Ordinary English contains different forms of quantification overobjects. In addition to the usualsingular quantification, asin

(1)

There is an apple on the table

there isplural quantification, as in

(2)

There are some apples on the table.

Ever since Frege, formal logic has favored the two singularquantifiers \(\forall{x}\) and \(\exists{x}\) over their pluralcounterparts \(\forall{xx}\) and \(\exists{xx}\) (to be read asfor any things \(xx\) andthere are some things\(xx)\). But in recent decades it has been argued that we have goodreason to admit among our primitive logical notions also the pluralquantifiers \(\forall{xx}\) and \(\exists{xx}\) (Boolos 1984 and1985a).

More controversially, it has been argued that the resulting formalsystem with plural as well as singular quantification qualifies as“pure logic”; in particular, that it is universallyapplicable, ontologically innocent, and perfectly well understood. Inaddition to being interesting in its own right, this thesis will, ifcorrect, make plural quantification available as an innocent butextremely powerful tool in metaphysics, philosophy of mathematics, andphilosophical logic. For instance, George Boolos has used pluralquantification to interpret monadic second-order logic^[1] and has argued on this basis that monadic second-order logicqualifies as “pure logic”. Plural quantification has alsobeen used in attempts to defend logicist ideas, to account for settheory, and to eliminate ontological commitments to mathematicalobjects and complex objects.

• 1. The Languages and Theories of Plural Quantification
  • 1.1 Regimenting plural quantification
  • 1.2 The Theories PFO and PFO+
• 2. Plural Quantification vs. Second-Order Quantification
  • 2.1 Plural quantification and monadic second-order logic
  • 2.2 Relations
  • 2.3 Modal contexts
  • 2.4 Higher levels of plural quantification?
• 3. The Logicality Thesis
• 4. Applications of Plural Quantification
  • 4.1 Establishing the Logicality of Monadic Second-Order Logic
  • 4.2 Logicism
  • 4.3 Set Theory
  • 4.4 Mathematical Nominalism
  • 4.5 Eliminating Complex Objects
• 5. Ontological Innocence?
  • 5.1 The set-theoretic argument
  • 5.2 The incorrect predication argument
  • 5.3 The direct argument
  • 5.4 Semantic values and ontological commitments
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. The Languages and Theories of Plural Quantification

The logical formalisms that have dominated in the analytic traditionever since Frege do not allow for plural quantification. Inintroductory logic courses students are therefore typically taught toparaphrase away plural locutions. For instance, they may be taught torender “Alice and Bob are hungry” as “Alice ishungry & Bob is hungry”, and “There are some apples onthe table”, as “\(\Exists{x} \Exists{y} (x\) is an appleon the table & \(y\) is an apple on the table & \(x \ney)\)”. However, not only are such paraphrases often unnatural,but they may not even be available. One of the most interestingexamples of plural locutions which resist singular paraphrase is theso-called Geach-Kaplan sentence:

(3)

Some critics admire only one another

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

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

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

In two important articles from the 1980s George Boolos challenges thistraditional view (Boolos 1984 and 1985a). He argues that it is simplya prejudice to insist that the plural locutions of natural language beparaphrased away. Instead he suggests that just as the singularquantifiers \(\Forall{x}\) and \(\Exists{x}\) get their legitimacyfrom the fact that they represent certain quantificational devices innatural language, so do their plural counterparts \(\Forall{xx}\) and\(\Exists{xx}\). For there can be no doubt that in natural language weuse and understand the expressions “for any things” and“there are somethings”.^[4] Since these quantifiers bind variables that take name (rather thanpredicate) position, they arefirst-order quantifiers, albeitplural ones.

1.1 Regimenting plural quantification

I will now describe a simple formal language which can be used toregiment plural quantification as it occurs in English and othernatural languages.

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

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

However, the language \(L_{\textrm{PFO}}\) has one severe limitation.We see this by distinguishing between two kinds of plural predication.A predicate \(P\) taking plural arguments is said to bedistributive just in case it is analytic that: \(P\) holds ofsome things \(xx\) if and only if \(P\) holds of each \(u\) such that\(u\prec xx\).^[5] For instance, the predicate “is on the table” isdistributive, since it is analytic that some things \(xx\) are on thetable just in case each of \(xx\) is on the table. A predicate \(P\)that isn’t distributive is said to benon-distributiveorcollective.^[6] For instance, the predicate “form a circle” isnon-distributive, since it is not analytic that whenever some things\(xx\) form a circle, each of \(xx\) forms a circle. Another exampleof non-distributive plural predication is the second argument-place ofthe logical predicate \(\prec\): for it is not true (let aloneanalytic) that whenever \(u\) is one of \(xx, u\) is one of each of\(xx\). It is therefore both natural and useful to consider a slightlyricher language:

The formal language \(L_{\textrm{PFO}+}\). The language\(L_{\textrm{PFO}+}\) allows non-distributive plural predicates otherthan \(\prec\). We do this by modifying the definition of\(L_{\textrm{PFO}}\) so as to allow predicates \(R^{n}_i\) that takeone or moreplural arguments. These predicates can be eitherlogical or non-logical.^[7]

Should we also allow predicates with argument places that take bothsingular andplural arguments? Lots of Englishpredicates work this way, for instance “… is/are on thetable”. So if our primary interest was to analyze naturallanguage, we would probably have to allow such predicates. However,for present purposes it is simpler not to allow such predicates. Wewill anyway soon allow pluralities that consist of just one thing.^[8]

For now, the formal languages \(L_{\textrm{PFO}}\) and\(L_{\textrm{PFO}+}\) will be interpreted only by means of atranslation of them into ordinary English, augmented by indices tofacilitate cross-reference (Boolos 1984: 443–5 [1998a:67–9]; Rayo 2002: 458–9). (More serious semantic issueswill be addressed inSection 4, where our main question will be whether our theories of pluralquantification are ontologically committed to any sort of“set-like” entities.) The two clauses of this translationwhich are immediately concerned with plural terms are

(4)

\(\Tr(x_i \prec xx_j) = \textrm{it}_i\) is one of them\(_j\)

(5)

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

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

(2″)

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

1.2 The Theories PFO and PFO+

We will now describe a theoryPFO of plural first-orderquantification based on the language \(L_{\textrm{PFO}}\). Let’sbegin with an axiomatization of ordinary first-order logic withidentity. For our current purposes, it is convenient to axiomatizethis logic as a natural deduction system, taking all tautologies asaxioms and the familiar natural deduction rules governing the singularquantifiers and the identity sign as rules of inference. We thenextend in the obvious way the natural deduction rules for the singularquantifiers to the plural ones. Next we need some axioms which forsuitable formulas \(\phi(x)\) allow us to talk about the \(\phi\)s. Inordinary English the use of plural locutions generally signals aconcern with two or more objects. But the existence of two or moreobjects may not be semantically required; for instance, “Thestudents who register for this class will learn a lot” seemscapable of being true even if only one student registers. It istherefore both reasonable and convenient to demand only that there beat least one object satisfying \(\phi(x)\). (Most people who write onthe subject make this concession.) This gives rise to thepluralcomprehension axioms, which are the instances of the schema

where \(\phi\) is a formula in \(L_{\textrm{PFO}}\) that contains“\(u\)” and possibly other variables free but contains nooccurrence of “\(xx\)”. (That is, if something is \(\phi\)then there are some things such that everything is one of them if andonly if it is \(\phi\).) In order fully to capture the idea that allpluralities are non-empty, we also adopt the axiom

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

LetPFO+ be the theory based on the language \(L_{PFO+}\)which arises in an analogous way, but which in addition has thefollowing axiom schema of indiscernibility, stating that coextensivepluralities satisfy the same formulas:

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

While the theories PFO and PFO+ represent what is today the mosttraditional form of plural logic, there are divergent views thatrestrict the plural comprehension schema. Even if a plurality isnothing over and above the objects it comprises, these objects do needto be circumscribed. And on certain metaphysical views, reality as awhole resists proper circumscription (Spencer 2012; Florio and Linnebo2020 and 2021, ch. 12; see also Hossack 2014). Based on considerationsof this sort, Florio and Linnebo (op. cit.) develop analternativecritical plural logic.

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

2. Plural Quantification vs. Second-Order Quantification

By “second-order logic” we understand a logic that extendsordinary first-order logic by allowing for quantification intopredicate position. For instance, from “\(a\) is anapple” we can in second-order logic infer“\(\Exists{F}\mstop Fa\)”. But the plural logics describedabove extend ordinary first-order logic in a different way, namely byallowing quantification intoplural argument position. Butpredicates and plural noun phrases belong to different syntactic andsemantic categories. For instance, the former consists of expressionsthat are unsaturated (in Frege’s sense) — that is, thatcontain gaps or argument places — whereas the latter consists ofexpressions that are saturated (Higginbotham 1998: sect. 7; Oliver andSmiley 2001 and 2016; Rayo and Yablo 2001: sect. X; Simons 1997;Williamson 2003: sect. IX; Yi 2005; Florio and Linnebo 2021).Accordingly, second-order quantification and plural quantification aregenerally regarded as different forms of quantification. In thissection I discuss some of the differences and similarities.

2.1 Plural quantification and monadic second-order logic

(Readers less interested in technical issues may want to skim thissection.) Boolos observed that it is possible to interpret monadicsecond-order logic in the theory PFO.^[9] LetMSO be some standard axiomatization of (fullimpredicative) monadic second-order logic in some suitable language\(L_{\textrm{MSO}}\) (Shapiro 1991: ch. 3; Booloset al.2002: ch. 22). Boolos first defines a translation \(\Tr'\) that mapsany formula of \(L_{\textrm{MSO}}\) to some formula of\(L_{\textrm{PFO}}\). This definition, which proceeds by induction onthe complexity of the formulas of \(L_{\textrm{MSO}}\), has as itsonly non-trivial clauses the following two, which are concerned withthe second-order variables:

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

By induction on derivations in MSO one easily proves that each theoremof MSO is mapped to some theorem of PFO. Moreover, it is easy todefine a “reverse” translation that maps formulas of\(L_{\textrm{PFO}}\) to formulas of \(L_{\textrm{MSO}}\) and to provethat this translation maps theorems of the former to theorems of thelatter. This shows that PFO and MSO are equi-interpretable. A similarresult can be proved about PFO+ and an extension MSO+ of MSO whichallows predicates of (first-level) concepts, provided MSO+ contains anaxiom schema to the effect that coextensive concepts areindiscernible.

It is important to be clear on what the equi-interpretability of PFOand PFO+ with respectively MSO and MSO+ does and does not show (Florioand Linnebo 2021: sect. 4.2). It shows that these two pairs oftheories are equivalent for most technical purposes. But by itself itdoes not show anything about these two pairs of theories’ beingequivalent in any of the more demanding senses that philosophers oftencare about (such as having the same epistemic status, ontologicalcommitments, or degree of analyticity). (For instance PFO isequi-interpretable with atomic extensional mereology, whichphilosophers tend to find much more problematic than PFO.) In order toshow that the two pairs of theories are equivalent in somephilosophically important respect \(F\), we would need to show thatthe above translations preserve \(F\)-ness.

2.2 Relations

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

This limitation can be overcome (at least for technical purposes) ifthere is a pairing function on the relevant domain, that is, if thereis a function \(\pi\) such that \(\pi(u, v) = \pi(u', v')\) just incase \(u = u'\) and \(v = v'\). For then quantification over dyadicrelations can be represented by plural quantification over orderedpairs. Moreover, by iterated applications of the ordered pair functionwe can represent \(n\)-tuples and thus also quantification over\(n\)-adic relations. The question is how this pairing function is tobe understood. One option is to proceed as in mathematics and simplypostulate the existence of a pairing function as an abstractmathematical object. But this option has the obvious disadvantage ofstepping outside of what most people are willing to call “purelogic”. A cleverer option, explored in the Appendix to Lewis1991 and in Hazen 1997 and 2000, is to simulate talk about orderedpairs using only resources that arguably are purely logical. It turnsout that talk about ordered pairs can be simulated in monadicthird-order logic, given some plausible extra assumptions.Monadic third-order logic can in turn be interpreted either in atheory which combines plural quantification with mereology (Lewis1991: ch. 3; Burgess and Rosen 1997: II.C.1) or in terms ofhigher-level plural quantification (Section 2.4).

2.3 Modal contexts

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

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

2.4 Higher levels of plural quantification?

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

What \(is\) relevant for present purposes is whether there is someform of “super-plural” quantification that stands toordinary plural quantification as ordinary plural quantificationstands to singular quantification. If so, let’s call thissecond-level plural quantification. More generally, we mayattempt to introduce plural quantification of any finite level. Thiswould result in a theory which for technical purposes is just like asimple type theory (Hazen 1997: 247; Linnebo 2003: sect. IV; Rayo2006).

It is fairly straightforward to develop formal languages and theoriesof higher-level plural quantification (Rayo 2006). For instance, wecan introduce variables of the formxxx to be thought of asranging over second-level pluralities and the relation \(xx \prec_2\)xxx to be understood in analogy with the relation \(x \precxx\). (See Linnebo and Rayo 2012 for extensions to transfinite levelsand comparison of the resulting theories with those of ordinary settheory.) But can these formal theories of higher-level pluralquantification be justified by considerations similar to those thatjustify the theories PFO and PFO+?

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

Both arguments are controversial. Concerning the first, it is notclear why ontology should be relevant to the legitimacy ofhigher-level plurals quantification. It should be sufficient that thebase-level objects can be organized in certain complex ways. Forinstance, the second-level plurality based on Cheerios organized asoo oo oo should be no moreontologically problematic than the first-level plurality based on thesame objects organized asoooooo,although the former has an additional level of structure orarticulation (Linnebo 2003: 87–8).

The second of the above two arguments is also problematic. To beginwith, the claim that there are no higher-level plural locutions innatural language is almost certainly false. In Icelandic, forinstance, the number words have plural forms which count, notindividual objects, butpluralities of objects that formnatural groups. Here is an example:

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

(12)

These people, those people, and these other people competeagainst each other. (Linnebo and Nicolas 2008)

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

Moreover, the very idea that the legitimacy of higher-level pluralquantification is decided by the existence or non-existence ofhigher-level plural locutions in English and other natural languagesis problematic (Hazen 1993: 138 and 1997: 247; Linnebo 2003: 87; Rayo2006). What really matters is presumably whether we can iterate theprinciples and considerations on which our understanding of ordinaryfirst-level plural quantification is based: if we can, thenhigher-level plural quantification will be justified in much the sameway as ordinary first-level plural quantification; and if not, thennot. Thus, even if there were no higher-level plural locutions innatural languages, this would provide little or no evidence for thestronger — and philosophically more interesting — claimthat there can be no iteration of the step from the singular to theplural inany language spoken by intelligent agents.Moreover, any evidence of this sort could be defeated by pointing toindependent reasons why higher-level plural locutions are scarce innatural languages. One such independent reason may simply be thatordinary speakers aren’t very concerned about their ontologicalcommitments and thus find it more convenient to express factsinvolving second-level pluralities by positing objects to representthe first-level pluralities (for instance by talking about twopairs of shoes) rather than by keeping track of additionalgrammatical device for second-level plurals (as in the above examplefrom Icelandic).

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

3. The Logicality Thesis

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

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

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

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

Perhaps the least controversial candidate for a defining feature oflogic is itstopic neutrality. A logical principle is validin any kind of discourse, no matter what kind of objects thisdiscourse is concerned with. For instance, modus ponens is valid notonly in physics and mathematics but in religion and in the analysis ofworks of fiction. Frege captures the idea nicely when he says that alogical principle is valid in “the widest domain of all;[…] not only the actual, not only the intuitable, buteverything thinkable” (Frege 1884: 21). Thus, whereas theprinciples of physics are valid only in the actual world and in worldsthat are nomologically similar to it, the principles of logic governeverythingthinkable. If one of these principles is denied,“complete confusion ensues” (ibid.).

Another feature widely believed to be defining of logic is itsformality: the truth of a principle of logic is guaranteed bytheform of thought and/or language and does not in any waydepend upon itsmatter. What this feature amounts to willobviously depend on how the distinction between form and matter isunderstood. The most popular explication of the distinction betweenform and matter derives from the widely shared view that no objectsexist by conceptual necessity (Field 1993; Yablo 2000). On this viewit is natural to regard anything having to do with the existence ofobjects and with their particular characteristics as belonging to thematter of thought rather than to its form. This gives rise to twofeatures that are often regarded as defining of logic. Firstly, logichas to beontologically innocent; that is, a principle oflogic cannot introduce any new ontological commitments (Boolos 1997;Field 1984). Secondly, the basic notions of logic must notdiscriminate between different objects but must treat them all alike.This latter idea is often spelled out as the requirement that logicalnotions must be invariant under permutations of the domain of objects(Tarski 1986).

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

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

4. Applications of Plural Quantification

I will now outline some applications of the theories PFO and PFO+. Inthe previous section three strains of the notion of logicality weredisentangled. Special attention will be paid to the question which ofthese three strains PFO and PFO+ must possess for the applications tosucceed.

4.1 Establishing the Logicality of Monadic Second-Order Logic

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

Some of the challenges facing the first step will be examined inSection 5. The second step too should not be underestimated. Perhaps thegreatest worry here is that Boolos’s translation rendersexpressions from one category (that of monadic predicates) in terms ofexpressions from another category (that of plural noun phrases). Forinstance, “… is an apple” is rendered as “theapples”. But these categories are very different (Section 2).

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

4.2 Logicism

Both Fregean and post-Fregean logicism make essential use ofsecond-order quantification. Frege defined the various objects of puremathematics as extensions of concepts, and his famous Basic Law Vstated that two concepts \(F\) and \(G\) have the same extension justin case they are co-extensive:

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

Philosophers have attempted to rescue some ideas of Fregean logicismby using axioms weaker than (V). One of the most important suchattempts is Bob Hale and Crispin Wright’sneo-logicism,which gives up Frege’s theory of extensions but holds on to thecentral idea of his definition of cardinal numbers, namely that thenumber of \(F\)s is identical to the number of \(G\)s just in case the\(F\)s and the \(G\)s can be one-to-one correlated. This has becomeknown asHume’s Principle, and can be formalized as

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

Even more modest is Boolos’ssub-logicism, whichrejects the idea (endorsed by both logicists and neo-logicists) thatthere are logical objects, but insists that Frege’s definitionof theancestral of a relation can be used to show, asagainst Kant, that at least some non-trivial mathematics is analytic(Boolos 1985b). Recall that a relation \(R\) stands to its ancestral\(\Rarel\) as the relationis a parent of stands tois anancestor of. (More precisely, \(\Rarel\) holds between twoobjects \(x\) and \(y\) just in case \(x\) and \(y\) are connectedthrough a finite sequence of objects each of which bears \(R\) to itssuccessor.) Frege gives a second-order definition of the ancestralrelation \(\Rarel\) by laying down that \(x\) and \(y\) are related by\(\Rarel\) just in case \(y\) has every property that is had by\(x\)’s \(R\)-successors and inherited under the\(R\)-relation:

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

It has been suggested that PFO be used to accommodate the post-Fregeanlogicists’ need for second-order quantification. Since theancestral of a dyadic predicate can be defined using only monadicsecond-order quantification, PFO does indeed serve the logical needsof Boolos’s sub-logicism.^[13] But since the neo-logicist definition of \(F\approx G\) usesdyadic second-order logic, PFO alone does not have sufficientexpressive power to accommodate the needs of neo-logicism. Theneo-logicist may attempt to solve this problem by regardingequinumerosity as a primitive logical quantifier or by simulatingdyadic second-order quantification in some suitable extension of PFO,as discussed inSection 2.2^[14] (see Boccuni 2013 for another option).

Which strains of the Logicality Thesis are needed for theseapplications to succeed? Since these logicists attempt to show thatparts of mathematics are analytic (or at least knowableapriori), this would require that PFO be analytic (or at leastknowablea priori), which in turn is likely to require thatPFO enjoy some form of cognitive primacy. Moreover, PFO would have tobe either ontologically innocent or committed only to entities whoseexistence is conceptually necessary (or at least establishableapriori).

4.3 Set Theory

Another application of the Logicality Thesis is concerned with settheory. One may for a variety of reasons want to talk about andquantify over collections of sets (Uzquiano 2003). For instance, onemay want to assert

(13)

There are some sets which are all and only the non-self-memberedsets.

If we formalize this as

how is the quantifier \(\exists{R}\) to be understood? It clearlycannot be taken to range over all sets, as this would lead straight toRussell’s paradox: (13′) would then assert the existenceof the Russell-set. Three other responses are prominent in theliterature.

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

(14)

There are some classes which are all and only thenon-self-membered classes.

What kind of entity would this collection of classes be? Asuper-class? If so, we will be forced to postulate higher and higherlevels of classes. Lewis (1991: 68) argues that Russell’sparadox is still inescapable because, when we consider all set-likeentities, we realize that the following is true:

(15)

There are some set-like things which are all and only thenon-self-membered set-like things.

However, Hazen (1993: 141–2) has pointed out that Lewis’sobjection violates essential type-restrictions. Classes of differentlevels belong to different logical types, just as concepts ofdifferent levels do. So Lewis’s attempt to talk about allset-like entities in one fell swoop involves an attempt to quantifyacross different logical types. But this violates type restrictions inthe same way as an attempt to quantify simultaneously over objects andconcepts of all different levels. Although we can quantify overeach level of classes, we can never quantify overalllevels simultaneously.

The second response is that(13) asserts the existence of a set \(R\), but that \(R\) isn’t inthe range of the quantifier \(\forall{x}\). This prevents us frominstantiating the quantifier \(\forall{x}\) with respect to \(R\),which means that we cannot draw the fatal conclusion that \(R\) is amember of itself just in case it isn’t. However, this responseentails that the quantifier \(\forall{x}\) cannot be chosen to rangeover absolutely all sets; for if it could be so chosen, we would notbe able deny that \(R\) is in this range of quantification. This meansthat the universe of sets has a certaininexhaustibility:whenever we have formed a conception of quantification over some rangeof sets, we can define a set which isn’t in this range (Dummett1981: ch. 15 and 1991: ch. 24; Glanzberg 2004; Parsons 1977). However,this response has been criticized for being, at best hard to state,and at worst self-refuting (Boolos 1998b: 30; Lewis 1991: 68;Williamson 2003: sect. V). (See also Rayo and Uzquiano 2006 for anumber of essays discussing whether absolutely general quantificationis possible.)

Because of the difficulties involved in the first two responses, athird response has become popular in recent years (Boolos 1984 and1985a; Burgess 2004; Cartwright 2001; Rayo and Uzquiano 1999; Uzquiano2003). This is that the quantifier \(\exists{R}\) is a pluralquantifier (and would thus be better written as \(\exists{rr})\) andthat plural quantification is ontologically innocent. Therefore(13) does not assert the existence of any “set-like” entityover and above the sets in the range of the quantifier \(\forall{x}\).But as we will see inSection 5, the claim about ontological innocence is controversial.

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

4.4 Mathematical Nominalism

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

One class of applications of plural quantification aim to makeontological bargains in the philosophy of mathematics. In particular,a number of philosophers have attempted to use plural quantificationas an ingredient of nominalistic interpretations of mathematics. Anice example is Geoffrey Hellman’s modal nominalism, accordingto which mathematical statements committed to the existence ofabstract objects are to be eliminated in favor of statements about thepossible existence of concrete objects. For instance, insteadof claiming, as the platonist does, that there exists an infinitecollection of abstract objects satisfying the axioms of Peanoarithmetic (namely the natural numbers), Hellman claims that therecould exist an infinite collection of concrete objectsrelated so as to satisfy these axioms (Hellman 1989 and 1996).However, even this modal claim appears to talk aboutcollections of concrete objects andrelations onthese objects. To forestall the objection that this smuggles inthrough the back door abstract objects such as sets, Hellman needssome alternative, nominalistically acceptable interpretation of thistalk about collections and relations. Plural quantification may offersuch an interpretation.

For this application of plural quantification to work, PFO must beapplicable to all kinds of concrete objects, and it must beontologically innocent, or at least not committed to any entities thatshare those features of abstract objects that are found to benominalistically objectionable. Moreover, in order to simulatequantification over relations, we will need not just PFO but a theorymore like monadicthird-order logic (Sections2.2 and2.4).

4.5 Eliminating Complex Objects

Another class of applications attempts to eliminate the commitments ofscience and common sense to (some or all) complex objects. Forinstance, instead of employing usual singular quantification overtables and chairs, it is proposed that we use plural quantificationover mereological atoms arranged tablewise or chairwise (Rosen andDorr 2002; Hossack 2000; van Inwagen 1990). For instance, instead ofsaying that there is a chair in one’s office, one should saythat there are some atoms in one’s office arranged chairwise. Inthis way one appears to avoid committing oneself to the existence of achair. Note that such analyses require PFO+, not just PFO, since thenew predicates “are arranged \(F\)-wise” arenon-distributive.

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

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

(16)

There are some chairs arranged in a circle.

We have already “used up” ordinary plural quantificationand predication to eliminate apparent commitment to individual chairs(Uzquiano 2004). So in order to analyze (16), we will need somethinglike “super-plural” quantification — quantificationthat stands to ordinary plural quantification as ordinary pluralquantification stands to singular — and correspondingnon-distributive predication. The legitimacy of such linguisticresources was discussed inSection 2.4.

There have also been attempts to use plural logic to eliminateapparent reference to groups. Instead of referring to the SupremeCourt, say, perhaps we might get by with referring plurally to itsmembers. An obvious challenge for this approach is that a group canhave different members at different moments of time and possibleworlds, while the plurality of its (current and actual) membersappears to be rigid, temporally as well as modally (Section 2.3). See (Horden and López de Sa 2021) for a defense of theproposed elimination against such challenges.

5. Ontological Innocence?

The traditional view in analytic philosophy has been that all plurallocutions should be paraphrased away, if need be, by quantifying oversets (Section 1). George Boolos and others objected that it is both unnatural andunnecessary to eliminate plural locutions. This led to the theoriesPFO and PFO+. Proponents of plural quantification claim that thesetheories allow plural locutions to be formalized in a way that isfundamentally different from the old set-theoretic paraphrases. Inparticular, they claim that these theories are ontologically innocentin the sense that they introduce no new ontological commitments tosets or any other kind of “set-like” entities over andabove the individual objects that compose the pluralities in question.Let’s call this latter claimOntological Innocence.

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

\((3''')\)

There are some critics such that any one of them admires anothercritic only if the latter is one of them distinct from theformer.

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

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

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

I will now discuss three arguments in favor of OntologicalInnocence.

5.1 The set-theoretic argument

The first argument begins by asking us to consider the claim

(17)

There are some sets which are all and only the non-self-memberedsets.

and admit that it is true. It continues by arguing that, if pluralexpressions were committed to collections or any other“set-like” objects, then the truth of (17) would leadstraight to Russell’s paradox. This is sometimes thought to be aknock-down argument in favor of Ontological Innocence (Boolos 1984:440–443 [1998a: 64–67]; Lewis 1991: 65–69; McKay2006: 31–32). But in fact it is less conclusive than it appears.For as we saw inSection 4.3, Russell’s paradox will follow only if two alternative views areruled out. Since these views cannot be dismissed out of hand, muchwork remains before this argument can be said to be conclusive.

5.2 The incorrect predication argument

The second argument is nicely encapsulated by Boolos’s remarkthat “It is haywire to think that when you have some Cheerios,you are eating aset” (1984: 448–9 [1998a: 72]).What Boolos is suggesting here is that analyses which deny OntologicalInnocence are likely to get the subject of plural predicationswrong.

The obvious response is to interpret plural predicates in a way whichensures that what we eat arethe elements of a set and notthe set itself. Consider the sentence:

(18)

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

When the direct object of the verb “ate” is plural, we canfor instance interpret the verb by means of the relationxate-the-elements-of y.

It will be objected that this response makes the verb“ate” ambiguous in an implausible way (Oliver and Smiley2001). For when the verb has a direct object that is singular, it willpresumably be interpreted by means of the ordinary relationx atey. But there is fairly strong evidence that the verb“ate” isn’t ambiguous in this way. For instance, oneeffect of an ambiguity is to disallow certain kinds of ellipsis. Anexample is the ambiguity of “make” in “makebreakfast” and “make a plan”, which disallows thefollowing ellipsis:

(*19)

Boolos made breakfast, but his guest, only a plan.

So if “ate” was ambiguous in the way just described, thefollowing ellipsis would be disallowed as well, which itisn’t:

(20)

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

However, it is far from clear that the above response toBoolos’s argument needs to be committed to any such problematicambiguities. We can for instance letall predicates takeplural entities as their arguments. The verb “ate” willthen always receive as its interpretation the relationthe-elements-of x ate-the-elements-of y, thus removing anyambiguity. Whether or not this response is ultimately acceptable, itshows that the argument in question remains inconclusive.

5.3 The direct argument

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

In this simple form the argument is vulnerable to the objection thatpeople’s intuitions provide a poor basis for settlingtheoretical disputes about ontological commitments. We have seen thatthere are competent speakers of English, such as Michael Resnik, whodon’t share these intuitions. Moreover, as Davidson’spopular analysis of action sentences in terms of events makes clear,ordinary people’s intuitions about ontological commitmentscannot always be trusted (Davidson 1967). For instance, someone maysincerely assert that John walked slowly, without being aware that hehas committed himself to the existence of an event (namely a walkingwhich was by John and which was slow).

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

(21)

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

This inference would no doubt be quite peculiar. This providesevidence that(18) isn’t committed to any kind of “set-like”entity.

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

(22)

Some students surrounded the building.

most speakers of English would be perfectly happy to infer that

(23)

A group of students surrounded the building.

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

However, let’s assume that the defenders of the Direct Argumentare right that(18) does not entail(21). What would follow? It would follow that(18) does not incur any additional ontological commitmentsof the sortthat can be incurred by singular first-order quantifiers. Butthis conclusion falls short of the argument’s desired conclusionthat(18) does not incur any additional ontological commitmentsof anysort. In order to get from the actual conclusion to the desiredone, we would in addition have to assume that all ontologicalcommitments are of the sort incurred by singular first-orderquantifiers. But there is an influential philosophical tradition thatdenies this assumption and instead holds that all kinds of quantifiersincur ontological commitments, not just singular first-order ones.^[15] The most famous exponent of this tradition is Frege, who claims thatsecond-order quantifiers are committed to concepts, just as singularfirst-order quantifiers are committed to objects. This tradition tiesthe notion of ontological commitment very closely to that of asemantic value. This will be the topic of the next and finalsubsection.

5.4 Semantic values and ontological commitments

In semantics it is widely assumed that each component of a complexexpression makes some definite contribution to the meaning of thecomplex expression. This contribution is known as thesemanticvalue of the component expression. It is also assumed that themeaning of the complex expression is functionally determined by thesemantic values of the component expressions and their syntactic modeof composition. This assumption is known ascompositionality.

According to Frege, the semantic value of a sentence is just itstruth-value, and the semantic value of a proper name is its referent(that is, the object to which it refers). Once we have fixed thesemantic values assigned to sentences and proper names, it is easy todetermine what kinds of semantic value to assign to expressions ofother syntactic categories. For instance, the semantic value of amonadic predicate will have to be a function from objects totruth-values. Frege calls such functionsconcepts.

As an example, let’s consider the simple subject-predicatesentence

(24)

Socrates is mortal.

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

(25)

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

(26)

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

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

(27)

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

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

What matters for present purposes is not the truth or falsity ofFrege’s claim about concepts but whether a cogent argument ofthis sort can be developed for plural expressions. To investigatethis, let’s consider a simple non-distributive pluralpredication such as

(28)

These apples form a circle.

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

(29)

[\(\mathbf{aa}] = a_1\) and … and \(a_n\) (where the\(a_i\) are all and only the apples demonstrated)

(30)

[\(\mathbf{C}] =\) the function \(g\) from pluralities totruth-values such that \(g(xx)\) is the true if \(xx\) form a circleand \(g(xx)\) is the false otherwise

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

(31)

[(28)] \(= [\mathbf{C}(\mathbf{aa})] = [\mathbf{C}]([\mathbf{aa}]) = g(a_1\) and … and \(a_n\)) = the true (if\(a_1\) and … and \(a_n\) form a circle) or the false(otherwise)

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

Assume that this analysis is correct and that each plural term thushas some objects as its semantic value, just as each singular term hasone object as its semantic value. What will this mean for the questionof Ontological Innocence? According to the Fregean tradition, whichconnects the notion of ontological commitment to that of a semanticvalue, this will mean that plural expressions incur commitment toplural entities, much as predicates incur commitment to concepts. Forto say that a sentence incurs a commitment to a plural entity is justto say that the truth of the sentence requires there to be somesemantic value of the sort appropriate to plural expressions. However,this line of reasoning will be resisted by other philosophers, whobelieve that the notion of ontological commitment should be tied (atmost) to singular first-order variables.

How can this disagreement be adjudicated? On the one hand, it maycount in favor of the Fregean tradition that their view is highlysystematic. There may be somethingad hoc about the idea thatsome sorts of semantic value give rise to ontological commitmentswhile other sorts don’t. On the other hand, it may count infavor of the alternative view that it does better justice to manypeople’s strongly felt intuition that plural locutions areontologically innocent.

Another possibility is that the whole controversy is ultimately just apseudo-disagreement (see especially Florio and Linnebo 2016, but alsoParsons 1990; Shapiro 1993; Linnebo 2003; Rayo 2007; and Linnebo andRayo 2012). If both parties agree that plural expressions havesemantic values, and if both agree that commitments toobjects are incurred only by singular first-order terms andvariables, then perhaps it does not matter whether other sorts ofterms and variables should be regarded as introducing their owndistinctive kinds of ontological commitment. Some philosophers speakabout a theory’sideological commitments and not justabout its ontological commitments. By this is meant the logical andconceptual resources that the theory employs. Perhaps philosopherswould be well advised to focus more on the metaphysical andepistemological questions raised by a theory’s ideologicalcommitments and worry less about whether these ideological commitmentsshould be also regarded as introducing a distinctive kind ofontological commitment. After all, the notion of an ontologicalcommitment is a theoretical one, not one that has any sharp contentoutside of philosophy. So perhaps we should regard the notion more asa means to the end of providing good philosophical explanations andless as a goal in itself.

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Acknowledgments

Thanks to Salvatore Florio, Allen Hazen, Frode Kjosavik, Tom McKay,David Nicolas, Agustín Rayo, and Gabriel Uzquiano fordiscussions and written comments on earlier drafts.