nominalism through de-nominalization

Nominalism Through De-Nominalization

A. Rayo

S. Yablo

draft of June 3, 1999 — comments much appreciated
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I. Introduction

Not all that long ago, second-order statements were thought to be committed twice over. The one commitment was to the members of a first-order domain D; the other was to the members of a second-order domain consisting of D's subsets and more generally the k-adic relations on D.

A good way to recapture the mood is to recall Quine's description of second-order logic as "set theory in sheep's clothing." What did Quine mean by this? His point was not that the semantics of second-order logic is set-theoretical; for the semantics of first-order logic is set-theoretical well. And Quine certainly doesn't think that first-order logic is set theory in sheep's clothing.

Nor could Quine's point have been that there are second-order statements whose validity goes with the truth-value of this or that set-theoretic hypothesis, for instance, the continuum hypothesis. There are first-order 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 infinite-domain 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, first-order logic is not, for Quine, set theory in sheep's clothing.

If the point was not that second-order semantics is set-theoretical, or that second-order sentences depend for their validity on the behavior of sets, what was it? Quine was concerned that second-order logic advanced theses which could not be true unless sets were counted into the range of their quantifiers. Almost any second-order thesis worth mentioning has this property for Quine; because as he sees it, a second-order quantifier is basically a first-order quantifier stipulated to range over certain sets. This makes clear why he second-order 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 second-order comprehension axiom (to take that example) cannot be true unless sets are counted into the range of its quantifiers, hence second-order 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 second-order 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 second-order 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 self-defeating; it would be committed to entities of the very sort that it explicitly rejects. But far from being self-defeating, (3) is an important and hard-won truth about sets; it is the clearest statement yet found of what separates the older, paradox-prone, concept of set from the (hopefully) non-paradoxical 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 second-order quantifiers like '∃G' in (1), gets no grip whatever on polyadic second-order 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. Second-order logic however makes essential use of dyadic quantifiers like the one in (4). A famous example is the second-order 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 one-one relation between the Fs and the Gs).

The most common response to this difficulty has been to look for ways of "coding" polyadic second-order 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 second-order 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 second-sentence that purportedly expresses this fact uses a dyadic quantifier, and so is committed to ordered pairs on the proposed scheme. Other set-theoretical 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 second-order 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 pair-surrogates could be found in the first-order 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 pair-surrogates 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 first-order 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 second-order languages do to be done without taking on any commitment to sets. It would be better if second-order 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 second-order logic - the idea that second-order quantifiers need not be construed as ranging over abstract objects. One reason for being interested in nominalism about second-order 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 second-order quantifiers interpreted in an ontologically unloaded fashion. That someone embraces sets (concepts, properties...) does not necessarily mean they believe second-order statements to quantify over sets - not any more than a belief in angels requires one to think that second-order quantifiers range over angels. But more important, even someone who likes the set-theoretic 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 would-be 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 class-sized relations as non-nominal talk about how things are related.

(B) Fregean "platonists" like Crispin Wright and Bob Hale attempt to deduce the existence of numbers from Fregean second-order logic plus definitions. An obvious objection is that since (Fregean) second-order 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 second-order quantifiers whereby they carried no new commitments would enable the Fregean platonist to answer the objection on its own terms.

(C) Advocates of Frege-style 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 truth-value. 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 second-order 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- committal-seeming bit of English. In step two one attempts to show that this bit of English is all that's needed to translate the relevant second-order 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 second-order meanings. First though let's look briefly at another reason for dissatisfaction with Boolos's proposal.

The second-order 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 first-order "there is a thing which...." in needing to be completed, not by a noun phrase, but something verb-ish: "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 second-order variable. There appears, indeed, to be no grammatical way of reading both the plurality and the predicativeness back into the initial 'F'. "There are some-things-of-which-one- 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 non-committal (or not- further-committal) English locution into which polyadic second-order 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 second-order 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 non-nominal 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. (10-11)

If predicates and the like needn't name to be meaningful -- to make their characteristic contribution to truth-value -- 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."

Non-nominal 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 red-haired" 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 non-nominal, from the words which introduce questions — the nominal "whoever" from "who," and the non-nominal "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 non-nominal quantifier is needed] is simply to extend the use of the "thing" quantifiers in a perfectly well-understood 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 non-nominally 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 non-nominal 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 non-nominal variable — phrases like "by treading on him," and "kind" - are not referential.[2] So non-nominal 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 self-undermining. 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 die-hard objectualist might try to construe the "something" in (ii) in terms of container-objects 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

Non-nominal quantifiers allow for something like anaphoric cross-referencing -- 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 non-nominals 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 pro-verbs or pro-adjectives but pro-adverbs, 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 second-order '∃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 second-order 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 second-order 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 non-nominal 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 R-related 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 is-or-isn'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 is-or-isn'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 is-or-isn'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 second-order 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')

(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. N-adic quantifiers with n > 2 present no additional problems, for the sort of cross-indexing 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 second-order 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_ix_j...' as 'an object is something_i such that...that_iis what it_j is ... .' Our translation of '∃P ∀x (Px -> Qx)' would be 'an object is something_i such that a thing is that_ionly 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 P-ly 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 first-order plural variables x₁, x₂, x₃, ..... (not to be confused with our existing second-order variables P₁, P₂, P₃, .....!!!!). 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 second-order plural variables R_i. These function grammatically as predicates taking first-order plurals ('the students) as arguments; so 'R_ix_jy_k' has the grammar of 'the soldiers have the students surrounded'. The translation rule, finally, is just what you'd expect:

(f) Tr('R_ix_jy_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

(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 non-nominals 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
first-order	x///'17'	x///'the prime numbers'
second-order	P/// '...is prime'	P///'...are co-prime'

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 second-order quantifiers if the background theory has ordered tuples; just lay it down that

(12) Take any n-tuples you like, things relate somehow_i such that x₁…x_n are so_i related iff <x₁,…,x_n > is one of the n-tuples.

Can plurals be used to fend off the "nice" interpretation even in the absence of n-tuples? 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 non-nominal second-order 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 non-nominal 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 non-nominals 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 non-nominals 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 - - non-nominals 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 second-order languages is extensional; it never happens that second-order 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 second-order variables as denoting "intensional" entities that are finer-grained than their extensions. That having been said, however, one might wonder whether the non-nominal interpretation is extensional or intensional.

Our first response is to say that the question makes no sense. One can't discuss the identity-conditions of the values of second-order variables unless the second-order 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 second-order 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 predicate-variables 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_iis that_jand 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 second-order quantification into modal contexts.

She might take the position that the values of predicate-variables 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 extensionality-minded, 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 non-modal 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 non-objectualist 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 non-objectualist 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 second-order languages is that it makes use only of full models, that is, ones whose second- order domain contains every subset of the first-order domain D, and for n-adic quantifiers every subset of Dⁿ. 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 truth-value of the continuum hypothesis. It is also what makes some commentators wonder what in our thought and practice could possibly rule out non-standard 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 truth-value of Goldbach's conjecture, the continuum hypothesis, and so on.

Is it really so clear, though, that (12) commits us to a standard-like 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 non-standard construal that second-order 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 de-nominalizer 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 standard-like semantics. Which is all to the good, really. The less the question of de-nominalizing has to do with other second-order controversies, the freer we are to decide it on its own terms. And here, the de-nominalizing 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 non-nominal 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 non-nominals 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, 243-248

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, 255-286

[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,' 'all-where,' 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 super-duper semantical values for them, say, functions from worlds to functions from worlds and n-tuples 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.