last step in the argument--that the objects
referred to by numerical singular terms satisfy the sortal concept of
number--invoked the following identity criterion for numbers:



> 
> *Hume's Principle*. The number of *F*s is identical
> with the number of *G*s if and only if there is a one-to-one
> correspondence between the *F*s and the *G*s. 
> 
> 
> 


Frege considered, but rejected, the idea that Hume's Principle, by
itself, could be taken to define the concept of number. His rejection
was based on what has come to be known as the "Caesar
Problem": Hume's Principle allows one to decide whether the
number of *F*s is identical with *q*, when *q* is
given in the form 'the number of *G*s', but it
fails to do so when *q* is not given in that form. Thus, Frege
(1884: SS56) requires that we be able to




>  decide by means of our definitions whether any
> concept has the number Julius Caesar belonging to it, or whether that
> same familiar conqueror of Gaul is a number or not.
> 
> 
> 


 Frege thus opted instead to define numbers explicitly in terms of
extensions: the number of *F*s = the extension of the concept
'equal to the concept *F*' (where *F is equal to
G* just in case the *F*s are in one-to-one correspondence
with the *G*s). These explicit definitions arguably allow Frege
to maintain the logicist thesis that arithmetic is logic: they define
numbers as *logical objects* in the sense that their existence
can be justified on the basis of (Frege's) logic alone. But, as is
well known, these explicit definitions foundered on the inconsistency
of Frege's underlying theory of extensions, in particular, the naive
comprehension principle embodied in his Basic Law V.


The *Neo-Fregean* seeks to defend arithmetical platonism and
logicism by taking a different tack: Hume's Principle by itself is
taken to provide a contextual definition of numbers. The viability of
this approach is secured by the result now called *Frege's
Theorem*: Hume's Principle, when added to a suitable system of
second-order logic, suffices for the proof of the basic laws of
arithmetic, Peano's axioms (see Wright 1983:
158-69). Arithmetical logicism is secured if Hume's Principle
can be shown to be analytic in an appropriate sense. Arithmetical
platonism, with numbers as logical objects, is secured because, when
logical truths are substituted for the right-hand side of Hume's
Principle, the resulting instance on the left-hand side is also
logically true, and explicitly refers to numbers.


For our purposes, the main interest of the Neo-Fregean approach is
that it leads to a conception of ontological commitment more liberal
than any yet considered. This conception arises from the
Neo-Fregean's particular way of understanding Hume's
Principle. First, since Hume's Principle is to *define* the
concept of number, the left- and right-hand sides of the principle
must at least have the same truth conditions. Second, sentences with
the same truth conditions have the same ontological commitments. This
second assumption amounts to a sense in which *being supervenes on
truth*: circumstances cannot differ with respect to what there is
without differing with respect to what is true. It follows from these
two assumptions that the left- and right-hand sides of Hume's
Principle have the same ontological commitments. But that still
leaves open three different ways that Hume's Principle can be viewed
(Wright 1990). First, one might *reject* it on the grounds
that the left- and right-hand sides clearly do not have the same
ontological commitments: the numerical identity on the left-hand side
is committed to numbers; the claim that two concepts are equal on the
right-hand side is not. On this view, only a conditionalized version
of Hume's Principle could be accepted: "if there are numbers,
then ..." (Field 1984). In response, the neo-Fregean can
say that this confuses "having the same truth conditions"
with some more fine-grained equivalence such as "expressing the
same thought". Only the latter requires that both sides refer
to the same objects, where reference is a kind of explicit
ontological commitment.[45]


A second response takes Hume's Principle to provide
an *ontological reduction* of numbers to concepts, and
relations between concepts. On this view, Hume's Principle is true,
but ontological commitments should be read off of the right-hand
side. Since the right-hand side, it is supposed, is not ontologically
committed to numbers, the left-hand side--being
truth-conditionally equivalent--cannot be ontologically committed
to numbers either; surface grammar is deceptive here. This response is
in line with the standard Quinean approach to paraphrase as a means of
eliminating ontological commitment. But, as noted
in SS4.2, it is unclear what justifies
the asymmetric attitude with respect to commitment. Granted, there is
no *explicit* ontological commitment to numbers by the
right-hand side; but once one allows that ontological commitment can
be implicit, it is unclear why one cannot say that it is the surface
grammar of the *right-hand* side that is deceptive with respect
to ontological commitment.


The third response, that of the Neo-Fregean, is to say that
ontological commitments should be read off of the left-hand side, and
that therefore both sides are ontologically committed to numbers.
This is no idle preference. It is backed up by the Fregean argument
considered above that the left-hand side is ontologically committed to
numbers in virtue of containing numerical singular terms. The symmetry
is broken.


The neo-Fregean thus endorses a sufficient condition for
ontological commitment much weaker than the Fregean account:



> 
> *Neo-Fregean Account.* Suppose that *t* is a
> singular term purporting to refer to objects of kind *K*. Then
> any theory truth-conditionally equivalent to a theory containing the
> term *t* is *ontologically committed*
> to *K*s.
> 
> 
> 


To illustrate: "there are exactly as many forks as
knives" is ontologically committed to numbers because it is
truth-conditionally equivalent to "the number of forks is
identical with the number of knives", which contains a singular
term 'the number of forks' which purports to refer to
numbers.


The liberality of the neo-Fregean account of ontological commitment
is best appreciated by considering its application to abstraction
principles for which, unlike Hume's Principle, the right-hand side
appears to be entirely about concrete entities, and so appears to be
acceptable to a nominalist. For example, consider the abstraction
principle for directions: the direction of line *a* is
identical to the direction of line *b* if and only
if *a* is parallel to *b* (where *a*
and *b* are concrete). According to the neo-Fregean account,
the claim that concrete lines *a* and *b* are parallel
is committed, by itself, to abstract objects,
directions.[46]
Moreover, it has this commitment whether or not any mind ever grasps
the concept of direction, or formulates the corresponding abstraction
principle. In general, commitments to *abstracta* are
concomitant with certain commitments to *concreta*, and
inseparable from them, regardless of our ways of
conceptualizing.[47]


## 6. Carnap and Neo-Carnapianism


### 6.1 Carnap


Carnap was a deflationist about traditional ontological disputes,
such as whether numbers exist, or physical objects exist. He held that
existence questions could be interpreted in two different ways,
internally or externally. When interpreted internally, the question
was to be answered relative to a linguistic framework. If the
framework was mathematical, existence questions, such as whether there
are prime numbers greater than 1000, were to be answered by means of
proof, given the axioms and rules of logic introduced by the
framework. If the framework was empirical, existence questions, such
as whether electrons exist, were to be answered through observation
according to the rules of confirmation and disconfirmation introduced
by the framework. When the existence questions involved the most
general kinds introduced by the framework, the answers were
trivial. In the framework of numbers, it is trivial that numbers
exist; in the framework of physical objects, it is trivial that
physical objects exist. For it is stipulated in introducing these
frameworks that the variables range over numbers, or physical
objects.


When existence questions are interpreted externally, they purport
to be about "reality" independent of any particular
linguistic framework. So interpreted, according to Carnap, these
questions--such as "do numbers exist?" or "do
physical objects exist?"--are pseudo-questions lacking any
cognitive meaning. Since metaphysicians engaging in ontological
disputes over these questions evidently have the external
interpretation in mind--they do not take them to be
trivial--much of traditional metaphysics for Carnap was
cognitively meaningless as well. The best one can do to make sense of
external existence questions is to re-construe them as pragmatic
questions as to which frameworks to accept. The question, "do
numbers exist", for example, could be re-construed as the
question, "is it useful to introduce the number
framework". A framework could be deemed useful, for example, if
it facilitated factual communication in ways that were
"efficient, fruitful, and simple" (Carnap 1950).


One might wonder whether it is coherent to interpret all meaningful
existence questions as internal: don't we need the concept of
existence to introduce and meaningfully discuss the frameworks
themselves, and their pragmatic virtues? Can it be frameworks
"all the way down"? But, for purposes of this article, it
suffices to ask how the notion of ontological commitment fits in to
Carnap's scheme. One might expect Carnap to tie talk of ontological
commitment to external questions, and reject it accordingly: if it is
cognitively meaningless to say that *K*s exist *in
reality*, then it is cognitively meaningless to say that a theory
presupposes or assumes that *K*s exist in reality. But that
is not what Carnap did. Rather, he endorsed Quine's quantifier
account of ontological commitment, applying it internally; his only
reservation had to do with Quine's use of the word
'ontological'.[48] 
For Carnap, a theory always comes
packaged with a linguistic framework for interpreting the theory. To
say that a theory is ontologically committed to *K*s, then, is
just to say that according to the rules of the framework-cum-theory,
it is true that *K*s exist. A philosopher can be said to
"accept a kind of entities" when he introduces a
linguistic framework that, first, introduces a general term satisfied
by all and only entities of that kind and, second, introduces
variables to range over all and only those entities (see Carnap 1950:
213).


How do Carnap's views relate to Quine's? First, it might well be
asked, which of the quantifier accounts canvassed above is Carnap
endorsing when he claims to be in agreement with Quine? Since Carnap
clearly allows a philosopher who has *not* accepted (for
pragmatic reasons) some kind of entity to attribute acceptance of that
kind to another philosopher, Carnap's view would appear to be closest
to the metalinguistic version of Quine's quantifier account. We are
making claims about linguistic frameworks, and what follows from the
rules of those frameworks; and these claims, at least for Carnap, do
not make problematic assertions about "reality". Second, a
consequence of Carnap's view is that theories with different
ontological commitments should not be thought of as conflicting. This
contrasts sharply with Quine. Because Quine has a single, all-purpose
logico-linguistic framework--austere first-order predicate
logic--within which all theories are couched for purposes of
examining their ontological commitments, theories with differing
existential presuppositions genuinely conflict; a philosopher must
choose which theory to accept. For Carnap, theories with differing
ontological commitments are *ipso facto* couched within
different linguistic frameworks. If there is disagreement, it can only
be over the relative pragmatic virtues of the frameworks in
question. In this way, Carnap's deflationary approach to ontology
carries with it a deflationary approach to ontological
commitment. Carnap is a *pluralist*; a philosopher can choose
to accept multiple theories, with differing ontological commitments,
without pain of contradiction. Let many flowers bloom (see Carnap
1950: 221).


### 6.2 Neo-Carnapianism: Quantifier Variance


According to Carnap, each linguistic framework comes with its own
unrestricted existential quantifier, and thus its own notions of
object and existence. No one of these quantifiers is logically or
metaphysically privileged over any of the others; there is no neutral
standpoint from which one framework can be judged correct and another
incorrect. When frameworks appear to disagree--as when one says
that arbitrary fusions exist, and another denies it--the dispute
is merely verbal; they are using quantifiers with different
meanings. This is a form of the doctrine of *quantifier
variance*. One might well wonder: don't we get a single,
privileged quantifier by combining all linguistic frameworks, and
considering a quantifier that ranges over the union of all the domains
of these frameworks? But the question presupposes, in effect, that the
notion of "all linguistic frameworks" is meaningful, and
there is no reason why Carnap would accept that. Any attempt to
introduce a single all-inclusive quantifier that ranges over
absolutely everything would be rejected by Carnap. For what does
everything mean here, if not "everything *in reality*?
And that, for Carnap, is cognitively meaningless.


Eli Hirsch (2002) is a neo-Carnapian proponent of the doctrine of
quantifier variance, but he differs from Carnap in at least two
respects (see also Putnam 1987). First, Hirsch applies the doctrine of
quantifier variance not just to the quantifiers in artificial
languages, as was Carnap's way, but to ordinary language
quantifiers. Instead of Carnap's many linguistic frameworks, we have
many possible languages including English and (Hirsch would say) the
somewhat bizarre variations on English spoken by some contemporary
metaphysicians. For example, Lewish, the language spoken by
mereologists (and named after its defender, David Lewis) quantifies
over arbitrary fusions, whereas English, Hirsch claims, does
not. Second, Hirsch's brand of quantifier variance is meant to be
compatible with realism, that there is a single reality in virtue of
which our theories, with their various quantifiers, are made true or
false. This reality cannot, in and of itself, be ontologically
structured; it cannot have a metaphysically privileged division into
objects, lest one quantifier be privileged over the others. Reality
consists rather of unstructured facts that we carve up differently
with our different quantifiers.


Hirsch's realism contrasts sharply with Carnap's anti-realism. For
Carnap, different linguistic frameworks are incommensurable. They
peacefully coexist, logically isolated from one another. Apparent
ontological disputes are meaningless for lack of a common subject
matter, a single "reality" that is being disputed. For
Hirsch, on the other hand, each side in an ontological dispute is
speaking truly of the one and only reality. The appearance of a
genuine dispute arises because they are speaking different languages
whose quantifiers have different meanings. Each side of the dispute
can translate the others claims into his own language to see that both
are speaking truly, and that the dispute is merely verbal. The
disagreement is over the meaning of words, not the nature of
reality.


This claim of intertranslatability is difficult to maintain (see
Hawthorne 2009). But suppose it can be made good: what would that say
about ontological commitment? Since theories that assert "there
are *K*s" and theories that assert "there are
no *K*s" need not conflict, one cannot simply look to the
quantifiers for ontological commitment. Moreover, since no language,
and no quantifier, is ontologically privileged, it would not help to
first translate into a single language. We could relativize the notion
of ontological commitment to languages; but given that reality itself,
for Hirsch, isn't ontologically structured, none of these relativized
notions of ontological commitment would be getting at something of
ontological significance (including the one that gave ontological
commitment relative to ordinary language, and ordinary language
quantifiers). All theories, it appears, are ontologically committed to
the single reality, and the unstructured facts that comprise it; but
that commitment does not come from quantification. Where it comes from
is not addressed.