Absolute Generality

Agustın Rayo

September 3, 2019

Absolutism, as I will understand it here, is the thesis that there is sense tobe made of absolutely general quantification: quantification over absolutelyeverything there is. The aim of this chapter is to clarify the metaphysicaldebate between absolutists and some of their detractors.1

1 The Role of Absolutism

Absolutism plays an important role in contemporary philosophy. Here aresome examples:

1.1 Ontology

Certain ontological claims are naturally seen as presupposing Absolutism(Williamson 2003). Take Mathematical Nominalism: the claim that thereare no mathematical objects. If Mathematical Nominalism is to have itsintended meaning, it can’t be understood as the claim that that there areno mathematical objects within the range of a restricted quantifier. A natu-ral way of ruling out this unintended interpretation is to take MathematicalNominalism to presuppose Absolutism and read it as the claim that abso-lutely everything is non-mathematical.

1There is a large literature on Absolutism, which has been surveyed elsewhere. (SeeFlorio 2014 and the introduction to Rayo & Uzquiano 2006.) I will not duplicate suchefforts here. I will also not engage with the debate within linguistics on how quantifierdomain restriction is best understood. (See, for instance, Stanley & Szabo 2000.)

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1.2 Meta-metaphysics

Absolutism can be used to guard against the complaint that a metaphysicaldispute is purely verbal.

Consider, for example, the dispute between a mereological nihilist, whothinks that everything is a mereological atom, and a mereological universalist,who thinks that any individuals have a fusion (van Inwagen 1990). Now imag-ine a critic who takes the debate to be purely verbal. She thinks that the ni-hilist and universalist accept the same propositions but that they use differentsentences to express those propositions because they use their quantifiers indifferent ways. For instance, when the nihilist asserts “¬∃xTable(x)” she ex-presses the same proposition that the universalist expresses when she asserts“¬∃x(Atom(x)∧Table(x))”. And when the universalist asserts “∃xTable(x)”she expresses the same proposition that the nihilist expresses when she as-serts “∃xxTableishly(xx)” (read: “some things are arranged table-ishly”).Notice, however, that the critic’s proposal cannot get off the ground whenthe nihilist and the universalist are interpreted as absolutist positions, whosequantifiers range over absolutely everything.

1.3 Philosophy of Logic

Absolutism is an important tool in the debate between singularists and plu-ralists in the philosophy of logic.2

The pluralist believes that plural terms like “them” or “they” might referto several objects at once. Consider, for example, “There are four workersand they carried the piano together”. The pluralist thinks that “they” inthis sentence should be taken to refer, collectively, to the four workers. Thesingularist disagrees. She thinks that “they” must be taken to refer to asingle object, though perhaps a single object of a special sort (e.g. the setthat has each of the four workers as members). Pluralists and singularistshave a corresponding disagreement about plural quantification. The pluralistthinks that a plural quantifier like “some things are such that . . . ” rangesover ordinary individuals in a special way (i.e. plurally). The singularistthinks that the only way to make sense of plural quantifiers is to see them asranging over special objects (e.g. sets) in the ordinary way (i.e. singularly).

2The seminal text is Boolos 1984. For more recent discussion, see Oliver & Smiley2013, Florio & Linnebo typescript.

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Absolutism can be used to generate a powerful family of arguments againstsingularism. Consider, for example, the sentence:

(1) There are some things that include all and only the individuals thatare not members of themselves.

It is natural to think that (1) is true. But when the plural quantifier is readas ranging singularly over, e.g. sets, it is equivalent to:

(2) There is a set that includes all and only the individuals that are notmembers of themselves.

which can be used to prove a contradiction by asking whether the relevant setis a member of itself. The singularist might point out that the contradictionwould be avoided if the relevant set is outside the range of “all” in (2). Butthe pluralist can deploy Absolutism to block this move by stipulating that“all” is to be read as ranging over absolutely everything.

1.4 Philosophy of Mathematics

Vann McGee (1997) has shown that the absolutist is in a position to givea categorical axiomatization of set theory (i.e. an axiomatization that canonly be satisfied by different mathematical structures when the structuresare isomorphic to one another). McGee’s starting point is an axiom systemthat Ernst Zermelo showed to be quasi-categorical (i.e. it can only be satisfiedby different mathematical structures of the same size when the structures areisomorphic to one another). McGee then enriches Zermelo’s system with afurther axiom, whose quantifiers are to be understood as absolutely general:

Urelement Set Axiom For any non-sets, there is a set that has exactlythose individuals as members.

With the new axiom in place, McGee is able to prove that there must bejust as many sets in the absolutely general domain as there are individuals,and therefore that any two structures satisfying the extended axiom systemmust be of the same size. So Zermelo’s quasi-categoricity result becomes acategoricity result.

We have considered one respect in which Absolutism is a boon to thephilosophy of mathematics. But it is worth noting that Absolutism can alsolead to trouble. For instance, Absolutism entails that there is good sense to

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be made of the question of how many individuals there are. But if there isan answer to this question, there must be consistent mathematical theoriesthat are, in fact, false. For however many individuals are in the absolutelygeneral domain, there will be consistent mathematical theories that can onlybe true if there are more.3 This generates a tension between Absolutism andstandard mathematical practice, since mathematicians are, by and large,happy to work with any consistent theory they find interesting.4

2 Anti-Absolutism

Why reject Absolutism? According to Timothy Williamson, “The most seri-ous concern about [unrestricted uses of quantifiers] is their close associationwith the paradoxes of set theory . . . However, one cannot generate [the rele-vant] contradiction just by using unrestricted quantifiers. The contradictionalways depends on auxiliary assumptions . . . [and we] can reject those as-sumptions without rejecting unrestricted quantification” (Williamson 2013,p 15).

I would like to suggest that this underplays the case against Absolutism.Although it is certainly true that the set-theoretic paradoxes have led somephilosophers to question Absolutism,5 I think the most important challengeto Absolutism is a particular conception of the relationship between ourlanguage and the world it represents. I shall refer to it here as RecarvingAnti-Absolutism.

3Here is a simple example. Let the individuals aa consist of every individual that infact exists. Then the following higher-order theory is consistent, but it is also unsatisfiable(for Cantorian reasons):

¬∃R(∀α∃a ≺ aa(Rαa) ∧ ∀α∀β∀a(Rαa ∧Rβa→ α = β))

4Here is David Hilbert: “if the arbitrarily given axioms do not contradict each otherwith all their consequences, then they are true and the things defined by them exist. Thisis for me the criterion of truth and existence”. (Letter to Frege, December 29, 1899; inFrege 1980b.) And here is Georg Cantor: “Mathematics is in its development entirely freeand only bound in the self-evident respect that its concepts must both be consistent witheach other and also stand in exact relationships, ordered by definitions, to those conceptswhich have previously been introduced and are already at hand and established.” (Cantor1883). Both quotations are drawn from Linnebo 2018, §1.3.

5Michael Dummett, in particular. See, for instance, Dummett 1963 and Dummett1991, 316–19. See also Parsons 1974, 1977, Glanzberg 2004.

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Recarving Ant-Absolutism is something of a household view. It has beena visible presence in analytic philosophy since Frege, and has been defended,in different forms, by many influential philosophers since.6 And yet it hasoften been ignored by absolutists.

I think it is hard to appreciate why Absolutism is such a substantial claim,and why it might be regarded as controversial, without having an alternativealong the lines of Recarving Anti-Absolutism firmly in view. For this reason,and because other aspects of the Absolutism debate have been adequatelysurveyed elsewhere,7 I will devote much of the remainder of this chapterto articulating Recarving Anti-Absolutism and clarifying its relationship toAbsolutism.

3 Recarving Anti-Absolutism

In this section, I will describe an especially straightforward version of theview:

Recarving Anti-Absolutism

A. To interpret a language is to assign a (coarse-grained) proposition toeach sentence.8 Any such assignment counts as an admissible inter-pretation of the language, as long as it respects logical entailments(and thereby respects the semantic structure that gives rise to thoseentailments).9

B. There is a definite fact of the matter about how the world is. (RecarvingAnti-Absolutism is therefore a form of Realism, in at least a limitedsense.)

C. The way the world is determines whether each sentence of an inter-preted language is true by determining whether the (coarse-grained)

6Here is a partial list: Carnap 1950, Dummett 1981, Wright 1983, Rosen 1993, Stalnaker1996, Sidelle 2002, Glanzberg 2004, Burgess 2005, Hellman 2006, Parsons 2006, Chalmers2009, Eklund 2009, Hirsch 2010, Thomasson 2015, Linnebo 2018.

7See, for instance, Williamson 2003 and Florio 2014.8A coarse-grained proposition is a way for the world to be. Here it will be modeled as

a set of metaphysically possible worlds.9In particular: if ψ is a logical consequence of φ, then the proposition expressed by

φ must entail the proposition expressed by ψ. For a more detailed specification of theneeded condition, see Rayo 2013, §1.3.

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proposition assigned to that sentence is true.

D. For a singular term of the language I am now speaking to refer isfor it to occur in an atomic sentence that expresses a (coarse-grained)proposition other than the absurd proposition ⊥.10

The best way to get clear about Recarving Anti-Absolutism is to contrast itwith an alternative picture:

The Metaphysical Conception of Language

A. There is a metaphysically privileged articulation of the world into con-stituents. Some of these constituents are objects and make up theworld’s metaphysically privileged ontology ; others are properties (or,more generally, relations) and make up the world’s metaphysically priv-ileged ideology.

B A metaphysical assignment of reference for a given language is a pairingof each singular term in the language with an object in the world’smetaphysically privileged ontology, and each n-place atomic predicatein the language with an n-place relation in the world’s ideology.

C. Let I be a function that assigns a proposition to each sentence of thelanguage. In order for I count as an admissible interpretation of thelanguage, there must be some metaphysical assignment of reference ρsuch that, for each atomic sentence pP (t1, . . . , tn)q of L, the individualsρ(t1), . . . , ρ(tn) are each part of w’s metaphysically privileged ontologyand are related (in that order) by ρ(P ) at w.

As an example, consider the language of first-order arithmetic, interpretedby the function A:

• A assigns the trivial proposition, > (i.e. the set of all worlds), to eacharithmetical sentence that is standardly taken to be true;

• A assigns the absurd proposition, ⊥ (i.e. the empty set of worlds), toeach arithmetical sentence that is standardly taken to be false.

10The absurd proposition is just the empty set of worlds. Why insist on a non-absurdproposition? See Section 3.3. Why restrict attention to the singular terms of one’s ownlanguage? See Section 3.5.

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An immediate consequence of the Metaphysical Conception is that A onlycounts as an admissible interpretation of the language if the world has theright metaphysically privileged constituents. Suppose, for example, that theworld’s metaphysically privileged ontology consists entirely of contingentlyexisting individuals. Then “0 = 0” cannot be assigned to the necessaryproposition, >, since any individual that might be selected as the referent of“0” will fail to be part of some world’s metaphysically privileged ontology.

According to Recarving Anti-Absolutism, in contrast, all it takes for anassignment of propositions to sentences to count as an admissible interpre-tation of the language is for the assignment to preserve logical entailments.And A certainly satisfies this condition. Notice, moreover, that A can bespecified without having to worry about whether the world’s metaphysicallyprivileged ontology contains numbers. It can be specified syntactically, forinstance:11

A(pt = sq) =

{>, if pt = sq can be syntactically reduced to “0 = 0”

⊥, otherwise

A(p¬φq) = >−A(φ)

A(pφ ∧ ψq) = A(φ) ∩ A(ψ)

A(p∃xφq) =

{>, if A(pφ[t/x]q) = > for some closed term t

⊥, otherwise

Notice, finally, that Recarving Anti-Absolutism entails that one can establishthe existence of numbers on the basis of purely linguistic considerations. Forsuppose that first-order arithmetic is a sublanguage of the language we speak.We can then use the following argument to show that the number 0 exists:

11I assume that t and s are closed terms, built from “0”, the successor function “ ′ ”,the addition function “+” and the multiplication function “×”. A syntactic reductionfrom pt = sq to “0 = 0” is a two-step transformation. The first step is to eliminateevery occurrence of “+” and “×” from t and s by repeated application of the followingtransformations:

pn+m′q→ p(n+m)′q pn+ 0q→ n

pn×m′q→ p(n×m) + nq pn× 0q→ “0”

The resulting terms, t∗ and s∗, are each of the form “0′...′”. The second step of the

transformation is to simultaneously eliminate an occurrence of “ ′ ” from each of t∗ and s∗

in pt∗ = s∗q until one of the terms is “0”. If the result is p0 = 0q, we say that pt = sq issyntactically reducible to “0 = 0”.

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Since A(“0 = 0”) = >, “0 = 0” is an atomic sentence of our lan-guage that expresses a non-absurd proposition. So, by RecarvingAnti-Absolutism, “0” refers. But if “0” refers, it refers to 0. So0 exists.

Analogous arguments can be used to establish the existence of any othernatural number, on the basis of linguistic considerations. How can we makesense of this?

It is useful to start with a slightly different example. Let p be the (coarse-grained) proposition that Socrates died and consider an interpretation onwhich “Socrates died” and “Socrates’s death occurred” are both assigned p.The Recarver has no reason to doubt the admissibility of this interpretation.12

So following Frege (1980a, §64), she might claim that p can be “carved up”in different ways. When we assert “Socrates died” we carve p up into theobject Socrates and the property of dying; when we assert “Socrates’s deathoccurred” we carve it up into the event of Socrates’s death and the propertyof occurring.

The case of first-order arithmetic is similar. Since, on interpretation A,every true sentence of first-order arithmetic expresses the necessary propo-sition >, the Recarver will think that each such sentence corresponds to acarving of >. When we assert “0 = 0”, for example, we carve > into 0 andthe identity relation; when we assert “¬(0 = 1)” we carve it into 0, 1, theidentity relation and the negation operation.

It is worth emphasizing that although Recarving Anti-Absolutism entailsthat one shouldn’t be too concerned about ontological issues, it doesn’t entailthat anything goes. The Recarver believes that one could introduce an objectcalled “Vulcan”, and do so cost-free. (For instance, one could expand one’slanguage to talk about modulo-one arithmetic and use “Vulcan” to referto the relevant mathematical object.) But Recarving Anti-Absolutism doesnot entail that one could use “Vulcan” to refer to the planet Vulcan. Forthat would require an assignment of truth-conditions according to which“∃x(x = Vulcan)” is only true at the actual world if a particular astronomicalcondition obtains. Which condition? I can use my own language to identifythe relevant coarse-grained proposition by uttering “there is a planet orbiting

12Note that I am not claiming that “Socrates died” and “Socrates’s death occurred”express the same proposition as they are actually used in English. The claim is only thatthe Recarver has no reason to question the admissibility of a regimentation of English onwhich the two sentences express the same proposition.

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the Sun within Mercury’s orbit”. A speaker of a different language mightidentify the same coarse-grained proposition differently (perhaps by utteringa version of “there are planitizing simples in orbit around the sunish simplesand within the orbit of the mercurish simples”). Regardless of the linguisticresources one uses to identify it, the condition itself will fail to be satisfiedat the actual world. So “∃x(x = Vulcan)” won’t be true. And, for reasonsdiscussed in Section 3.3, this ensures that “Vulcan” won’t refer.

3.1 Why Accept Recarving Anti-Absolutism?

In ordinary speech, we take a decidedly carefree attitudes towards expansionsof our domain of discourse. We do not hesitate to go from an event-freestatement like “they started arguing” to an event-loaded counterpart, like“an argument broke out”. We routinely expand our domain to keep track ofour social practices: we talk about commitments, memes, Internet accounts,votes, and cabinet positions. We routinely expand our domain to streamlineour descriptions of the natural world—as when we talk about orbits or tides—or to abstract away from irrelevant differences between objects—as when wetalk about particle-types or letter-types. We do not hesitate to introducetalk of mathematical objects when it is expedient to do so. And so forth.

Recarving Anti-Absolutism captures this important phenomenon, sinceit entails that the project of expanding one’s domain of discourse is nothostage to a certain kind of metaphysical fortune. The Recarver agrees that“an argument broke out” won’t be true unless a certain worldly conditionobtains—unless someone started arguing—but she insists there is no need toworry about whether a further ontological condition is satisfied: the condi-tion that the world’s metaphysically privileged ontology include argumentsalongside people who argue.

The Metaphysical Conception of Language, in contrast, entails that “anargument broke out” cannot be true unless this additional condition obtains.Whatever the merits of such a position form a metaphysical point of view,it strikes me as bizarre as an account of our linguistic practice. An ordinaryspeaker might choose to assert “an argument broke out” rather than “a fewpeople started to argue” on the basis stylistic considerations, or in orderto achieve the right emphasis, or in order to create the right context for afuture assertion. But it would be preposterous to suggest that her usage ofsuch sentences, in marketplace contexts, turns on her views about whetherarguments figure amongst the world’s ultimate furniture.

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I certainly do not mean to suggest that this is enough to settle the debatein favor of the Recarver. But I hope it suffices to highlight some of thereasons philosophers have found Recarving Anti-Absolutism attractive.13

3.2 A Languge-Relative Conception of Objecthood

The Recarver thinks that we can carve a fact into objects by using singular-term-containing sentences to describe it. But she thinks there is no sense tobe made of a language-transcendent domain, such as the one presupposed bythe Metaphysical Conception of Language. In other words, she thinks thereis no sense to be made of an articulation of the world into constituents that issignificant from a purely metaphysical point of view and therefore significantindependently of how the world happens to be represented.

As a result, the Recarver espouses a language-relative conception of ob-ject. In slogan form: an object (relative to the language I am now speaking)is the referent of one of my singular terms, or the value of one of my singu-lar variables.14 And what are the values of my variables? I can accuratelyanswer this question by using my variables. I can assert “Any x is suchthat x is the value of a variable”. I can go on to summarize the language-relative conception of object, as it applies to my language, by saying “Anyx is such that x an object”—or, indeed, “absolutely everything is an ob-ject”. I can then give an account of the quantifiers of my language by saying“unrestricted quantifiers range over a domain consisting of absolutely every-thing; restricted quantifiers range over a subdomain of the absolutely generaldomain”. In saying this, however, I do not mean that every individual in alanguage-transcendent domain is an object, or that my quantifiers range overthe individuals in such a domain. There is, after all, no sense to be madeof a language-transcendent domain. All I mean is that every individual intowhich the world gets carved up by my language is an object.

Consider an example. In Empiricism, Semantics and Ontology, Car-nap considers the “thing language”, which is concerned with “the spatio-temporally ordered system of observable things and events”. The thinglanguage carves the world into objects with spatiotemporal locations. So

13For a sustained defense of Recarving Anti-Absolutism, see Rayo 2013.14Does this mean that there couldn’t be objects without language? No. The objects

there could have been (relative to the language I am now speaking) are the values of myactual singular variables at different worlds, regardless of the linguistic resources availableat those worlds. (Compare Linnebo 2018, §2.2.)

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a speaker of the thing language can use “everything has a spatiotemporallocation”—or, indeed, “absolutely everything has a spatiotemporal location”—to expresses a true proposition. But now suppose she extends her languagewith arithmetical vocabulary. The enriched language delivers a carving ofthe world that yields both numbers and spatiotemporal objects. And sincethe new language subsumes the old, our speaker can now use “not everythinghas a spatiotemporal location” to express a true proposition.

One could, if one wanted, describe the situation by saying that the subjecthas “extended” the range of her quantifiers by moving to a richer language.But it is important to be clear that such a claim would have to be made fromthe point of view of a language that subsumes both the thing language andthe language of arithmetic. The claim can be made from the point of viewof the language that the subject has moved to, or from the point of viewof our own language. But there is no sense to be made of the “thing on-tology” from the perspective of a language that does not subsume the thinglanguage. And, crucially, there is no such thing as a “neutral” point of view,from which one can compare the ontologies delivered by these different lan-guages. Since the only intelligible conception of object is language-relative,ontological questions can only be assessed from within some language orother.

A critic might use this point as an objection against Recarving Anti-Absolutism, by claiming that it underscores the Recarver’s expressive limi-tations. She might complain that the Recarver is unable to state a suitablygeneral version of her own view. This is an important point, which I willaddress in Section 3.5.

3.3 Empty Names

Recarving Anti-Absolutism allows for empty names. But it imposes theconstraint that any atomic sentence containing an empty name must expressthe absurd (coarse-grained) proposition, ⊥.

The Recarver’s view of singular terms is based on two theses, both of themsensible. The first is that an atomic sentence pP (a1, . . . , an)q can only betrue when its singular terms refer,15 and, more generally, that the following

15Wait! Aren’t there contexts in which one can correctly assert e.g. “Sherlock Holmesis a detective”? The issue is controversial, but here I will follow Williamson 2013, §4.1in presupposing that there are no contexts in which it is both the case that “SherlockHolmes is a detective” is true and “Sherlock Homes” is an empty name. For example, when

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is a logical truth:

P (a1, . . . , an)→ ∃x1 . . . ∃xn(x1 = a1 ∧ · · · ∧ xn = an)

The second is the Kripkean thesis that “granted that there is no SherlockHolmes, one cannot say of any possible person that he would have beenSherlock Holmes, had he existed”, (Kripke 1980, 158) and, more generally,that p♦ ∃x(x = n)q fails to be true whenever n is an empty name.

When the two theses are combined, they deliver the following:16

Name Principle

♦(P (a1, . . . , an))→ ∃x1 . . . xn(x1 = a1 ∧ · · · ∧ xn = an)

The Name Principle ensures that a singular term of my language will referwhenever it is part of an atomic sentence that expresses a coarse-grainedproposition other than ⊥. To get from here to the Recarver’s view of refer-ence, all we need is the claim that satisfying such a condition is constitutiveof reference: what it is for a singular term of my language to refer is for itto be part of an atomic sentence that expresses a coarse-grained propositionother than ⊥.

“Sherlock Homes is a detective” is used in Conan Doyle’s writings (assuming he actuallyused that sentence), it used as part of a story. So there is no presumption that the sentenceis true (as opposed to true according to the story). Maybe you think that “Sherlock Holmesis a detective” can be true as used outside the story, perhaps in the context of a lecture onBritish crime fiction. In such a context, however, “Sherlock Homes” is not an empty name:it refers to a fictional character (as opposed to a human, which what Holmes is accordingto the stories). The same idea applies to more complex cases. Consider, for example,“James Bond isn’t the only example of a famous nonexistent murdering spy”, which washelpfully suggested by a referee for this volume. The most natural way of getting a truereading from this sentence is to interpret it as roughly equivalent to the following: “JamesBond (the fictional character) isn’t the only example of something that is: (a) famous, (b)a fictional character, and (c) according to the relevant fiction, a murdering spy”. I haveno idea how one might generate such a reading semantically, with no need for pragmaticrepair. If the reading cannot be generated semantically, it is not a threat. If it can, we’llhave a true sentence without an empty name, since “James Bond” will refer to a fictionalcharacter.

16Proof: Suppose p♦(P (a1, . . . , an))q is true. By the first thesis,p♦(∃x1 . . . ∃xn(x1 = a1 ∧ · · · ∧ xn = an))q is true. So, by the second thesis, a1, . . . , an arenon-empty names. So p∃x1 . . . xn(x1 = a1 ∧ · · · ∧ xn = an)q is true.

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3.4 Question-Begging

A critic might complain that by adopting a language-relative conception ofobject, the Recarver is simply changing the subject. The critic might claim,in particular, that the Recarver isn’t really talking about objects in the sensethat is relevant to metaphysicians. Here is an argument to that effect:

The Recarver’s conception of reference is a poor substitute forgenuine reference: just because a singular term has been shownto refer by the Recarver’s lights, there is no reason to think thatit really refers. As a result, her conception of object, which istied up with her conception of reference, is a poor tool for gen-uine ontological inquiry: just because one speaks a language thatcontains the syntactic string “0 = 0” and is interpreted in such away that the string is paired with a consistent proposition, thereis no reason to think that the world’s ontology really containsnumbers.

Our critic would be begging the question. According to Recarving Anti-Absolutism, what it is for a singular term of one’s language to really refer isfor it to figure in an atomic sentence that expresses a non-absurd proposition.And all it takes for me to really speak truly when I say that 0 is an objectis for my language to be interpreted in such a way that “0 = 0” expresses aconsistent proposition.

It is useful to consider an analogy. Suppose that there no sense to bemade of the claim that ascots are “objectively” fashionable, as opposed tofashionable by the lights of some community or other. Then the notion ofobjective fashionability is irreparably misguided. The only notion of fashion-ability that deserves a place in our cognitive lives is the community-relativenotion: to be fashionable just is to be fashionable relative to the relevantcommunity. The Recarver sees a language-transcendent conception of objectas similarly misguided. So she thinks that the only notion of object thatdeserves a place in our cognitive lives is the language-relative notion: to bean object just is to be an object relative to the relevant language. From thepoint of view of the Recarver, the only way of really talking about objects isto talk about objects in the language-relative sense.

The critic might try a different line of attack. She might claim that theRecarver doesn’t really avoid commitment to a language-transcendent con-ception of object. For she is committed to the conception of object that

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results from considering the referents of singular terms of every possible lan-guage.

Once more, our critic would be begging the question. For just likethe anti-absolutist thinks that there is no sense to be made of a language-transcendent notion of object, she thinks there is no sense to be made of alanguage-transcendent notion of reference. So even if she were to grant thatthere is sense to be made of “all possible languages”,17 she would think thereis no neutral perspective from which one can make sense of the combinedontology that all possible languages would deliver. Let me explain.

Suppose that Recarving Anti-Absolutism is true and suppose that I wantto make sense of the claim that the singular term t refers. If t is a term in mylanguage, there is no problem. For if t does indeed refer, some atomic sentencepP (t, a1, . . . , an)q expresses a non-absurd (coarse-grained) proposition. So,by the Name Principle of the preceding section, p∃x(x = t)q is true. So I canconclude that t refers to something, and go on to assert p‘t’ refers to tq.

Now suppose that t is not a term in my language. If t refers, it mustrefer to something—and therefore to some object. But the only conceptionof object I am able to understand is the one that is linked to the terms andvariables of my own language. In other words: I can only make sense of anobject insofar as it is one of the constituents into which my own languagecarves up the world.18 So there is no guarantee that I will be able to finda suitable referent for t, even if the user of some other language would bejustified in treating it as referential. (Notice, in contrast, that from the pointof view of the Metaphysical Conception of Language, I can make good senseof the claim that t refers. For as long as I am able to make sense of ametaphysically privileged articulation of the world into constituents, I canunderstand the claim that t refers as the claim that it is suitably paired witha member of the world’s metaphysically privileged ontology.)

17For relevant discussion, see Eklund (2006) and the debate between Eklund 2014 andRayo 2014.

18Could we assess reference claims from the point of view of the “maximal” languagethat results from combining the linguistic resources of “all possible” languages? Accordingto Recarving Anti-Absolutism, there can be no such thing as a maximal language. Forif you give me a supposedly maximal language, I can use its resources to dream up anon-trivial extension. For example, I can introduce the notion of a “hyper-set”, with thestipulation that for any objects countenanced by the given language, there is a hyper-setwith exactly those objects as hyper-members.

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3.5 Anti-Absolutism Beyond One’s Language

The Recarver’s views about reference are restricted to “the language I amnow speaking”. Recall Part D of the definition of Recarving Anti-Absolutismin Section 2:

Reference ThesisFor a singular term of the language I am now speaking to refer isfor it to occur in an atomic sentence that expresses a non-absurd(coarse-grained) proposition

The importance of this restriction is brought out by a point I made earlier.Let t be an alien singular term. In order to make sense of the claim that trefers, I must be able to make sense of the claim that it refers to something—and therefore to some object. But there is no guarantee that I will be able tofind a suitable object amongst the constituents into which my own languagecarves up the world. So there is no guarantee that I will be able to makesense of the claim that t refers.

This raises the question of whether the Recarver is in a position to stateher view in a way that is general enough to deserve being described as anaccount of the relationship between language and the world it represents, asopposed to simply an account of one particular language.

The answer, I think, is that she can get part-way there. The Recarver iscommitted to thinking that one cannot make sense of reference for arbitrarylanguages. But she takes herself to have a recipe for explaining what refer-ence consists in, and she thinks that this recipe can be used by speakers ofalien languages to make sense of a language-relative conception of languagecentered on their own language. (In the case of our language, the recipeyields the Reference Thesis; in the case of an alien language, it yields ananalogue of the Reference Thesis for that language.)

Consider an analogy. Unlike dialetheists, I am not able to make sense ofa way for the world to be with true contradictions. I nonetheless have a goodenough understanding of paraconsistent logic to describe a formalism thatcould be used by a dialethiest to make sense of a scenario in which, e.g. theLiar Sentence is both true and false. So although I cannot make it all the wayto understanding a true contradiction, I can get part-way there. For I am ina position to articulate a recipe that could be used by dialetheists to makesense of a situation in which the Liar is both true and False. The Recarvertakes herself to be in an analogous position with respect to reference. Even if

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she can’t make sense of a notion of reference for a sufficiently alien language,she is able to articulate a recipe that could be used by speakers of thatlanguage to make sense of the relevant notion.19

4 Strong Absolutism

At the beginning of the chapter, I characterized Absolutism as the thesis thatthere is sense to be made of absolutely general quantification: quantificationover absolutely everything there is.

Thus stated, Absolutism is not as restrictive a thesis as it first appears.Notice, in particular, that the Recarver has no objection to quantificationthat is absolute, in the sense of being unrestricted. Recall, for example, ourspeaker of Carnap’s thing language. When she uses “absolutely everythinghas a spatiotemporal location” to express a true proposition, she is quanti-fying over absolutely every object without restriction, in the only sense ofobject that is well-defined from her point of view. So quantifier restriction isnot really what’s at issue.20

At the same time, it is clear that Recarving Anti-Absolutism is in tensionwith Absolutism, as Absolutism tends to be used in the literature. One wayto see this is to think back to the examples I gave in Section 1 to illustratethe role of absolute generality in contemporary philosophy. Consider, forinstance, an absolutist statement of Mathematical Nominalism, accordingto which absolutely everything is non-mathematical. It would be perverseto suggest that the speaker of Carnap’s “thing language” is a nominalist inthe intended sense, even though she accepts “absolutely everything has aspatiotemporal location”. For if she later enriched her language with arith-metical vocabulary and accepted “there are numbers” she would not takeherself to have abandoned a metaphysical position she previously held. Shewill merely take herself to have adopted a different representational systemwith which to describe the world.

19Some anti-absolutists might hope to go beyond the part-way strategy I am offeringhere by developing theoretical tools that can do some of the work that absolutists think canbe played by absolutely general quantifiers. One strategy is to use schemas (Parsons 1974,2006, Glanzberg 2004, Lavine 2006). Another strategy is to use a modal construction.Fine (2006), for instance, argues that necessarily, for any range of objects, there couldbe a set-like entity not among them whose members are exactly those objects. (See alsoHellman 2006).

20For related discussion, see Fine 2006, Hellman 2006, Parsons 2006.

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I would like to suggest that what is really at issue between Recarving Anti-Absolutism, on the one hand, and the forms of Absolutism that tend to bedefended in the literature, on the other, is not the question of whether quan-tification can be absolutely unrestricted, but the question of whether there issense to be made of a language-transcendent domain: an articulation of theworld into constituents that is significant from a purely metaphysical pointof view, and therefore significant independently of how the world happensto be represented. Accordingly, I suspect that contemporary metaphysicianswho think of themselves as absolutists are often presupposing the following:

Strong Absolutism There is sense to be made of a language-transcendentdomain, which consists of all objects. For a quantifier to be absolutelygeneral is for it to range over every entity in this domain.

Strong Absolutism creates a clear dividing line. Views like Recarving Anti-Absolutism, which deny the intelligibility of a language-transcendent domain,are on one side of the line; views that support applications of absolutelygeneral quantification of the sort we discussed in Section 1 are on the otherside of the line.

5 Quantifying over everything

Neither Absolutism nor Strong Absolutism entails that absolute generalityis, in fact, attainable. Absolutism is the claim that there is sense to be madeof absolutely general quantification and Strong Absolutism is the claim thatthere is sense to be made of absolutely general quantification over a language-transcendent domain. Each of these views allows for the concept of absolutelygeneral quantification to be intelligible but unrealized in practice.

On the other hand, it is unlikely that anyone would want to accept Ab-solutism, on either of its forms, while denying that absolutely generality is,in fact, attainable. Notice, in particular, that it is not clear that the view inquestion could be endorsed without creating an instability. For it is hard tosee how one might explain what one means by absolute generality withoutpurporting to quantify absolutely generally. (For instance, the statement ofAbsolutism we have been using here glosses absolutely general quantificationas “quantification over absolutely everything there is”.) Notice, in contrast,that there is nothing unstable about denying Absolutism, on either of itsforms: the claim is simply that there is no sense to be made of expressions

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like “absolutely everything”, as they are intended by one’s opponent. So, paceWilliamson (2003, §V), I do not think that Anti-Absolutists face a specialdifficulty in stating their view.

When anti-absolutist views of the kind we have considered in this chapterare ignored, discussions of absolutism are open to a certain kind of pitfall.They run the risk of taking it for granted that there is sense to be made ofabsolute generality and proceed on the assumption that the only open ques-tion is whether absolute generality is, in fact, attainable in practice. DavidLewis (1991), for example, wonders whether a foe of absolutism would needto resort to the idea that “some mystical censor stops us from quantifyingover absolutely everything without restriction” (p. 68). This suggests thatLewis is thinking of the anti-absolutist as someone who agrees that there issense to be made of an absolutely general domain and is worried only aboutwhether our quantifiers can be stretched wide enough to encompass all of thedomain at once. I think a more interesting target is an ant-absolutist whodenies that there is sense to be made of an absolutely general domain.21

21I was lucky enough to have Ari Koslow as my research assistant on this project andam grateful for her excellent work. I am also grateful to Salvatore Florio, Nick Jones andan anonymous referee, for their many helpful comments.

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