Plural Quantification and Classes. Gabriel Uzquiano Department of Philosophy University of Rochester Rochester, NY USA

Transcription

1 Plural Quantification and Classes Gabriel Uzquiano Department of Philosophy University of Rochester Rochester, NY USA August 30, 2001

2 1. George Boolos has extensively investigated plural quantification in Boolos [1984] and Boolos [1985]. He observed that such locutions as the Geach and Kaplan s sentence There are some critics who admire only one another are not adequately formalized in the language of first-order predicate logic, but rather in the language of monadic second-order logic. Indeed, Boolos showed in Boolos [1984] that plural quantification is interdefinable with second-order quantification. This is an indisputable technical result, but its philosophical significance depends on the ontological status of plural quantification. What is crucial for Boolos is that plural quantification requires neither metalinguistic ascent nor ontological commitment with sets, classes or Fregean concepts. Boolos persuasively argued that plural quantification is not singular quantification in grammatical disguise. The Geach and Kaplan s sentence involves ontological commitment with critics, but not with sets, classes, or Fregean concepts under which critics fall. The observation that second-order quantification is interdefinable with plural quantification is of special interest for the philosophy and the foundations of set theory. There are important reasons to want to make use of the vocabulary of classes in set theory. The terminology of classes enables us to express as single sentences some assertions that cannot but receive partial expression as first-order schemata of the language of first-order set theory. Two important examples are the axiom schemata of separation and replacement of standard set theory, i.e., Zermelo-Fraenkel set theory plus choice (ZFC). For example, the axiom schema of separation: x y z(z y z x φ), where φ is a formula in which no occurrence of y is free, is succinctly reformulated with a class variable as a single sentence of the language of classes: C x y z(z y z x z C). The trouble with classes, however, is that there is considerable pressure to dispense with them. The subject matter of set theory is presumably more than an ill-defined ensemble of collections, it comprises all the collections there are or at least all the well-founded collections there are. There are no well-founded collections that lie outside of its realm, and, in particular, no collections that fail to form a set, i.e., no proper classes. There is, as a consequence, no distinction to be made between sets and classes. There is no proper class of all sets. Nor is there a proper class of all ordinals. And, in general, one should not take at face value locutions that suggest that there are proper classes as well as sets. 1

3 Second-order quantification emerges as a promising alternative to the vocabulary of classes. For Boolos reminded us that the axioms of separation and replacement are formalized as single sentences in the language of second-order set theory. The axiom of separation, for example, is succinctly formulated with the help of a second-order variable: X x y z(z y z x Xz). And likewise for other first-order schemata such as replacement. What is much more controversial is that second-order logic is available in contexts in which the domain of discourse includes all sets. The standard interpretation of second-order quantification takes second-order variables to range over sets of individuals in the domain. But some instances of second-order comprehension are bound to fail when we take our domain to include all sets. The Russell sentence X y (Xy y / y) will be false, since there is no set of all and only those sets that are not members of themselves. Nor is it an option, for present purposes, to take the second-order variables of the language of second-order ZFC to range over classes. For we considered the ascent to second-order logic as an alternative to the vocabulary of classes. The move to second-order logic would accomplish very little for us in this context, if, in the end, it turned out to involve commitment to proper classes. The situation changes drastically when we notice Boolos observation that plural quantification and monadic second-order quantification are interdefinable. When the second-order quantifier X and Xx are then read as: There are zero or more objects X and x is one of the Xs, respectively. 1, the ascent to second-order logic becomes unobjectionable even in a context in which the domain of discourse includes all sets. For the Russell sentence X y (Xy y / y), for example, is then read as the truism: There are zero or more sets such that a set is one of them if and only if it is not a member of itself. And similar remarks apply to all other instances of second-order comprehension. My purpose in this paper is to argue for two further claims. First, I want to argue that the vocabulary of classes provides for more compact and, in some cases, perhaps irreplaceable formulations of large cardinal hypotheses that are prominent in much very important and very interesting work in recent set theory. I want to argue, in the second place, that there is no need to relinquish the vocabulary of classes. For it is open to us to continue to use the vocabulary of classes within set theory, but warn that its grammatical form is not ontologically transparent. Quantification over classes is covert plural quantification over sets, and uses of 2

4 the noun class in set theory are best regarded as elliptical for plural noun phrases that refer to sets. 2. There is little doubt that set theorists make abundant uses of the vocabulary of classes. Some of these uses are casual and need not be taken at face value. Other uses are eliminable, but heuristically valuable. There are technical reformulations that dispense with classes, but it is doubtful that anyone would have arrived to them without the detour through proper classes. In some other cases, it is much more difficult to dispense with what appears to be reference to proper classes. A typical example of a use of classes that requires foundational clarification concerns reflection arguments for large cardinal axioms. Some set theorists seem to regard it as plausible to suppose that V, the universe of all (pure) sets, is structurally undefinable, and thus that structural properties of V are reflected lower down in some level V κ of the cumulative hierarchy. 2 Reflection is the main heuristic advanced for various large cardinals, but it is not quite formalizable in the language of Zermelo-Fraenkel set theory. 3 Reflection principles approach the cumulative hierarchy from above, as they project properties of the entire universe to lower stages of the cumulative hierarchy. There is a different, but complementary approach from below in which it appears to be crucial to be able to refer to proper classes. Some authors regard it as plausible to suppose that set theory is a formal extension of facts in finite set theory into the transfinite. 4 What one would like to do on this approach is to project properties of V ω, a stage that is low down in the cumulative hierarchy, into V, the universe of all sets. The focus of this section, however, concerns certain uses of the vocabulary which are sometimes eliminable, but which nevertheless seem heuristically indispensable. These uses involve the model-theoretic concept of an elementary embedding of the universe into a submodel of Zermelo-Fraenkel set theory. By an elementary embedding, I mean, as usual, an injective map, j, of the universe, V, into some inner model, M, of ZFC that preserves satisfaction for first-order formulas of the language of ZFC. 5 M is an inner model of ZFC if M is a transitive -model of ZFC which includes ON. j is called non-trivial if it not the identity map. The critical point of a non-trivial elementary embedding j is the least ordinal α moved by j. Dana Scott pioneered the research on elementary embeddings when he proved in Scott 3

5 [1961] that if κ is the critical point of an elementary embedding j : V M other than identity, then κ is a measurable cardinal. Much very important and very interesting work in set theory ever since Scott proved his result has been concerned with the hierarchy of large cardinal principles that result from the imposition of additional closure conditions on the inner model M. For example, we call κ γ-strong if κ is the critical point of an elementary embedding j : V M such that V γ M; we call κ strong if it is γ-strong for every γ < κ. We call κ superstrong if V j(κ) M. The stronger the closure conditions imposed on M, the stronger the large cardinal principle that results. (The existence of a superstrong cardinal implies the existence of a strong cardinal, but the converse is not true.) These are the first stages of a hierarchy that has been intensely studied in the last decades. In later stages of the hierarchy lie compact, supercompact, and huge cardinals. Kenneth Kunen set an upper limit on the hierarchy when he proved in Kunen [1971] that there is no elementary embedding j : V V (other than identity), whose existence had been conjectured by William Reinhard. 6 The formalization of large cardinal hypothesis in terms of the existence of an elementary embedding takes us beyond the language of first-order ZFC. For at least two reasons. One is the use of the satisfaction relation for formulas of ZFC, which is not formalizable within ZFC on account of Tarski s result on the undefinability of truth. 7 The other is the assertion of the existence of a map from the set-theoretic universe into an inner model of ZFC. It is a theorem of first-order ZFC that there is no set of all sets. Nor is there, according to ZFC, a set which contains all ordinals as members, much less a set that maps all the sets there are into some other sets. Much effort and ingenuity on the part of set theorists has been invested in the search for equivalent formulations of some of these principles within the language of first-order ZFC. There are, as a result, set-theoretic equivalent formulations of most large cardinal principles in the hierarchy. For a paradigmatic instance of this, consider the equivalence between the existence of a non-trivial elementary embedding j : V M and the existence of a witnessing ultrafilter for a measurable cardinal κ. This is undoubtedly a source of comfort, but it doesn t change the fact that the interest of the relevant large cardinal principles stems more often from their model-theoretic characterization than from their technical formulations within first-order ZFC. 4

6 There is, in addition, no a priori guarantee that all arbitrary hypotheses stated in terms of the existence of a non-trivial elementary embedding admit of an equivalent formulation within first-order ZFC. There is, for example, little reason to expect the existence of such equivalent formulations when we reverse the arrow, and consider the question of whether there are non-trivial elementary embeddings, j : M V, from some inner model M into V. 8 In view of this, set theorists sometimes develop the theory of large cardinal principles in the hierarchy within an extension of the language of ZFC that results when one supplements it with a functional symbol, j. To each large cardinal principle in the hierarchy, there corresponds a theory framed in the language {, j} whose axioms are all of the axioms of ZFC plus the schema φ(x 1,..., x n ) φ(j(x 1 ),..., j(x n )), designed to ensure that j represents an elementary embedding, and axioms to the effect the elementary embedding j represents satisfies certain closure conditions. This is helpful, but it is not nearly as perspicuous as it would be to develop the study of the large cardinal principles in the hierarchy within a first-order theory of sets and classes, in which one is free to quantify over arbitrary maps of the universe into some subclass of it. 9 Indeed, set theorists often begin to work within an informal theory of sets and classes, and then search for technical formulations within either ZFC or some schematic extension thereof. 3. I hope to have made it plausible to assume that there are uses of the vocabulary of classes, which, if made available, one should be reluctant to do without. I now would like to suggest that it is open to us to exploit Boolos thesis that plural quantification is not covert singular quantification over sets, classes or Fregean concepts to provide an unobjectionable interpretation of the vocabulary of classes. I would like to propose that we construe reference to classes not as singular reference to gigantic collections other than sets, but rather as plural reference to sets. This point of view is not unprecedented; it certainly echoes Bertrand Russell s distinction between classes as one and classes as many in Russell [1903]. 10 The main obstacle faced by Russell s distinction appears to stem from the fact that the use of the singular number makes it difficult to avoid the impression that a class as many is some object which somehow should not be conceived of as one, but as many. But how, one may now ask, is it possible for one entity not to be one, but many? 11 5

7 I think this difficulty can be overcome without much cost to Russell s distinction, if we take advantage of an observation made by Helen Cartwright in Cartwright [1993]. Cartwright suggested that, in some uses at least, a term serves only to singularize a plural nominal, and thereby affords the means of referring in the singular to what can also be referred to in the plural. 12 The obvious suggestion is that we construe Russell s use of the noun class as many as grammatically singular, but warn that its semantic value is not singular, but plural. In this use, class as many would invariably be elliptical for some plural noun phrase, and quantification over classes as many would be covert plural quantification over sets. The crucial question for us is not whether we can make sense of Russell s notorious distinction, but rather whether we can treat uses of the noun class in contemporary set theory as a purely singularizing device in the spirit of Cartwright s observation. Once this suggestion is in place, we will be in a position to take the truth of sentences that, on the face of it, appear to involve singular reference to classes to reduce to the truth of sentences that involve plural reference to sets. Thus the truth of: (1) There are some sets that are such that no one of them is a member of itself and such that every set that is not a member of itself is one of them, would, on the current proposal, be sufficient for the truth of the sentence: (2) There is a class of sets such that no one of them is a member of itself and such that every set that is not a member of itself is one of the class. The main syntactic difference between (1) and (2) is that (1) contains an occurrence of are some where (2) contains an occurrence of is a class of. But on the proposal under consideration, the purpose of the noun class is just to enable us to refer in the singular to what is perhaps more usual to refer in the plural. This is of course not an invitation to give up the vocabulary of classes as used in (2). For as it will soon become plain, plural paraphrases quickly become unwieldy and difficult to parse. The proposal is rather that we continue to use the vocabulary of classes in the context of set theory, but warn that its grammatical form is not ontologically transparent. The grammatical number of noun phrases in which the word class occurs in sentences like (2) might at first glance suggest that its semantic value should be singular, but this impression is immediately 6

8 corrected when we paraphrase such sentences into sentences like (1). Paraphrases like (1) are useful because they exhibit the truth conditions of sentences like (2) more perspicuously than the original sentences. This is helpful, but it leaves a number of important questions open. What is, on the current proposal, the relation between sets and classes? It is common to assume that sets are special cases of classes. The identification of sets with classes, is, however, incompatible with the current proposal. 13 An important difference between sets and classes is that, while sets are members of other sets, classes cannot enter into the membership relation with other classes or sets. The membership relation takes two unmistakably singular arguments, but class is no longer a semantically singular noun. The ordinals form a class, but, as there is more than one of them, they cannot be a member of a set. And, likewise, nothing can be a member of them even though every ordinal is one of them. But there is another relation sets bear to classes. Let us say that a set x and a class X correspond just in case for all sets y, y is a member of x if and only if y is one of the Xs. Thus a set x corresponds to the class X of its members, but some classes, i.e., proper classes, such as the class of all ordinals or that of all sets do not correspond to a set. It is of interest to notice that the proposal to eliminate occurrences of the term class in favor of plural noun phrases does not iterate. The present proposal cannot make sense of such uses of the noun class as in: There is a class of non-self-membered classes. What would seem to be required to make sense of such uses of the noun class is a form of plurally plural quantification, that is, plural quantification not just over sets, but over pluralities of them. Unfortunately, one who advocates that there is such phenomenon as plurally plural quantification faces some very difficult questions. Is there, for one, a distinction to be drawn between a plurality of pluralities of sets and a plurality of sets? The answer to this question would certainly seem to be negative; a plurality of pluralities of sets is nothing over and above a plurality of sets, some sets, that is. But then there is no reason to think that we should be able to successfully refer to one, but not to the other. Nor should we be able to quantify over pluralities of pluralities of sets without thereby quantifying over pluralities of sets. That is a very welcome consequence of the present account, since standard usage of term class in the context of set theory doesn t iterate either. As it will be plain in a moment, standard theories of classes for sets to be members of classes, but not for classes to be members 7

9 of other classes. Unlike other accounts of the distinction between sets and classes, we are in a position to offer a principled explanation of the reason classes should not be allowed to enter into the membership relation with other classes. 3. Two standard theories that admit sets and classes are von Neumann-Bernays-Gödel (NBG) class theory and Morse-Kelley (MK) class theory. In their two-sorted formulation, their language is a first-order language, L, with variables x, y,... for sets and variables X, Y,... for classes, and a two-place predicate letter. The logic of the two theories is the classical first-order predicate logic with identity, and the notion of theorem of the theory is perfectly standard. Two axioms of NBG are explicitly concerned with classes: Class Extensionality: X Y z((z X z Y ) X = Y ) Predicative Comprehension: X y(y X φ), where no bound class variable occurs in φ. The two other axioms of NBG that require class variables for their formulation are separation and replacement: Separation: X x y z(z y z x z X) Replacement: X(F unc X x y z(z y w(w x w, z X))). The rest of axioms of first-order NBG are uneventful transcriptions of axioms of ZF such as pair set, infinity, union, power set, replacement, and foundation for sets. NBG is a conservative extension of ZFC: no formula φ of the language of ZFC that is not a theorem of ZFC is a theorem of NBG. The reason is that a model M of ZFC can be expanded into a model of NBG whose sets are exactly those of M. 14 NBG enables us to formalize certain semantic relations such as satisfaction for formulas of first-order ZFC by a set structure. This is, in fact, sufficient for NBG to be able to formalize some large cardinal principles discussed above. 15 Unfortunately, NBG is lacks the resources necessary to formalize the relation of satisfaction for class structures. This is an immediate consequence of the fact that NBG is a conservative extension of ZFC by the Gödel-Tarski undefinability of truth. 8

10 The satisfaction relation for class structures is definable in Morse-Kelley set theory (MK), a theory that results from NBG when we remove the restriction to formulas that contain no bound class variables in the axiom of Predicative Comprehension. The result is an axiom of Impredicative Comprehension: Impredicative Comprehension: X y(y X φ), where X doesn t occur free in φ. Morse-Kelley is not a conservative extension of first-order ZFC or NBG, but permits one to develop the model theory of the previous two theories. The focus of the next section will be to exploit the resources of plural reference to provide an unobjectionable interpretation of the considerably stronger Morse-Kelley class theory. 4. Morse-Kelley is an interpreted first-order formal theory; their set variables are supposed to range over all sets, and a formula x y is true (relative to an assignment of values to the variables) when the set assigned to the variable x is a member of the set assigned to the variable y. The trouble is that, in order to complete the interpretation, we must specify both the range of its class variables and the conditions under which a formula x Y is true relative to an assignment of values to the variables. The proposal outlined above will treat the class variables of Morse-Kelley class theory as plural variables that range over the universe of all sets. Thus the class quantifier X will be read as a plural quantifier: There are zero or more sets. The other item of business is to specify the conditions under which a formula of the form x X is to be evaluated as true (relative to an assignment of values to the variables). The answer is that x X will be true relative to an assignment when the set assigned to the set variable x is one of the sets assigned to the class variable X. There is one technical point that deserves mention. The identity symbol = is sometimes flanked by two class variables in the language of classes. The trouble is that it would be a mistake to take =, when flanked by class (and hence plural) variables, to stand for identity, since identity is a relation an individual bears to itself and to nothing else. Fortunately, we are not required to treat =, when flanked by class variables, as part of the primitive vocabulary. Instead, since there is just one position in which class variables appear in the language of two-sorted Morse-Kelley, it is open to us to treat a formula of the form 9

11 X = Y as an abbreviation for: x(x X x Y ). 16 suggests itself as an explicit definition of X = Y : The axiom of Class Extensionality Class Extensionality: x(x X x Y ) X = Y. It is a useful exercise to look at the conditions under which the axioms of Morse-Kelley class theory are true on the plural interpretation just now outlined. Let me start with the axiom schema of Impredicative Comprehension: Impredicative Comprehension: X x(x X φ(x)), where φ doesn t contain the variable X free. This axiom will now read: There are zero or more sets X such that a set x is one of the sets X if and only if φ(x). A little reflection shows that all instances of comprehension are true. How could an instance of comprehension fail to be true? There would need to be a formula φ such that it is neither the case that there are zero sets that satisfy it nor is it the case that there is one or more sets that satisfy it. The axioms of separation and replacement read much like their second-order counterparts: Separation: X x y z(z y z x z X). Separation will be true if and only if no matter what some sets the Xs are, if x is a set, then there is a set y such that a set z is a member of y if and only if z is both a member of x and one of the Xs. Replacement: X(F unc X x y z(z y w(w x w, z X))). Replacement will be true just in case no matter what some ordered pairs the Xs are, if they form a function, 17 then if x is a set, there is a set y such that a set z is a member of y if and only if z appears as a second component of one of the Xs with a member of x as a first component. The rest of axioms of the two-sorted theory of classes will retain their customary interpretation, as no class variables or quantifiers occur in them. The plural interpretation of Morse-Kelley class theory resembles Boolos s interpretation of second-order ZFC. There is, however, an important difference between the two theories. While the underlying logic of second-order ZFC is axiomatic second-order logic, the logic of 10

12 Morse-Kelley class theory is the classical first-order predicate calculus. Thus, unlike secondorder ZFC, Morse-Kelley class theory is a first-order theory that is satisfied in a variety of countable models. None of these models uses plurals to interpret the formulas of Morse-Kelley, but they bear witness to important deducibility facts. This difference might be obscured by two-sorted formulations of Morse-Kelley class theory. The two-sorted notation is convenient, but it disguises the fact that the logic of the theory is the classical first-order predicate calculus. A more perspicuous notation uses only class variables, X, Y, Z,..., and distinguishes sets from classes by the fact that the former but not the latter satisfy the formula Y (X Y ). In this context, it is useful to explicitly define a predicate Set X as an abbreviation for: Y (X Y ). The crucial question is whether it is possible to provide a plural interpretation of onesorted formulations of Morse-Kelley class theory. The answer is almost trivial, since there is a syntactic transformation from the one-sorted to the two-sorted formulation of Morse-Kelley class theory. All is required for the transformation is to replace: X Y by: x ( y (y x y X) x Y ). 18 This transformation, in combination with the plural interpretation of two-sorted Morse-Kelley, yields a plural interpretation of one-sorted Morse-Kelley class theory. The class quantifier X of one-sorted Morse-Kelley class theory will be read as a plural quantifier: There are zero or more sets. But the transformation from X Y to: x ( y (y x y X) x Y ) suggests that we take X Y to be true (relative to an assignment) if and only if there is a set that corresponds to the Xs and that set is one of the Y s. As in two-sorted Morse-Kelley, we need not treat identity as part of our primitive stock of symbols. If we introduce X = Y as an abbreviation for: Z((Z X Z Y ) Z(X Z Y Z)), then, when formulated in primitive notation, the axiom of Class Extensionality will tell us that we can infer the second conjunct from the first. This should convince us that there is an unobjectionable, perfectly intelligible interpretation of both two- and one-sorted axiomatizations of first-order Morse-Kelley. Effective as it is when the domain is taken to encompass all the sets there are, it is important to note that the plural interpretation of Morse-Kelley is not obligatory. It is, in particular, not forced upon us in circumstances in which the domain of the interpretation constitutes a set. Thus if the domain of the set variables is V κ, for a suitable κ, then it still is open to us to take the 11

13 uppercase variables of the language to be individual variables that range over the members of V κ The apparatus of plural reference would seem to be necessary for the interpretation of Morse-Kelley class theory, but it is not required for the interpretation of the predicative (twosorted) NBG. In fact, once reference to classes is construed as plural reference to sets, there appears to be no reasonable motive to stop at Predicative Comprehension. I should like to close with the observation that there is an alternative interpretation of the vocabulary of classes that is admirably adapted for the interpretation of NBG. This interpretation takes the class quantifiers of NGB as substitutional quantifiers ranging over a specific set of formulas of first-order ZFC. There are at least two substitutional interpretations of predicative theories of classes in the literature, which have been developed, respectively by W.V.O. Quine in Quine [1974] and by Charles Parsons in Parsons [1971]. A substitutional interpretation of first-order (two-sorted) NGB begins with the stipulation that, if φ(x) is a formula of the language of first-order ZFC with exactly one free variable, then we rewrite φ(t) as: t {x : φ(x)}. The next step is to take X, Y,... to be variables that range over expressions of the form {x : φ(x)} for φ(x) a formula of first-order ZFC. Then, a contextual definition of X = Y is given by: x(x X x Y ). And the interpretation is completed by the introduction of a substitutional quantifier ΠX, which ranges over formulas of the language of first-order ZFC with just one free variable. On the substitutional interpretation a sentence of the form ΠXψ(X) is true if and only if all sentences gotten from the schema ψ(x) when X is substituted by an expression of the form {x : φ(x)}, for φ(x) a formula of first-order ZFC with exactly one free variable, are true. As a consequence, we can rewrite the axiom of separation of NGB as: ΠX x y z(z y z x z X), which will be true just in case all sentences obtained by substituting a term of the form {x : φ(x)}, where φ(x) is a formula of first-order ZFC with exactly one free variable, for X in the formula x y z(z y z x z X) are true. In order for the substitutional versions of separation and replacement of NGB to match the strength of their schematic counterparts of first-order ZFC, one must allow for formulas with more than one free variable (or parameters) in the substitution class. Charles Parsons 12

14 devised in Parsons [1971] a generalization of substitutional quantification that can be used to accommodate these formulas. Parsons noticed that, given the usual definition of satisfaction for formulas of first-order ZFC, we can stipulate that a sequence s satisfies a formula of the form ΠXψ(X) if and only if s satisfies every formula obtained by substituting a term of the form {x : φ(x, y 1,..., y n )}, where φ(x, y 1,..., y n ) is a formula of first-order ZFC containing variables y 1,..., y n free, for X in ψ. This generalization of substitutional quantification is discussed by Charles Parsons in Parsons [1971] and in Parsons [1974]. What is much more doubtful is that substitutional formulations of some of the large cardinal axioms discussed above are invariably able to match the strength of their plural counterparts. Even if that this is sometimes the case, as a recent result by Akira Suzuki 19 in Suzuki [1999] would seem to suggest, it is plain that it should be possible for a large cardinal principle to require the existence of an arbitrary class that need not be definable by a formula of first-order ZFC with parameters. The substitutional interpretation of class quantification is not available for Morse-Kelley class theory. For what would be the range of the class variables of Morse-Kelley on a substitutional interpretation of class quantification? There is certainly no hope for an interpretation that takes its class variables to range over expressions {x : φ(x)}, for φ(x) a formula of firstorder ZFC, to verify all instances of impredicative comprehension. One reason is that the class of (Gödel codes of) true formulas of first-order ZFC can be defined as the extension of a formula of the form ΣX T r(x, x)), where T r(x, x) contains no class variables other than the variable X. There is, as a consequence, an instance of Impredicative Comprehension, ΣZ x(x Z ΣX T r(x, x)), that is bound to fail on account of Tarski s theorem on the undefinability of truth. 20 What if we allow substitutions by formulas that contain themselves substitutional quantifiers? A sentence of the form ΠXφ(X) derives its truth conditions from the truth conditions of its instances, but there will then be no reason to expect the substitution instances of a sentence to be simpler than the sentence itself. The result will be that there will be sentences whose evaluation will lead us into circularities. 21 What I would like to conclude from this is that substitutional quantification is not an alternative to plural quantification when it comes to provide a perfectly unobjectionable interpretation of Morse-Kelley class theory, a theory within which it is possible to develop 13

15 much recent and very interesting work in set theory. Notes 1 The English plural construction There are some objects is sometimes taken to mean There is one or more objects and some other times it is taken to mean There are two or more objects. Each of these interpretations seem different from There are zero or more objects, which is the one that corresponds to second-order quantification. Whatever plural quantifier one takes as primitive, it is not difficult to provide definitions of the other two in terms of it. 2 A reflection argument for the existence of strong inaccessibles would start with the observation that On is strongly inaccessible if λ is an ordinal, then 2 λ is an ordinal, and the limit of arbitrary sequences of ordinals of length < ON is certainly an ordinal, and infer that, by a principle of reflection, there must be a level of the cumulative hierarchy, V κ, with κ strongly inaccessible. 3 To the limited extent to which reflection is formalized in first-order ZF as the principle: α β > α x 1,..., x n V β (φ(x 1,..., x n ) φ Vβ (x 1,..., x n )) it is a theorem of ZF. 4 Friedman [1997] contains a partial development of this approach. A transfer principle is an assertion to the effect that a certain fact of finite set theory can be generalized into the transfinite. Friedman [1997] explores connections between functions on ω and functions on the entire class of ordinals, On to isolate plausible transfer principles of the form: If for all suitable functions f 1,..., f p from N k N, A(f 1,..., f p ), then for all suitable functions f 1,..., f p from On k On, A(f 1,..., f p ), for appropriate existential sentences A(f 1,..., f p ). 5 That is, V = φ(x 1,..., x n ) if and only if M = φ(j(x 1 ),..., j(x n )), for each formula φ(v 1,..., v n ) of first-order ZFC. 14

16 6 Kunen proved his result within Morse-Kelley set theory, an impredicative theory of sets and classes on which more later. 7 This, however, is not the principal obstacle for their formalization. To overcome it, it would be sufficient to supplement the language of first-order ZFC with an implicit definition of satisfaction for formulas of first-order ZFC. 8 Vickers and Welch have recently studied some of these principles in Vickers and Welch [2001]. 9 This is not to deny that, for certain purposes, the former course of action may be preferable. In Corazza [2001], Paul Corazza mentions some reasons to think that this course of action is preferable to the use of classes for the purpose of identifying the source of the inconsistency Kunen revealed in his proof that there is no elementary embedding of the universe into itself. 10 Russell s distinction appears in Russell [1903], pp Of special interest for present purposes is Russell s suggestion that it is correct to infer an ultimate distinction between a class as many and a class as one, to hold that the many are only many, and are not also one on p As Russell stated the difficulty in Russell [1903]: Is a class that has many terms to be regarded as itself one or many? Taking the class as equivalent simply to the numerical conjunction A and B and C and etc, it seems plain that it is many; yet it is quite necessary that we should be able to count classes as one each, and we do habitually speak of a class. Thus classes would seem to be one in one sense and many in another. 12 Cartwright [1993]. She suggested some uses of the noun collection to be a case in point. 13 Modulo the concern that if x is a set, then (the class which consists of) the sets that are identical with x would seem to be indistinguishable from the set x itself even though not, of course, from the singleton of x. 15

17 14 Thus if M is a model of ZFC that is a model of φ, then it can be expanded into a model of NBG in which φ is satisfied. 15 Akira Suzuki has recently shown in Suzuki [1999] that NBG formalizes j : V V is an elementary embedding. The reason is a technical lemma to the effect that j : V V preserves formula φ iff: it preserves the formula v 0 Ord Sat(V v0, v 1, v 2 ), where Sat(A, φ, a) expresses the satisfaction relation by a set structure. 16 General logic tells us that, in the presence of a finite number of non-logical symbols, we can dispense with = as a primitive symbol. Thus we can dispense with = even when this symbol is flanked by set variables. x = y would then abbreviate: z(z x z y) z(x z y z) Z(x Z y Z). When formulated in primitive notation, Set Extensionality would license the inference from the first to the last two conjuncts. 17 That is, if the ordered pairs X are such that no two of them differ in their second component but not in their first component. 18 Thanks here to an anonymous referee. For the converse direction, it is sufficient to replace x...x... by: X(Set(X)...X...). 19 This result is mentioned in note 15 above. 20 There is no formula of the language of ZFC, that is coextensive with ΣX T r(x, x). 21 This problem, however, is by not specific to the language of classes, but rather it is a perfectly general difficulty: whatever one s language, if one expects to make unambiguous use of substitutional quantification, one should not allow for the occurrence of substitutional quantifiers within the substituted terms. References George Boolos. To Be is To Be the Value of a Variable (or Some Values of Some Variables). The Journal of Philosophy, 81: ,

18 George Boolos. Nominalist Platonism. Philosophical Review, 94: , Helen M. Cartwright. On Plural Reference and Elementary Set Theory. Philosophical Review, 96: , Paul Corazza. Reports That The Existence of a j : V V Is Inconsistent Have Been Grealy Exaggerated. URL pcorazza/papers/spectrumofj.pdf. Unpublished manuscript, Harvey Friedman. Transfer Principles in Set Theory. Unpublished manuscript, Kenneth Kunen. Elementary Embeddings and Infinite Combinatorics. The Journal of Symbolic Logic, 36: , Charles Parsons. A Plea for Substitutional Quantification. Journal of Philosophy, 68: , Charles Parsons. Sets and Classes. Noûs, 8:1 12, W.V.O. Quine. The Roots of Reference. Open Court, B. Russell. The Principles of Mathematics. George Allen & Unwin Ltd., Dana Scott. Measurable cardinals and constructible sets. Bulletin of the Polish Academy of Sciences, 7: , A Suzuki. No Elementary Embedding from V into V is Definable from Parameters. The Journal of Symbolic Logic, 64: , J. Vickers and P. D. Welch. On Elementary Embeddings from an Inner Model to the Universe. The Journal of Symbolic Logic, 66,