Nominalism Through DeNominalization
A. Rayo
S. Yablo
draft of June 3, 1999 — comments much
appreciated
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I. Introduction
Not all that long ago, secondorder statements were
thought to be committed twice over. The one commitment was to the
members of a firstorder domain D; the other was to the members of
a secondorder domain consisting of D's subsets and more generally
the kadic relations on D.
A good way to recapture the mood is to
recall Quine's description of secondorder logic as "set theory in
sheep's clothing." What did Quine mean by this? His point was not that the semantics
of secondorder logic is settheoretical; for the
semantics of firstorder logic is settheoretical well. And Quine certainly doesn't think that firstorder logic is set theory in sheep's clothing.
Nor could Quine's point
have been that there are secondorder statements whose validity
goes with the truthvalue of this or that settheoretic
hypothesis, for instance, the continuum hypothesis. There are firstorder statements too which the
semantics classifies differently according to how matters stand in
the world of sets. Suppose that no sets have infinitely many members. Then there are no infinitedomain models, and so the
semantics finds
(0) ∀x [x ≠ s(x) & o≠s(x) & ∀y≠x s(y)≠s(x)],
(essentially the first few Peano Postulates for arithmetic) to
be logically inconsistent. So, the negation of (0) is valid unless
there are infinite sets. (As ZF's Axiom of Infinity explicitly
requires there to be.) And once again, firstorder logic is not, for Quine, set theory in sheep's clothing.
If the point was not that secondorder semantics is settheoretical, or that secondorder
sentences depend for their validity on the behavior of
sets, what was it? Quine was concerned that secondorder logic advanced
theses which could not be true unless sets were counted into the range of their quantifiers. Almost any secondorder thesis worth mentioning has this property for Quine; because as he sees it, a
secondorder quantifier is basically a firstorder
quantifier stipulated to range over certain sets. This makes clear
why he secondorder logic strikes him as really mathematics. A statement
that cannot be true unless Xs are counted into the range of its quantifiers is by
definition — Quine's definition  ontologically
committed to Xs. The secondorder comprehension axiom (to take that example) cannot be true unless sets are counted into the range of its quantifiers, hence secondorder logic is
committed to sets.
So much is to remind ourselves of what Quine and others used to think about the ontological commitments carried by secondorder quantifiers. Over the last couple of decades the Quinean consensus has been breaking down, due to an observation of George Boolos's. What Boolos noticed is that at least many secondorder quantifiers can be construed in terms of phrases
of English that carry no "extra" commitments, that is, no
commitments not incurred already at the first order. A simple but
representative example is Boolos's interpretation of
(1) ∃G (Ga & ¬Gb).
This is read as saying that
(2) There are some things such that a is one
of them and b is not one them.
Of course, it needs to be shown that that the plural
quantifier "there are some things ..." is committed just to the
things and not their set. But a prima facie case is not hard to make. Suppose the second commitment were there. Then
(3) There are some things that are too many to form
a set
would be selfdefeating; it would be committed to
entities of the very sort that it explicitly rejects. But far from being selfdefeating, (3) is an important and hardwon truth about
sets; it is the clearest statement yet found of what separates the older, paradoxprone, concept
of set from the (hopefully) nonparadoxical concept in use today.
II. Ontological Issues
Boolos's victory over the Quinean consensus has not been total, however. One reason is that Boolos's interpretation scheme, effective as it is
with monadic secondorder quantifiers like '∃G' in (1), gets no grip whatever on
polyadic secondorder quantifiers like '∃R' in
(4) ∃R (Rab & ∀x ∃y ¬Rxy)
One can't read the initial quantifier here as "there
are some things....," because that leaves out the relationality;
it makes no sense to say of "the things" that a
bears them to b. Secondorder logic however makes essential
use of dyadic quantifiers like the one in (4). A famous example is
the secondorder formulation of what has come to be known as "Hume's principle": the number of
Fs = the number of Gs iff (∃R)(R is a oneone relation
between the Fs and the Gs).
The most common response to this difficulty
has been to look for ways of "coding" polyadic secondorder
quantifiers monadically. One idea is to construe dyadic
'∃R' as a (disguised) monadic
quantifier over pairs of domain elements.
(4) could
then be read as
(5) There are some pairs such that <a,b> is one of them, and for all x there
is a y such that <x,y> is not one of
them.
This approach is least objectionable when one is
working with a domain held to be closed under the operation of
pairing — when one is doing secondorder set theory, for
instance. Even here though there is cause for unhappiness. That there are exactly as many even numbers as odds looks like a statement about numbers and nothing else. But the
secondsentence that purportedly expresses this fact uses a dyadic
quantifier, and so is committed to ordered pairs on the proposed scheme.
Other settheoretical statements do, of course, carry a commitment
to ordered pairs. But that we're working in a theory that makes pairs available hardly means that a sentence like "there are as many evens as odds" can avail itself of them with no change of subject matter.
A more serious problem is that one uses second
order logic in connection with all sorts of domains, many of which have not much to offer in the ordered pair department. What
if my intended subject matter is material objects, and only them?
Must we treat the claim that "there as many left shoes as right in
my closet" as covertly concerned with pairs? It would be an enormous letdown if secondorder statements, having just been cleared of the "serious" charge of being committed to all manner of sets, must plead guilty to the
"lesser" charge of being committed to the special sets that
are ordered pairs.
The "more serious"
problem would not arise if pairsurrogates could be found in the
firstorder domain — say, the domain of material objects 
to which we count ourselves already committed. (A version of the first problem would remain.) The project of looking for concrete pairsurrogates is taken up by Lewis, Burgess, and Hazen in Parts of
Classes, "Appendix on Pairing." Because there
are no "natural" concrete codes for ordered pairs, this approach inevitably involves a certain amount of ad hocery (as the authors do not dispute). But that is not
what concerns us.
Our concern is that the approach does not even get off
the ground unless certain cosmological conditions are met. Lewis
sums up the state of play in 1990 by saying that we need to have "infinitely
many atoms [and] not too much atomless gunk" (PC, 121). Hazen in later work improves the condition to: there are infinitely many atoms or some atomless gunk. But the fact remains that depending on empirical circumstances, and on our chosen firstorder
domain, the trick may well not work. This is a discouraging
result. It is one thing to say that we can arrange, in
fortunate conditions, for the work secondorder languages
do to be done without taking on any commitment to sets. It would
be better if secondorder languages were construable as
already uncommitted to sets, and uncommitted regardless of
the cosmological circumstances.
III. Why bother?
A word about motivation. "Nominalism" in the paper's title refers not to the general ontological thesis  the rejection of abstract objects  but rather nominalism about secondorder logic  the idea that secondorder quantifiers need not be construed as ranging over abstract objects. One reason for being interested in nominalism about secondorder logic is that one is a nominalist in ontology. But there are other reasons, and it is not only the ontological nominalist who would (should) like to see secondorder quantifiers interpreted in an ontologically unloaded fashion. That someone embraces sets (concepts, properties...) does not necessarily mean they believe secondorder statements to quantify over sets  not any more than a belief in angels requires one to think that secondorder quantifiers range over angels. But more important, even someone who likes the settheoretic construal ought still to be interested in the expressive advantages offered by alternative construals.
Just in case there are doubts on this score, consider some applications of the nominalizing project of potential interest to platonists. You may or may not find them interesting; that's not the point. The point is that the first two are of no use whatever to anyone but a platonist, while the third is as much use to a platonist as to anyone else.
(A) Boolos suggests "there are some sets that are too many to form a set" as a restatement of "there are some wouldbe sets that are too big to be a set." The people whose doctrine Boolos is offering to clarify here are not nominalists; the nominalist doesn't think there are too many sets to form a set, because s/he doesn't think there any sets. Likewise it will be the platonist who benefits if we can reconstrue singular talk about classsized relations as nonnominal talk about how things are related.
(B) Fregean "platonists" like Crispin Wright and Bob Hale attempt to deduce the existence of numbers from Fregean secondorder logic plus definitions. An obvious objection is that since (Fregean) secondorder quantifiers range over concepts, they are at best deriving one sort of abstract entity from another. How is that better than what Zermelo did when he reduced arithmetic to set theory? A construal of secondorder quantifiers whereby they carried no new commitments would enable the Fregean platonist to answer the objection on its own terms.
(C) Advocates of Fregestyle semantics say that predicates refer, but not to objects; that is, not to anything of the sort that a singular term can refer to. This leads to the paradox of the concept "horse" and threatens to make the advocated semantical theory inexpressible. To say that predicates do not refer  to say in other words what is literally true for Frege  escapes the paradox, but leaves us with no way to explain predicates' systematic contribution to truthvalue. Not so fast, though. Systematic semantics has to quantify, but not necessarily over anything. The way out is to do our semantics in a nominalistic secondorder metalanguage of the sort about to be explained. Examples (borrowing on the later explanation): "Susan is kind" is true because Susan is something that all and only the the kind things are. "Edinburgh is north of London" is true because things relate somehow such that "is north of" is true of x and y iff they are so related, and Edinburgh is so related to London.
IV. Grammatical Issues
One can think of Boolos's argument in the monadic
case as having two steps. Step one is the provision of a non
committalseeming bit of English. In step two one attempts to show
that this bit of English is all that's needed to translate the relevant
secondorder quantifiers.
Existing attempts to extend the Boolos argument have
focused entirely on the second step; they have tried to
render the polyadic in plural terms. Our suggestion is going to be
that we focus instead on the first step. Rather than trying
to read everything in terms of the English plural, we should
look instead for another English locution, one that does
better than the plural at capturing polyadic secondorder
meanings. First though let's look briefly at another reason for
dissatisfaction with Boolos's proposal.
The secondorder quantifier is supposed to be a
predicative quantifier. The positions it governs are built
for predicative expressions; and it is predicative expressions one
needs to "plug in" to obtain a grammatical substitution instance.
Plural quantifiers by contrast have a distinctly nominal
feel. To say "there are some things which..." feels like a way of
talking about things, not what things are like or how they are
related. This intuitive judgment is confirmed by the fact that the
expressions Boolos's quantifier binds are the nominal "they" and
"them." Also by the fact "there are some things which... " is just
like the firstorder "there is a thing which...." in needing to be
completed, not by a noun phrase, but something verbish: "swim(s)"
rather than "the swimmer."
One can and does avoid the problem in practice by
smuggling in appropriate connecting material. '∃F Fb' is read not as "there
are some things such that they b" (!!) but "there are some
things such that they includeb" or "... b
is one of them." The 'F' is seen as somehow sprouting the
connecting material en route from its original position in the
quantifier to its later position in the matrix. Because 'F' is not born predicative, but has predicativeness thrust upon it, the informal practice, to the extent that it concerns itself with predicativeness at all, is without a consistent interpretation of the
secondorder variable. There appears, indeed, to be no grammatical
way of reading both the plurality and the predicativeness back into the initial 'F'. "There are somethingsofwhichone
is..." is as close as we've been able to come, and apart from not
being sufficiently general, it just doesn't scan.
V. Connections
Our claim so far is that there are two ways in which
Boolos's approach can seem not to go far enough. These suggest in turn two
desiderata, one ontological and one linguistic.
The first thing we want is a noncommittal (or not
furthercommittal) English locution into which polyadic
secondorder quantifiers can be translated. The second is that
that locution should respect the predicative, or at least non
nominal, character of these quantifiers, and for that matter
monadic secondorder quantifiers as well.
These two desiderata — the ontological and the
linguistic  are not as unrelated as they may seem. Quine
in "On what there is" was already clear that adjectives and other
nonnominal phrases are ontologically innocent. Speaking of McX's
view that "'There is an attribute' follows from 'There are red
houses, red roses, red sunsets'" (10), Quine says that
the word 'red'...is true of each of sundry
individual entities which are red houses, red roses, red sunsets;
but there is not, in addition, any entity whatever, individual or
otherwise, which is named by the word 'redness' ... McX cannot
argue that predicates such as 'red'...must be regarded as names
each of a single universal entity in order that they be meaningful
at all. For we have seen that being a name of something is a much
more special feature than being meaningful. (1011)
If predicates and the like needn't name to
be meaningful  to make their characteristic contribution to
truthvalue  then we have it seems no reason to regard them as
presupposing entities at all. And this indeed appears to be
Quine's view. But now he goes on to say something prima facie
puzzling:
One may admit that there are red houses,
roses, and sunsets, but deny, except as a popular and misleading
manner of speaking, that they have anything in common (10).
Quine is right, let's assume, that "there are red
houses, roses, and sunsets" is not committed to anything beyond
the houses, roses, and sunsets, and that one cannot infer that "there is a property of redness that they all share." But it's not clear why "they have something in common" — or better, "there is
something that they all are" — should be seen as therefore
misleading. If predicates are noncommittal, one might
well think, the quantifiers binding predicative positions
are not committal either.
Why would one think this? The commitments of a
quantified claim "Qv...v..." are normally expected to line up with
the commitments of its substitution instances. Subtleties aside, existential
generalizations "SOMEv ...v..." are seen as less (or no more) committal
than their instances "... a...," and a given instance is in turn less (or no more) committal than its universal generalization "ALLv...v..." But "these things are red" is not committal on the
predicative side at all . So its existential generalization
"there is something that these things all are" can't be considered
committal either.
VI. Quantifiers
It's clear enough where Quine is coming
from. To existentially generalize on "they are red"
would be misleading if every quantifier was nominal
in character; because then the generalization would have to be along the lines of "there is a property, viz. redness, that they have." But Quine gives no evidence that quantifiers are per se nominal. And
in the present case the assumption seems clearly wrong, since one plugs in not
the noun "redness" but the predicate "red."
Nonnominal quantification has not been much
discussed by philosophers, with the shining exception of Arthur
Prior. Here is what Prior says in "Platonism and Quantification"
(chapter II of Objects of Thought):
If we start from an open sentence such as "x
is redhaired" and ask what the variable "x" stands for..., the
answer depends on what we mean by "stands for." The variable may
be said...to stand for a name (or to keep a place for a name) in
the sense that we obtain an ordinary closed sentence by replacing
it by a name, ...say, "Peter"...the variable "x" may be said in a
secondary sense to "stand for" individual objects or persons such
as Peter. It "stands for" any such object or person in the sense
that it stands for (keeps a place for) any name that stands for
(refers to) an object or person. If we now consider the open
sentence "Peter φ's Paul," it is equally easy to say what..." φ's" "stands for" in the
first sense — it keeps a place for any transitive verb, or
any expression doing the job of a transitive verb. The question
what it "stands for" in the second sense....is senseless, since
the sort of expression for which it keeps a place is one which
hasn't got the job of designating objects...(35)
But does idiomatic English contain
quantifiers governing variables like this  variables that don't
(in the second sense) "stand for" anything? Prior thinks it does:
we form colloquial quantifiers, both nominal
and nonnominal, from the words which introduce questions —
the nominal "whoever" from "who," and the nonnominal "however,"
"somehow," "wherever," and "somewhere" from "how" and "where" [1]....no grammarian would
count "somehow" as anything but an adverb, functioning in "I hurt
him somehow" exactly as the adverbial phrase "by treading on his
toe" does in "I hurt him by treading on his toe"...What is [also]
done in English [when a nonnominal quantifier is needed] is
simply to extend the use of the "thing" quantifiers in a perfectly
wellunderstood way, as in "He is something that I am not —
kind" .... "something" here is quite clearly adjectival rather
than nominal in force. (37)
According to Prior, whether we quantify
nominally or nonnominally makes all the difference where
ontological commitment is concerned. A propos "I hurt him
somehow," he notes that
we might also say "I hurt him in some way,"
and argue that by so speaking we are "ontologically committed" to
the real existence of "ways"; but...there is no need to do
it this way.... (37);
and the implication is that if we don't, we
take on no more commitment to ways than would be incurred with "by
treading on him." Similarly, we could say "he exemplifies
something that I don't: kindness," and in that case we are prima
facie committed to properties. But there is no need to do it this
way, and if we stick with the adjectival "something," then we
incur no more commitment than we would with an adjective like
"kind."
VII. Commitment
The claim is that nonnominal
quantifiers — quantifiers like "somehow" and adjectival
"something"  encode no commitments. What is the evidence for
this? The first argument is implicit in what has already been
said.
Argument from Instances: Use of a
quantifier commits one to the entities it has to be taken as ranging over for the sentence to come out true. The entities a quantifier ranges over are at most the entities capable of being referred to by phrases substituting in for the variable. The phrases substitutable in for a nonnominal variable — phrases like "by treading on
him," and "kind"  are not referential.[2] So nonnominal
quantifiers carry no commitments.
A second argument avoids semantical assumptions and works
entirely at the level of commitment:
Argument from Entailment: Suppose
that "I hurt him somehow" were committed to entities beyond those
presupposed by "I hurt him by stepping on him," that is, me and
him and (maybe) my foot. Then "I hurt him somehow" would not be trivially entailed by "I hurt him by treading on him" — because
it is not a trivial matter whether these additional entities exist.
"I hurt him somehow" is, however, trivially entailed by "I hurt him by treading on him." So there is no additional commitment. Likewise, "he is
something that I am not" follows trivially from "he is kind
and I am not." The inference would not be trivial if "he is something that
I am not" were committed to entities other than him and me. So it
isn't committed to entities other than him and me.
A third argument is adapted from Boolos.
Argument from Consistency: Suppose
that the "something" in " a is something that b is
not" carried a commitment to BLAHs — properties, say, or
sets. Then to say " a is something that b is not,
but there are no properties or sets to witness the fact" would be
selfundermining. And in general it isn't. Sometimes indeed the
claim is importantly true: " a is something that b
is not, viz. not a member of itself, but as we know from Russell's
paradox there is no witnessing set." Likewise it is quite
consistent to say that " a and b are related
somehow, in that a is a member of b, but the things
thusly related are too many to fit into a set."
Our final argument tries to make use of the "reason" why
Russell's contradiction arises:
Argument from Cardinality: By Cantor's theorem,
a domain's pluralities always outnumber its members;
there are always going to be objects x, y,
z,....such that no domain element contains all and only
them. Another way to put it is that the following is
mathematically impossible:
(i) take any objects you like, there's an object containing them
and nothing else.
But it is not at all impossible — it is
on one reading quite true  that
(ii) take any objects you like, they are something that the rest
of the objects are not.
A diehard
objectualist might try to construe the "something" in (ii) in
terms of containerobjects somehow eluding the grasp of the
initial "any objects you like." But this escape hatch can be
closed by stipulating that the initial quantifier is absolutely
universal. Not only does this stipulation fail to make
(ii) look any less consistent, (ii) continues to look true. It could not be true on the stipulated reading if "something" had ontological import.
VIII. The Interpretation
Nonnominal quantifiers allow for something
like anaphoric crossreferencing  the kind we see in "That bat is mine, but you can use
it" and "Someone came into the store, and
she demanded satisfaction." The reason for inserting the "like" is that cross
reference is a privilege reserved to referring, so presumably
nominal, phrases. We can avoid these issues by claiming instead that nonnominals allow for anaphorical cross
indexing. This has been pointed by, among others, Belnap and Grover:
Anaphors do not always occupy nominal positions. There are, for example, proverbial uses of 'do.' 'Do' is used as a...quantificational proverb: "Whatever Mary did, Bill did," "Do whatever you can do." "Such" and "so" can be used anaphorically as proadjectives: "The pointless lances of the preceding day were certainly no longer such" (Scott), "To make men happy and to keep them so" (Pope).
Our strategy will make use not of proverbs or proadjectives but proadverbs, such as
likewise in "I walk quickly, and she walks likewise," and
thusly in "he did it by breaking the window, and we did it thusly too," and (especially)
so in "they are related as brother and sister, and we are so related as well."
Suppose we paraphrase "Connecticut is larger than Delaware" by the more
cumbersome "Connecticut is related to Delaware in that the former
is larger than the latter." Then we can say that
(6) Connecticut is related to Delaware in
that the former is larger than the latter; Texas is so (thus,
likewise) related to Nebraska.
From (6) it is a short step to
(7) Connecticut is related to Delaware
somehow such that Texas is so related to Nebraska.
This gives us the materials to begin thinking about how to render secondorder '∃R ...Rxy..'. A natural thought given (7) is: 'something is related to something somehow such that ... x is so related to y.' This has two problems, however, one logical and one relating more to readability.
The logical problem (to
put it in objectualist terms) is that there is one way of
being related such that it is not the case that something
is related in that way to something. The proposal in a word overlooks the
empty relation. As a result, the secondorder logical truth
'∃R
∀x
∀y
¬Rxy' is translated into the obvious falsehood that
'something is related to something somehow such that no objects
are so related.'
Boolos faced an analogous difficulty when
constructing his translation scheme in terms of plurals; the most
straightforward scheme takes '∃P ∀x ¬Px,' a secondorder
logical truth, to the falsehood 'there are some things such that
nothing is one of them.' Boolos's response was to add a disjunct, interpreting '∃P Px' as 'there are some things such that x is one of them, or else x ≠ x'. It would be
easy enough to follow his lead. But, leaving aside from the unnaturalness of the extra disjuncts, it forgoes one of the main advantages of nonnominal quantification over plural: its amenability to an intensional interpretation in which things could have been related somehow even though nothing is so related in actual fact. (See section X.) Run through the envisaged scheme, the formula '(∃R)[¬(∃x)(∃y)Rxy & <>(∃x)(∃y)Rxy]' that ought to express the (mere) possibility of Rrelated things winds up affirming instead the possibility of objects distinct from themselves.
A different approach to the empty relations problem starts by noting that quantificational adverbs can "reach inside" negation contexts and attach themselves to an embedded verb. One says of a world traveler that "there must be somewhere he hasn't been," meaning by this not the negation of "he's been somewhere," but the existential generalization of "he hasn't been to Tasmania." Just so, one might say of people seemingly related every which way "surely they are not related somehow" and mean, not the negation of "they are related somehow" (that would be silly), but the existential generalization of "they are not related as brother and sister." But then, we can read '∃R ...Rxy..' as 'something isorisn't related to something somehow such that .... x is so related to y.' And now the logical truth
'∃R
∀x
∀y
¬Rxy' goes not into a falsehood but another obvious truth:
something isorisn't related to something somehow such that no objects
are so related.
That leaves the readability problem: it hurts readability to be putting objects that are somehow related front and center, when our real concern is not with them but the way they're related. Putting the "somehow" first, as in "somehow an object isorisn't related to an object....," helps a little, especially with a Yiddish intonation pattern: "somewhere you left it, that's all you can say?" A different approach compresses rather than relocating the material in front of the "somehow." Suppose we use "things relate somehow" as short for "an object and an object are (or are not) related somehow." Then our interpretation scheme for dyadic secondorder logic is as follows:
(a) Tr('R_{i}(x_{j},
x_{k})') = 'it_{j} is
so_{i} related to it_{k}'
(b) Tr('¬φ') = 'it is not the case that' ^
Tr('φ')
(c) Tr('φ & ψ') = Tr('φ')^ 'and' ^
Tr('ψ')
(d) Tr('∃x_{i} φ') = 'some
object_{i} is such that' ^ Tr('φ')
(e) Tr('∃R_{i} φ') = 'things relate somehow_{i} such that'
^^{ }Tr('φ')
So, for example, '∃R ∀x ∃y Rxy' is to be read as saying that
(8) Things relate somehow such that
everything is so related to something.
The reader can try his/her hand at interpreting other dyadically quantified formulae; as far as we can see no special difficulties arise, once one gets past the initial awkwardness of using "things relate thusly" to express not their being thusly related but their classifiability as thusly related or not. (A similar usage: "people relate by marriage, regardless of whether any are in fact married; numbers do not relate in this way.")
So much for the dyadic case. Nadic
quantifiers with n > 2 present no additional
problems, for the sort of crossindexing used above works with them too. Example: Utah is intermediate in
size between Nevada and Colorado; and Alabama is intermediate
between Georgia and Mississippi. So, Utah, Nevada, and Colorado
are related somehow such that Alabama, Georgia, and Mississippi
are thusly related as well. Now finally for the case that Boolos started, and finished, with: the case where n=1.
IX. Monadic Quantifiers
Note before we begin that it wouldn't be
too much of a problem if no suitable translation of monadic
'∃P
...Px...' could be found. This is because (speaking like an
objectualist) for any P, there's a relation R such that the Ps are
the things that bear R to themselves; and for any R, there's a P
such that the things bearing R to themselves are exactly the Ps.
Thus one can always mimic the effect of monadic '∃P ...Px...' with a
diagonal dyadic construction '∃R ...Rxx... .' But while it's
nice that we are ready with an excuse if monadic secondorder
quantifiers should prove untranslatable, it would be nicer if we could just translate them.
Picking up on the discussion above of
adjectival 'something', one idea is to read '∃P_{i}...P_{i}x_{j}...' as 'an
object is something_{i} such that...that_{i}is
what it_{j} is ... .' Our translation of '∃P ∀x (Px > Qx)'
would be 'an object is something_{i} such that a thing
is that_{i}only if it is also Q' , or more colloquially,
'some object is something_{i} which_{i} only Qs
are '.
But suppose that we want a treatment more
along the lines of what was suggested for the dyadic case. The
strategy that then suggests itself is to introduce a verb
'determined' ('comported'? 'complected'?) on the understanding
that to be determined Ply is the same as being P. Then when each
of two objects is red, we can say that 'a is
determined redly, and b is so determined too'. Next we
explain that 'a is determined somehow' stands to 'a
is determined redly' exactly as 'a and b are related
somehow' stands to 'a and b are related in that
a is larger than b'. This gives us all we need. A
formula like '∃
P ∀x (Px > Qx)' translates as
'an object is determined somehow such that only Qs are so
determined.'
X. Interactions with Plurals
An interesting feature of the translation
scheme in VII. is how easily it extends to Boolosian
plurals. Whenever it makes sense to say, in the singular, that
'a and b are related somehow_{i}' and
'it_{j} is so_{i} related to it_{k}', it
can also be said, in the plural, that 'the Fs are related to the
Gs somehow' and 'they_{j} are thusly_{i} related to
them_{k}'. ('The soldiers are somehow related to the students  they have them surrounded  and the students for their part are thusly related to the administrators.')
So, let's try it. Begin by adding to the
formal language firstorder plural variables
x_{1}, x_{2}, x_{3},
..... (not to be confused with our existing secondorder
variables P_{1}, P_{2}, P_{3}, .....!!!!).
Taking a leaf from Boolos, we translate '∃x_{i}....y_{i}∈x_{i} ... ' as 'there are some
things_{i} such that...y_{i} is one of
them_{i} ... '. Next
come secondorder plural variables R_{i}.
These function grammatically as predicates taking firstorder
plurals ('the students) as arguments; so
'R_{i}x_{j}y_{k}' has
the grammar of 'the soldiers have the students surrounded'. The
translation rule, finally, is just what you'd expect:
(f)
Tr('R_{i}x_{j}y_{k}')
= 'they_{j} are so_{i} related to
them_{k}'.
(g) Tr('∃R_{i}')
= 'things relate somehow_{i}
such that,'where this last is an abbreviation for readability of 'some things are related (or not) to some things somehow_{i} such that.'
The elaborated scheme takes
(8) ∃R ∃x
∀y Rxy
to
(9) Things relate somehow_{i}
such that, take any objects_{j} you like, there are
objects_{k} to which they_{j} are so_{i} related.
What needs to be explained now is why anyone should
care about the possibility of combining plurals with non
nominals in this way.
One advantage of combining plurals with
nonnominals is that it helps us to disentangle two distinctions.
According to us, the distinction between the first and second orders
is so far from lining up with the singular/plural distinction that
the two in fact cut across each other:
variable///substituend

singular

plural

firstorder

x///'17'

x///'the prime numbers'

secondorder

P/// '...is prime'

P///'...are coprime'

A second advantage is that the combination helps us
to fend off an unintended interpretation of '∃P'  the one that
says that '∃P (P a &P b)' is true only if a
and b have something "nice" in common; they are both green,
say, as opposed to both being grue. Such an interpretation is
bound to invalidate the claim that
(10) ∀x ∃P ∀y (Py ↔ y∈x),
that is,
(11) Take any objects you like, there is
something that they and only they are.
One can secure the same result for polyadic secondorder
quantifiers if the background theory has ordered tuples; just lay
it down that
(12) Take any ntuples you like, things relate
somehow_{i} such that
x_{1}…x_{n} are
so_{i} related iff
<x_{1},…,x_{n} > is
one of the ntuples.
Can plurals be used to fend off the "nice" interpretation even
in the absence of ntuples? If they can, we haven't been able to
figure out how. It should be stressed, however, that that
is the intended result; there is no reason why nonnominal
secondorder quantifiers should not cover the same extensional
ground as gets covered on the familiar nominal interpretation. How
much ground does get covered is controversial; the point is
just that nonnominal quantifiers are plausibly in the same boat
as nominal ones. (See section X.)
A third advantage is that we get a cleaner formulation of the
Argument from Cardinality (section VI). Suppose we were to
construe '∃P' in (10) as an
objectual quantifier over, say, sets. Then the meaning of (10)
would be given not by (11) but
(14) Take any objects you like, there is a
set containing exactly them.
And we know that (14) is false; it's in direct
contradiction with Cantor's theorem. So to the extent that (10)
seems correct, an objectual interpretation of '∃P' cannot be right.
A fourth advantage of combining plurals with nonnominals is the
gain in expressive power. This is suggested already by (10), but a
clearer case is
(13) ¬ ∃R ∀y ∃x ∀z (z∈y ↔ Rxz),
which says that the objects cannot be paired off with the
pluralities  in effect that there are more pluralities of things
than things. This could be expressed using just plural quantifiers
if we had ordered pairs, but more complicated examples can be
given where the combination of plurals with nonnominals seems
essential even given the ordered pairs. (One can "say," e.g., that
there are more pluralities of pluralities than there are
pluralities.)
A fifth advantage is that where plurals are rigidly extensional 
 if x is one of them, then it is one of them necessarily 
 nonnominals are flexible as between extensional and intensional
readings (see the next section). The combination thus provides a
natural setting in which to study interactions between extensional
and intensional styles of classification.
XI. Comparison with the standard semantics
The "standard semantics" for secondorder
languages is extensional; it never happens that secondorder
variables are assigned different values despite being true of the
same things. Nothing much hangs on the extensionality
assumption, it should be noted; one could equally construe the
secondorder variables as denoting "intensional" entities that are
finergrained than their extensions. That having been said,
however, one might wonder whether the nonnominal interpretation
is extensional or intensional.
Our first response is to say that the
question makes no sense. One can't discuss the identityconditions
of the values of secondorder variables unless the secondorder
variables have values. And on our interpretation, they
don't.
But maybe that is too quick. The question of
extensionality, one might say, is really a question about whether
variables true of the same objects are thereby identical in their
total contribution to truth value. Are secondorder variables
extensional in this sense?
It depends; one has to look what other
semantic machinery is present in the language. Usually in
discussions of these matters, we are thinking of ordinary second
order predicate calculus unsupplemented with any clever devices.
If this is the sort of language at issue, then which objects P is
true of does indeed fix its semantic potential. Our "semantics" is
thus fully extensional.
But suppose the language contains modal
operators. Then it's compatible with everything that we've said
that predicatevariables true of the same things should fail to be
intersubstitutable in all contexts. It could happen, for example,
that
(15) ∃P ∃Q [ ∀x ((Px↔Qx) & <>∃y (Py & ¬Qy))],
comes out true, because
(16) There is something_{i} that my
cat is — a creature with a kidney  and
something_{j} that my dog is — a creature with a
heart — such that everything that is that_{i }is
that_{j }and vice versa; but there could be a thing
that was that_{i} and not that_{j} .
Before rushing to any conclusions, notice that it's also
compatible with everything your typical objectualist says
that (17) should come out true. Your typical objectualist, after
all, has no particular view about how to deal with secondorder
quantification into modal contexts.
She might take the position that the
values of predicatevariables should continue to be extensions
when modal operators are introduced; and she might think that sets
have their members essentially; and she might conclude from all
this that (15) is false. But she might equally think that when the
logic goes modal, we should take the variables to stand for
intensions (functions from worlds to extensions); in that case
(15) will almost certainly come out true, since intensions which
coincide on one world need not coincide on others. Either way, if
she is extensionalityminded, she will probably want to lay it
down as an axiom that
(17) ∀P []∀x (Px ↔ []Px)
 because (17) is true in all models, if
she takes the first view, or because she wants to restrict
attention to the models that make (17) true, if she takes the
second. From (17) it follows, given the usual sorts of
assumptions, that the extensionality scheme
(18) ∀P ∀Q [( ∀x (Px ↔Qx) > (...P... ↔...Q..)]
holds in full generality, that is, for modal
and nonmodal contexts alike. Depending on the application, of
course, the assumption of extensionality may be unwelcome and out
of place. But in that case the objectualist is free to simply
refrain from imposing (17) or any similar condition.
The reason for mentioning all of this is
that the nonobjectualist would seem to be in exactly the
same position. If and when she is attracted to extensionality, she
can lay it down that (17). It is true that (17) means something
different in her mouth; it means, simplifying some, that
(19) whatever_{i} a thing is, it is
necessarily, and vice versa.
But the effect is the same. (17) assures the
nonobjectualist too that extensionality reigns. Should she
encounter an application where extensionality is not wanted, she
is just as free as the objectualist to ditch (17), thus opening
the door to an object being something that it might not have been.
A second and much more controversial feature
of the "standard semantics" for secondorder languages is that it
makes use only of full models, that is, ones whose second
order domain contains every subset of the firstorder
domain D, and for nadic quantifiers every subset of
D^{n}. This is what gives standardly interpreted second
order languages their stunning logical power: the power to pin
down the standard model of arithmetic, for example, and to settle
the truthvalue of the continuum hypothesis. It is also what makes
some commentators wonder what in our thought and practice could
possibly rule out nonstandard interpretations. Couldn't our
quantifiers be ranging over an approximation to all the
subsets such that creatures like ourselves could form no notion of
what was missing?
Someone might say: your acceptance of (12),
which says that given any objects there is something that they
alone are, suggests that you are committed to a standard
like interpretation in which (to put it a bit
paradoxically) '∃P' ranges over all the things that domain elements
can be. You therefore take on the burden of explaining how it
could be that statements we understand and know to be true settle
the truthvalue of Goldbach's conjecture, the continuum
hypothesis, and so on.
Is it really so clear, though, that (12)
commits us to a standardlike interpretation? Only to the extent
that (12)'s plural quantifier '∀x' is
assured of full coverage. It could be, and has been, argued that
plural quantifiers are themselves vulnerable to the sort of
nonstandard construal that secondorder skeptics have laid so
much weight on. All we really need, though, is that they are not
obviously invulnerable.[3] And it seems clear that they are not. Why should a
slight change in wording  'take whatever members of D you like'
as opposed to 'take whatever subset of D you like'  be enough to
scare the skeptic back to his/her cave?
Where does this leave us? The denominalizer
is most likely just as unclear as anyone else how worried to be
about the skeptical challenge. Probably then she has no idea
whether to see herself as committed to a standardlike
semantics. Which is all to the good, really. The less the question
of denominalizing has to do with other secondorder
controversies, the freer we are to decide it on its own terms. And
here, the denominalizing approach looks like it offers clear
advantages.
XII. Parting methodological shot
A certain kind of philosopher is going
to react as follows: I see what your nonnominal quantifiers are
supposed not to be. They are not supposed to be objectual,
you've made that clear. And they don't appear to be substitutional
either, for it may well be true to say 'x and y are
related somehow such that no predicate S of English or any other
language is such that things are so related
iff they satisfy S.' But it's one thing to say that there's a kind
of quantifier that doesn't range "over" anything, or signal
disjunction/conjunction with respect to a fixed class of
substituends; it's another thing to make out in positive
terms what the alternative is. Until you do that, your "theory" is
just so much wishful thinking and obscurantism.
It's an open question in our minds how much truth there is to
this accusation. There really does seem to be a distinction
between, as Quine somewhere puts it, "clarity" and "fluency." And
it may well seem that what we have with nonnominals is just
fluency. What it really means to say that there is
something x is that y isn't remains desperately
unclear; and there is no way to make it clear but through an
objectual semantics that reintroduces all the original
paraphernalia.
That anyway is the objection, and there may be something to it.
But let's not lose sight of a point stressed by Quine: logical
formalisms are explained in natural language  what else?  and
the best we can hope for is to base our formalisms in fragments of
the language whose logical properties are relatively transparent,
or can be made transparent by regimentation and stipulation. Quine
happened to be of the opinion that singular objectual
quantification was the only "generality device" transparent
enough not to need explanation in other terms. But that is a
separate claim and a debatable one, as we see from the acceptance
on their own terms of Boolosian plurals. Why should not non
nominals be taken on in the same spirit? Any plausible reply must
start by clarifying the rules by which a natural language locution
is judged clear enough to speak for itself.
Bibliography
Grover 1992, A Prosentential Theory of Truth (Princeton: Princeton U Press)
Hazen 1997, "Relations in Lewis's framework without atoms," Analysis 57.4, 243248
Lewis 1990, Parts of Classes (London:Blackwell)
Prior 1971, Objects of Thought (Oxford: Oxford U Press)
Quine 1953, From a Logical Point of View (Cambridge, MA: Harvard U Press)
Wright 1983, Frege's Conception of Numbers as Objects (Aberdeen: Aberdeen University Press)
Yablo 1996, "How in the World?" Philosophical Topics 24.1, 255286
[1] The OED gives many examples of such phrases,
including on the existential side 'somehow,' 'somewhen,'
'somewhere,' 'somewhy,' 'somewhence,' 'somewhither,' 'somewise,'
and and on the universal side 'everyhow,' 'however,' 'everyway,'
'everyways,' 'everywhen,' 'whenever,' 'whichever,' 'everywhence,'
'everywhither,' 'anyhow,' 'anyway,'
'anywhat,' 'anywhen, 'anywhence,' 'anywhere''anywhither,'
'anywise,' 'allwhat,' 'allwhere,' and 'allwhither.'
[2]A comment on "referential." In asking, "is this phrase referential?" I mean not, are there Montague grammarians or other formal semanticists somewhere who have cooked up superduper semantical values for them, say, functions from worlds to functions from worlds and ntuples of objects to truth values? (The answer to that is yes almost no matter what part of speech you're talking about.) I mean: are they referential in the way that singular terms are, so that someone using the could reasonably be said to be talking about its referent, or purporting to talk about its purported referent? [3] Charles Parsons tells us that Boolos did not
think that the plural construction was especially resistant to
skeptical reinterpretation.