Determinate Necessity, NecessaryDeterminacy, Penumbral Determinacy,Necessary and Determinate Existence∗

Jon Erling Litland

February ,

Abstract

Keywords: Vagueness, Vague Existence, Supervaluationism, IndeterminacyArgument, Determinacy, Penumbral Connections, Metaphysical Modality

Introduction

Metaphysical modalities—necessity, possibility, actuality—and modalities ofvagueness—determinacy, indeterminacy—have both been extensively studied,but only separately. But many philosophical issues turn on the interplay of thesemodalities. In this paper I make good on that claim, and begin systematicallystudying the interplay between these modalities.

My approach is broadly supervaluational. More specifically, I work in anextension of the local supervaluationism of (Asher, Dever, and Pappas, ).Apart from showing how to deal with metaphysical modality in a supervalua-tionistic framework I also provide a new way of thinking of vague existence in asupervaluational framework: unlike several recent authors I don’t think there’sany problem making sense of vague existence on a “precisificational” accountof vagueness. I also provide a novel argument that the logic of determinacycannot be S. Most opponents of S take the bad axiom to be —that if φis determinate, then φ is determinately determinate—and the problem to behigher-order vagueness. My argument is directed at the B-axiom: if it’s notdeterminately not determinately the case that φ, then it’s the case that φ. Andthe arguments turns on the idea that even though it might be indeterminatewhat exists, there cannot be any objects that indeterminately exist.

∗Material from this paper was presented at the CSMN Colloquium, Jan . Thanks to themembers of the audience there. Special thanks to Sam Roberts and Michael Morreau. I owe aspecial debt of gratitude to Ned Hall who helped guide my thinking about these matters from thevery beginning.

Formally, my account takes the form of a quantified bimodal logic (pre-sented model theoretically). The model theory itself does not constrain theinterpretation of the operators for the modalities of vagueness very much; if oneis unhappy with the supervaluational gloss given the determinacy operatorsother interpretations—epistemic, metaphysical—are possible. The frameworkitself, and the distinctions one can draw within it, should be of interest also tothose not of a supervaluationist persuasion.

In the first part of the paper §§ to I discuss the interplay of necessity anddeterminacy in the propositional case. I argue that it is plausible that certainmetaphysical entailments hold, but it’s indeterminate which entailments hold.This phenomenon is important if one wants to formulate supervenience thesesin vague language. This part ends by considering what it would take to ensurethat determinate necessity coincided with necessary determinacy.

In the second part of the paper §§ to , I consider the interplay betweendeterminacy and necessity in the setting of quantified modal logic. This allowsus shed some new light on the vexed question whether existence can be vague.In particular, we’ll be able to throw some new light on the Barcan and Con-verse Barcan Formulae for the determinacy operator. Putting all this togetherwe’re able to throw some more light on the question whether everything existsnecessarily or not.

I would be remiss in not saying something about the relationship betweenthis theory and the theory of (Barnes and Williams, ). Barnes and Williamspresent a model theory for languages containing both modal and determinacyoperators in the course of developing a theory of metaphysical indeterminacy.The present model theory is more general and the approach is more standard.In particular: (i) All the work is done by having two accessibility relations;there is no need for “halos” and “selection-functions”. If one likes them theycan, in any case, be defined (halos) and reintroduced (selection functions). (ii)This allows for a much smoother treatment of logics for determinacy that areweaker than S. (iii) I distinguish various determinacy operators, some whichare S and some which are not. (iv) I also discuss in rather more detail theinterplay between determinacy and modality and the interplay of both withquantification. (v) The treatment of vague existence is much smoother. I will gointo somewhat more detail on these points as they arise.

Penumbral Connections

. Supervaluationism recalled

Suppose that a particular color shade lies in the middle of a sorites seriesbetween red and orange. Then, let’s suppose, it’s neither determinately red nordeterminately orange. It is nevertheless true that if it is a shade of red then itis not a shade of orange; and it is not true that if it is a shade of red then it is

I should note that this is very much in the spirit of Asher, Dever, and Pappas’s () suggestionthat supervaluationism is best seen as a modal theory.

a shade of orange. While such penumbral connections make notorious troublefor treatments of vagueness by means of many-valued logic, they are one of themain selling points of supervaluationism.

The idea behind supervaluationism, recall, is that a vague predicate admitsdifferent precisifications. Relative to a domain D, a vague predicate P has bothan extension E+—the class of objects determinately falling under it—and ananti-extension E−—the class of objects determinately not falling under it. Aprecisification of the predicate P is a pair 〈F+,F−〉 such that E+ ⊂ F+, E− ⊂ F−,F+ ∩F− = ∅ and F+ and F+ ∪F− =D.

Penumbral connections like the above—that nothing red is orange andvice versa—constrain the space of precisifications. Consider a language L.An admissible precisification of L consists in a precisification of all the vaguevocabulary in L that respects all the penumbral connections. Let φ be asentence containing some vague vocabulary. We say that φ is supertrue if φis true on every admissible precisification of the vocabulary occurring in φ.Similarly, we say that φ is superfalse if it is false on all admissible precisificationsof the vocabulary occurring in φ.

What penumbral connections are there? An important consequence ofsupervaluationism is that excluded middle holds in general—indeed, this is oneof the main selling points of supervaluationism. Take again our color shade:since every precisification is classical, on each precisification either the shade isred according to that precisification or the shade is not red according to thatprecisification; but that means that the shade is either red or not red accordingto each precisification. Thus it’s supertrue that the shade is either red or notred. More generally, every classical validity is supervaluationally valid. Whatother penumbral connections are there?

. Penumbral Entailment

We should accept the following supervenience claim: no difference in vague termswithout a difference in precise terms. This gives rise to an important type ofpenumbral connection. Consider again the borderline case of red, and let G bea complete description of this shade in precise terms. Then we should acceptthat it’s either necessary that for all b if b is G then b is red or it’s necessary thatfor all b if b is G then b is not red. Otherwise there would be a difference inWe have to consider precisifications of all the vague vocabulary in L, since we want to capture

internal as well as external penumbral connections. “If the blob is to be red, it is not to be pink; ifceremonies are to be games, then so are rituals; if sociology is to be a science, then so is psychology.”(Fine, , p. )Whether one captures classical consequence is a thornier question. On the local definition of

consequence I’m be employing one does.This might be a bit too strong: couldn’t there be fundamental facts stated in vague language?

Maybe so. But if there are, I take it that facts about which objects are red aren’t basic facts in thatsense. The relevant thesis, then, is that there can be no difference in how things are with the rednessof the shade without there being a difference in which precise facts obtain. For the possibility offundamental indeterminacy see (Barnes, in progress)

redness between two shades without a precise difference between the shades.

Now one might wonder how this can be compatible with a’s being a border-line case: am I not now saying that the precise facts about the shade determinewhether the shade is red or not? And how can that be compatible with theshade’s being a borderline case? Isn’t a borderline case one where the precisefacts don’t determine whether the predicate P applies or not?

In fact, there is no problem here. The sense in which it fails to be determinatewhether the predicate P applies to x is not to be cashed out in terms of theprecise facts failing to entail both that P applies to x and that P fails to apply tox.

The key is to note that facts about what metaphysically entails what canalso be vague; we should thus expect the supervaluationist strategy to apply tocases of metaphysical entailment as well. We will have a situation analogous tothe one we have with excluded middle. It can fail to be supertrue both that φmetaphysically entails ψ and that φ metaphysically entails not ψ, even thoughit is supertrue that φ either metaphysically entails y or φ metaphysically entailsnot ψ. I’ll shortly develop this idea rigorously; for now, I just want to point outthat this move is available to the supervaluationist and that one should find itas (un)problematic as the acceptance of excluded middle in the first place.

Similar points apply to supervenience. In every world in which the shade isexactly as it is in this world, the shade is either red or not red. Since we acceptthe supervenience claim that there is no difference in how the shade is withouta difference in how the shade is in precise terms we have to accept that either allthe worlds in which the shade is exactly as it is in this world are worlds in whichthe shade is red, or else that all the worlds in which the shade is exactly as it isin this world are worlds in which the shade isn’t red. What’s not determinate iswhich case obtains.

.. Digression: grounding

One might reasonably think that how the world is in vague terms dependson how the world is in precise terms in a finer sense than the ones describedby either metaphysical entailment or supervenience. For one wants it to bethe case that the precise facts explain why the facts stated in vague languageobtain. And, as has been extensively argued, metaphysical entailment isn’t anexplanatory notion. Perhaps the claim one wants to make is that the precisefacts ground the facts stated in vague language. One would then find oneselfin a similar situation. The supervaluationist accepts excluded middle for anystatement, but one should also accept that a true disjunction is grounded in itstrue disjuncts. Suppose that φ is such that neither φ nor ¬φ is supertrue. Then

I’ll by and large talk about entailment instead of (global) supervenience. While it is true thatentailment, in general, is stronger than global supervenience, I’ll assume that the conditions underwhich they coincide are met. That is, I’ll assume that the supervenience base is closed undernegation and that the class of propositions is closed under (infinitary) boolean operations. I’vechosen to talk of entailment instead of supervenience in order not to have quantification overworlds in the object language.See e.g., Fine, a; Schaffer, ; Audi, , forthcoming; Trogdon, forthcoming.

one is committed to its being the case that “(φ grounds φ∨ψ)∨ (ψ grounds φ∨ψ)” is supertrue while neither “φ grounds φ∨ψ” nor “ψ grounds (φ∨ψ)” issupertrue.

It seems that there shouldn’t be any problems developing a supervaluationistsemantics on top of Fine’s State Space Semantics for grounding (Fine, b),but this is a task for elsewhere.

Determinacy

So far the discussion has proceeded in terms of the semantic notion of su-pertruth. In particular, I have noted that we can have

�(φ→ ψ)∨�(φ→¬ψ) ()

supertrue without having either

�(φ→ ψ) ()

or�(φ→¬ψ) ()

supertrue.We want to express the vagueness or non-vagueness of φ in the object-

language. To that end we’ll—following standard supervaluationistic practice—introduce a determinacy operator, Det, into the object-language.

Corresponding to () we would have:

Det(�(φ→ ψ)∨�(φ→¬ψ)) ()

But just like neither () nor () we are supertrue, we cannot assert:

Det�(φ→ ψ)∨Det�(φ→¬ψ) ()

The crudest way of introducing a determinacy operator is by simply “push-ing” the semantic notion of supertruth down in the object language. We wouldthen have the clause:

• Detφ is true at precisification w if φ is supertrue

With this understanding of D we clearly get the possibility of () being correctwhilst () isn’t.

Abstaining now from considerations of metaphysical modality, the problemwith this type of determinacy operator is that whether Detφ is true at a nodew doesn’t depend on the node w; if Detφ is true at a node w, then it is true atany node v. And similarly, if Detφ is false at a node w, then Detφ is false at anynode v. We are therefore committed to the following:That one might take the grounding of a disjunction in one of its disjuncts as a penumbral

truth is overlooked by in Fine’s () otherwise illuminating discussion of the interplay betweensupervaluationism and grounding.

(i) Detφ→DetDetφ

(ii) ¬Detφ→Det¬Detφ

Because of item (i) we can’t express any higher order vagueness; and because ofitem (ii) we can’t give an adequate account of vague existence (this will becomeclear in section .).

The problem here is that the above operator is meant to reflect the meta-linguistic notion of supertruth. And if we are in the meta-language, usingclassical logic, sentences have exactly one of three statuses: they’re supertrue,superfalse or neither. That “admissible precisification” is itself vague—andit is—is neither here nor there. For presumably the vagueness of “admissibleprecisification” will itself be dealt with along supervaluationist lines; and thenit will still be the case that on every admissible precisification of “admissibleprecisification” each sentence will be either supertrue, superfalse or neither.

There is another way of thinking about determinacy. We can think of theoperator Det as belonging to the vague language itself. So rather than having thedeterminacy operator reflect the semantic (and external) notion of supertruththe determinacy operator is itself one of the terms that gets precisified. Weshould think of this as follows. When we consider precisifications we don’t onlydetermine whether a term such as “red” applies to an object x, we also determinewhether it’s essential to English that the term “red” applies to the object x. Oneof the things which is up for precisification is whether it’s mandated by therules of English that “red” applies in exactly these circumstances.

In the model theory we’ll have ways of expressing both notions. In a situationwhere we have vague existence, they end up playing very different roles.

Models

What follows will be an extension to the bi-modal case of the local superval-uationist framework of (Asher, Dever, and Pappas, ). More specifically,our framework will be a quantified (bi)modal logic with modal operators D, Sand � for determinacy and metaphysical modality respectively. The space of“worlds” here might be thought of as follows.

Think of a world as consisting of a specification of the precise facts togetherwith a precisification of the vague language. Since a precisification of the vagueterms together with a specification of the precise facts determine the truth-values of every sentence I’ll just identify worlds with the sentences that are truein them. We’ll be able to recover the difference between the true sentences thatare true because they are correspond to precise facts and the true sentences thatare true as the result (merely) of precisification by imposing conditions on theaccessibility relations. To avoid cumbersome formulations I’ll abuse notationand say that when two worlds agree on all the precise facts but differ only in

Later, in section . we’ll introduce actuality operators ↑D ,↓D for determinacy; and in section we’ll introduce actuality operators ↑,↓ for metaphysical modality.

how the vague language has been precisified that one world is a precisificationof the other.

The intended reading of the S-operator is that Sφ is true if the non-semanticfacts ensure that φ is true.

For definiteness let’s assume that we’re given a language L with atomic pred-icates P, P, . . . and logical constants E,∧,∀,¬,�,D,S. Here “E” is an existencepredicate.Definition .. A 〈D,S,�〉-pre-model is a tupleM = 〈K,R�,RD ,@,D,D,I〉. HereK is the space of worlds; R� is an equivalence relation and RD is a reflexiverelation on K ; @ is function from K to subsets of K . D : K 7→ D is the domainassignment function. And I : K ×Pred×D<ω 7→ {,} is an interpretation of theatomic predicates of L.We define the notionM,u,g |= φ—that formula φ is true at world u in modelMwith respect to assignment g—by recursion on the complexity of φ as follows.If g is an assignment, g[x/a] is the assignment that agrees with g every exceptthat a is assigned to variable x.

• M,ug |= P (x,x, . . . ,xn) iff I (u,P ,g(〈x,x, . . . ,xn〉)) =

• M,u,g |= Ea iff a ∈ D(u)

• M,u,g |= φ∧ψ iffM,u,g |= φ andM,u,g |= ψ.

• M,u,g |= ¬φ iffM,u,g 6|= φ.

• M,u,g |= ∀xφ iffM,u,g([x/a]) |= φ, for all a ∈ D(u).

• M,u,g |=Dφ iffM,v,g |= φ for all v such that uRDv

• M,u,g |= Sφ iffM,v,g |= φ for all v ∈@(u).

• M,u,g |= �φ iffM,v,g |= φ for all v such that uR�v.

When there is no danger of confusion I’ll suppress mention of the modelM andsay instead that φ is true at u (u |= φ). And I’ll say that φ(a) is true instead ofsaying that φ(x) is true with respect to the assignment g that assigns a to thevariable x.

I define consequence locally. Let Γ ,φ be sets of sentences. Γ |= φ iff for allpre-modelsM and all w ∈ KM, ifM,w |= Γ thenM,w |= φ.

. Truth simpliciter?

So far we’ve defined truth-at-a-world and consequence, but I haven’t definedtruth simpliciter. Defining truth simpliciter is the big problem for “modal”approaches to vagueness like the above. What one takes as an adequate defini-tion of truth will depend on what one thinks the nature of indeterminacy orvagueness is. Since I in this paper merely intend to set up a general frameworkfor discussing the interplay between determinacy and modality I will leave this

issue open here. That being said, I think that there is a very natural positionone can adopt on which this is no worry at all. For one might take a somewhatinstrumentalist attitude towards the model-theory. Instead of thinking that themodel theory tells us what the meanings of the operators D, S and � are, theseoperators are rather to be taken as primitive. The model theory then serves asa useful tool for studying the inferential relationships between these variousoperators, but the meaning of the operators isn’t given by the model theory.

If one adopts this approach one obviously has to say much more aboutthe meanings of these operators than I have done here; moreover, it is veryimportant to find a proof-theoretic presentation of the logics that have onlybeen model-theoretically presented here.

Setting this aside, let’s move on.

. Further constraints

The idea behind RD is that if uRDv then v is an admissible precisification ofthe vague language from the point of view of u. The idea behind @ is thatwhat’s common to @(u) specifies all the non-semantic facts. Since an admissibleprecisification doesn’t change the precise facts and we want the vague facts to bedetermined by the precise facts we have to lay down the following constraints.

uRDv ∧u , v→¬uR�v ()

u ∈@(v)∧u , v→¬uR�v ()

If the pre-modelM satisfies eqs. () and () then we callM a 〈D,S,�〉-model.If we want S to give us all the non-semantic facts we should, extending

(Asher, Dever, and Pappas, , p. ), impose the following constraint aswell:

∀w(w ∈ RD [D]→∃w′(w ∈@(w) such that for all P and all x,x, . . . ,xn,

I (w,P ,〈x,x, . . .xn〉) = I (w′ , P ,〈x,x, . . .xn〉)) ()

(Here RD [D] is the range of RD .) This ensures that the following principle iscorrect:

• ∀x,x, . . . ,xn(Sφ→Dφ)

In section . we’ll consider constraints that ensure that D�φ↔ �Dφ.Three things to note: (i) We don’t demand that u ∈ @(u). How are we to

make sense of this? I’ll address this worry in the next section. (ii) Notice that inthe clause for the atomic sentences we don’t demand that for P (a) to be true at u,that a all exist at u. Some might be worried about this since it allows objects to

I’m here taking the line of (Asher, Dever, and Pappas, ).Naturally, this doesn’t prejudge the issue whether D and S are to be interpreted as semantic,

epistemic or metaphysical operators.I hope to deal with the latter task elsewhere.

have properties in worlds in which they don’t exist. For a convincing argumentthat one shouldn’t be worried about this, see (Einheuser, ). (My reasonsfor proceeding in this way are those of (Fine, ).) (iii) This is a variabledomain semantics, both along RD and R�. Some might have scruples aboutthis. I should admit upfront that this necessitates treating the model theory ina somewhat instrumentalist spirit. There are, however, special problem aboutmaking sense of variable domains along RD ; I’ll discuss these in section .

. The relationship to (Barnes and Williams, )

In Barnes and Williams’s framework each world is associated with a “halo” ofprecisificational alternatives to that world. The space relevant to metaphysicalmodality is the space of halos, not the space of worlds as such. The models arealso equipped with selection-functions, that is, functions from halos to worlds.More formally, Barnes and Williams’s models are tuples: M = 〈W,U ,D,Σ,I .Here W is the space of worlds, U the space of halos, Σ the collection of selectionfunctions, D the domain and I the interpretation function. They make the(reasonable) demands that for each σ ∈ Σ and each U ∈ U , σ (U ) ∈ U and thatfor each U ∈ U and each w ∈ U , there is σ ∈ Σ such that σ (U ) = w. They thendefine the notion of truth relative to halo, selection-function, assignment andmodel. The relevant semantic clauses are:

• M,U,σ ,g |=Dφ iffM,U,σ ′ , g |= φ for all σ ′ ∈ Σ.

• M,U,σ ,g |= �φ iffM,U ′ ,σ ,g |= φ for all U ∈ U .

If we insist that the RD be an equivalence relation we get the effect of their halos.By insisting that R� is an equivalence relation we capture their clause for �. Ihave no need for selection functions.

. D and S: penumbral determinacy

I noted above that the way to read the S-operator is as follows. u |= Sφ iff φ issettled true by the non-semantic facts. This allows us to make sense of u <@(u).What this says, in effect, is that u has settled some semantic facts in a way thatisn’t determined by what the non-semantic facts are.

Since we now have two determinacy operators, there are now two ways ofstating that it’s not vague whether φ.

Sφ∨S¬φ ()

andDφ∨D¬φ ()

There is a sense in which () doesn’t really express that φ isn’t vague. We cansee this as follows. Suppose that φ isn’t settled true by the non-semantic factsI’ve adapted their notation to make it more in line with the present one.We need to add an “actual” halo and an actual selection function as well, but I’m leaving this

out since it doesn’t affect the point I’m making.

and not settled false by the non-semantic facts. Then there are precisificationsaccording to which φ is the case and precisifications according to which φ isn’tthe case. But this is perfectly compatible with (i) every precisification in whichφ is the case being such that it only has RD-access to precisifications in whichφ also the case; and (ii) every precisification according to which ¬φ is the casebeing such that it only has RD-access to precisifications according to which ¬φis the case. We might say thatDφ∨D¬φ expresses that every precisification willbe dogmatic about what essential to the language. Let’s call such φ penumbrallydeterminate.

Are there such penumbrally determinate φ? Arguably, the concept being anexisting object is one. The second part of the paper is devoted to bringing outthe consequences of this for the idea that it might be vague what exists.

Let’s now return to the idea of indeterminate entailment.

Entailment but not determinate entailment

Let Det stand ambiguously for D or S. Let’s construct a formal model where wehave

Det(�(φ→ ψ)∨�(φ→¬ψ)) ()

and¬Det�(φ→ ψ)∨¬Det�(φ→¬ψ) ()

We have four worlds w− ,w+ ,w

− ,w

+ . In the w-worlds φ does not obtain; in the

w-worlds φ does obtain. In the w worlds ψ does not obtain; in w+ , ψ does

obtain whilst in w− ψ does not obtain. We now put @(w+ ) = @(w− ) = {w+

,w− },

@(w+ ) = @(w+

) = {w− ,w+ }. We let the RD relation be symmetric and reflexive

and put w+RDw

− and w+

RDw− . We put w+

R�w+ and w−R�w

− . figure depicts

the essentials of the model. It is helpful to think of the “signed” worlds asbeing the result of evaluating different precisifications with respect to thespecifications of the precise facts given by the “unsigned” worlds w,w.

This model nicely illustrates the case of the borderline shade of red discussedabove. Take φ to be a completely precise description of all the features of theshade of color that are relevant to which color it is and take ψ to be the statementthat the shade is red. Now consider an arbitrary world w in which φ does notobtain. Let suppose that the color shade isn’t red in this world. Now one of thethings that has to be precisified is whether it’s necessary that if the shade is likeφ says it is, then the shade is red. According to the precisification representedby w+

it is indeed necessary that if φ is the case then ψ is the case—that is, theshade is red. For w+

doesn’t think that w− is metaphysically possible. Accordingto the other precisification w− it is instead necessary that if φ is the case then¬ψ is the case—that is, the shade is not red. For from the point of view of w− ,w+ isn’t metaphysically possible.

But since the shade is a borderline case of red, w+ and w− recognize each

other as admissible precisifications of “red” and so since w− thinks that it’s not

w+ |= ψ

D

��

�

��

w |= φ

w− |= ¬ψ

OO

��

w− |= ¬ψ

D

��

�

ZZ

��

w |= ¬φ

w+ |= ψ

]]

OO

Figure : Failure of determinate entailment

necessary that if φ then ψ, we don’t have w+ |= Det�(φ→ ψ). And since w+

thinks that it’s not necessary that if φ then ¬ψ, we don’t have w− |= Det�(φ→¬ψ).

An interesting feature of the examples considered so far is that we’re dealingwith borderline cases. In the case of the shade, we have ¬Detφ∧¬Det¬φ truein w. In this case one might explain the indeterminacy of the metaphysicalentailment by the indeterminacy in what is entailed. And one might wonderwhether this is always the case? In fact, this is not so. There are cases whereboth what is entailed and what is entailing are such that it’s necessarily deter-minate whether it obtains or not, but there nevertheless is indeterminacy in theentailment! Before we illustrate this possibility, it’ll be useful to consider therelationship between Det and � in general.

. Determinate necessity, necessary determinacy

Determinacy certainly does not entail necessity. Obama is determinately theUS president, but it’s not necessary that he is the US president. And the follow-ing simple argument shows that necessity had better not entail determinacy.Consider a borderline bald man. The supervaluationist will insist on excludedmiddle, so either he’s bald or he isn’t. So either he’s actually bald or he is actuallynot bald. But if he’s actually bald, he’s necessarily actually bald; and if he’sactually not bald he’s necessarily actually not bald. So either he’s necessarily ac-

tually bald or he’s necessarily actually not bald. If necessity entails determinacy,we get that either he’s determinately actually bald or determinately actually notbald, and hence we get that he’s determinately bald or he’s determinately notbald. That contradicts that he’s borderline bald.

But what about determinate necessity and necessary determinacy: how dothey relate? Should we demand that

Det�φ↔ �Detφ ()

be valid? With Det understood at D? With Det understood as S?The first thing to note is that the model theory developed so far does not

validate (); indeed () fails in both directions. If we want to enforce thevalidity of () (with Det interpreted at D) we only have to demand that thefollowing diagram commutes.

wD //

�

��

w

�

��w

D // w

In order to ensure S�φ→ �Sφ, we have to impose a similar condition.

Is this a reasonable demand? Let’s consider the two directions separately.First, let’s consider

�Detφ→Det�φ ()

This direction is dubious. Det�φ makes a claim about modal space itself. Inthe terminology of possible worlds, it makes the claim that it’s determinatethat all possible worlds are φ-worlds. �Detφ, on the other hand, makes aclaim about each possible world individually: it says that each possible worldis determinately a φ-world. But the claim about the determinacy of the wholeof modal space doesn’t obviously follow from the determinacy of each part.Indeed, if put in terms of possible worlds the failure of () just is the failure ofthe Barcan Formula for D.

Indeed, there are reductionist views about modality on which we shouldexpect () to fail. Suppose we reduce modality to quantification over possibleworlds and that we have the following type of reductionist story about whatpossible worlds there are. The whole space of possible worlds is a collection ofobjects satisfying a certain class of conditions. Now it might be indeterminateboth what the conditions are and what it would be to satisfy them, so it mightnot be determinate what the whole space of possible worlds is. Within anysuch space, every world might be a world in which determinately φ, within adifferent space some worlds might be non-φ worlds.

Exercise for the reader.Exercise for the reader.The view of (Sider, ) would be an example.The point that one must consider precisifications of the entirety of modal space at once, and

not just precisify the worlds individually is well made in (Barnes and Williams, , CITE!).

The other direction,Det�φ→ �Detφ ()

is rather more plausible. In the possible worlds parlance this is an instance ofthe Converse Barcan Formula for Det. It says that if it’s determinate that everyworld is a φ-world then every world is determinately a φ-world. I’ll defendthe Converse Barcan Formula for D, so if we understand the modal operatorsas quantifiers over worlds, it seems that we have to accept (). If we don’tunderstand the modal operators as quantifiers over worlds we cannot use theConverse Barcan Formula to argue for (); nevertheless, () seems to be ingood shape.

. Indeterminate entailment again

Let’s now return to the question above: can we have indeterminate entailmentwithout indeterminacy in what entails or what is entailed? If we accept () wecannot; indeed, it suffices to accept ().

Somewhat surprisingly the answer is yes if we don’t impose this requirement.The following claims are consistent. (Here I again let Det stand ambiguouslyfor S and D.)

�(Detφ∨Det¬φ)∧�(Detψ ∨Det¬ψ) ()

Det(�(ψ→ φ)∨�(ψ→¬φ)) ()

Det�θ→ �Detθ ()

¬Det�(ψ→Detφ)∧¬Det�(ψ→Det¬φ) ()

A simple modification of the model used to show indeterminate entailmentestablishes this. The model is exactly like the one in figure except that w+

does not have RD access to w− and that we put @(w+

) = w+ and @(w− ) = w− .

However, it matters that ψ does not obtain inw. We can prove the following.() and () entail

ψ→ (Det�(ψ→Detφ)∨Det�(ψ→Det¬φ)) ()

Here’s a semantic proof for the case where Det isD. Suppose that u |=D(�(φ→ ψ)∨�(φ→¬ψ)),and suppose that u |= �(φ→ ψ). (The cases are symmetric.) Now let uRDv. And let vR�v. Andsuppose that v |= φ. Then there is u with uR�u, uRDv. We then have u |= φ→ ψ. Sinceu |= �(Dφ∨D¬φ). We have to have u |= φ. Hence u |= ψ and thus u |=Dψ. But then v |= ψ, andsince v was arbitrary v |= �(φ→ ψ), and since v was arbitrary, u |= D�(φ→ ψ). The case for S isanalogous.I’ll give a semantic proof. Let u be given and suppose that ψ obtains in u, then u |= Dψ. We

have u |=Dφ∨D¬φ. Without loss of generality suppose that u |=Dφ. Now let uRDv. Since u |=Dψ,v |= ψ. By () v |= �(ψ→ φ). Since v was arbitrary we get that u |=D�(ψ→ φ). The proof for S isessentially the same.

w+ |= ψ

�

��

w |= φ

w− |= ¬ψ

��

w− |= ¬ψ

D

��

�

ZZ

��

w |= ¬φ

w+ |= ψ

]]

OO

Figure : Determinacy in the entailed does not entail determinacy of entailment

Now what if φ is only penumbrally determinate, that is, what if we don’thave Sφ∨S¬φ? In that case while we can derive () with Det interpreted asD we cannot derive

ψ→ (S�(ψ→Dφ)∨S�(θ→D¬φ)) ()

This is easily seen by modifying the model in figure as follows. We put@(w+

) = @(w− ) = {w+ ,w

− }.

. Why does actuality matter?

There is a big difference between the case where the entailing fact obtains andwhere it does not. If the entailing fact obtains the determinacy of the entailedand the entailing guarantees the determinacy of the entailment.

The reason is that if the entailing fact does not obtain it does not help toknow that it and the entailed facts are necessarily determinate. For this onlyensures that either all the φ-worlds will be ψ-worlds or that all the φ-worldswill be ψ-worlds. But it doesn’t guarantee that it is determinate what outcomewe get.

. � is vague

If there are cases where the indeterminacy in entailment cannot be traced backto indeterminacy in what it is for the entailed or the entailing to obtain, the

source of the vagueness must be the notion of metaphysical necessity itself. Itis important not to mistake this for metaphysical vagueness, whatever exactlythat might mean. The claim is simply that there is vagueness in �. That makesthe vagueness neither more nor less metaphysical than the vagueness in “red”.(One is of course free to treat both as metaphysical vagueness if one so desires.)

What if one thinks of the � as a defined by a quantifier over worlds? Thereare well known arguments that we cannot precisify the quantifiers. Soon(section ) I will take issue with Sider’s (, ) argument that this doesn’tmake sense, but for now I would just stress that whether that be so or not, itposes no problem. For let’s grant that there is no vagueness in the unrestrictedquantifier; there can still be vagueness in “possible world”, and when one—wrongly in my view—tries to analyze modality in terms of quantification overworlds one is of course using a restricted quantifier.

Vague Existence

So far I’ve only considered the propositional aspects of the interplay betweendeterminacy and metaphysical modality. As is often the case, the most inter-esting aspects come up in the quantified case. I will focus here on the case ofvague existence.

For me a case of vague existence is a case in which there isn’t anythingsuch that it’s indeterminate whether it is an F, but it’s nevertheless neitherdeterminate that there is an F nor determinate that there isn’t an F. Let’s define∇φ as ¬Dφ∧¬D¬φ. Then a case of vague existence is a case such that

∇∃xφ∧¬∃x∇φ ()

The situation is well put by Hawley: “What’s crucial about vague existence isthat, where it occurs, it is at once true that it is indeterminate whether somethingis F, and yet not true that there is a thing such that it is indeterminate whetherthat thing is F.” (Hawley, , p. )

In other words, for me a case of vague existence is a case in which the BarcanFormula for ∇—that is, the “it’s indeterminate whether”-operator—fails. Notethat ∇ is defined using D. In this setting it turns out that the difference betweenD and S will matter enormously.

Let’s start by discussing some cases of vague existence. Some remarkson my use of cases are in order here. I am personally quite attracted to themetaphysical theses I’ll use as illustrations here. Nevertheless, it would be agreat mistake to think the importance of the distinctions that will be drawnturns on the correctness of the relevant metaphysical theses. There are othermetaphysical views on which similar distinctions can be drawn. I want tobring forth how there are several very natural views of the metaphysics ofmaterial objects, such that vagueness in existence comes about naturally on

Thanks to Einar Bøhn for pressing me on this point.See also Inwagen, , ch. .

many of them. Nevertheless, it’s important to free oneself from the relianceon mereological cases. Once one realizes that there is vagueness even in theabstract realm one needs a general account: the ad hoc moves made in themereological cases simply won’t do. Third, vagueness in existence can be soradical—“undomesticated” as I’ll say—that even the profligate ontology ofWilliamson’s () necessitism can’t quite get it under control.

. Mereological cases

Consider the notorious Special Composition Question (Inwagen, , pp. –):

• Under which circumstances does a class of objects C compose a (further)object x?

The moderate answer “sometimes” has been argued to lead to vagueness inexistence in the sense that there are circumstances where it is indeterminatewhether a class C composes some object x, where there is no “fact of the mat-ter” about whether there is some object x such that the class C composes x;restrictions on composition which give plausible moderate answers seem to bevague.

Does this immediately lead to vagueness in existence? No. Whether it doesso or not depends on one’s views about the metaphysics of material objects. Butthere are several views on which it does lead to vague existence.

To fix ideas, let us consider a particular moderate view about composition.Suppose, following (Inwagen, ), that a class C composes some object y iffthe activity of the members of C constitutes a life. Consider now the case of adying dromedary. It is dying alone in the desert; let’s assume that there are noother objects weighing over lb within a mile radius of the point r wherethe dromedary is; call this area A. When the dromedary is alive it’s determinatethat there is something weighing over lb in A; when the dromedary is dead,it’s determinate that there is nothing weighing over lb in A. (Not beingalive, the corpse of the dromedary doesn’t exist, a fortiori, doesn’t weigh lb.) But it is indeterminate when death occurs, so throughout some interval it isindeterminate whether there is something weighing over lb in A.

However, it is not the case that there is something such that it is indetermi-nate whether it weighs more than lb. Such an object cannot be the putativeanimal, for were the animal to exist it would not be indeterminate whether itweighs over lb: it would be determinate that it weighs over lb. And bysupposition any other putative object determinately weighs less than lb. Sothere cannot be such an object.

One doesn’t get vague existence only on views like van Inwagen’s. Forinstance, they arise naturally if one distinguishes sharply between constitutionand identity. Consider, again, the dromedary. Consider now the predicate “x isa dromedary”. In the above scenario it will still be true that it’s indeterminateThe idea that the restrictions on composition have to be vague is forcefully stated in (Lewis,, pp. –; see also Sider, , , ). It is accepted by some moderates (Inwagen, ,pp. –; Hawley, ) (For a differing view see Nolan, ; Merricks, ; Smith, ).

whether something is a dromedary, even though there isn’t something such thatit’s indeterminate whether it is dromedary.

However, being a moderate about composition and accepting that it’s vaguewhen composition occurs does not by itself force us to accept that there is vagueexistence: for if the Barcan Formula for ∇ held we would be on safe grounds.Some moderates about composition have argued that the Barcan Formula for ∇should be accepted. I’ll say that those who accept the Barcan Formula for ∇ tryto domesticate vague existence.

Let me first say that there clearly are cases where the Barcan Formula holds:these are the humdrum cases where it’s vague what exists. Suppose, e.g., thatthere is a rock, r say, which is a borderline case of belonging to the Kilimanjaro.Then this is case where it’s indeterminate whether there exists a mountain thathas r as its part. Here of course there does exist something, viz., the Kilimanjaro,such that it is indeterminate whether it is a mountain that has r as a part. What’sat issue isn’t whether the Barcan Formula sometimes holds; what’s at issue iswhether is always holds. And this is what these philosophers claim (at least forthe mereological case).

For instance, (Carmichael, forthcoming) argues that the Barcan Formulahas to be accepted if we want a “precisificational” account of vagueness. Sincehe accepts that it can be vague whether composition occurs, he posits theexistence of “proto-objects”. In the case of the dromedary where I say that it’sindeterminate whether there is any composite object whatsoever, Carmichaelholds that there definitely is an object x but that it is indeterminate whether thesimples compose x and whether x has the simples as parts.

Wake (forthcoming) argues that supersubstantivalists—those who believethat material objects are identical to the regions of space-time they occupy—have the resources to block the argument. In a case where it’s indeterminatewhether there is a fusion of the class C, a subersubstantivalist could say thatthere is something x—viz., the region R of space-time which is the union ofthe regions of the objects in C—such that x definitely exists but such that it isindeterminate whether x is the fusion of the class C.

Shortly, I’ll argue that we can give a precisificational account of vaguenesswhilst allowing for vague existence; but let’s set that aside. An immediateproblem with such views is that it is hard to believe that there are objectsof the right sort. In particular, could there be objects with the right modalor essential properties? A dromedary is essentially a dromedary; a region ofspace-time is not essentially a dromedary. If it’s indeterminate whether there isa dromedary, it’s indeterminate whether there is something which is essentiallya dromedary. One might have thought that something that isn’t essentially a

One has to distinguish this claim from the claim that there is something that constitutessomething that is alive. It is of course correct that there is something—to wit, the matter—such thatit is indeterminate whether it constitutes something that is alive. I’m assuming for now that if thedromedary is dead, then it is not around to have properties. In section I’ll consider views thatdeny this.(Noonan, ) argues along similar lines but does not try to give an account of what these

objects are. (Donnelly, ) also argues for a similar conclusion.

dromedary determinately fails to be a dromedary.My purpose in this paper isn’t to settle such issues in the metaphysics of

ordinary objects, so let’s set this aside. Let’s consider vague existence in theabstract realm: there’s plenty of it. The least controversial cases, however,behave quite differently from the above mereological cases.

Consider some property P such that it is not determinate whether a has P ,for some particular object a. For definiteness, consider Bob and the property ofbeing bald. Bob, let’s assume, is a borderline case of being bald. Now considerthe particularized property (trope) of a’s P -ness. (Or consider the fact that ais P .) Then it is indeterminate whether there is such a thing as a’s P -ness. (Orit is indeterminate whether there is a fact that a is P .) But it does not followfrom this that there is something x such that it is indeterminate whether x is a’sP -ness (is the fact that a is P )(See Williams, , p. ; and Barnes, , formore arguments along these lines)

One could object that one shouldn’t think that tropes are abundant in thisway. But the reply to that surely is that if there is a sense in which tropes aresparse, why shouldn’t there be a sense of trope on which they are abundant?And even if there isn’t, it shouldn’t be the business of a general account of vagueexistence to rule them out.

These examples are subtler than the mereological ones: there is a scope-ambiguity hidden in them. This means that they—unlike the mereologicalexamples of vague existence above—cannot shoulder the burden of providingcounterexamples to B. We’ll return to this issue in section ..

Such examples can be multiplied. The focus on only one area of discourse—the mereological—has perhaps blinded us to how prevalent vague existence is.If one wants to domesticate it, one needs a general approach. The necessitistposition of (Williamson, ) promises to have the resources for universaldomestication. Before we go on to consider that position, however, it is time toconsider what sense we can make of variable domains. This will allow us todraw some surprising conclusions about the logic of D.

Variable domains: the indeterminacy argument

. The external indeterminacy argument

Suppose one thinks that it’s indeterminate what the existing objects are. Thenthere should be a precisification w according to which an object x exists andanother precisification v according to which x doesn’t exist. But it’s hard tomake sense of this. Sider’s “indeterminacy argument” illustrates the difficultyin an exemplary way (Sider, , p. ).

Sider argues as follows. Suppose there were two alternative precisificationsP and P of the existential quantifier, then we can argue that (at least) one ofthe precisifications is unintended as follows. Since P differs from P there hasto be an object x which is in one, but not in the other—it’s in P but not in P,say. But that shows that there is an object which P doesn’t range over. So P is

not an admissible precisification of the unrestricted existential quantifier.I call this the “external” indeterminacy argument because the precisifica-

tions P, P are considered from a point of view external to either. (Later I’llconsider an internal version of the argument.) Is this external argument anygood? Varzi objects that it begs the question:

The reason is that the phrase ‘there is’. . . is ambiguous. Why shouldwe take it to range over a domain that includes x? To do so would beto identify this phrase with the existential quantifier as precisifiedby P. But why so? On pain of circularity, there is no ‘Archimedeanpoint’—as Sider himself puts it—from which to claim that P is arestricted quantifier. There is genuine competition between P andP, and this competition affects the meaning to be attached to thephrase ‘there is’.. (Varzi, , p. )

But this objection is no good. We give the formal semantics in a particularmeta-language and there can be no question about the interpretation of themeta-language: we acquiesce in it. ‘There is’ means there is. If P and P differ inthe way sketched above then there is something in the range of the quantifiersof one which is not in the range of the other, and that means that one of thequantifiers is not unrestricted. We cannot escape this conclusion by sayingthat the competition between the two precisifications affects the metalinguistic‘there is’.

How can we respond to Sider’s argument? We get a clue that something iswrong with the argument by noting that we could run a parallel argument inthe case of metaphysical modality. For suppose that two worlds w,w differedin that the domain of quantification of w contained an object x that was not inthe domain of quantification of w. Then the quantifier that only ranged overthe objects in w would not be the unrestricted quantifier.

But this argument is no good. It only shows that we have to take theoperators for metaphysical modality as primitive and not to be explained interms of quantification over possible worlds and possible individuals. In sayingthis I’m not suggesting that there isn’t a serious question about whether therecould have been objects that don’t actually exist. That question is seriousenough; the point is just that this is a question that is best posed in a languagewith primitive modal operators.

This suggests that the problem may go away if we instead of talking aboutprecisifications talk in terms of a primitive precisificational modality. So insteadof saying that there is a precisification according to which x is “red” we shouldrather say that it’s possible to precisify in such and such a way. In order to getaround the indeterminacy argument we would also have to allow precisificationsto be “ontologically innovative”. By this I mean that it needn’t follow from the

For a supervaluationist, this move is especially hopeless. For suppose the genuine competitionaffects the meta-linguistic ‘there is’; then this vagueness can be treated by supervaluationisticmethods in a meta-meta-language. The problem is that the reasoning of Sider’s indeterminacyargument goes through on any precisification of the meta-linguistic ‘there is’ so the conclusion thatthe quantifier has a restricted interpretation on one of P, P is supertrue..

possibility of precisifying in such a way that there is an x which is F that thereis an object x such that it’s possible to precisify in such a way that x is F. If onetreats the notion, “it’s possible to precisify such that” as a modal operator P,what this means is that the Barcan Formula for P fails.

It is perhaps worth remarking that a lot of the trouble one gets into when onethinks about precisifying quantifiers comes from thinking that one precisifies aquantifier by precisifying the domain. But this is quite wrong. Quantifiers aresecond-level concepts: functions from concepts to truth-values. If one thinksof concepts as intensional entities, one should not be so surprised that thereare different precisifications of the second-level concept corresponding to theexistential quantifier.

The case of particularized properties already provides us with an exampleof ontological innovation. Suppose that Bob is borderline bald. Then if Bobis bald there is such a thing as Bob’s baldness, the particularized property ofBob’s being bald. If Bob isn’t Bald there isn’t such a thing as Bob’s baldness. Ifwe precisify “bald” in one way, we get the existence of Bob’s baldness; if weprecisify in another way, there isn’t anything which is Bob’s baldness.

Sider, of course, would not be happy with such a precisificational modality.But the indeterminacy argument is a problem even for those who take a veryrelaxed view about what objects there are. And such philosophers might behappy to avail themselves of such a precisificational modality. Much morehas to be said about how one should understand this modality, but instead ofdoing that, I’ll bring out one way in which it is very different from metaphysicalmodality.

. The internal indeterminacy argument

The above primitivist move is harder to pull off than one might initially think.For suppose that it’s possible to precisify in such a way that there is an x that isF. From the point of view of this precisification x isn’t an object that indeter-minately exists, so from the point of view of that possibility it’s determinatethat the object x exists. Any precisification that doesn’t recognize the object x isunintended because it doesn’t recognize the object x, and so the interpretationof the quantifier according to that precisification is unintended. One mightthink that it’s a simple truth that there are no objects straddling the dividebetween being and non-being, and so that it’s a simple truth that the followingholds

∀xDEx ()

More generally, one should expect the Converse Barcan Formula for D to hold.

∃x¬D¬φ→¬D¬∃xφ ()

For if there is an object a such that it’s not determinate that a isn’t F, thenit can’t be determinate that there isn’t an object which is F—a witnesses theimpossibility of this.

As is well known we enforce the validity of () by insisting on the followingcondition.

• for all w and all a, if a ∈ D(w), then for all u such that wRDu, a ∈ D(u).

The situation here is markedly dissimilar from the case of metaphysicalmodality. On contingentist assumptions there is an object x that possibly doesn’texist; one had better not take that to mean that it’s possible that there is anobject that doesn’t exist! The reason that () holds but the claim ∀x�Ex doesn’twould seem to be the following. In considering whether φ is metaphysicallynecessary, one has to consider different circumstances. And it’s possible thatcircumstances could have differed in such a way that an object a that exists inthe present circumstances doesn’t exist in those circumstances.

When one considers whether it is determinate that φ, on the other hand, onedoesn’t consider how things are in different circumstances. That things couldhave been different in such a way that the object a doesn’t exist, is completely ir-relevant to the question whether a determinately exists. In considering whetherit is determinate that φ one consider different ways of interpreting our language,but one of the thing that an interpretation cannot do is change which objectsthere are.

One might also think that from the point of view of a precisification thatdoesn’t introduce an object x, a precisification that introduces an object x wouldalso be inadmissible, since it would mistakenly think that the object x exists.This move, however, is much more questionable. In making this move oneassumes that a precisification not only takes a view on the objects that existaccording to it—that they determinately exist—but also that the objects thatexist according to it are, determinately, the only objects that exist. And thereis little reason to impose this requirement—indeed, if we want to make senseof vagueness in existence we had better not impose this requirement. For ifwe impose that requirement we commit to the Barcan Formula for D and theBarcan Formula for D together with the Converse Barcan Formula entail theBarcan Formula for ∇.

But if we have to accept the Converse Barcan Formula for D and want tohold that there can be vagueness in existence, the logic for D cannot be anextension of KTB.

. Against B

In presenting the model theory above I only demanded that RD should bereflexive. (Whatever else we know about determinacy, we know that if it’sdeterminate thatφ, then it’s the case thatφ.) If there are cases of vague existence,then these cannot be modeled using a symmetric relation RD ; but we can in factshow something stronger. There are counterexamples to the B-axiom, i.e., to¬D¬Dφ→ φ.

I don’t mean to suggest that this is especially difficult or even surprising. Similar phenomenahave long been noticed in modal set theories.

Observation .. If ∇∃xFx, ¬∃x∇Fx and D∀xD(Fx → DFx) then ∃xDFx, is acounterexample to B.Proof: I give a semantic proof. I here exploit the fact that KT is sound and com-plete for reflexive frames. The use of the model theory for standard quantifiedmodal logic is just for convenient proof. What matters is what sentences of theobject-language are true. Counterpart theory is thus an irrelevance.

Suppose that w |= ∇∃xFx and hence that w |= ¬D¬∃xFx. Then there is aworld w such that wRw with w |= ∃xFx. Hence there is a ∈ D(w) such thatw |= F(a). But then since w |= D∀xDEx) we have w |= DFa, and hence thatw |= ∃xDFx. Since w |= D∀xD(Fx → DFx), w |= D∃xDFx. That means thatw |= ¬D¬D∃xDFx. Suppose, then, that w |= ∃xDFx. Then there is b ∈ D(w) suchthat w |=DFb. Now let w be RD-accessible from w. Then we have w |= Fb andsince b ∈ D(w) we havew |= ∃xF. But that means thatw |=D∃xF, contradictingthat |= ∇∃xF. Hence u 6|= ∃xDF, giving us our counterexample to B. �

. A bias towards existence (and determinacy)?

If there are cases of vague existence and B indeed fails, it might seem as if incases of vague existence there is a bias towards existence. It might be vaguewhether a certain type of object exists, but it can never be vague, of a givenobject, whether it exists. And one might worry that this makes indeterminacywell-nigh impossible.

For consider again the case of particularized properties. Suppose, again,that Bob is bald. Then there exists his baldness. His baldness determinatelyexists. Moreover it’s determinate that if his baldness exists then Bob is bald.But that seems to ensure that Bob is determinately bald. And thus, since Bobis a borderline case, that Bob isn’t bald. But if we are able to prove that Bobisn’t bald, how can he then be a borderline case? Indeed, the situation is evenworse. For suppose that Bob isn’t bald, then there exists his non-baldness, andan exactly similar argument would show that Bob is bald. Contradiction.

We have to block this reasoning! What we have to do is to note that there is ascope-ambiguity in the talk of Bob’s Baldness. While it is determinate that Bob’sBaldness is the particularization of the property that is the result of precisifying“bald” in the way in which we did to get “Bob is bald” to come out true, it’snot determinate that Bob’s Baldness is the particularization of a the (possiblydifferent) property that is the result of precisifying bald in a different way.

We can introduce scope operators ↑,↓ to help us mark this distinction. Wenow define truth relative to a world, an assignment and a sequence of worlds σ .

• w,σ |=↑ φ iff w,σ ∗ 〈w〉 |= φ

• w,〈 〉 |=↓ φ iff w〈 〉 |= φ

• w,σ ∗ 〈u〉 |=↓ φ iff u,σ |= φ

In order to get a model of the above situation we introduce the second-levelpredicate P (x,B) which is to be read as saying that x is the particularization of

Baldness with respect to Bob. Now letMmodels that Bob is borderline bald. Inevery world w ofM, if Bob is bald according to w, we introduce one and onlyone object bw such that P (bw,B); and we make sure that bw , bw′ if w = w′ .

In such a model we have all of:This models that the vagueness of “being Bob’s baldness” is perfectly corre-

lated with the vagueness of “Bob is bald”. Suppose that x is Bob’s baldness onone precisification of “bald”; on a precisification where “bald” gets a slightlydifferent meaning x no longer counts as Bob’s baldness.

Letting Bb be the statement that Bob is bald, in this model we have:

D(Bb↔∃xP (x,B)) ()

∀x ↑ (D ↓ (P (x,B))∨D ↓ ¬P (x,B)) ()

∇∃xP (x,B) ()

What we don’t have is:

∀x(D(P (x,B))∨D¬P (x,B)) ()

This is not enough to give us the counterexample to B. Now, the reasonthese scope distinction are forced on is because the vagueness of “being theparticularization of the property P ” and P are perfectly aligned. That meansthat we have to deny that what is the particularization of P determinately isthe particularization of P . Particularized properties have too rich essences. Theprecisification of “bald” affects the identity identity of what is to serve as theparticularized property of being bald.

Now, we don’t have something similar in the mereological cases. This isespecially clear in van Inwagen’s case. The precisification of “weighs more than lb.” does not affect the identity of the object the existence of which (at agiven time) is indeterminate. The constitution case can be given a similar spin.For one might think that “dromedary” isn’t vague; in that case the precisifi-cations of dromedary doesn’t affect the identity of the object the existence ofwhich (at a given time) is indeterminate.

These cheap cases of vague existence in the abstract therefore don’t afford usa counterexample to the Barcan Formula for determinacy. Nevertheless, thereseems to be a difference between these cases and the humdrum cases of vagueexistence discussed above. For consider the humdrum case of vague existenceabove.

For we might that if something isn’t the particularized property of Bob’sbeing bald, then it’s determinate that it isn’t the particularized property of Bob’sbeing bald. (And similarly for Bob’s non-baldness.) We have little inclination,however, to say that if the rock r isn’t part of Kilimanjaro it’s determinately notpart of the Kilimanjaro.

. S and D again

At this point we should stress the difference between S and D. For the aboveargument in favor of the Converse Barcan Formula doesn’t go through for S.We can have an object that neither S-determinately exists nor S-determinatelyfails to exist.

Unlike the claims that ∃x∇Ex, the claim ∃x(¬SEx ∧¬S¬Ex) is not prob-lematic. Indeed, the claim S∃x(¬SEx∧¬S¬Ex) is not problematic. It is onlyproblematic if we don’t remember how we should think of the S-claims. It’s notas if there is this thing which is in this suspended state between being and non-being. The claim that there exists something which is in an S-indeterminatestate of being just is the supervaluationist way of saying that the non-semanticfacts are consistent with several ways of precisifying existential quantifier.

This sheds some light on (some) lightweight conceptions of objects. Supposewhat it takes for there to be a dromedary in area A just is for matter to bein a certain way. But it’s vague exactly in what way the matter has to be inorder for there to be a dromedary there. Whether there is a dromedary there,or not, then is partly a semantic matter, and it isn’t settled by settling the non-semantic facts. It’s in that sense that we should interpret the claim that it’s notS-determinate that there is and not S-determinate that there isn’t a dromedaryin A. However, relative to having settled the semantic facts in such a way thatthere is a dromedary in A, it is determinate that that thing exists.

The distinction between S and D helps shed some light on Barnes’s ()account of how we can precisify the quantifiers. . Her account is supposed todovetail with a theory of metaphysical indeterminacy. Her idea is that whenwe precisify the unrestricted quantifiers the different precisifications are quasi-quantifiers. Roughly, a quasi-quantifier is something that would have been aquantifier if only the domain had been as the quasi-quantifier represents it asbeing. For definiteness, we might take these quasi-quantifiers to be differentsecond-order concepts. If there is genuine metaphysical indeterminacy, thenthere is no one unique domain, and so there are several equally good quasi-quantifiers.

I find Barnes’ idea quite congenial, but given my acceptance of the ConverseBarcan Formula for D I cannot accept it as it stands. For on Barnes’s proposalit’s not determinate that there isn’t objects that don’t determinately exist. Never-theless, if couched in terms of S, I’m quite willing to accept Barnes’s proposal.Per the above, the claim that there is an object that does not determinately exist,should not be taken as a claim about a particular object. Rather, it should just betaken to be the supervaluationist expression of the vagueness of the existentialquantifier.

There is an interesting parallel between the difference between S and D and the difference be-tween metaphysical necessity and actual metaphysical necessity. Salmon’s () counterexamplesto the necessity of necessity are directed at metaphysical necessity, and not actual necessity.

Variable domains: metaphysical modality

We’ve seen that if one wants to domesticate vague existence one needs objectswith quite rarefied modal properties. One might be tempted to appropriate theview of (Williamson, a, , , ; Williamson, ; and Linskyand Zalta, , ). These authors hold that everything exists necessarily:let’s call them necessitists. Their opponents, who hold that some things existonly contingently, we can call contingentists. In the cases where a contingentistwould say that an object exists only contingently, the necessitist will say thatthe object is only contingently concrete.

If one were to accept the necessitist position one has a general strategyfor domesticating vague existence. In the above mereological cases of vagueexistence one would say that what’s indeterminate is whether there is a concreteobject x which satisfies φ. The consequent of the Barcan Formula is thenunproblematic since there is an object x such that it is indeterminate whether xis concrete and satisfies φ.

There is no problem with these objects having the wrong type of essence. Inthe case of the dying dromedary, the object which is such that it is indeterminatewhether it is a concrete dromedary, is a dromedary! Indeed, it might as wellbe the dromedary itself. An advantage of this approach is that it is completelygeneral; moreover, the ontology isn’t motivated by the desire to domesticatevague existence.

Which combinations of fixed versus variable domains along RD and R� arepossible. Well, they are all possible: any combination of Variable/constantdomains along RD can be combined with any combination of Variable/Constantdomains along R�. That being said, there does not seem to be any plausibleview taking the domains to be fixed along RD but variable along R�. For thisamounts to taking all vague existence to be domesticated, and it’s hard to getthe right types of domesticating objects without invoking something like thecontingently non-concrete.

Of the remaining views, which should be accepted? My personal view isthat we should allow variable domains along R�. Irrespective of that, I thinkthat a strong case can be made that we have to allow to allow variable domainsalong RD . There are limits to domestication.

. The limits of domestication

It’s worth noting, first, that there is a sense in which most vague existence isdomesticated for the contingentist too. For even though there might not beanything such that it’s indeterminate whether it is F, it might nevertheless be

In the sense that it is a dromedary when concrete.To see that we can have constant domains along R� while having variable domains along RD

just take a model where we have a failure of the Barcan Formula for D, and then make that worldonly have R� access to itself.One could of course help oneself to “just enough” of the contingently non-concrete to help with

domestication; but such a view is clearly unmotivated.

possible that there is something so that it’s actually indeterminate whether it isF. Admittedly, in the presence of determinacy operators it’s not entirely clearhow to deal with a standard actuality operator; what, after all, is the actualworld? Fortunately, the scope-indicating devices ↑,↓ introduced in section .above will serve here as well.

The contingentist domesticating principle is then:

∇∃xφ→↑^∃x ↓ ∇φ ()

In order to ensure that our models satisfy this principle we have to impose thefollowing condition.

• If uRDv and a ∈ D(v) then there is w, uR�w with a ∈D(w).

But there are, perhaps, limits to domestication. Suppose one holds the viewthat sets have their members determinately and necessarily and that sets existnecessarily. Is this compatible with its being vague what sets there are? If itwas we would have the most undomesticated form of vagueness in existence:for in such cases we not only don’t have something such that it’s indeterminatewhether it is the relevant set, but it’s also impossible for there to be somethingsuch that it’s actually indeterminate whether it is this set.

There are two ways in which such an indeterminacy could come about. First,one might think that it’s vague how “high” the hierarchy of sets extends. In thiscase it might be vague whether there is, say, a measurable cardinal, but thereplausible isn’t anything such that it is vague whether it is a measurable cardinal.

Against this idea, however, a Williamsonian necessitist may have a reply.Williamson (b) has argued that there are several almost indistinguishableset-like concepts, and that what happens in a case like this is that a putativemeasurable cardinal counts as a “set” on some of these concepts and not others.In such a case one at least has a story allowing one to deny that everything iseither determinately a set or determinately not a set.

But one can also hold—or at least seriously entertain the possibility of(Koellner, )—that the hierarchy is indeterminate as to “width”. One might,e.g., hold that our concept of set is indeterminate between two conceptions—onone conception of set CH holds, on another CH doesn’t hold. It seems to me thatthe Williamsonian story is even less plausible in this case. For the indeterminacywould here be over what onto functions from the natural numbers to sets ofsets of natural numbers there are, and what would something such that it’sindeterminate whether it is a function from the natural numbers onto a set ofsets of natural numbers be?

We have to take seriously the idea that there is undomesticable vagueness inexistence.

Conclusions

I hope to have persuaded the reader that many issues in metaphysics turn on theinterplay between determinacy and metaphysical modality. My hope is that the

present framework will prove useful in studying these matters. Let me indicateone avenue of further research. Sider’s indeterminacy argument argues againstthe idea that one can precisify unrestricted quantifiers by precisifying thedomain. I mention that quantifiers shouldn’t be identified with their domains,but should rather be thought of as functions from concepts to truth-values.The place to study whether an unrestricted quantifier is determinately uniquewould be in a higher-order (at least third order) logic. For in such a logic onecould quantify over possible quantifiers and one might be able to give preciseexpression to the claim that any two unrestricted quantifiers are identical.

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