To Be is to Be the Object of a Possible Act of Choice (Studia Logica)

Studia Logica (2010) 96: 289–313DOI: 10.1007/s11225-010-9282-2 © Springer 2010

M. Carrara

E. Martino

To Be is to Be the Object

of a Possible Act of Choice

Abstract. Aim of the paper is to revise Boolos’ reinterpretation of second-order monadic

logic in terms of plural quantification ([4], [5]) and expand it to full second order logic.

Introducing the idealization of plural acts of choice, performed by a suitable team of agents,

we will develop a notion of plural reference. Plural quantification will be then explained in

terms of plural reference. As an application, we will sketch a structuralist reconstruction

of second-order arithmetic based on the axiom of infinite a la Dedekind, as the unique

non-logical axiom. We will also sketch a virtual interpretation of the classical continuum

involving no other infinite than a countable plurality of individuals.

Keywords: Second-order logic, Second-order arithmetic, Arbitrary reference, Plural quan-

tification, Plural reference, Fictionalism.

Introduction

Boolos ([4], [5]) has proposed an interesting reinterpretation of second-ordermonadic logic in terms of plural quantification. In Boolos’ perspective,second-order monadic logic is ontologically innocent: second-order variablesdo not range over sets of individuals but over individuals plurally. Boolos’view, although very attractive, has met many criticisms. The aim of thepaper is to propose a new approach to plural quantification, which shouldsupersede these criticisms. We think that the role of plural quantificationin logic (and mathematics) can be better understood within the frame ofa highly idealized notion of plural reference. Our approach to second-orderlogic will start from the observation that the possibility, in principle, of refer-ring to any individual of the universe of discourse of a mathematical theoryis essentially, although implicitly, presupposed in mathematical reasoning.In order to make such presupposition explicit and to clarify the sense of thelocution “in principle”, we will introduce a team of ideal agents, which aresupposed to have direct access to any individual. Plural reference to certainindividuals will be realized through an act of plural choice, i.e. an act ofchoosing an individual performed simultaneously by each agent. By meansof this device, we will obtain an idealized version of Kripke’s notion of direct

Special Issue: Philosophy of MathematicsEdited by Andre Fuhrmann, Ivan Kasa and Manfred Kupffer

292 M. Carrara and E. Martino

reference, more basic than any sort of reference by description. Plural quan-tification will be explained in terms of such notion of reference. We thinkthat our framework provides a new perspective for understanding the natureof plural reference and plural quantification. Pluralities of individuals arenot second-order entities: speaking of a plurality of individuals is nothingbut a linguistic device for referring to the individuals selected by an act ofchoice. Moreover, our team of agents will provide an intuitive means forsimulating relations, so that the full second-order logic can be reconstructedwithout any further device. We think that a careful distinction between actsand entities can vindicate the ontological innocence of second-order logic.

As an application, we will sketch a structuralist reconstruction of second-order arithmetic based on the axiom of infinite a la Dedekind, as the uniquenon-logical axiom. We will also sketch a virtual interpretation of the clas-sical continuum involving no other infinite than a countable plurality ofindividuals.

1. Boloos on second order monadic logic

According to Boolos, both first order variables and second-order monadicvariables range over individuals, the former singularly, the latter plurally.So, no second order entity is involved in second order monadic logic.

Boolos’ basic idea consists of interpreting the atomic formulas of the formXy, as “y is one of the X s”, and the existential formulas of form ∃X . . . as“There are some individuals X s such that . . . ”. The universal quantifier ∀X

is expressible in terms of the existential one in the usual way.1

Boolos gives no explanation of how to refer to an arbitrary pluralityof individuals. He treats directly plural existential quantification taking asprimitive the locution “there are some objects such that . . . ” in natural lan-guage. But the meaning of this locution is somewhat ambiguous, strictlydepending on the context of discourse. In some contexts, its meaning isthe same as the first-order quantification “there is at least an object suchthat . . . ”. And when such locution is not reducible to a first-order quantifi-cation, as in the famous Geach-Kaplan’s proposition “Some critics admire

1For instance, consider the second-order definition of an ancestor:

a is an ancestor of b =df. ¬∃X(Xb ∧ ∀x∀y(Xx ∧ Pyx → Xx) ∧ ¬Xa),

where Pyx is the relation “y is a parent of x”. (For simplicity, b himself is included amonghis ancestors.) In Boolos’ interpretation, this is to be read, “There are no individuals X ssuch that (i) b is one of the X s, (ii) every parent of one of the X s is one of the X s, (iii) ais not one of the X s”.

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To Be is to Be the Object of a Possible Act of Choice 293

only one another”, it may seem a sloppy way of referring to some class of in-dividuals. This proposition, not formalizable in first order language, seems,according to Resnik, hardly interpretable without resorting to classes. Howcould we understand “one of them” without referring to a certain class andagreeing that the referent of “one” belongs to it? [16]. Besides, the naturallanguage does not suggest a direct interpretation of the universal quanti-fier: “for each individuals . . . ” is ungrammatical and “for all individuals” isindistinguishable from the first-order quantification “for each individual”.2

Boolos disregards the problem and defines universal quantification in termsof existential and negation in the usual way.

Truly, Boloos provides also a formal semantics for his language in Nom-

inalistic Platonism [5]. He gives a semantic for second order logic restatingthe Tarskian truth definition by modifying the notion of assignment as fol-lows. Given a domain D of individuals, he defines as an assignment anybinary relation R between variables and individuals that correlates a uniqueindividual with every first order variable, while it is subject to no constraintfor second order variables. So R may correlate a second order variable withno, one or (possibly infinitely) many individuals. The satisfiability relationis inductively defined as usual, with the following clauses for atomic formulasand second order existential quantification:

(i) R satisfies the atomic formula Fx iff the correlate of x is one of thecorrelates of F ;

(ii) R satisfies ∃FA iff there is a relation R’, differing from R at most forthe correlates of F, such that R’ satisfies A. (The universal quantifier isdefined in terms of the existential one).

Truth is then defined, as usual, in terms of satisfaction. So the set ofthe correlates of F is not involved in the definition of truth. This makes thenotion of plural quantification precise and shows how it yields an alternativesemantics for second order logic. This semantics turns out to be equivalentto the usual one, according to which the values of second order variables areall sets of individuals. And since the notion of value of a variable can bemade precise only by the definition of assignment, in Boolos’ perspective theproposed reformulation shows that, using Quine’s slogan that “to be is to

2Lewis has suggested an interpretation of the universal plural quantifier in terms of“whenever there are some things, then . . . ” [12, p. 11]. But, we have some doubt that suchlocution is appropriate. “Whenever there are some numbers, then . . . ” sounds somewhatstrange and seems to suggest that it might happen sometimes that there are no numbers.

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be the value of a variable”, there is no commitment in second order logic toany entities but individuals.

Of course, the new definition of assignment presupposes the notion ofset of individuals if relations are understood set-theoretically. But a relationcan in turn be understood in terms of plural reference to certain orderedpairs (taking for granted the notion of ordered pair).

However, the use of plural reference in the meta-language begs the cru-cial question, whether plural reference involves surreptitiously the notion ofset. Boolos’ view, although very attractive, has met the criticism of sev-eral philosophers of mathematics. The main criticism rests on the suspicionthat speaking of pluralities of individuals is just a rough manner of speakingof sets. Take, for example, Parsons’ criticism to Boolos [15]. Although heacknowledges to Boolos the merit of throwing new light on the old notionof manifold, he argues that his interpretation of second order logic is notontologically noncommittal:

The great interest of his reading, [ . . . ], is that he breathes new lifeinto the older conception of pluralities or multiplicities. As a sourceof second order logical forms, the plural and plural quantification arerightly distinguished from what was so much emphasized by Frege,predication and, more generally, expressions with argument places. Inparticular, if it is the idea of generalization of predicate places thatwe appeal to in making sense of second-order logic, then the mostnatural interpretations will be relative substitutional or by semanticascent, and these will not license impredicative comprehension, andit is hard to see how that will be justified. But if one views examplessuch as Boolos’s as involving ‘pluralities’, they are more like setsas understood in set theory in that no definition by a predicate isindicated, so that one need not expect them to be definable at all.Thus no obstacle to the acceptance of impredicative comprehensionis removed.

An advocate of Boolos’ interpretation in an eliminative structural-ist setting could grant my claims about ontological commitment, butthen take a position analogous to the Fregean: second-order vari-ables indeed have pluralities as their values, but these are not objects.It does not seem to me to have the same intuitive force as Frege’sposition, since there is no analogue to the regress argument that canbe made if one views the reference of a predicate as an object. Therewill still be, just as with Frege’s concepts, the irresistible temptationto talk of pluralities as if they were objects, as we have already noted

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above. The only gain this interpretation offers over the Fregean isa more convincing motivation of impredicativity. [15, p. 328]

Quine’s old claim that second-order logic is set theory in disguise (metapho-rically a wolf in sheep clothing) does not seem to have lost all its advocates.We propose to supersede the above criticism with a new approach to pluralquantification. We will argue that plural quantification in logic (and math-ematics) can be better understood within the frame of a highly idealizednotion of plural reference.

2. Plural reference: some background

The idea of explaining the notion of a set by means of plural reference canbe found, in nuce, in Black’s famous paper The elusiveness of sets [3]. Blackstarts from the consideration that, although the basic notions of set theoryare nowadays very familiar to all mathematicians, the very nature of a set,conceived of as a well-determined entity built up from its members, is quitemysterious:

Beginners [of set theory] are taught that a set having three membersis a single thing, wholly constituted by its members but distinct fromthem. After this, the theological doctrine of the Trinity as “three inone” should be child’s play. [3, p. 616]

Black criticizes at length Cantor’s well-known definition of set:

“By a “set” we understand every comprehension S of determinate

well-distinguished objects s of our intuition or thinking into a whole”

as well as all other attempts to explain what a set, as an abstract entity, is.He regards such attempts as a misleading mystification and recommendsavoiding any question about the nature of sets. What is important, accordingto Black, is rather to look at the use of plural reference in natural languageas the appropriate starting point for learning to master the sophisticatedlanguage of mathematical set theory:

The notion of “plural” or simultaneous reference to several things atonce is really not at all mysterious. Just as I can point to a singlething, I can point to two things at once, using two hands, if necessary;pointing to two things at once need be no more perplexing thantouching two things at once. Of course it would be a mistake tothink that the rules for “multiple pointing” follow automatically from

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the rules for pointing proper; but the requisite conventions are almosttoo obvious to need specification. The rules for “plural reference” areno harder to elaborate. [3, p. 629]

Black’s conclusion can be satisfactory for a working mathematician who isonly interested in the correct use of a set-theoretical talk. But it is unsat-isfactory for a philosopher interested in understanding how to think of theobjects, if any, that set theory seems to speak of. We think that, in a philo-sophical perspective, “the requisite conventions for multiple pointing” arefar from being obvious and need an adequate specification, which we wantto propose in the sequel.

An important insight to obtain an adequate specification of plural refer-ring comes from Kitcher, who has elaborated a conception of set theory interms of ideal actions (see [10]). Kitcher has clearly realized the importanceof an idealized notion of act vs. that of object. He has proposed a recon-struction of Mill’s arithmetic and of ZF set theory in terms of the acts ofcollecting of an ideal agent, free of the empirical limitations of real humanbeings:

One central ideal of my proposal is to replace the notions of abstractmathematical objects, notions like that of a collection, with the notionof a kind of mathematical activity, collecting . . . In its most rudimen-tary form, collecting is tied to physical manipulation of objects. . . Welearn how to collect by engaging in this type of activity. However,our collecting does not stop here. Later we can collect the objectsin thought without moving them about. We become accustomed tocollecting objects by running through a list of their names, or byproducing predicates which apply to them. Naively, we may assumethat the production of any predicate serves to collect the objects towhich it applies. (This naive assumption is implicit in nineteenth-century analysis, and it was made explicit by Cantor.) Thus, ourcollecting becomes highly abstract. We may even achieve a hierarchyof collectings by introducing symbols to represent our former collec-tive activity and repeating collective operations by manipulating thissymbols. . . As I construe it, the notation ‘{. . . }’ obtains its initial sig-nificance by representing first-level collecting of objects, and iterationof this notation is itself a form of collective activity. [10, p. 110]

So Kitcher’s idealized collecting seems to be the product of an extrapolationfrom several ways of collecting in our real life. Unfortunately, he deliberatelydoesn’t explain what the ideal act of collecting consists in:

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Obviously, the ideal subject is an idealization of ourselves, but I ex-plicitly reject the epistemological view that we can know a priori theways in which the idealization should be made. [10, p. 111]

We think, however, that the rejection to pinpoint the idealization in questionprecludes the possibility of supplying a well-determined conception of themathematical infinite and of justifying the adoption of classical logic. Howcan the ideal agent collect infinitely many objects? Is this suggested by ourmanipulations of concrete objects? This familiar activity may suggest that, ifwe were free of spatio-temporal limitation, we could indefinitely collect moreand more objects. But could an act of collecting consist of an indefinitelyproceeding process? To admit an infinite process of collecting, step by step,new arbitrarily chosen objects would make undetermined whether any givenobject will be collected or not in the course of the process. But Kitcherassumes, in accordance with classical logic, that it is well determined, forany collective operation, which objects are collected. Nor can an infinitecollection be conceived of as the result of singling out infinitely many objectsby means of a common property: collecting should be a primitive notion notgrounded on the highly problematic notion of property. Kitcher is aware ofthis fact and holds that a collective operation is performed independently ofany property we use for describing the collected objects. But then, the mereassumption that the ideal agent is free of empirical limitations is far fromgiving any account of how the infinitely many actions of the ideal agent areto be imagined. Concerning this point, Brown observes:

Not surprisingly, the ideal agent is faster than a speeding bullet,stronger than a mighty locomotive, and able to leap tall buildings ina single bound. This much idealization is rather harmless and Kitcheris certainly entitled to help himself to it. (Superman, after all, is stilla finite being). But when it comes to infinite operations, we mustsurely object. This cannot be passed off as merely overcoming an“accidental limitation” that the rest of us humans have. A platonicrealm is not half so mysterious or implausible. [6, pp. 11–12]

In order to supersede this difficulty, we will introduce, in section 4, a teamof infinitely many agents.

Kitcher regards his philosophy of mathematics as a philosophical natural-

ism, since it starts from considering certain usual actions of real life. SarahHoffman [9] observes that, since the ideal agent does not exist, Kitcher’sphilosophy would be better understood as a form of fictionalism. We think,however, that a fictional aspect, together with the naturalistic one, is al-ready implicit in Kitcher’s perspective, since any idealization is fictional in

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the obvious sense that idealized entities do not exist and idealized actionsare not performable by real human beings.3

But if one wants to go fictionalist, why should one not prefer a fic-tionalism about mathematical objects rather than about operations? Onemight argue that the former shares with Platonism the advantage of takingmathematical language at face value, while the latter seems to be somewhatbizarre. Hoffman defends the ideal-agent-fictionalism as follows:

The ideal-agent formulation takes mathematics to be about opera-tions rather than abstract mathematical objects. If mathematics isreally about abstract objects then it is entirely false, since there arenone of these at all. If mathematics is a set of stories about oper-ations, however, then a portion of it that is about operation thatactually exist really is true. . . Operations are kind of things that re-ally exist; numbers are not. . .Mathematics is not only a tool, it isalso a theory of what mathematicians do. [9, p. 14]

This kind of argument is not convincing. We don’t think that the nonexistence of mathematical objects outside of our imagination makes math-ematics false. If mathematical talk is about fictional objects, then math-ematical truth is to be understood as concerning only the fictitious worldof such objects. The truth of the proposition “there are infinitely manyprime numbers” by no means implies that prime numbers are non-fictionalobjects. In general, no mathematical proposition entails that the existenceof mathematical objects is independent of our game of make-believe. As tothe concrete mathematical activity, one could object that it does not con-sist of singling out and ordering concrete objects. Mathematicians do notperform such actions, they rather reason on notions grasped by reflecting onsuch kind of actions. Mill’s and Kitcher’s interpretations of the mathemat-ical talk as about actions is certainly somewhat artificial, since it is plainthat mathematical talk, taken at face value, is about objects. However, wewant to argue in the next section that mathematical reasoning about objectsimplicitly presupposes a highly idealized possibility of referring arbitrarilyto them, and that the introduction of ideal agents can serve the purposeof making such an ideal reference explicit. In this sense, we want to em-brace a sort of fictionalism about objects that involves a fictionalism aboutideal agents.

3For an introduction to fictionalism in the philosophy of mathematics see [1], [2], [24],[20], [21].

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3. Arbitrary Reference

Mathematicians use in their reasoning expressions of the kind “let a bean arbitrary object of the universe of discourse,” for instance “let a be anarbitrary real number”. Observe that there is no link between the letter“a” and the number that it is supposed to be indicating. However, afterintroducing a with that locution, a working mathematician, accustomed toargue informally through meaningful propositions, reasons about a as if “a”

designated a well-determined individual. This means that he reasons underthe implicit fictional assumption that someone has associated to letter “a” acertain (unknown) individual. For instance, she reads a formula of form F (a)as “the individual a falls under the concept F”. Lacking that assumption,F (a) would be meaningless. If so, mathematical reasoning would seem topresuppose, at least ideally, the possibility of indicating any object of theuniverse of discourse, even when, as in the case of real numbers, not everyobject has a name in the language.4

A number of arguments in the literature seem to be in disagreement withthis observation. The most obvious is perhaps due to a misunderstandingof the notion of arbitrariness. One may argue that considering an arbitrarynumber is nothing but a way of speaking, which by no means involves thepossibility of actually singling out such a number because, for the very samearbitrariness, it is irrelevant which number one is speaking of. Indeed, whenreasoning about an arbitrary number a, there is no need to know it. Yet,ignorance of the number one is referring to has the desired effect of grant-ing generality to the reasoning: what is provable for a completely unknownnumber holds necessarily for all numbers. That is right. However, the lackof information about a cannot avoid the assumption that the letter “a” des-ignates a precise number; lacking that assumption, it would make no sensetalking about a, not even to say it is unknown. Perhaps one could objectthat speaking of an arbitrary number amounts to speaking of all numberssimultaneously. But this is not the case. When reasoning about a, a mathe-matician can exploit the assumption that “a” has a well-determined referent,which is kept fixed in the whole course of the reasoning. For instance, whenreasoning about a, a mathematician can distinguish the case that a is ratio-nal and the case that it is irrational. But, of course, that is not the same asdistinguishing the case that all numbers are rational and the case that allnumbers are irrational. So, speaking of a is not speaking of all numbers. Be-sides, when, for instance, arguing about a, one proves F (a) and later ¬F (a),

4On the same topic see [14].

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she gets a contradiction. But, of course, no contradiction would follow if thetwo occurrences of a might designate different individuals. For these reasonsa is to be thought of as a well-determined individual. Nor could F (a) beunderstood as expressing the fact that F is instantiated by some individual(without any reference to a particular individual), like in Frege’s interpre-tation of the existential quantifier. For, according to such interpretation,¬F (a) would say, in turn, that the complement of F in instantiated as well.Again, no contradiction would follow from F (a) and ¬F (a). Similar con-siderations occur in Russell’s explanation of the difference between “each”and “any”.5

The possibility, in principle, of referring to any individual of the universeof discourse is not confined to informal reasoning. It is also required forjustifying the quantification rules in a formal system of natural deduction.The introduction rule of the universal quantifier (I∀) allows the inference of∀xAx from the premise Ab in the usual way:

.

.A(b)

.∀xA(x)

where “b” is an arbitrary name (or a free variable) not occurring in anyassumption on which A(b) depends. The soundness of the rule is groundedon the consideration that if one has proved that b satisfies the formula A,

5“The general enunciation tells us something about (say) all triangles, while the partic-ular enunciation takes one triangle and asserts the same thing of this one triangle. But thetriangle taken is any triangle, not some one special triangle; and thus, although, through-out the proof, only one triangle is dealt with, yet the proof retains its generality. If wesay: “Let ABC be a triangle, then the sides AB and AC are together greater than theside BC”, we are saying something about one triangle, not about all triangles; but theone triangle concerned is absolutely ambiguous, and our statement consequently is alsoabsolutely ambiguous. We do not affirm any one definite proposition, but an undeterminedone of all the propositions resulting from supposing ABC to be this or that triangle. Inthe case of Euclid’s proofs, this is evident: we need (say) some one triangle ABC to reasonabout, though it does not matter what triangle it is. The triangle ABC is a real variable;and although it is any triangle, it remains the same triangle throughout the argument.But in the general enunciation the triangle is an apparent variable. If we adhere to theapparent variable, we cannot perform any deduction, and this is why in all proofs realvariables have to be used”. [18, pp. 156–157]

Russell’s distinction between real and apparent variables corresponds to the today dis-tinction between free and quantified variables; and his considerations are in accordancewith the use of variables in natural deduction (see later).

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without any specific piece of information about b, then any individual en-joys the property in question. This justification clearly presupposes that b

can denote any individual in the range of the variables (otherwise the uni-versal quantifier should be restricted to the denotable individuals). Similarconsiderations hold for the elimination rule of the existential quantifier (E∃).

The problem arises: how can one refer to an arbitrary individual? Per-haps, one might think, by means of some characterizing property, but that,unfortunately, would involve a problematic universe of properties, suitablefor characterizing any individual. Besides, this option faces the problem ofhow to refer to an arbitrary property. Therefore, it seems that the notionof reference to an arbitrary individual, presupposed in mathematical rea-soning, is more basic than any linguistic notion of reference via a definitedescription. We think that the most appropriate idealization for justifyingarbitrary reference should be grounded on the ideal possibility of a direct

access to any individual. We shall invoke an ideal agent who is supposed tobe able, by means of an arbitrary act of choice, to single out any individualby ostension. In such a conceptual frame, the introduction rule of the uni-versal quantifier (I∀) is justified as follows. Let us imagine an ideal agent

who arbitrarily chooses an individual b about which we have no informationat all. If we are able, just by reasoning about b, to conclude that it satis-fies the formula A(x), because, as far as we know, any individual could bethe chosen one, we can conclude that each individual has the property inquestion and, therefore, infer ∀xAx.

Since mathematical entities are often conceived of as abstract entities,one may wonder if it is possible, even in principle, to single out such anentity by ostension. In fact, our argument for arbitrary reference brings for-ward a difficulty of understanding mathematical entities as abstract. Thedifficulty can be superseded embracing a structuralist conception of math-ematics, where the role of mathematical entities is played by idealized, butconcrete individuals. In the sequel we will adopt an idealization, whereindividuals are humans and sets of individuals are simulated by means ofsimultaneous acts of choice.

Now, consider the usual set-theoretical semantic of second-order logic.Since second-order variables are intended to range over all classes of individ-uals, the problem arises: How can the ideal agent have direct access to anyclass of individuals? Since all we know about an arbitrary class is that it isan entity determined by its members (or by absence of members, in the caseof the empty set), it seems that one can have access to a class only throughits members. And since a class may be infinite, it seems that a single agent isunable to perform a simultaneous choice of all members of any class, unless

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he is conceived of as a horrible monster with infinitely many fingers. Wedon’t like that! On the other hand, a choice of infinitely many individuals,one at a time, would face the hard problem of dealing with undeterminedclasses (as it happens in intuitionistic mathematics). Black himself describesa “plural pointing” as a “simultaneous reference to several things at once”.Kitcher seems to disregard this difficulty (see [10, cap. 6]). He assumes thathis ideal agent is able to collect any infinity of objects. Moreover, as al-ready observed, he outlines a reconstruction of set theory by assuming thatthe ideal agent is able to iterate his collecting operations over and over re-producing the whole transfinite hierarchy of types required by the iterativeconception of set.

What does it mean to collect some actions of collecting? Kitcher’s answeris that, once certain collecting acts have been performed, one can introducesome symbols for representing them and perform collecting acts of suchsymbols. So one should understand the collecting of acts as a collectingof representative of acts. This explanation is certainly useful for didacticpurposes: it can help a student to understand what is a set of sets. But itis inadequate from a philosophical point of view. Moreover, it is misleadingwithin Kitcher’s framework. In fact, that explanation suggests to the studentthat one can collect sets because sets are in turn objects. That is trueaccording to the usual conception of set, but Kitcher’s collecting acts fail tobe objects. Kitcher himself maintains that an operation exists only insofaras it is performed. So the expedient of introducing symbols as representativeof acts cannot give any account of how acts can be in turn collected. In ourframework there will be no room for collecting acts.

4. A team of agents for a semantic of second-order logic

We propose to overcome the discussed difficulties by extending the idealiza-tion of a single agent introducing an infinite team of agents, consisting of anumber of agents equal to the number of individuals. Yet, for our purposes,we can take as individuals the agents themselves. Our agents are supposedto be able to perform the following actions:

1. Singular selecting choice (s.s.c.): one of the agents chooses an individualad libitum;

2. Plural selecting choice (p.s.c.): it is performed by all agents simultane-

ously : each agent chooses an individual ad libitum (independently oneof the others) or refrains from choosing;

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3. Plural relating choice (p.r.c.) of degree n � 2: it is performed by allagents simultaneously: each agent chooses n (not necessarily distinct)individuals in a certain order or refrains from choosing. (Abstentionfrom choosing serves the purpose of introducing empty pluralities andrelations).

We imagine that the team is guided by an ideal leader, who can order at willthe execution of one of the foregoing actions. By means of such actions, hecan refer to a single individual or to a plurality of individuals or to a plural

relation among individuals, without submitting to abstract entities the jobof collecting and correlating individuals.

A locution as “Let X be an arbitrary plurality of individuals” is to berephrased as “Suppose that the leader has ordered a p.s.c and calls X thechosen individuals”. So the locution in question is to be understood as adescription of an act of reference performed by the leader, relative to a certainplural choice. Similarly, a universal quantification “for every plurality X ” isto be read as “however a p.s.c. of certain individuals X is performed”; anexistential quantification “there is a plurality such that” is to be read as “itis possible that such a p.s.c. of certain X s be performed that”.

Formally we use a full second-order language L with identity, with first-order variables x, y, z and second-order variables Xn, Y n, Zn (of any degreen � 1). We omit the superscripts for variables of degree 1.

We will explain the semantics of acts of choice.We say that the leader makes an assignment to a formula A if he orders,

for every free variable v (of any sort) in A, an appropriate act of choice, i.e.,a s.s.c. for every first-order variable, a p.s.c. for every second-order variableof degree 1, a p.r.c. of degree n for every variable of degree n � 2. Withreference to an assignment to a formula A, if v is a free variable of A of anysort, we indicate by v∗ the relative act of choice.

We define inductively the truth of a formula relative to an assignment:

(i) x = y is true if x∗ and y∗ choose the same individual;

(ii) Xy is true if the individual chosen by y∗ is one of the individuals chosenby X∗;

(iii) Xny1 . . . yn is true if the individuals chosen respectively by y∗1 . . . y∗n

arechosen in the order by Xn

∗;

(iv) usual clauses for the propositional connectives;

(v) ∀vB is true if, however the assignment may be extended to B by anappropriate act of choice v∗ for v, B turns out to be true;

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(vi) ∃vB is true if it is performable an act of choice v∗ for v such that B

turns out to be true.

Now, it is clear that a p.s.c. does not create any entity that collects thechosen individuals. Speaking of pluralities as if they were genuine entities isa mere facon de parler, paraphrasable in terms of plural choices. Thus, pluralreference does not involve the notion of class. On the contrary, singularreference to a class presupposes plural reference to its members. In thissense, Boolos’ claim that second-order logic, interpreted in terms of pluralquantification, does not involve second-order entities is vindicated.

The fictional Platonist flavour of our theory consists in the fact thatindividuals (i.e. our agents) are treated as if they actually existed. That isessential in order to make acts of plural simultaneous choices performable.In this respect, our perspective could be labelled as a “fictional Platonismabout agents”. In contrast, acts of choice, unlike agents, are to be understoodin a mere potential way: there is no realm of possible acts; an act exists onlyinsofar as it is performed. In this respect, our perspective could be labelledas a “fictional constructivism about acts of choice”. Concerning this point,it is worth clarifying how to understand the notion of possibility involvedwhen speaking of possible acts of choice. As already observed, we don’tassume any ontology of possible acts. If possible acts were understood asentities of a realm of possibilia, then the problem of arbitrary reference tothe objects of the universe of discourse would be simply reduced to the evenharder problem of arbitrary reference to possibilia. The force, if any, of ourapproach rests essentially on the view that acts of choice are no entities at all,neither actual nor possible. The possibility in point is merely combinatorial

and non-epistemic: it is determined by the rule of the play that establisheswhat every agent is allowed to do when performing an act of choice. Thisnotion of possibility is perfectly compatible with the use of classical logic.For, one can recognize, by induction on the complexity of a formula, thatthe truth-value of a formula, relative to an assignment, is well-determined bythe truth clauses. As an example, let us consider the case of an existentialformula ∃XB. An instance of clause (vi) says that:

∃XB is true if it is performable a p.s.c. X∗ for X

such that B turns out to be true.

By the induction hypothesis, however a p.s.c. for X may be performed,it determines a truth value of B. The combinatorial possibilities concerningthe performance of a p.s.c. are determined by the play rule governing pluralchoices, according to which every agent is allowed to choose an individual

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ad libitum. Hence it is well determined if the possibility is left that a p.s.c.

be performed in such a way that B turns out to be true. Thus, ∃XB has awell-determined truth-value. Besides, our notion of combinatorial possibilityjustifies immediately, without any circularity, the comprehension principleof second-order logic:

(CP) ∃Xn∀y1 . . . yn(Xny1 . . . yn ↔ A)

For, in virtue of the arbitrarity of choices allowed by the choice rule, nothingcan prevent the possibility of a p.r.c. (or a p.s.c.) such that the chosen n-tuples are just the ones satisfying A. Possible occurrences in A of second-order quantifications cannot produce any circularity, since acts of choice,unlike properties, are all independent one of the other.

As we saw, our notion of truth is absolutely non-epistemic. Instead,on this respect, Kitcher’s perspective seems to be rather ambiguous. Hecriticizes the intuitionistic view of mathematics as based on a certain confu-sion between the ontological and the epistemic aspect [10, p. 143]. Kitcherclaims that his proposal wants to develop the constructive ontological thesis,while repudiating the epistemic one. Intuitionism aims to avoid the classicalnotion of truth by grounding the meaning of mathematical propositions onthe notion of assertability. However, Kitcher observes, assertability involvesthe notion of verification, which, in turn, involves the notion of truth. Weentirely agree with this objection. But then, according to this objection,Kitcher’s notion of truth would be expected to be non-epistemic. However,comparing his ideal agent with Brouwer’s creative subject, Kitcher justifieshis use of classical logic by removing the intuitionistic limitations to theknowledge of the creative subject and adopting a more liberal idealizationabout the epistemic power of the ideal agent [10, pp. 144–145]. This movesuggests that his conception of mathematical truth is still, after all, of anepistemic nature. As for the Brouwerian intuitionist, a proposition is trueonly if it is verifiable by the ideal agent.

In contrast, our leader, unlike the Brouwerian creative subject, is notexpected to have any mathematical competence: he has no job other thanordering acts of choice to his team and referring to the chosen individuals,in particular to make assignments. Once an assignment to a formula hasbeen made, the truth-value of the formula is well-determined by the clausesformulated above. Such value may not be known by any real or ideal subject.In this sense, our notion of truth is classical and presupposed by the notionof proof. Mathematical proofs are made by the working mathematician, whotries to discover truths by reasoning about individuals; the help of the idealagents in the proof consists in the fact that the working mathematician,

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in the course of his reasoning, can exploit, though implicitly, the abilityof the ideal leader of referring, singularly and plurally, to individuals. Soour fictionalism is in agreement with the requirement that mathematicaltalk be taken at face value. In our framework, mathematical talk is aboutindividuals (and not about actions, as in Kitcher’s and Mill’s interpretations)and uses ideal actions as a means of referring to them.

5. Comparison with other approaches

Our approach to pluralities is, to a certain extent, in accordance with Ste-nius’ treatment of the notion of a set-of in contrast with the notion ofa set-as-thing. This distinction goes back to the Russellian one betweena “class as one” and a “class as many”([17, p. 76]). Stenius defines a set-ofas follows:

If we start from a universe of discourse given in advance, then we maydefine a set-of things as being many different things in the universeof discourse or just one thing or even nothing if we want to introducethis way of speaking. [19, p. 169]

Stenius, as well as Black, finds highly problematic Cantor’s definition of aset-as-thing. However, we have no evidence whether, in Stenius’ perspective,a set-of is, after all, some sort of entity (different from a thing) or no entityat all. To say that a set of three things is the three things in questionforces the English grammar and seems to re-propose the mystery of thetrinity. Moreover, he seems to regard a set-of as existing in itself, quiteindependently of any act of referring to its elements. In fact, he rejectsthe following principle, which, as he himself observes, is often considered asimplicit in Cantor’s definition of set:

“In order that certain objects form a set, they are to be specified or singled

out”.

Stenius disagrees with Black, who seems to accept this principle for thenotion of set-of, when he claims that all one has to do in order that cer-tain things form a set is to single out such things. Stenius observes thatmathematicians, when speaking, for instance, of all subsets of a set, by nomeans require that the elements of each subset be specified in order for thesubset to exist. He concludes that the specification of the elements of a setis required only for the specification of the set but not for its existence. Forthe latter, he lays down the principle:

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“If certain things ‘exist’, then the set of these things exists too. Thata set-of things exists does not mean anything else than that the thingsforming it exist”. [19, p. 171]

We cannot agree with this principle. It explains how to understand theexistence of a set by identifying it with the existence of its elements. Thisexplanation is expressed in terms of the notion of forming a set. But, havingstripped this notion of the grounding idea of an act of collecting certainthings, the question arises: what does it mean that three things form a set?Stenius fails to provide any answer, and we are puzzling with the mysteryof the trinity again!

In our perspective, the mystery of the Trinity is solved by denying thata plurality of three elements is an entity of any sort. We use the wordplurality as a mere linguistic device for referring to the individuals selectedby a certain plural choice. What is a single thing is the word “trinity” takenas a collective name of the three elements.

Anyway, even if one agrees that the mere existence of a set-of doesn’trequire the specification of its members, he must concede that – as we haveshown in our discussion of arbitrary reference – the mathematical treatmentof sets-of requires that the members of any set-of are, at least in principle,specifiable. Just because of this requirement, we have engaged our team ofagents.

Our notion of reference can be regarded as a mathematical idealizationof Kripke’s explanation of the notion of direct reference for natural language:

A rough statement of the theory might be the following: an initial“baptism” takes place. Here the object may be named by ostension,or the reference of the name may be fixed by a description. When thename is “passed from link to link,” the receiver of the name must, Ithink, intend when he learns it to use it with the same reference asthe man from whom he heard it. [11, p. 96]

Kripke seems to be aware that the theory of descriptions presupposes aprimitive notion of reference by ostension:

Once we realize that the description used to fix the reference of aname is not synonymous with it, then the description theory can beregarded as presupposing the notion of naming or reference.[ . . . ] Notall description theorists thought that they were eliminating the notionof reference altogether. Perhaps some realized that some notion ofostension, or primitive reference, is required to back it up. CertainlyRussell did. [11, p. 97, note 44]

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In our perspective, a p.s.c. is just a way of referring to certain individualsby ostension.

Boolos’ approach to plural quantification provides an interpretation onlyof monadic second-order logic, because his appeal to natural language doesnot help to simulate relations. Lewis, in turn, extends the interpretation tofull second-order logic, by using mereological sums of individuals for codify-ing ordered pairs of individuals. An advantage of our approach is that we donot need mereology in order to recover full second-order logic, because ourteam of agents provides obvious means to simulate relations of any degree.A p.r.c. of any degree is not more problematic than a p.s.c., since choosingn individuals in a certain order is a finite operation not more problematicthan choosing a single individual. In this way, one can better appreciate thepower of our device of plural choices.

Our interpretation of second-order logic cannot be extended to third-order logic. In fact, since pluralities are not objects and have no actualexistence, there is no room for referring, through an act of choice, to anarbitrary plurality of pluralities of individuals. One can try, however, to takeseriously Kitcher’s suggestion of collecting pluralities by collecting certainsymbols representative of them. More precisely, for any binary p.r.c., one cantake each individual as a representative of all individuals correlated with it.Then one can wonder if a p.r.c. is possible that introduces a representativeof every possible plurality. But the answer is negative, according to thefollowing plural version of Cantor’s theorem:

Theorem 5.1 (Cantor). ¬∃X2∀Y ∃x∀y(Y y ↔ X2xy)

Proof. Suppose, by way of contradiction, that X2 is such that ∀Y ∃x∀y(Y y

↔ X2xy). By (CP) there is a plurality Z such that ∀x(Zx ↔ ¬X2xx). Letz such that ∀x(Zx ↔ X2zx). We get the contradiction (Zz ↔ ¬Zz).

The theorem shows that, although, through a binary p.r.c., the leadercan refer to an infinity of pluralities, he cannot refer to all possible pluralities.In other words, the potential existence of pluralities cannot be completelyactualized: no act of choice can give an actual simultaneous existence to allpossible pluralities.

6. Second-order arithmetic

As is well known, second-order arithmetic is derivable in second-order logicfrom the Hume principle as a single axiom:

(HP) #F = #G iff F and G are equinumerous.

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In the standard interpretation F , G range over second-order entities (asconcepts or classes), while # is a function mapping second-order entities ontoindividuals. The neologicism attempts to revise Frege’s logicism by replacingFrege’s ill-famed law V with (HP). In fact, Frege exploits his law V only toderive (HP), which is of the same kind of law V but, unlike the latter, turnsout to be consistent with second-order arithmetic. Crispin Wright ([22], [23])and other neologicists6 hold the highly controversial thesis that (HP) countsas a logical definition of number. The main argument against this thesisis that (HP) has ontological consequences: the existence of infinitely manynumbers. Boolos argues that pure logic should have no ontological conse-quences. We should not derive the existence of anything from considerationsof meaning alone. If so, there is no hope to derive arithmetic from logicalprinciples: arithmetic requires the existence of infinitely many individuals,whose existence cannot be guaranteed by logic.

We want to consider, however, an innocent use of Hume principle inour framework of plural logic. Assuming that the universe of individualsis infinite, we will formulate a virtual interpretation of second-order arith-metic, where the talk about numbers is understood as the use of a numerical

terminology for talking about pluralities of individuals.Given any plurality X, we introduce the symbol #X that we conven-

tionally read “the number of the Xs”. Besides, we introduce the locution“the number of the Xs is identical to the number of the Y s, in symbols#X = #Y , as synonymous of “X and Y are equinumerous” i.e. “there is a1-1-p.r.c. mapping X onto Y ”. So our virtual version of Hume principle:

(VHP) #X = #Y iff X and Y are equinumerous

is nothing else than a facon de parler. No individual or any other sortof entity is supposed to be denoted by the term #X. Intuitively, pluralitiesare regarded as numbers, whenever we are interested only in their sizes, sothat equinumerous pluralities are regarded as indistinguishable. Of course,in order to represent all numbers by means of pluralities, we need pluralitiesof any finite size, so that we need infinitely many individuals. Of course,such infinity of individuals cannot be assured by (VHP). So we will assumethat the universe of the individuals is Dedekind infinite:

(Axiom of infinite) There is a 1-1 p.r.c. that maps the universe U ofindividuals onto a proper part.

6For an introduction to neologicism see Linsky and Zalta [13].

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Let F be such a p.r.c. and let u be an individual not belonging to theimage F (U) of F . Define the chain C determined by F and u as the least plu-rality X such that (i) Xu and (ii) X is F -closed, i.e. if Xx then XF (x). Fromthis definition, it follows straightforwardly the principle of C -induction:

(C -induction) If X is any F -closed subplurality of C such that Xu, thenX = C.

Define a well-ordering on C by putting:

x � y if Xy for every F -closed X such that Xx.

If Cx, let x∗ be the plurality of all y < x.

Definition 6.1. A plurality is finite if it is equinumerous with x∗, for somex in C.

By using C -induction, it is easily provable that a finite X is equinumerouswith a unique x∗ (with x in C ) and that it is Dedekind-finite.

If X is finite, we say that #X is a natural number. We use letters m,n,

for natural numbers.

If X is finite, then X �= U so that there is a y not belonging to X ; we saythat m = #(X ∪ {y}) (where {y} is the plurality formed by y alone) isthe successor of n = #X and write m = Sn. If Λ is the empty plurality,we define 0 = #Λ. In particular:

0 = #x∗ and, for x in C, S(#x∗) = #((x∗) ∪ {x}).

If X is a subplurality of C, we introduce the symbol X# and say that X#

is the plurality of the natural numbers #x∗, for x such that Xx. Inparticular C# is said the plurality N of all natural numbers.

We can translate the principle of C -induction into the principle of N-induc-tion:

(N-induction) If X# is a plurality of natural numbers such that (i) X#0and (ii) if X#n then X#S(n), then X# = N.

It is a matter of routine to prove the other Peano axioms.We think that virtual arithmetic has a certain logicist flavour. For, if

one agrees that proper logic cannot have ontological consequences, it followsthat arithmetic must rest on some non-logical axiom adequate to assure theexistence of infinitely many entities. But, once the existence of infinitely

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many individuals has been postulated, pure logic alone is able to constructarithmetic. Of course, one can take (HP) as the unique non-logical principle,as Boolos did. But we believe that our virtual arithmetic has the virtue ofbetter enlightening certain aspects of logic and of mathematics. For, ourtheory of plural choices shows how second-order logic can avoid any exis-tential commitment to second-order entities, so that it merits the appealof logic. As to arithmetic, (VHP) brings forward a sense in which one cansee Humes principle as a genuine definition. It doesn’t define numbers ascertain special objects, but it introduces a numerical terminology that ex-plains how to regard pluralities as numbers. According to this view, theinfinitely many objects presupposed by arithmetic are not numbers, but ar-bitrary individuals that one can use for forming and comparing pluralities.In this respect, our theory recovers Mill’s and Kitcher’s ideas of understand-ing number theory as a talk about actions rather than about mathematicalobjects.

In this framework one can also get a structuralist interpretation of num-ber theory by taking C itself as the plurality N of natural numbers, whereu = 0 and F is the successor function. In this case, if X is finite, #X canbe defined as the individual x such that Cx and X is equinumerous with x∗.

7. Virtual real numbers

As it is well known, the continuum is definable in second order arithmetic.Rational numbers are representable by ordered pairs of integers whose com-ponents are relatively prime; and these, in turn, are codifiable by naturalnumbers and the order of rational numbers is definable arithmetically. Areal number can be identified with the lower class of a Dedekind cut andhence, adopting the set-theoretical semantic of second order logic, with a setof natural numbers.

This suggests that in our framework, dealing with pluralities (which areno entities) instead of sets, one can develop a virtual interpretation of thecontinuum, where speaking of real numbers as if they were objects is a facon

de parler, translatable in terms of pluralities.We sketch here how to modify the set-theoretical interpretation.We use the plurality C, introduced in section 5, as a model of natural

numbers, so that we can identify rational numbers with certain Cs andsimulate the lower class of a Dedekind cut by a certain plurality of Cs.Precisely, call a plurality of rational numbers a real-number-generator if it islimited above, has no maximum, and is closed toward below under the orderrelation of rational numbers. Then one can introduce for such pluralities

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the language of real numbers and define the usual operations between realnumbers.

Observe that even the original Dedekind theory of real numbers is, inour sense, virtual. He says:

Whenever, then, we have to do with a cut (A1, A2) produced byno rational number, we create a new, an irrational number α, whichwe regard as completely defined by this cut (A1, A2); we shall saythat the number α corresponds to this cut or that it produces thiscut. For now on, therefore, to every definite cut there correspondsa definite rational or irrational number, and we regard two numbersas different or unequal always and only when they correspond toessentially different cut. [7, p. 15]

It is clear from this passage that Dedekind doesn’t define a real numberas a class of equivalent cuts, as is done in the nowadays set-theoretical treat-ments. He conceives of an irrational number as a new creation of our mind.But these new numbers have no autonomous life; they are parasitic to thecuts. What is actually created is nothing but a linguistic device, accordingto which “we shall say that number α corresponds to this cut” [italics ours].Any talk about real numbers can be rephrased in terms of cuts. In thesame vein, we say conventionally that a real-number-generator determinesa real number. So every talk about real numbers is rephrasable in terms ofpluralities.

Of course, there is no room, in our framework, for arbitrary pluralitiesof real numbers (our theory of choices cannot be extended to third orderlogic, because we don’t admit choices of choices of individuals). However,we can introduce arbitrary countable sequences {αn}n∈N

of real numbers.Such a sequence is to be understood as a binary p.r.c., where the objectscorrelated with any natural number n form the nth real-number-generator.In this way, a countable sequence of real numbers doesn’t involve any choiceof choices but only a plural choice of individuals (where each agent choosestwo individuals in a certain order).7 Then one can prove the completenesstheorem in the form: every sequence of real numbers, limited above, has anupper-bound.

In this way, the virtual continuum requires only the ontological commit-ment to countably many individuals.

Perhaps one may think, at first sight, that the ontological status of realnumbers is the same as that of individuals, since both are, after all, fictional

7Observe that, in order to obtain an uncountable sequence of real numbers by meansof a binary p.r.c., one should assume the existence of uncountably many agents.

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entities. Of course, one can rightly regard our real numbers as fictionalentities introduced by our conventional terminology. But there is a substan-tial difference between our fictionalism about individuals (our agents) andour fictionalism about real numbers. For, our theory is committed (thoughfictionally) to individuals: these are the values of our variables, we reasonabout them essentially exploiting their supposed existence. Their role is es-sential for our theory, as a fictional character is essential for a novel abouthim. Telling a story of a fictional character is not a facon de parler. Incontrast, the theory of real numbers, as here proposed, by no means exploitstheir existence nor any assumption about them, since, as we saw, what thetheory says is nothing but a new facon de parler about individuals.

8. Some further remarks

Perhaps one might think that our commitment to plural choices be as strongas the traditional commitment to classes. That is not so, however. First,as already observed, the commitment to classes presupposes plural referenceto individuals, while acts of choice by no means presuppose that the chosenindividuals are collected by some second-order entity. Second, the infinityof classes of individuals is larger than that of individuals. Of course, if thepossible plural choices should be performed all at once, we should invoke aninfinity of teams of agents as large as the infinity of set-theoretical classes.But that is by no means required in our framework: acts of choice areperformable along the time by the same team of agents that constitutes therange of singular and plural quantification. Thus, our approach to second-order logic does not involve any infinity higher than that of the team ofagents. In particular, a countable team of agents is sufficient to reconstructthe classical second-order arithmetic and the classical continuum.

Finally, a word about our use of the Hume principle (VHP) in our virtualarithmetic. We think that the Hume principle can be considered as a legiti-mate definition of numbers insofar as it is used as a mere act of abstraction.It explains how to regard a plurality as a number. If X is a plurality, the actof abstraction at issue is performed by merely disregarding the very natureof the Xs by taking into account only how many they are, so that X isidentified with any other equinumerous plurality Y . But this identificationconcerns pluralities of the given individuals and, pace the neologicists, hasnothing to do with the alleged existence among the given individuals of cer-tain special ones that should be the genuine numbers. No act of abstractioncan create new entities nor witness the existence of special entities. In thissense, numbers are parasitic to pluralities. Similarly, we can formulate an

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innocent version of Frege’s ill-famed law V by introducing a set-theoreticalterminology for pluralities. We stipulate that any plurality X determinesa set whose members are the Xs. So we can speak of pluralities using theset-theoretical language, but this has nothing to do with the alleged exis-tence of sets among the individuals. In this sense, Hume principle is in goodcompany with law V and all other principles of abstraction.8

Acknowledgements. We would like to thank the referee for the helpfulcomments and suggestions.

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25 (2002), 72–102.

Massimiliano Carrara

Department of PhilosophyUniversity of PaduaP.zza Capitaniato 3Padova, [email protected]

Enrico Martino

Department of PhilosophyUniversity of PaduaP.zza Capitaniato 3Padova, [email protected]

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To Be is to Be the Object of a Possible Act of Choice (Studia Logica)