The Model-Theoretic Argument against Quantifying over Everything

Abstract

A variant of Hilary Putnam's model-theoretic argument against metaphysical realism appears to show that our quantifiers do not determinately range over absolutely everything. This paper argues that some recent attempts to respond to the quantificational skeptic are unsuccessful and offers an alternative response: the key to answering the skeptic is not to refute her argument but to realize that the argument's setup prevents it from being convincing to those it is directed at.

Do our quantifiers determinately range over absolutely everything? A variant of Hilary Putnam's model-theoretic argument against metaphysical realism appears to show that they don't.¹ In this paper, I shall propose a skeptical resolution of this skeptical possibility, showing that on the very assumptions the skeptic has to make in order to set up her argument, it does not have any force with its target audience.

1. The model-theoretic argument

Suppose we speak a countable first-order language, first-order English (which I shall refer to simply as ‘English’). In some situations, like when pursuing ontological inquiries, we intend to talk about everything. So the intended interpretation of English is a structure with domain E – the collection of everything.² Assume that E is uncountable. By the Downward Löwenheim-Skolem theorem (DLST), E has a countable elementary substructure . Then the sentences of English which are true about (everything and its structure) are also true about (a tiny portion of everything and its structure). Furthermore, the domain S of can be chosen such that any countable collection of objects we point to is included in it. Since we can only name countably many things, it is argued that there is nothing we can say or do to force our quantifiers to range over everything rather than just some small subset of everything.

To put the same point in terms that highlight the crucial premises of the model-theoretic argument: if the interpretation of English – whose domain constitutes the range of the English quantifiers – is taken to be whatever structure meets the two conditions

(C1)  satisfies all true sentences of English, and

(C2)  respects the meaning of the nonlogical constants of English in so far as it is effectively specified by speakers of English³

then it is indeterminate what the English quantifiers range over, simply because there is, by the DLST, more than one structure that meets conditions C1 and C2. With David Lewis, we may call the view that C1 is all that constrains adequate interpretations of English global descriptivism (Lewis 1984). C2 provides a certain amount of anchoring of adequate interpretations, making sure that they hook up to the world in ways we think they should.

The most fruitful way to understand the skeptical conclusion is as follows: if an outside interpreter were to interpret English and the only constraints on his interpretation are that it preserve the truth-values of English sentences and that it respect the above constraints, then there is no unique collection that he must take the quantifiers to be ranging over. There is nothing that the speakers of English can say to force the outside interpreter to understand any particular instance of their quantifiers as ranging over absolutely everything.

One way to counter the skeptic is to deny her semantical assumptions. If English doesn't have a standard first order, or non-standard second order semantics, her argument won't get off the ground.⁴ Here, I will consider the prospects for countering the skeptic while granting her assumptions.

2. Universal quantification is truly universal

Vann McGee presents a cluster of promising arguments against the skeptical conclusion, all of which are based on the observation that natural language is “constituted . . . by the rules and practices followed by its speakers” (McGee 2000). While arbitrary reinterpretation of formal languages might be acceptable, there are constraints imposed by actual linguistic practice when it comes to natural languages.

The argument from learnability. Suppose that some instances of the universal quantifier range over S rather than over everything. To learn the rules for S-quantification speakers of English have to be able to distinguish S's from non-Ss. But if they were able to distinguish between S's and non-S's, then there would be a perceivable difference between quantifying over S and quantifying over everything, contrary to the assumption of the skeptic who claims that without realizing it speakers of English could be quantifying over S only. Given the skeptic's assumption, then, S-quantification is unlearnable and so speakers of English don't ever use the universal quantifier to range over S only.

The argument from uniqueness. Speakers of English learn to use the universal quantifier with whatever meaning the relevant rules bestow on it. The rules which govern universal quantification, namely

(R1) {(∀x)ϕ(x)} ├ϕ(τ) for any closed term τ,

and

(R2) If Γ├ϕ(c) and the individual constant c appears neither in Γ nor in ϕ, then Γ├ (∀x)ϕ(x)

satisfy Belnap's uniqueness condition, that is, no two logical operators could satisfy R1 and R2 and yet fail to play the same inferential role (Belnap 1962). So rules R1 and R2 are sufficient to pin down the meaning of the universal quantifier unambiguously in the following sense: if there are two quantifiers, ‘∀₁’ and ‘∀₂’ in a first-order language both governed by rules R1 and R2, then, whenever ϕ₁ and ϕ₂ are first-order sentences differing from each other at most in that ϕ₁ has occurrences of ‘∀₁’ where ϕ₂ has occurrences of ‘∀₂’, then ϕ₁ and ϕ₂ are interderivable using the laws of classical logic. Since ‘∀₁’ and ‘∀₂’ are logically indistinguishable, there is an important sense in which they mean the same. When speakers of English learn to use the universal quantifier by learning the relevant deduction rules, they acquire a unique logical operation.

The argument from naming and predication. Finally, it needs to be seen that the unique logical operation speakers of English acquire when they learn the rules for the universal quantifier is indeed quantification over everything. McGee argues that once the “whole apparatus of naming and predication” of natural language is taken into account, universal quantification over less than everything fails to satisfy the relevant rules of inference. First, note that, at least in principle, speakers of English can name anything there is and can refer, using a predicate, to any countable collection of (n-tuples of) objects that exist. Now suppose, for reductio, that quantification in English was not over everything but over some proper countable subset S of everything. Let ‘c’ designate some object not in S and let ‘P’ have S as its extension. Now consider the quantifier-rule R1 instantiated with ‘P’ and ‘c’:

• 

According to rule R1, speakers of English are allowed to infer ‘P(c)’ from ‘(∀x)P(x)’, but this particular instance is false given the interpretations of ‘P’ and ‘c’. Thus, if quantification is over less than everything, it will infringe rule R1. So the inference rules which speakers of English acquire when they learn to use the quantifiers ensure that their quantifiers range over everything.

3. Salvaging determinate quantification over everything?

We can distinguish between a quantifier's logical meaning and its extension. The logical meaning of a quantifier is given by its inferential role, which in turn is determined by inference rules such as the ∀-elimination and -introduction rules R1 and R2. These particular inference rules determine a unique inferential role (relative to a language) and so any quantifier governed by them is logically unambiguous. The extension of a quantifier is the collection it ranges over. If the quantifier is not explicitly or tacitly restricted, its extension is taken to be the domain of discourse, that is, the domain of the language's model-theoretic interpretation.

Logical univocality of English quantifiers. The model-theoretic argument concerns the extension of the English quantifiers, while the result concerning Belnap uniqueness only shows that a quantifier governed by rules R1 and R2 is logically univocal. However, the logical meaning of a quantifier does not determine its range in any absolute sense: The DLST shows that the inferential relations among the sentences of a theory can be preserved throughout drastic changes of the underlying ontology.

It is instructive to compare this situation with the semantics of indexicals. The linguistic rules fix the reference of the indexical ‘I’ to be the utterer. But it is the context which supplies the referent. We might construct an analogue of the uniqueness-result for the first-person pronoun, ‘I’: suppose that ‘I’ was ambiguous between I₁ and I₂, where the semantic rules for I₁ and I₂ are the same: I_i as uttered in a (non-quotational) context is to refer to the utterer of the given context. Then any pair of sentences ϕ₁ and ϕ₂ uttered in the same context and such that ϕ₁ and ϕ₂ differ from one another only in that the one has occurrences of ‘I₁’ where the other has occurrences of ‘I₂’ will coincide in truth-conditions. So I₁ and I₂ mean the same. But certainly that does not imply that the reference (extension) of ‘I’ is fixed by the rules. Both in the case of the quantifiers as in the case of indexicals the meaning-fixing rules are not sufficient to determine the extension of the respective expressions.

Learning S-quantification. The argument from learnability presupposes that a quantifier as used by a speaker of English can be taken to range exclusively over S only if the speaker has learned to use it so. But this is not so: by acquiring the inference rules R1 and R2, neither of which require the ability to tell S's from non-S's, the speaker of English has learned to use the universal quantifier with a unique logical meaning. But, again, the quantifier's logical meaning does not determine its extension. Any instance of the universal quantifier used by the speaker with the logical meaning determined by the rules can be taken to range over S.

Quantification, naming and the rules of inference. The argument from naming suggests that if the quantifiers of English are taken to always range over less than everything, then the next extension of the language could invalidate the rules of inference. The rules of inference, however, are meant to be completely general and hence to hold throughout all extensions of the language. This open-endedness constraint, call it ‘G’, can, so it seems, only be met if the universal quantifier is interpreted as ranging over everything.

But now the skeptic argues: consider , the language which contains all the names English speakers will actually ever introduce. Thus, is the limit of all actual extensions of English. Its intended model is some appropriate structure whose domain E is the collection of everything. Assuming that E is uncountable, and given that English speakers will actually only introduce at most countably many names, we can apply the DLST to obtain a countable submodel . Let be the restriction of to English. Then is a countable model of English which contains all the objects which speakers of that language and its extensions will ever name. Thus, the inference rules will never actually be violated upon future extensions of English, despite the fact that the quantifiers of English range over less than everything.

The defender of determinate quantification will complain that the skeptic's reading of G is too weak: the rules of inference are meant to be truth-preserving not merely throughout all actual but throughout all mathematically possible countable first-order extension of English (McGee 2000 and 2006). So although only the objects in S will actually ever be named, objects that are in E but not in S could be named, and since for every such object there is a mathematically possible extension of English in which that object is named, the rules will fail to hold for that extension of English. Constraint G therefore has to be strengthened. The inference rules hold throughout all mathematically possible countable first-order extensions of English. At this point, the quantificational skeptic makes a move paralleling Putnam's ‘just more theory’ reply: she points out that constraint G, if true at all, is just another true sentence of English which an adequate interpretation has to preserve. But of course, the DLST still applies, and so constraint G can be satisfied in such a way that the quantifiers still do not range over absolutely everything.

This lands us in a familiar dialectical tangle, for now the quantificational determinist insists that it is not enough to satisfy constraint G by making it come out true under one's interpretation of English. The interpretation also has to conform to the constraint: its domain has to preserve the inference-rules throughout all mathematically possible extensions of English, not just a countable subset of them (Lewis 1984).

But how is this to help? Suppose you believe that the unrestricted English quantifiers do not determinately range over absolutely everything on account of the skeptic's argument. Another speaker of English points out to you that constraint G holds: for every possible extension of English, the quantifier rules will continue to hold. Even if you believe that what he said is true, you know that the skeptic's argument still applies. The problem is that what looks like conformity from a speaker's point of view may be mere satisfaction from the outside interpreter's point of view.

That puts us in a curious dialectical situation: if G was another constraint, alongside C1 and C2, on adequate interpretations of English – in which case global descriptivism would be false – then the DLST could not generate ‘small’ non-standard interpretations. But if a speaker of English believed his language to be in the grip of the DLST, then being told, in English, that adequate interpretations conform to G will not alleviate the skepticism. Once we accept the skeptic's semantic premises, appeal to linguistic rules and practices won't help us evade the skeptical conclusion.

4. Contexts for determinate quantification

Agustín Rayo suggests that Gricean cooperation principles might provide the means to secure determinate quantification at least in some contexts (Rayo 2003). Suppose we invite the philosophically trained and fully cooperative Susan to explicitly explain to us what she intends her quantifiers to range over. She states that her quantifiers are to range over everything that is self-identical, indicating no intention to have her quantifiers range over less than everything. We may assume that Susan's utterance is in conformity with the Gricean norm of maximal informativeness and so pragmatics dictates that we take her to intend to not restrict her quantifiers. So her quantifiers ought to be interpreted as ranging over absolutely everything. Note that what, on this proposal, is supposed to secure an absolutely universal interpretation is not just what Susan says, namely that she wants to quantify over everything, but also what she doesn't say, namely that she wants to restrict her quantifiers in some way or other.

Is this sufficient? Suppose that an interpreter has several universal quantifier expressions at his disposal, with some being more inclusive than others. Suppose further that he has no reason to think that Susan can tell the domains apart or can specifically single out objects in some of the more inclusive domains. Then if his interpretation is constrained merely by C1, C2 and the Gricean cooperation principles, he will be at a loss as to how to translate Susan's ‘everything’. The point here is that quantifier restriction is always relative to the domain of discourse of the language they occur in. Typically, we take the domain of discourse of a language to be more or less fixed, interpret some instances of the quantifiers as contextually restricted and others as unrestricted within that domain. An instance of a quantifier may be interpreted as unrestricted within a given domain of discourse without thereby being unrestricted with respect to all possible domains. If there are various candidate domains, domains that the interpreter but not the interpretee can distinguish, then Gricean principles alone do not prescribe a unique course of interpretative action.

With David Lewis we may take the main lesson of the model-theoretic argument to be that purely ‘voluntaristic’ accounts of meaning are inadequate. That's why what Susan says and thinks it not sufficient to enforce an absolutely universal interpretation. But more than that, even if a speaker can convey, by not mentioning and not thinking of any restriction, an intention to be interpreted as unrestricted, that will at most constrain a cooperative interpreter to take the speaker's quantifiers to range, unrestrictedly, over her language's domain of discourse. And if the interpreter has no reason to believe that that domain coincides with what he takes his own most inclusive quantifiers to range over unrestrictedly, then, once again, he has no reason to interpret the speaker's quantifiers as, from his point of view, absolutely general.

5. The model-theoretic argument reconsidered

This is how the problem was set up: if global descriptivism is true, then whatever English speakers think or say is insufficient to force their quantifiers to be interpreted as ranging over absolutely everything. Therefore, their quantifiers do not determinately range over absolutely everything. I have attempted to show that if an English speaker believes himself to be in that predicament, then the anti-skeptical strategies considered above will not lay his worries to rest. But maybe English speakers should not believe themselves to be in that predicament in the first place.

Start by noting that there is something rather odd about the skeptic's argument in that, from the point of view of an English speaker, it's conclusion –‘ “Everything” does not mean everything’ or ‘There is an object that is not in the range of my quantifiers’– amounts to a pragmatic paradox. If your quantifiers always range over less than everything, then I can express that fact. You cannot. So there appears to be some obstacle for an English speaker to see his own quantifiers as always restricted.

Suppose the skeptic wants to convince Bill, a speaker of English, that his quantifiers can be interpreted as ranging over less than everything and are thus indeterminate. The direct route to convince Bill of the existence of non-standard interpretations of his language – exhibition of a suitable incomplete domain together with a witness to the incompleteness – fails: it would enable Bill to name the witness and thus reject the skeptic's interpretation as inadequate.

The skeptic has to find a way to convince Bill without disclosing the small-domain interpretation she has in mind. The model-theoretic argument promises to provide such a way. In it, she presents a mathematical proof that there are such interpretations. Let us be more explicit about the conceptual resources the skeptic has to assume Bill to possess. First, Bill's language is a countable first-order language. Second, the principles of reasoning he employs include at least a complete set of deduction rules for the first- order predicate calculus. Finally, Bill needs to have at his disposal the machinery required to prove the DLST. For definiteness, let us suppose that his conceptual resources are characterized by a first-order deductive theory T which includes ZFC. Then an interpretation of Bill's language has to satisfy T.

If Bill's conceptual resources are limited to T, then he cannot appreciate the fact that there are small-domain interpretations of his language, since that would require him to derive, from T

(D) There is an interpretation I of my language such that on that interpretation my quantifiers range over a countable collection and for all sentences ϕ of my language: ϕ is true according to I if and only if ϕ is true.

Since English, being a classical first-order language, lacks the resources to express its own concept of truth, Bill would need to augment his language with a truth-predicate to obtain English′, add a theory of truth for English to his assumptions (extending the axiom schemata he accepted for the predicates of English to the newly introduced truth-predicate) thereby updating his theory T to T′, and then carry out the application of the DLST to English in English′ using the resources provided by T′. But then the conclusion of that application of DLST will not be that Bill's language has unintended countable models,⁵ for his language is now English′, while the DLST was applied to English, the language he used to speak.

Maybe, then, Bill should reason by analogy: if after passing to a richer language he realizes that the language he spoke before has countable models, he can predict that when he passes to a yet richer language and makes assumptions stronger than the ones he currently makes,⁶ he will be able to prove that his current language has countable models. Now we run into another obstacle: suppose Jill who speaks some language other than English proves D. The problem is that Bill can legitimately conclude that D is true only if he can show that Jill only proves truths. But if he can show this, then T, the system from which he draws the resources to prove this fact about Jill, is proof-theoretically stronger than the set of sentences that captures the assumptions and principles of reasoning that Jill employs when proving D. But then Jill could not have proved D as it applies to Bill's language after all. By contraposition, Bill cannot reason by analogy that D is true.

But maybe it is sufficient, for appreciating the skeptic's argument, that Bill derive the weaker

(D^-) If there is an interpretation I of my language which makes all my beliefs (T) true, then there is an interpretation I′ with a countable domain which makes all of my beliefs true.⁷

Since T does prove the conditional that if T has any models at all, it has countable models, Bill can indeed prove D^-. But note that Bill can also see, because T can prove, that these countable models are non-standard – they pass off some countable collections as uncountable by omitting the mappings that witness the countability of these collections. But that means that to the extent that he can appreciate that his language might have small-domain interpretations, Bill can tell that they would be non-intended and can therefore reject them as inadequate. We saw before that the skeptic cannot persuade Bill by exhibiting a particular small collection that his unrestricted quantifiers may range over: Bill would be able to recognize such an interpretation as unintended and so rule it out. Similarly, the skeptic cannot persuade Bill that, if his beliefs were consistent, his unrestricted quantifiers could range over some countable collection obtainable via the DLST that Bill could not tell apart from his intended quantifier range: Bill would be able to tell that such interpretations, too, were unintended, precisely because he can tell that they are countable. The very fact he needs to grasp to appreciate the skeptic's argument puts him in a position to distinguish the skeptic's proposed small-domain interpretation from the intended interpretation.⁸

Where does this leave us? The arguments put forth by McGee and Rayo succeed in undermining the skeptic only if they are made from a stance that is assumed not to be subject to the skeptic's assumptions, and in particular a stance relative to which the quantifiers do range over absolutely everything. From such a stance, a reading of the constraints can be enforced that does ensure that any adequate interpretation of English takes its quantifiers to range over absolutely everything. To the extent that one is concerned about the skeptical argument it seems preferable to try to resist it without pre- supposing that there is a meta-linguistic stance immune to the argument's conclusion. The argument offered here does not assume that such a stance is available. Rather, it grants the skeptic the assumptions about English needed for the DLST to apply and shows that, under these assumptions, speakers of English cannot be persuaded by the skeptical argument because in so far as they understand it, they recognize the candidate interpretations as unintended and therefore inadequate.*

Footnotes

1. ¹ See Putnam (1980) for the original model-theoretic argument and McGee (2000) for the suggestion that the argument can be taken to cast doubt on the range of our quantifiers. Many other considerations have been put forward to suggest that the quantifiers might not range over absolutely everything. See the editors' introduction in Rayo and Uzquiano (2006) for very helpful discussion of the issues surrounding absolutely universal quantification. This paper will focus entirely on the model-theoretic argument.

2. ² Most instances of the English quantifiers, however, will be restricted as most of our everyday verbal transactions are much more limited in scope.

3. ³ Speakers of English can effectively specify an object to be in the extension of a predicate by (a) singling it out either by description or by pointing to it, or (b) by laying down an effective rule which, when successively applied to objects already effectively specified, yields that object. Given the means of effective specification, speakers of English can actually name only countably many things and determine only countably much of the extension of countably many predicates.

4. ⁴ This kind of approach was recently taken by Rayo and Williamson (2004).

5. ⁵ If it has infinite models at all. This qualification will be suppressed in the remainder of the discussion.

6. ⁶ And these additional assumptions are obtained in an acceptable way, much as adding a consistency statement or a theory of truth to an acceptable theory yields a theory that is still acceptable and in fact makes explicit what is already implicit in the acceptance of the theory. This qualification is needed, because one can always extend one's theory such as to prove any statement by simply adding a contradiction.

7. ⁷ Thanks to an anonymous referee for pointing out that this provides a less demanding route to appreciation of the skeptical argument.

8. ⁸ For a similar argument in a slightly different context, see Parsons (2008), sections 48–49. Parsons considers non-standard interpretations of the language of arithmetic which may be thought to undermine our confidence that we refer to anything definite when we talk about “the” natural numbers. He points out that if speakers came to understand such interpretations, they would be in a position to recognize them as unintended (p. 288). “[T]he positive reason we might have to believe that the natural numbers are not unique”, he diagnoses, “is quite unconvincing” (p. 279).

9. * Thanks to Peter Koellner and Vann McGee for very helpful discussion of previous drafts. Two anonymous referees and several editors for dialectica have provided multiple rounds of extremely valuable comments that helped improve the paper.