## 1. The Languages and Theories of Plural Quantification



The logical formalisms that have dominated in the analytic tradition
ever since Frege do not allow for plural quantification. In
introductory logic courses students are therefore typically taught to
paraphrase away plural locutions. For instance, they may be taught to
render "Alice and Bob are hungry" as "Alice is
hungry & Bob is hungry", and "There are some apples on
the table", as "\(\Exists{x} \Exists{y} (x\) is an apple
on the table & \(y\) is an apple on the table & \(x \ne
y)\)". However, not only are such paraphrases often unnatural,
but they may not even be available. One of the most interesting
examples of plural locutions which resist singular paraphrase is the
so-called Geach-Kaplan sentence:



(3)
Some critics admire only one another


This sentence provably has no singular first-order paraphrase using
only the predicates occurring in the sentence
 itself.[2]



How are we to formalize such sentences? The traditional view, defended
for instance by Quine, is that all paraphrases must be given in
classical first-order logic, if necessary supplemented with set
theory. In particular, Quine suggests that
 (3)
 should be formalized as

 

\[\tag{\(3'\)}\label{ex3prime}
\kern-5pt\Exists{S}(\Exists{u}\mstop u \in S \amp \Forall{u}(u\in S \rightarrow Cu) \amp \Forall{u}\Forall{v}(u\in S \amp \textit{Auv} \rightarrow v\in S \amp u\ne v))
\]


(1973: 111 and 1982:
 293).[3]



In two important articles from the 1980s George Boolos challenges this
traditional view (Boolos 1984 and 1985a). He argues that it is simply
a prejudice to insist that the plural locutions of natural language be
paraphrased away. Instead he suggests that just as the singular
quantifiers \(\Forall{x}\) and \(\Exists{x}\) get their legitimacy
from the fact that they represent certain quantificational devices in
natural language, so do their plural counterparts \(\Forall{xx}\) and
\(\Exists{xx}\). For there can be no doubt that in natural language we
use and understand the expressions "for any things" and
"there are some things".[4]
 Since these quantifiers bind variables that take name (rather than
predicate) position, they are *first-order* quantifiers, albeit
*plural* ones.


### 1.1 Regimenting plural quantification



I will now describe a simple formal language which can be used to
regiment plural quantification as it occurs in English and other
natural languages.




*The formal language \(L\_{\textrm{PFO}}\)*. Let the formal
language \(L\_{\textrm{PFO}}\) (for *Plural First-Order*) be as
follows.


1. \(L\_{\textrm{PFO}}\) has the following terms (for every natural
number \(i\)):
	* singular variables \(x\_i\)
	* plural variables \(xx\_i\)
	* singular constants \(a\_i\)
	* plural constants \(aa\_i\)
2. \(L\_{\textrm{PFO}}\) has the following predicates:
	* two dyadic logical predicates = and \(\prec\) (to be thought of as
	*identity* and the relation *is one of*)
	* non-logical predicates \(R^{n}\_i\) all of whose argument places
	are singular (for every adicity \(n\) and every natural number
	\(i)\)
3. \(L\_{\textrm{PFO}}\) has the following formulas:
	* \(R^{n}\_i (t\_1,\ldots, t\_n)\) is a formula when \(R^{n}\_i\) is an
	\(n\)-adic predicate and \(t\_j\) are singular terms
	* \(t \prec T\) is a formula when \(t\) is a singular term and \(T\)
	a plural term
	* \(\neg \phi\) and \(\phi \amp \psi\) are formulas when \(\phi\)
	and \(\psi\) are formulas
	* \(\Exists{v}\mstop \phi\) and \(\Exists{vv}\mstop \phi\) are
	formulas when \(\phi\) is a formula and \(v\) is a singular variable
	and \(vv\) a plural one
	* the other connectives are regarded as abbreviations in the usual
	way.




In \(L\_{\textrm{PFO}}\) we can formalize a number of English claims
involving plurals. For instance,
 (2)
 can be formalized as

 

\[
\tag{\(2'\)}\label{ex2prime}
\Exists{xx} \Forall{u} (u\prec xx \rightarrow Au \amp Tu)
\]


And the Geach-Kaplan sentence
 (3)
 can be formalized as 

 

\[
\tag{\(3''\)}\label{ex3pprime}
\Exists{xx} [\Forall{u}(u\prec xx \rightarrow Cu) \amp \Forall{u}\Forall{v}(u\prec xx \amp \textit{Auv} \rightarrow v\prec xx \amp u\ne v)].
\]


However, the language \(L\_{\textrm{PFO}}\) has one severe limitation.
We see this by distinguishing between two kinds of plural predication.
A predicate \(P\) taking plural arguments is said to be
*distributive* just in case it is analytic that: \(P\) holds of
some things \(xx\) if and only if \(P\) holds of each \(u\) such that
\(u\prec
 xx\).[5]
 For instance, the predicate "is on the table" is
distributive, since it is analytic that some things \(xx\) are on the
table just in case each of \(xx\) is on the table. A predicate \(P\)
that isn't distributive is said to be *non-distributive*
or
 *collective*.[6]
 For instance, the predicate "form a circle" is
non-distributive, since it is not analytic that whenever some things
\(xx\) form a circle, each of \(xx\) forms a circle. Another example
of non-distributive plural predication is the second argument-place of
the logical predicate \(\prec\): for it is not true (let alone
analytic) that whenever \(u\) is one of \(xx, u\) is one of each of
\(xx\). It is therefore both natural and useful to consider a slightly
richer language:



> 
> *The formal language \(L\_{\textrm{PFO}+}\)*. The language
> \(L\_{\textrm{PFO}+}\) allows non-distributive plural predicates other
> than \(\prec\). We do this by modifying the definition of
> \(L\_{\textrm{PFO}}\) so as to allow predicates \(R^{n}\_i\) that take
> one or more *plural* arguments. These predicates can be either
> logical or
>  non-logical.[7]
> 



Should we also allow predicates with argument places that take both
*singular* and *plural* arguments? Lots of English
predicates work this way, for instance "... is/are on the
table". So if our primary interest was to analyze natural
language, we would probably have to allow such predicates. However,
for present purposes it is simpler not to allow such predicates. We
will anyway soon allow pluralities that consist of just one
 thing.[8]



For now, the formal languages \(L\_{\textrm{PFO}}\) and
\(L\_{\textrm{PFO}+}\) will be interpreted only by means of a
translation of them into ordinary English, augmented by indices to
facilitate cross-reference (Boolos 1984: 443-5 [1998a:
67-9]; Rayo 2002: 458-9). (More serious semantic issues
will be addressed in
 Section 4,
 where our main question will be whether our theories of plural
quantification are ontologically committed to any sort of
"set-like" entities.) The two clauses of this translation
which are immediately concerned with plural terms are



(4)
 \(\Tr(x\_i \prec xx\_j) = \textrm{it}\_i\) is one of them\(\_j\)


(5)
\(\Tr(\Exists{xx\_j}\mstop \phi) =\) there are some things\(\_j\)
such that \(\Tr(\phi)\)


The other clauses are obvious, for instance: \(\Tr(\phi \amp \psi) =
(\Tr(\phi)\) and \(\Tr(\psi))\). This translation allows us to
interpret all sentences of \(L\_{\textrm{PFO}}\) and
\(L\_{\textrm{PFO}+}\), relying on our intuitive understanding of
English. It is useful to consider some examples. Applying \(\Tr\) to
(\ref{ex2prime}), say, yields:



(2'')
There are some things\(\_1\) such that for everything\(\_2\) (if
it\(\_2\) is one of them\(\_1\), then it\(\_2\) is an apple and it\(\_2\)
is on the table). 

### 1.2 The Theories PFO and PFO+



We will now describe a theory *PFO* of plural first-order
quantification based on the language \(L\_{\textrm{PFO}}\). Let's
begin with an axiomatization of ordinary first-order logic with
identity. For our current purposes, it is convenient to axiomatize
this logic as a natural deduction system, taking all tautologies as
axioms and the familiar natural deduction rules governing the singular
quantifiers and the identity sign as rules of inference. We then
extend in the obvious way the natural deduction rules for the singular
quantifiers to the plural ones. Next we need some axioms which for
suitable formulas \(\phi(x)\) allow us to talk about the \(\phi\)s. In
ordinary English the use of plural locutions generally signals a
concern with two or more objects. But the existence of two or more
objects may not be semantically required; for instance, "The
students who register for this class will learn a lot" seems
capable of being true even if only one student registers. It is
therefore both reasonable and convenient to demand only that there be
at least one object satisfying \(\phi(x)\). (Most people who write on
the subject make this concession.) This gives rise to the *plural
comprehension axioms*, which are the instances of the schema



\[
\tag{Comp}
 \Exists{u} \phi(u) \rightarrow \Exists{xx} \Forall{u} (u\prec xx \leftrightarrow \phi(u))
\]


where \(\phi\) is a formula in \(L\_{\textrm{PFO}}\) that contains
"\(u\)" and possibly other variables free but contains no
occurrence of "\(xx\)". (That is, if something is \(\phi\)
then there are some things such that everything is one of them if and
only if it is \(\phi\).) In order fully to capture the idea that all
pluralities are non-empty, we also adopt the axiom 

 

\[
\tag{6}
\Forall{xx} \Exists{u} (u \prec xx).
\]


(That is, for any things, there is something that is one of them.)




Let *PFO*+ be the theory based on the language \(L\_{PFO+}\)
which arises in an analogous way, but which in addition has the
following axiom schema of indiscernibility, stating that coextensive
pluralities satisfy the same formulas: 

 

\[
\tag{7}
\Forall{xx}\Forall{yy} [\Forall{u}(u \prec xx \leftrightarrow u \prec yy) \rightarrow(\phi(xx) \leftrightarrow \phi(yy))]
\]


(That is, for any things\(\_1\) and any things\(\_2\) (if something is
one of them\(\_1\) if and only it is one of them\(\_2\), then they\(\_1\)
are \(\phi\) if and only if they\(\_2\) are \(\phi)\).) This is a
plural analogue of Leibniz's law of indiscernibility of identicals,
and as such needs to be restricted to formulas \(\phi\) that don't set
up intensional contexts.



While the theories PFO and PFO+ represent what is today the most
traditional form of plural logic, there are divergent views that
restrict the plural comprehension schema. Even if a plurality is
nothing over and above the objects it comprises, these objects do need
to be circumscribed. And on certain metaphysical views, reality as a
whole resists proper circumscription (Spencer 2012; Florio and Linnebo
2020 and 2021, ch. 12; see also Hossack 2014). Based on considerations
of this sort, Florio and Linnebo (*op. cit.*) develop an
alternative *critical* plural logic. 



*Note on terminology*. For ease of communication we will use
the word "plurality" without taking a stand on whether
there really exist such entities as pluralities. Statements involving
the word "plurality" can always be rewritten more
longwindedly without use of that word. For instance, the above claim
that "all pluralities are non-empty" can be rewritten as
"whenever there are some things \(xx\), there is something \(u\)
which is one of the things \(xx\)". Where an ontological claim
\(is\) made, this will be signalled by instead using the locution
"plural entity".


## 2. Plural Quantification vs. Second-Order Quantification



By "second-order logic" we understand a logic that extends
ordinary first-order logic by allowing for quantification into
*predicate* position. For instance, from "\(a\) is an
apple" we can in second-order logic infer
"\(\Exists{F}\mstop Fa\)". But the plural logics described
above extend ordinary first-order logic in a different way, namely by
allowing quantification into *plural argument* position. But
predicates and plural noun phrases belong to different syntactic and
semantic categories. For instance, the former consists of expressions
that are unsaturated (in Frege's sense) -- that is, that
contain gaps or argument places -- whereas the latter consists of
expressions that are saturated (Higginbotham 1998: sect. 7; Oliver and
Smiley 2001 and 2016; Rayo and Yablo 2001: sect. X; Simons 1997;
Williamson 2003: sect. IX; Yi 2005; Florio and Linnebo 2021).
Accordingly, second-order quantification and plural quantification are
generally regarded as different forms of quantification. In this
section I discuss some of the differences and similarities. 


### 2.1 Plural quantification and monadic second-order logic



(Readers less interested in technical issues may want to skim this
section.) Boolos observed that it is possible to interpret monadic
second-order logic in the theory
 PFO.[9]
 Let *MSO* be some standard axiomatization of (full
impredicative) monadic second-order logic in some suitable language
\(L\_{\textrm{MSO}}\) (Shapiro 1991: ch. 3; Boolos *et al.*
2002: ch. 22). Boolos first defines a translation \(\Tr'\) that maps
any formula of \(L\_{\textrm{MSO}}\) to some formula of
\(L\_{\textrm{PFO}}\). This definition, which proceeds by induction on
the complexity of the formulas of \(L\_{\textrm{MSO}}\), has as its
only non-trivial clauses the following two, which are concerned with
the second-order variables:

 

\[
\tag{8}
\Tr'(X\_jx\_i) = x\_i \prec xx\_j
\]

 

\[
\tag{9}
\Tr'(\Exists{X\_j}\mstop\phi) = \Exists{xx\_j}\mstop\Tr'(\phi) \lor Tr'(\phi\*)
\]


where \(\phi\*\) is the result of substituting \(x\_i \ne x\_i\)
everywhere for \(X\_j x\_i\). The idea behind these two clauses is to
replace talk about concepts (or whatever entities one takes the
monadic second-order variables to range over) with talk about the
objects that fall under these concepts. Thus, instead of saying that
\(x\_i\) falls under the concept \(X\_j\), we say that \(x\_i\) is one of
\(xx\_j\). The only difficulty is that some concepts have no instances,
whereas all pluralities must encompass at least one thing. But the
possibility that a concept be uninstantiated is accommodated by the
second disjunct on the right-hand side of (9).



By induction on derivations in MSO one easily proves that each theorem
of MSO is mapped to some theorem of PFO. Moreover, it is easy to
define a "reverse" translation that maps formulas of
\(L\_{\textrm{PFO}}\) to formulas of \(L\_{\textrm{MSO}}\) and to prove
that this translation maps theorems of the former to theorems of the
latter. This shows that PFO and MSO are equi-interpretable. A similar
result can be proved about PFO+ and an extension MSO+ of MSO which
allows predicates of (first-level) concepts, provided MSO+ contains an
axiom schema to the effect that coextensive concepts are
indiscernible.



It is important to be clear on what the equi-interpretability of PFO
and PFO+ with respectively MSO and MSO+ does and does not show (Florio
and Linnebo 2021: sect. 4.2). It shows that these two pairs of
theories are equivalent for most technical purposes. But by itself it
does not show anything about these two pairs of theories' being
equivalent in any of the more demanding senses that philosophers often
care about (such as having the same epistemic status, ontological
commitments, or degree of analyticity). (For instance PFO is
equi-interpretable with atomic extensional mereology, which
philosophers tend to find much more problematic than PFO.) In order to
show that the two pairs of theories are equivalent in some
philosophically important respect \(F\), we would need to show that
the above translations preserve \(F\)-ness.


### 2.2 Relations



Although plural quantification provides a fairly natural
interpretation of quantification over (monadic) *concepts*, it
provides no natural interpretation of quantification over (polyadic)
*relations*.



This limitation can be overcome (at least for technical purposes) if
there is a pairing function on the relevant domain, that is, if there
is a function \(\pi\) such that \(\pi(u, v) = \pi(u', v')\) just in
case \(u = u'\) and \(v = v'\). For then quantification over dyadic
relations can be represented by plural quantification over ordered
pairs. Moreover, by iterated applications of the ordered pair function
we can represent \(n\)-tuples and thus also quantification over
\(n\)-adic relations. The question is how this pairing function is to
be understood. One option is to proceed as in mathematics and simply
postulate the existence of a pairing function as an abstract
mathematical object. But this option has the obvious disadvantage of
stepping outside of what most people are willing to call "pure
logic". A cleverer option, explored in the Appendix to Lewis
1991 and in Hazen 1997 and 2000, is to simulate talk about ordered
pairs using only resources that arguably are purely logical. It turns
out that talk about ordered pairs can be simulated in monadic
*third-order* logic, given some plausible extra assumptions.
Monadic third-order logic can in turn be interpreted either in a
theory which combines plural quantification with mereology (Lewis
1991: ch. 3; Burgess and Rosen 1997: II.C.1) or in terms of
higher-level plural quantification
 (Section 2.4).


### 2.3 Modal contexts



Another way in which plural quantification and second-order
quantification come apart emerges in modal contexts. It is often a
contingent matter whether an object falls under a concept. Although I
am wearing shoes, I might not have done so. So there is a concept
\(F\) under which I fall but might not have fallen. In contrast, it
seems that being one of some objects is never contingent. Consider the
people \(aa\) who are all and only the people currently wearing shoes.
Then not only am I one of these people, but this seems to hold of
necessity (assuming the existence of the relevant objects). Removing
me from this plurality of people would just result in a different
plurality. For the plurality \(aa\) to be the plurality it is, it must
include precisely the objects that it in fact includes. So in any
world in which the objects \(aa\) exist at all, I must be one of them.
True, I might not have been wearing shoes. But even so I would have
been one of \(aa\), only then \(aa\) would not have been all and only
the people wearing shoes. Plural names and variables thus seem to be
rigid in a way that is analogous to the familiar rigidity of singular
names and variables: in any world in which a plural term denotes at
all, it denotes the same objects. In particular, it is widely believed
that pluralities are subject to the following two principles:



\[
\tag{10}
u \prec xx \rightarrow \Box(\EExists xx \rightarrow u \prec xx)
\]

 

\[
\tag{11}
\neg(u \prec xx) \rightarrow \Box(\EExists u \amp \EExists xx \rightarrow \neg(u \prec xx))
\]


where \(\EExists xx\) and \(\EExists u\) are suitable formalizations
of the claims that respectively \(xx\) and \(u\)
 exist.[10]


### 2.4 Higher levels of plural quantification?



One way of going beyond PFO+ would be by allowing quantification into
predicate positions, including those of predicates taking plural
arguments. Doing so would result in an extension which stands to PFO+
as ordinary (singular) second-order logic stands to ordinary
(singular) first-order logic. Such extensions will not be considered
here: for whether they are legitimate, and if so what axioms they may
support, has less to do with plurals and plural quantification than it
does with predication and quantification over the semantic values of
 predicates.[11]



What \(is\) relevant for present purposes is whether there is some
form of "super-plural" quantification that stands to
ordinary plural quantification as ordinary plural quantification
stands to singular quantification. If so, let's call this
*second-level* plural quantification. More generally, we may
attempt to introduce plural quantification of any finite level. This
would result in a theory which for technical purposes is just like a
simple type theory (Hazen 1997: 247; Linnebo 2003: sect. IV; Rayo
2006).



It is fairly straightforward to develop formal languages and theories
of higher-level plural quantification (Rayo 2006). For instance, we
can introduce variables of the form *xxx* to be thought of as
ranging over second-level pluralities and the relation \(xx \prec\_2\)
*xxx* to be understood in analogy with the relation \(x \prec
xx\). (See Linnebo and Rayo 2012 for extensions to transfinite levels
and comparison of the resulting theories with those of ordinary set
theory.) But can these formal theories of higher-level plural
quantification be justified by considerations similar to those that
justify the theories PFO and PFO+?



Boolos and many other philosophers deny that higher-level plural
quantification can be thus justified. Two kinds of arguments are given
for this view. Firstly, it is argued that a plurality is always a
plurality of *things*. But since plural quantification is
ontologically innocent, there are no such things as pluralities. There
is thus nothing that can be collected into a second-level plurality
(McKay 2006: 46-53 and 137-139). Secondly, ordinary plural
quantification is justified by the fact that it captures certain
quantificational devices of English and other natural languages. But
English and other natural languages contain no higher-level plural
quantification (Lewis 1991: 70-71).



Both arguments are controversial. Concerning the first, it is not
clear why ontology should be relevant to the legitimacy of
higher-level plurals quantification. It should be sufficient that the
base-level objects can be organized in certain complex ways. For
instance, the second-level plurality based on Cheerios organized as
oo oo oo should be no more
ontologically problematic than the first-level plurality based on the
same objects organized as oooooo,
although the former has an additional level of structure or
articulation (Linnebo 2003: 87-8).



The second of the above two arguments is also problematic. To begin
with, the claim that there are no higher-level plural locutions in
natural language is almost certainly false. In Icelandic, for
instance, the number words have plural forms which count, not
individual objects, but *pluralities* of objects that form
natural groups. Here is an example:





|  |  |  |
| --- | --- | --- |
| *einn skor* | means | one shoe |
| *einir skor* | means | one pair of shoes |
| *tvennir skor* | means | two pairs of shoes |




This allows us to talk about pairs of shoes as a second-level
plurality rather than as a first-level plurality of objects such as
*pairs*. (More examples involving foreign languages can be
found in Grimau 2021a.) For an English example, consider a video game
in which any number \(n\) of teams can compete in an \(n\)-way
competition. Then the following sentence seems to involve a
superplural term:



(12)
 These people, those people, and these other people compete
against each other. (Linnebo and Nicolas 2008)


(See also Oliver and Smiley 2004: 654-656, 2005: 1063, and 2016,
ch. 15; Ben-Yami 2013; Simons 2016; Florio and Linnebo 2021, ch. 9,
and Grimau 2021a and 2021b)



Moreover, the very idea that the legitimacy of higher-level plural
quantification is decided by the existence or non-existence of
higher-level plural locutions in English and other natural languages
is problematic (Hazen 1993: 138 and 1997: 247; Linnebo 2003: 87; Rayo
2006). What really matters is presumably whether we can iterate the
principles and considerations on which our understanding of ordinary
first-level plural quantification is based: if we can, then
higher-level plural quantification will be justified in much the same
way as ordinary first-level plural quantification; and if not, then
not. Thus, even if there were no higher-level plural locutions in
natural languages, this would provide little or no evidence for the
stronger -- and philosophically more interesting -- claim
that there can be no iteration of the step from the singular to the
plural in *any* language spoken by intelligent agents.
Moreover, any evidence of this sort could be defeated by pointing to
independent reasons why higher-level plural locutions are scarce in
natural languages. One such independent reason may simply be that
ordinary speakers aren't very concerned about their ontological
commitments and thus find it more convenient to express facts
involving second-level pluralities by positing objects to represent
the first-level pluralities (for instance by talking about two
*pairs* of shoes) rather than by keeping track of additional
grammatical device for second-level plurals (as in the above example
from Icelandic).



Suppose we accept higher-level plural quantification. How should this
be implemented? One option is to introduce a separate set of variables
for each level of such quantification, say "\(xxx\)", etc.
for second-level plurals (Rayo 2006). Another option is to use a
single set of "all-purpose" variables whose values can be
pluralities of any level (Oliver and Smiley 2016, ch. 15; Simons 2016;
and Florio and Linnebo 2021, sect. 11.7). 


## 3. The Logicality Thesis



It is often claimed that the theories PFO and PFO+ qualify as
"pure logic". We will refer to this (admittedly vague)
claim as *the Logicality Thesis*. Since the corresponding
languages are interpreted by the translation \(\Tr\) into ordinary
English, this is a claim about the logicality of certain axioms and
inferences rules of ordinary
 English.[12]



Even before the Logicality Thesis is made more precise, it is possible
to assess its plausibility for at least some of the axioms and
inference rules of PFO and PFO+. First there are the tautologies and
the inference rules governing identity and the singular quantifiers.
There is broad consensus that these qualify as logical. Next there are
the inference rules governing the plural quantifiers. Since these
rules are completely analogous to the rules governing the singular
quantifiers, it can hardly be denied that they too qualify as logical.
Then there are the indiscernibility axioms and the axiom that all
pluralities are non-empty. These axioms are unproblematic because they
can plausibly be taken to be analytic. What remains are the plural
comprehension axioms, where things are much less clear. For these
axioms have no obvious singular counterparts, and their syntactical
form indicates that they make existential claims. So it is not obvious
that these axioms can be taken to be purely logical.



This is not to say that it has not *struck* people as obvious
that the plural comprehension axioms are purely logical. For instance,
Boolos asserts without argument that the translation of each plural
comprehension axiom into English "expresses a *logical*
truth if any sentence of English does" (Boolos 1985b: 342
[1998a: 167]; his emphasis). 



In order to assess the Logicality Thesis in a more principled way,
more will have to be said about what it might mean for a theory to be
"purely logical". So I will now survey some of the
features commonly thought to play a role in such a definition.
Although people are free to use the word "logic" as they
please, it is important to get clear on what different usages entail;
in particular, theories that qualify as purely logical are often
assumed to enjoy a variety of desirable philosophical properties such
as epistemic and ontological innocence. In the next section, where
various applications of plural quantification will be discussed, I
will carefully note which strains of the notion of logicality our
theories PFO and PFO+ must possess for the various applications of
them to succeed.



Perhaps the least controversial candidate for a defining feature of
logic is its *topic neutrality*. A logical principle is valid
in any kind of discourse, no matter what kind of objects this
discourse is concerned with. For instance, modus ponens is valid not
only in physics and mathematics but in religion and in the analysis of
works of fiction. Frege captures the idea nicely when he says that a
logical principle is valid in "the widest domain of all;
[...] not only the actual, not only the intuitable, but
everything thinkable" (Frege 1884: 21). Thus, whereas the
principles of physics are valid only in the actual world and in worlds
that are nomologically similar to it, the principles of logic govern
everything *thinkable*. If one of these principles is denied,
"complete confusion ensues" (*ibid*.).



Another feature widely believed to be defining of logic is its
*formality*: the truth of a principle of logic is guaranteed by
the *form* of thought and/or language and does not in any way
depend upon its *matter*. What this feature amounts to will
obviously depend on how the distinction between form and matter is
understood. The most popular explication of the distinction between
form and matter derives from the widely shared view that no objects
exist by conceptual necessity (Field 1993; Yablo 2000). On this view
it is natural to regard anything having to do with the existence of
objects and with their particular characteristics as belonging to the
matter of thought rather than to its form. This gives rise to two
features that are often regarded as defining of logic. Firstly, logic
has to be *ontologically innocent*; that is, a principle of
logic cannot introduce any new ontological commitments (Boolos 1997;
Field 1984). Secondly, the basic notions of logic must not
discriminate between different objects but must treat them all alike.
This latter idea is often spelled out as the requirement that logical
notions must be invariant under permutations of the domain of objects
(Tarski 1986).



A third feature which is often regarded as defining of logic is its
(alleged) *epistemic primacy*. Primitive logical notions can be
grasped without relying on non-logical notions; and logical truths, if
knowable, can be known independently of non-logical truths. Assume,
for instance, that certain set-theoretic notions or truths must be
regarded as extra-logical. Then our grasp of the primitive logical
notions, or knowledge of logical truths, cannot depend on or involve
any of this set-theoretic material.



The key question, then, is whether the plural comprehension axioms
enjoy these features of logicality. As we will see shortly,
substantial doubts have been raised, especially concerning these
axioms' ontological innocence and epistemic primacy.


## 4. Applications of Plural Quantification



I will now outline some applications of the theories PFO and PFO+. In
the previous section three strains of the notion of logicality were
disentangled. Special attention will be paid to the question which of
these three strains PFO and PFO+ must possess for the applications to
succeed. 


### 4.1 Establishing the Logicality of Monadic Second-Order Logic



As we saw in
 Section 2.1
 Boolos defined an interpretation of the theory MSO of monadic
second-order logic in the theory PFO of plural quantification. He
sought to use this translation to establish the logicality of MSO.
Doing so will require two steps. The first step is to argue that PFO
is pure logic, that is, to establish the full Logicality Thesis
(however exactly it is interpreted). The second step is to argue that
the interpretation of MSO in PFO preserves logicality.



Some of the challenges facing the first step will be examined in
 Section 5.
 The second step too should not be underestimated. Perhaps the
greatest worry here is that Boolos's translation renders
expressions from one category (that of monadic predicates) in terms of
expressions from another category (that of plural noun phrases). For
instance, "... is an apple" is rendered as "the
apples". But these categories are very different
 (Section 2).



However, since the thesis that MSO is pure logic is very abstract,
much of its cash value will lie in its applications. And given the
equi-interpretability of MSO and PFO, it is likely that many
applications of the logicality of the former can be served equally
well by the logicality of the latter. This reduces somewhat the
importance of carrying out the second step.


### 4.2 Logicism



Both Fregean and post-Fregean logicism make essential use of
second-order quantification. Frege defined the various objects of pure
mathematics as extensions of concepts, and his famous Basic Law V
stated that two concepts \(F\) and \(G\) have the same extension just
in case they are co-extensive:

 

\[
\tag{V}
u\mstop Fu = u\mstop Gu \leftrightarrow \Forall{u}(Fu\leftrightarrow Gu)
\]


But as is well known, Russell's paradox shows that the
second-order theory with (V) as an axiom is inconsistent.



Philosophers have attempted to rescue some ideas of Fregean logicism
by using axioms weaker than (V). One of the most important such
attempts is Bob Hale and Crispin Wright's *neo-logicism*,
which gives up Frege's theory of extensions but holds on to the
central idea of his definition of cardinal numbers, namely that the
number of \(F\)s is identical to the number of \(G\)s just in case the
\(F\)s and the \(G\)s can be one-to-one correlated. This has become
known as *Hume's Principle*, and can be formalized as



\[
\tag{HP}
Nu.Fu = Nu\mstop Gu \leftrightarrow F\approx G
\]


where \(F\approx G\) says there is a relation that one-to-one
correlates the \(F\)s and the \(G\)s. The second-order theory with
(HP) as an axiom is consistent and allows us to derive all of ordinary
(second-order Peano-Dedekind) arithmetic, using some very natural
definitions (see the entry on
 Frege's Logic, Theorem, and Foundations for Arithmetic).



Even more modest is Boolos's *sub-logicism*, which
rejects the idea (endorsed by both logicists and neo-logicists) that
there are logical objects, but insists that Frege's definition
of the *ancestral* of a relation can be used to show, as
against Kant, that at least some non-trivial mathematics is analytic
(Boolos 1985b). Recall that a relation \(R\) stands to its ancestral
\(\Rarel\) as the relation *is a parent of* stands to *is an
ancestor of*. (More precisely, \(\Rarel\) holds between two
objects \(x\) and \(y\) just in case \(x\) and \(y\) are connected
through a finite sequence of objects each of which bears \(R\) to its
successor.) Frege gives a second-order definition of the ancestral
relation \(\Rarel\) by laying down that \(x\) and \(y\) are related by
\(\Rarel\) just in case \(y\) has every property that is had by
\(x\)'s \(R\)-successors and inherited under the
\(R\)-relation:

 

\[
\tag{Def \(\Rarel\)}
x\Rarel y \leftrightarrow \Forall{F}[\Forall{u}(x\Rrel u \rightarrow Fu) \amp \Forall{u}(Fu \amp u\Rrel v \rightarrow Fv) \rightarrow Fy]
\]


Using this definition, Frege 1879 proves some non-trivial mathematical
truths, such as that the ancestral \(\Rarel\) is transitive and that,
for any functional relation \(R\), the \(R\)-ancestors of any object
are \(\Rarel\)-comparable (that is, he proved: Functional \((R) \amp
x\Rarel y \amp x\Rarel z \rightarrow y\Rarel z \lor z\Rarel y\)).



It has been suggested that PFO be used to accommodate the post-Fregean
logicists' need for second-order quantification. Since the
ancestral of a dyadic predicate can be defined using only monadic
second-order quantification, PFO does indeed serve the logical needs
of Boolos's
 sub-logicism.[13]
 But since the neo-logicist definition of \(F\approx G\) uses
*dyadic* second-order logic, PFO alone does not have sufficient
expressive power to accommodate the needs of neo-logicism. The
neo-logicist may attempt to solve this problem by regarding
equinumerosity as a primitive logical quantifier or by simulating
dyadic second-order quantification in some suitable extension of PFO,
as discussed in
 Section 2.2[14]
 (see Boccuni 2013 for another option).



Which strains of the Logicality Thesis are needed for these
applications to succeed? Since these logicists attempt to show that
parts of mathematics are analytic (or at least knowable *a
priori*), this would require that PFO be analytic (or at least
knowable *a priori*), which in turn is likely to require that
PFO enjoy some form of cognitive primacy. Moreover, PFO would have to
be either ontologically innocent or committed only to entities whose
existence is conceptually necessary (or at least establishable *a
priori*).


### 4.3 Set Theory



Another application of the Logicality Thesis is concerned with set
theory. One may for a variety of reasons want to talk about and
quantify over collections of sets (Uzquiano 2003). For instance, one
may want to assert



(13)
 There are some sets which are all and only the non-self-membered
sets.


If we formalize this as

 

\[
\tag{\(13'\)}
\Exists{R} \Forall{x} (Rx \leftrightarrow x\not\in x),
\]


how is the quantifier \(\exists{R}\) to be understood? It clearly
cannot be taken to range over all sets, as this would lead straight to
Russell's paradox: (13') would then assert the existence
of the Russell-set. Three other responses are prominent in the
literature.



The first response is that \(\exists{R}\) ranges over *classes*
but that some classes are too large (or otherwise unsuited) to be
sets. In particular,
 (13)
 asserts the existence of the Russell-class, which isn't a set.
This response has been found problematic because it postulates the
existence of different kinds of "set-like" entities
(Boolos 1984: 442 [1998a: 66] and 1998b: 35). It has also been
objected that this response only postpones the problem posed by
 (13).
 For it would also be true that



(14)
 There are some classes which are all and only the
non-self-membered classes.


What kind of entity would this collection of classes be? A
super-class? If so, we will be forced to postulate higher and higher
levels of classes. Lewis (1991: 68) argues that Russell's
paradox is still inescapable because, when we consider all set-like
entities, we realize that the following is true:



(15)
 There are some set-like things which are all and only the
non-self-membered set-like things.


However, Hazen (1993: 141-2) has pointed out that Lewis's
objection violates essential type-restrictions. Classes of different
levels belong to different logical types, just as concepts of
different levels do. So Lewis's attempt to talk about all
set-like entities in one fell swoop involves an attempt to quantify
across different logical types. But this violates type restrictions in
the same way as an attempt to quantify simultaneously over objects and
concepts of all different levels. Although we can quantify over
*each* level of classes, we can never quantify over *all
levels simultaneously*.



The second response is that
 (13)
 asserts the existence of a set \(R\), but that \(R\) isn't in
the range of the quantifier \(\forall{x}\). This prevents us from
instantiating the quantifier \(\forall{x}\) with respect to \(R\),
which means that we cannot draw the fatal conclusion that \(R\) is a
member of itself just in case it isn't. However, this response
entails that the quantifier \(\forall{x}\) cannot be chosen to range
over absolutely all sets; for if it could be so chosen, we would not
be able deny that \(R\) is in this range of quantification. This means
that the universe of sets has a certain *inexhaustibility*:
whenever we have formed a conception of quantification over some range
of sets, we can define a set which isn't in this range (Dummett
1981: ch. 15 and 1991: ch. 24; Glanzberg 2004; Parsons 1977). However,
this response has been criticized for being, at best hard to state,
and at worst self-refuting (Boolos 1998b: 30; Lewis 1991: 68;
Williamson 2003: sect. V). (See also Rayo and Uzquiano 2006 for a
number of essays discussing whether absolutely general quantification
is possible.)



Because of the difficulties involved in the first two responses, a
third response has become popular in recent years (Boolos 1984 and
1985a; Burgess 2004; Cartwright 2001; Rayo and Uzquiano 1999; Uzquiano
2003). This is that the quantifier \(\exists{R}\) is a plural
quantifier (and would thus be better written as \(\exists{rr})\) and
that plural quantification is ontologically innocent. Therefore
 (13)
 does not assert the existence of any "set-like" entity
over and above the sets in the range of the quantifier \(\forall{x}\).
But as we will see in
 Section 5,
 the claim about ontological innocence is controversial.



A entirely different application of plural logic to set theory seeks
to use plurals to explain sets. Kurt Godel famously wrote of the
operation "set of", which can be applied to any
"well-defined objects" to form their set and whose
iterated application yields the cumulative hierarchy of sets (1964:
180). A key question is how to understand the notion of some
"well-defined objects". Suppose we accept that any objects
\(xx\) qualify as such. Then any objects \(xx\) give rise to a set
\(\{xx\}\). Since the cumulative hierarchy contains no universal set,
it follows that there cannot be a universal plurality comprising
absolutely all objects. Florio and Linnebo 2020 and 2021, ch. 12
develop a *critical* plural logic that is suited for being
combined with such an unrestricted "set of" operation.
Thus, this approach forsakes the power of this logic to talk about
arbitrary collections of sets and instead uses plurals to account for
the nature of, and our knowledge of, sets. 


### 4.4 Mathematical Nominalism



Some of the most popular applications of the plural quantification are
concerned with ontological economy. The idea is to pay the ontological
price of a mere first-order theory and then use plural quantification
to get for free (a theory with the force of) the corresponding monadic
second-order theory. That would obviously be an ontological bargain.
Applications of this sort fall into two main classes, which will be
discussed in this sub-section and the next.



One class of applications of plural quantification aim to make
ontological bargains in the philosophy of mathematics. In particular,
a number of philosophers have attempted to use plural quantification
as an ingredient of nominalistic interpretations of mathematics. A
nice example is Geoffrey Hellman's modal nominalism, according
to which mathematical statements committed to the existence of
abstract objects are to be eliminated in favor of statements about the
*possible* existence of concrete objects. For instance, instead
of claiming, as the platonist does, that there exists an infinite
collection of abstract objects satisfying the axioms of Peano
arithmetic (namely the natural numbers), Hellman claims that there
*could exist* an infinite collection of concrete objects
related so as to satisfy these axioms (Hellman 1989 and 1996).
However, even this modal claim appears to talk about
*collections* of concrete objects and *relations* on
these objects. To forestall the objection that this smuggles in
through the back door abstract objects such as sets, Hellman needs
some alternative, nominalistically acceptable interpretation of this
talk about collections and relations. Plural quantification may offer
such an interpretation.



For this application of plural quantification to work, PFO must be
applicable to all kinds of concrete objects, and it must be
ontologically innocent, or at least not committed to any entities that
share those features of abstract objects that are found to be
nominalistically objectionable. Moreover, in order to simulate
quantification over relations, we will need not just PFO but a theory
more like monadic *third-order* logic (Sections
 2.2
 and
 2.4).


### 4.5 Eliminating Complex Objects



Another class of applications attempts to eliminate the commitments of
science and common sense to (some or all) complex objects. For
instance, instead of employing usual singular quantification over
tables and chairs, it is proposed that we use plural quantification
over mereological atoms arranged tablewise or chairwise (Rosen and
Dorr 2002; Hossack 2000; van Inwagen 1990). For instance, instead of
saying that there is a chair in one's office, one should say
that there are some atoms in one's office arranged chairwise. In
this way one appears to avoid committing oneself to the existence of a
chair. Note that such analyses require PFO+, not just PFO, since the
new predicates "are arranged \(F\)-wise" are
non-distributive.



Let's set aside purely metaphysical worries about such analyses
as irrelevant to our present concern. What \(we\) would like to know
is what demands these analyses put on the theory PFO+, in particular,
which strains of the Logicality Thesis are needed. The most obvious
demands are that PFO+ be applicable to all kinds of simple objects and
that it be ontologically innocent, or at least not committed to
complex objects of the sort to be eliminated.



A less obvious demand has to do with the need to analyze ordinary
plural quantification over complex objects, for instance



(16)
 There are some chairs arranged in a circle.


We have already "used up" ordinary plural quantification
and predication to eliminate apparent commitment to individual chairs
(Uzquiano 2004). So in order to analyze (16), we will need something
like "super-plural" quantification -- quantification
that stands to ordinary plural quantification as ordinary plural
quantification stands to singular -- and corresponding
non-distributive predication. The legitimacy of such linguistic
resources was discussed in
 Section 2.4.



There have also been attempts to use plural logic to eliminate
apparent reference to groups. Instead of referring to the Supreme
Court, say, perhaps we might get by with referring plurally to its
members. An obvious challenge for this approach is that a group can
have different members at different moments of time and possible
worlds, while the plurality of its (current and actual) members
appears to be rigid, temporally as well as modally
 (Section 2.3).
 See (Horden and Lopez de Sa 2021) for a defense of the
proposed elimination against such challenges. 


## 5. Ontological Innocence?



The traditional view in analytic philosophy has been that all plural
locutions should be paraphrased away, if need be, by quantifying over
sets
 (Section 1).
 George Boolos and others objected that it is both unnatural and
unnecessary to eliminate plural locutions. This led to the theories
PFO and PFO+. Proponents of plural quantification claim that these
theories allow plural locutions to be formalized in a way that is
fundamentally different from the old set-theoretic paraphrases. In
particular, they claim that these theories are ontologically innocent
in the sense that they introduce no new ontological commitments to
sets or any other kind of "set-like" entities over and
above the individual objects that compose the pluralities in question.
Let's call this latter claim *Ontological Innocence*.



Other philosophers question Ontological Innocence. For instance,
Michael Resnik expresses misgivings about the alleged ontological
innocence of the plural formalization (\ref{ex3pprime}) of the
Geach-Kaplan sentence
 (3).
 For when (\ref{ex3pprime}) is translated into English as instructed,
it reads:



\((3''')\)
 There are some critics such that any one of them admires another
critic only if the latter is one of them distinct from the
former.


But \((3''')\), Resnik says, 



> 
> 
> seems to me to refer to collections quite explicitly. How else are we
> to understand the phrase "one of them" other than as
> referring to some collection and as saying that the referent of
> "one" belongs to it? (Resnik 1988: 77) 
> 
> 
> 



Related worries have been expressed in Hazen 1993, Linnebo 2003,
Parsons 1990, and Rouilhan 2002; see also Shapiro 1993.



I will now discuss three arguments in favor of Ontological
Innocence.


### 5.1 The set-theoretic argument



The first argument begins by asking us to consider the claim



(17)
 There are some sets which are all and only the non-self-membered
sets.


and admit that it is true. It continues by arguing that, if plural
expressions were committed to collections or any other
"set-like" objects, then the truth of (17) would lead
straight to Russell's paradox. This is sometimes thought to be a
knock-down argument in favor of Ontological Innocence (Boolos 1984:
440-443 [1998a: 64-67]; Lewis 1991: 65-69; McKay
2006: 31-32). But in fact it is less conclusive than it appears.
For as we saw in
 Section 4.3,
 Russell's paradox will follow only if two alternative views are
ruled out. Since these views cannot be dismissed out of hand, much
work remains before this argument can be said to be conclusive.


### 5.2 The incorrect predication argument



The second argument is nicely encapsulated by Boolos's remark
that "It is haywire to think that when you have some Cheerios,
you are eating a *set*" (1984: 448-9 [1998a: 72]).
What Boolos is suggesting here is that analyses which deny Ontological
Innocence are likely to get the subject of plural predications
wrong.



The obvious response is to interpret plural predicates in a way which
ensures that what we eat are *the elements* of a set and not
the set itself. Consider the sentence:



(18)
 George Boolos ate some Cheerios for breakfast on January 1,
1985.


When the direct object of the verb "ate" is plural, we can
for instance interpret the verb by means of the relation *x
ate-the-elements-of y*.



It will be objected that this response makes the verb
"ate" ambiguous in an implausible way (Oliver and Smiley
2001). For when the verb has a direct object that is singular, it will
presumably be interpreted by means of the ordinary relation *x ate
y*. But there is fairly strong evidence that the verb
"ate" isn't ambiguous in this way. For instance, one
effect of an ambiguity is to disallow certain kinds of ellipsis. An
example is the ambiguity of "make" in "make
breakfast" and "make a plan", which disallows the
following ellipsis:



(\*19)
 Boolos made breakfast, but his guest, only a plan.


So if "ate" was ambiguous in the way just described, the
following ellipsis would be disallowed as well, which it
isn't:



(20)
 Boolos ate some Cheerios, but his guest, only an apple.


However, it is far from clear that the above response to
Boolos's argument needs to be committed to any such problematic
ambiguities. We can for instance let *all* predicates take
plural entities as their arguments. The verb "ate" will
then always receive as its interpretation the relation
*the-elements-of x ate-the-elements-of y*, thus removing any
ambiguity. Whether or not this response is ultimately acceptable, it
shows that the argument in question remains inconclusive.


### 5.3 The direct argument



Perhaps the most popular argument for Ontological Innocence is the one
to which I now turn. In its simplest form, this argument is based on
our intuitions about ontological commitments. When you assert
 (18),
 you don't have the feeling that you are committing yourself
ontologically to a collection or to any other kind of
"set-like" object. Nor do you have any such feeling when
you assert the Geach-Kaplan sentence or any other translation of a
sentence of PFO or PFO+ into English. Or so the argument goes.



In this simple form the argument is vulnerable to the objection that
people's intuitions provide a poor basis for settling
theoretical disputes about ontological commitments. We have seen that
there are competent speakers of English, such as Michael Resnik, who
don't share these intuitions. Moreover, as Davidson's
popular analysis of action sentences in terms of events makes clear,
ordinary people's intuitions about ontological commitments
cannot always be trusted (Davidson 1967). For instance, someone may
sincerely assert that John walked slowly, without being aware that he
has committed himself to the existence of an event (namely a walking
which was by John and which was slow).



Although this objection has force, the argument can be sharpened by
undertaking a more careful study of what forms of existential
generalization are warranted on a sentence containing plural
expressions (Boolos 1984: 447 [1998a: 70]; McKay 2006: ch. 2; Yi 2002:
7-15 and 2005: 469-472). For instance, we may ask whether
the following can be inferred from
 (18):



(21)
 There is an object such that Boolos ate all of its elements (or
constituents) for breakfast on January 1, 1985.


This inference would no doubt be quite peculiar. This provides
evidence that
 (18)
 isn't committed to any kind of "set-like"
entity.



However, this evidence is not incontestable. For there are analogous
inferences that seem quite natural. For instance, from



(22)
 Some students surrounded the building.


most speakers of English would be perfectly happy to infer that 



(23)
 A group of students surrounded the building.


So perhaps the peculiarity of the inference from
 (18)
 to
 (21)
 is a pragmatic rather than a semantic phenomenon. Perhaps it has to
do with the fact that it is less natural to regard some Cheerios as a
set (or some other sort of plural entity) than it is to regard some
students as a group. 



However, let's assume that the defenders of the Direct Argument
are right that
 (18)
 does not entail
 (21).
 What would follow? It would follow that
 (18)
 does not incur any additional ontological commitments *of the sort
that can be incurred by singular first-order quantifiers*. But
this conclusion falls short of the argument's desired conclusion
that
 (18)
 does not incur any additional ontological commitments *of any
sort*. In order to get from the actual conclusion to the desired
one, we would in addition have to assume that all ontological
commitments are of the sort incurred by singular first-order
quantifiers. But there is an influential philosophical tradition that
denies this assumption and instead holds that all kinds of quantifiers
incur ontological commitments, not just singular first-order
 ones.[15]
 The most famous exponent of this tradition is Frege, who claims that
second-order quantifiers are committed to concepts, just as singular
first-order quantifiers are committed to objects. This tradition ties
the notion of ontological commitment very closely to that of a
semantic value. This will be the topic of the next and final
subsection.


### 5.4 Semantic values and ontological commitments



In semantics it is widely assumed that each component of a complex
expression makes some definite contribution to the meaning of the
complex expression. This contribution is known as the *semantic
value* of the component expression. It is also assumed that the
meaning of the complex expression is functionally determined by the
semantic values of the component expressions and their syntactic mode
of composition. This assumption is known as
*compositionality*.



According to Frege, the semantic value of a sentence is just its
truth-value, and the semantic value of a proper name is its referent
(that is, the object to which it refers). Once we have fixed the
semantic values assigned to sentences and proper names, it is easy to
determine what kinds of semantic value to assign to expressions of
other syntactic categories. For instance, the semantic value of a
monadic predicate will have to be a function from objects to
truth-values. Frege calls such functions *concepts*.



As an example, let's consider the simple subject-predicate
sentence



(24)
Socrates is mortal.


The logical form of (24) is \(\mathbf{M}(\mathbf{s})\), where
\(\mathbf{M}\) is the predicate "is mortal" and
\(\mathbf{s}\) is the singular term "Socrates".
Let's write [\(\mathbf{E}\)] for the semantic value of an
expression \(\mathbf{E}\). In accordance with the previous paragraph,
the semantic values relevant to (24) are as follows:



(25)
\([\mathbf{s}] = \textrm{Socrates}\) 


(26)
 \([\mathbf{M}] =\) the function \(f\) from objects to
truth-values such that \(f(x)\) is the true if \(x\) is mortal and
\(f(x)\) is the false otherwise


The truth-value of
 (24)
 is thus determined as



(27)
 [(24)] \(= [\mathbf{M}(\mathbf{s})] = [\mathbf{M}] ([\mathbf{s}])
= f(\textrm{Socrates}) =\) the true (if Socrates is mortal) or the
false (otherwise).


Frege took the connection between semantic values and ontological
commitments to be a very close one. For on the above analysis,
 (24)
 supports two kinds of existential generalizations: not just to
\(\Exists{x}\mstop\mathbf{M}(x)\) (which is true just in case there
exists some object which is mortal) but also to \(\Exists{F}\mstop
F(s)\) (which is true just in case there exists some concept under
which Socrates falls). According to Frege, this shows that sentences
such as
 (24)
 are ontologically committed not just to an object but also to a
concept.



What matters for present purposes is not the truth or falsity of
Frege's claim about concepts but whether a cogent argument of
this sort can be developed for plural expressions. To investigate
this, let's consider a simple non-distributive plural
predication such as



(28)
 These apples form a circle.


The logical form of (28) appears to be \(\mathbf{C}(\mathbf{aa})\),
where \(\mathbf{C}\) is the predicate "form a circle" and
\(\mathbf{aa}\) is the plural term "these apples". (If you
think complex plural demonstrative have internal semantic structure,
use instead some plural name stipulated to refer directly to the
apples in question.) The natural view will then be as follows.



(29)
 [\(\mathbf{aa}] = a\_1\) and ... and \(a\_n\) (where the
\(a\_i\) are all and only the apples demonstrated)


(30)
 [\(\mathbf{C}] =\) the function \(g\) from pluralities to
truth-values such that \(g(xx)\) is the true if \(xx\) form a circle
and \(g(xx)\) is the false otherwise


The truth-value of
 (28)
 will then be determined as



(31)
 [(28)] \(= [\mathbf{C}(\mathbf{aa})] = [\mathbf{C}]
([\mathbf{aa}]) = g(a\_1\) and ... and \(a\_n\)) = the true (if
\(a\_1\) and ... and \(a\_n\) form a circle) or the false
(otherwise)


which is what one would expect, given the syntactic similarity between
 (24)
 and (28).



Assume that this analysis is correct and that each plural term thus
has some objects as its semantic value, just as each singular term has
one object as its semantic value. What will this mean for the question
of Ontological Innocence? According to the Fregean tradition, which
connects the notion of ontological commitment to that of a semantic
value, this will mean that plural expressions incur commitment to
plural entities, much as predicates incur commitment to concepts. For
to say that a sentence incurs a commitment to a plural entity is just
to say that the truth of the sentence requires there to be some
semantic value of the sort appropriate to plural expressions. However,
this line of reasoning will be resisted by other philosophers, who
believe that the notion of ontological commitment should be tied (at
most) to singular first-order variables.



How can this disagreement be adjudicated? On the one hand, it may
count in favor of the Fregean tradition that their view is highly
systematic. There may be something *ad hoc* about the idea that
some sorts of semantic value give rise to ontological commitments
while other sorts don't. On the other hand, it may count in
favor of the alternative view that it does better justice to many
people's strongly felt intuition that plural locutions are
ontologically innocent.



Another possibility is that the whole controversy is ultimately just a
pseudo-disagreement (see especially Florio and Linnebo 2016, but also
Parsons 1990; Shapiro 1993; Linnebo 2003; Rayo 2007; and Linnebo and
Rayo 2012). If both parties agree that plural expressions have
semantic values, and if both agree that commitments to
*objects* are incurred only by singular first-order terms and
variables, then perhaps it does not matter whether other sorts of
terms and variables should be regarded as introducing their own
distinctive kinds of ontological commitment. Some philosophers speak
about a theory's *ideological commitments* and not just
about its ontological commitments. By this is meant the logical and
conceptual resources that the theory employs. Perhaps philosophers
would be well advised to focus more on the metaphysical and
epistemological questions raised by a theory's ideological
commitments and worry less about whether these ideological commitments
should be also regarded as introducing a distinctive kind of
ontological commitment. After all, the notion of an ontological
commitment is a theoretical one, not one that has any sharp content
outside of philosophy. So perhaps we should regard the notion more as
a means to the end of providing good philosophical explanations and
less as a goal in itself.