If you have two similar knives, each consisting of ahandle and a blade, you can imagine a knife consistingof the handle of the knife on the right and the blade ofthe knife on the left. Even though no such knife is evermade, there seems to be a particular possible knife thatyou are imagining, because your description is specificenough to answer the question ‘Which possible knifeare you thinking about?’. You might even give it aname. If there are merely possible knives, many otherkinds have merely possible members too. What placeis there, if any, for merely possible Fs? Until we havean adequate answer to the question, we shall notproperly understand the nature of possibility and itsdual, necessity. Even the simplest logical characteristicsof the phrase ‘possible F’ are often misunderstood.

The concatenation ‘possible F’ is structurallyambiguous. On one reading, ‘possible diamond’ isanalogous to ‘South African diamond’. The latter isnaturally understood as ‘diamond which is SouthAfrican’; ‘possible diamond’ would then be understoodas ‘diamond which is possible’. Thus ‘x is a SouthAfrican diamond’ is equivalent to ‘x is a diamond andx is South African’; ‘x is a possible diamond’ wouldbe equivalent to ‘x is a diamond and x is possible’. Callthat the

predicative reading. On another reading,‘possible diamond’ is analogous to ‘alleged diamond’.The latter is naturally understood as ‘item such that itis alleged that it is a diamond’, not as the ill-formed‘diamond which is alleged’; ‘possible diamond’ wouldthen be understood as ‘item such that it is possible thatit is a diamond’ (by contrast, ‘item such that it is SouthAfrican that it is a diamond’ is ill-formed). Thus ‘x isan alleged diamond’ is equivalent to ‘it is alleged thatx is a diamond’; ‘x is a possible diamond’ would beequivalent to ‘it is possible that x is a diamond’. Callthat the attributive reading.

The attributive reading uses the modal sentencefunctor ‘it is possible that’, which as usual we formalize

by

e. In this context we read ‘possible’ as ontic ratherthan epistemic; thus ‘it is possible that’ is equivalentto ‘it is or could have been the case that’. TheAristotelian-sounding phrase ‘potential F’ conveyssomething of this reading. But what should we makeof ‘x is possible’ in the predicative reading? The naturalsuggestion is that, as an adjective of individuals,‘possible’ means possibly existent: ‘x is possible’ isequivalent to ‘it is possible that x exists’. With anexistence predicate E we therefore formalize ‘x ispossible’ as eEx. No tacit assumptions will be madeabout the interaction of E with the quantifiers that wouldnot be equally true of any other predicate.

We can now distinguish the two readings formally:

x is a possiblepredicative F =df Fx & eEx

x is a possibleattributive F =df eFx

Since the actual is possible, every possiblepredicative F isa possibleattributive F, independently of the second conjunctin the definition and the value of ‘F’. But unless every-thing that could have been an F is an F, not everypossibleattributive F is a possiblepredicative F. For example, ifyou could have won the lottery but did not in fact doso, you are a possibleattributive lottery winner without beinga possiblepredicative lottery winner.

On both readings, ‘At least one possible F exists’follows from ‘At least one F exists’. For we can derive‘There is at least one possiblepredicative F’ and ‘At leastone possibleattributive F’ from the latter by prefixing e to‘exists’ and ‘F’ respectively. Since at least one catexists, at least one possible cat exists. Thus, for manyvalues of ‘F’, many possiblepredicative Fs and manypossibleattributive Fs uncontroversially exist. To reach deepwater, we must consider the merely possible.

‘It is merely possible that P’ is equivalent to theconjunction of the negation of ‘P’ with ‘It is possiblethat P’. We can replace ‘P’ by ‘x exists’ and ‘x is an F’

The Necessary Framework of Objects* Timothy Williamson

Topoi

19: 201–208, 2000. 2000 Kluwer Academic Publishers. Printed in the Netherlands.

for the predicative and attributive readings respectively.Thus the two readings of ‘x is a merely possible F’corresponding to the two readings of ‘x is a possible F’are these:

x is a merely possiblepredicative F =df Fx & ¬Ex & eEx

x is a merely possibleattributive F =df ¬Fx & eFx

Thus possiblepredicative Fs are either existent Fs or merelypossiblepredicative Fs; possibleattributive Fs are either Fsor merely possibleattributive Fs. Since ‘x is a merelypossiblepredicative F’ is the conjunction of Fx and ¬Ex &eEx, the latter of which we can express as ‘x is merelypossiblepredicative’, ‘merely possiblepredicative F’ is simply therestriction of F to a fixed type of possibilia, the merelypossiblepredicative objects, those that contingently fail toexist. By contrast, no class of possibilia fixed indepen-dently of the value of ‘F’ plays a role in the definitionof ‘merely possibleattributive F’; the definition uses nopredicate or other expression for existence.

Trivially, nothing is both a possiblepredicative F and amerely possibleattributive F, for by definition every possi-blepredicative F is an F and no merely possibleattributive is anF. A fortiori, nothing is both a merely possiblepredicative Fand a merely possibleattributive F. For a given value of ‘F’,we have two quite different theses to consider:

(1) There is at least one merely possiblepredicative F.(2) There is at least one merely possibleattributive F.

For many values of ‘F’, (2) is reasonably uncon-troversial. For example, if you could have won thelottery but did not in fact do so, you are a merelypossibleattributive lottery-winner, so there is at least onemerely possibleattributive lottery-winner. Indeed, since youexist, at least one merely possibleattributive lottery-winnerexists. By contrast, (1) is not uncontroversially true forany value of ‘F’. For a merely possiblepredicative F is bydefinition nonexistent; thus each instance of (1) entailsthat there is at least one nonexistent, contradicting theQuinean “actualist” doctrine that everything exists. Forthe same reason, (1) cannot consistently be strengthenedto the claim that at least one merely possiblepredicative Fexists. The defender of (1) must make sense of quan-tification over nonexistent items. In contrast, for manyvalues of ‘F’, the defender of (2) is not as such com-mitted to any special view about the nature of existence(although a defender of (2) is committed to quantifica-tion over the nonexistent if ‘F’ is ‘existent’). Forsome values of ‘F’ neither (1) nor (2) is defensible:

since there are no round squares, there are no merelypossiblepredicative round squares, and since there could nothave been any round squares, there are no merelypossibleattributive round squares.1 Our interest is in philo-sophical strategies that invoke some instances of (1) or(2), not the wild idea that (1) or (2) is valid withoutrestriction on ‘F’.

We can develop the contrast between merelypossiblepredicative Fs and merely possibleattributive Fs byexploring their use in defending the Barcan formulaagainst apparent counterexamples. We take the formulain this form:

(BF) e

∃xFx

⊃ ∃xeFx

Informally, if there could have been an F then there issomething that could have been an F. From a formalperspective, (BF) plays an attractively simplifying rolein quantified modal logic and semantics (Cresswell,1991; Williamson, 1990, 1998). Unfortunately, itappears to have counterintuitive consequences.

For an apparent counterexample to (BF), read Fx as‘x is a golden mountain’. We may legitimately assumethat no past, present or future golden mountain existsanywhere, and that a golden mountain could haveexisted somewhere. Therefore, according to (BF), thereis something that could have been a golden mountain.But what is it? No doubt there is a mereological sumof gold atoms that could have constituted a goldenmountain; still, the claim that that mereological sumcould have been identical with a golden mountain (asrequired by the consequent of (BF)) implies an implau-sibly reductive view of the metaphysics of mountains.Mereological sums of atoms survive scattering; moun-tains do not. Equally, we can assume that no actualmountain could have been golden (if need be, let‘golden’ mean always made entirely of gold). No actualmountain could have had a constitution and history sodifferent from its actual constitution and history. Sincean initial survey produces no plausible candidate, on theface of it there is nothing that could have been a goldenmountain. If so, that instance of (BF) is false.

Would the relevant instance of (1) help? A merelypossiblepredicative golden mountain would be a goldenmountain that contingently failed to exist. It wouldmake the consequent of (BF) true; indeed, (1) impliesthat something not only could have been but is a goldenmountain. The obvious problem with this claim is thatit is inconsistent with the assumed fact that there areno golden mountains. There are no special mountains,

202 TIMOTHY WILLIAMSON

golden or otherwise, with the property of contingentlyfailing to exist. Nothing can literally be a mountainwithout being in some place at some time in theordinary way; to think otherwise is to lose one’s grip onwhat it is to be a mountain.2 Someone could tell a storyaccording to which there was a golden mountain; wemight even say that they had created a fictionalmountain, but a fictional mountain is not a mountainof a special kind. Fictional characters are cultural arti-facts of a special kind, like poems, representations of asort, although of course it is not usually part of the storythat the characters in it are such cultural artifacts (seevan Inwagen, 1977 and Thomasson, 1999). As a culturalartifact, the fictional golden mountain originates byhuman agency when and where the story is first told(more or less), even if, according to the story, the goldenmountain originated without human agency a millionyears ago on Mars. Since fictional mountains arenot mountains, the predicative reading of ‘fictionalmountain’ as ‘mountain which is fictional’ is incorrect.3

However widely we quantify, there is no goldenmountain. This difficulty with the particular instanceof (1) goes beyond that of making sense of the quan-tification over nonexistent items intrinsic to any instanceof (1).

Quite generally, if there is a possiblepredicative Fwhenever there could have been an F then not only (BF)but the much stronger principle e∃xFx ⊃ ∃xFx is true.Since the converse is automatic, the distinction betweenpossibility and actuality collapses for ‘there is an F’.In particular, it collapses for ‘there is an existent F’ bythe instance e∃x(Ex & Fx) ⊃ ∃x(Ex & Fx). Indeed, thedistinction collapses more generally. For given asentence p, the schema e∃xFx ⊃ ∃xFx has the specialcase e∃x(x = x & p) ⊃ ∃x(x = x & p), which entailse∃x(x = x & p) ⊃ p. Thus whatever could have beenso when there was something actually is so. The non-emptiness qualification will usually be compatible withp when ep is true; since the friends of (1) are com-mitted to anti-actualist quantification, they will probablyhold that it is always met, in which case the modalcollapse is perfectly general. Be that as it may, theschema e∃x(x = x & p) ⊃ p has many clearly falseinstances. For example, although there could have beensomething such that it was self-identical and Cavourdied in 1880, Cavour did not die in 1880. Thus a generaldefence of (BF) invoking (1) whenever the antecedentof (BF) is true is hopeless. By invoking enoughpossiblepredicative Fs to validate (BF) as a schema, one not

only commits oneself to an obscure and implausibleontology but obliterates virtually all modal distinctionsof interest.

Would a merely possibleattributive F be of more aid tothe defence of (BF)? The consequent of (BF) says pre-cisely that there is a possibleattributive F. If ∃xFx is true,(BF) is unproblematic because both its antecedent andconsequent are true; we can insert e before or after thequantifier. Given that ∃xFx is false, the consequent of(BF) says in effect that there is a merely possibleattributive

F, and (BF) is equivalent to the claim that if there couldhave been an F although there is not an F then (2) holds.Thus merely possibleattributive Fs are just what (BF) needs.But how plausible is it that there are such things?

Since there is no golden mountain, (BF) needs amerely possible golden mountain: something that is nota golden mountain, but could have been. Space and timeappear to contain no such thing. What remains open isthat there is a merely possibleattributive golden mountainnot in space or time. Such a thing would not violate theprinciple that necessarily every mountain is in space andtime, for being a merely possibleattributive golden mountainis compatible with not being a mountain at all. A merelypossibleattributive golden mountain could have been agolden mountain, but then it would have been in spaceand time. It contingently possesses the properties of notbeing a golden mountain and of not being in space andtime. If it had been a golden mountain, it would havecontingently possessed the properties of being a goldenmountain and of being in space and time.4 Thusdefending (BF) this way involves denying some essen-tialist claims (Parsons, 1995). But not all: it does notthreaten the claim that necessarily the conditionalproperty of being a golden mountain if in space and timeis an essential property of any possibleattributive goldenmountain.5

The invocation of merely possibleattributive Fs maysound reminiscent of David Lewis’s modal realism(1986). According to the modal realist, since there couldhave been a golden mountain, we can truly say (with aquantifier not restricted to inhabitants of our ownworld): there is a golden mountain in some world(roughly: in some space-time system). Since there isno golden mountain in our world, there is a goldenmountain in some other world. More simply, we canomit the information about worlds and truly say (withthe unrestricted quantifier): there is a golden mountain.By contrast, if one underpins (BF) with merelypossibleattributive Fs, one can consistently assert that there

THE NECESSARY FRAMEWORK OF OBJECTS 203

is a merely possibleattributive golden mountain and denythat there is a golden mountain, however unrestrictedlyone quantifies. Unlike modal realism, that has notendency to assimilate what could have been so to whatis so elsewhere. The modal realist’s merely possible Fsshould be understood as merely possiblepredicative Fs, withthe predicate E restricted to inhabitants of the currentworld. But the modal realist is not committed to allinstances of the disastrous schema e∃xFx ⊃ ∃xFx evenwhen the quantifier is not tacitly restricted by E, forsome values of ‘F’ will contain world-relative elements.Lewis’s own system of counterpart theory invalidates(BF) (1983, p. 36). To validate (BF), one needs merelypossible Fs in a more modest sense than Lewis’s: merelypossibleattributive Fs.

The question will still be asked: what kind of thingis a merely possibleattributive golden mountain? Thequestion assumes that ‘possibleattributive golden mountain’is not itself an adequately informative kind term.Why not? It carries some information; presumablypossibleattributive golden mountains are not possibleattributive

silver valleys. The questioner may respond ‘I want toknow what it is, not what it could be’. A nonmodal kindis wanted. But we should not assume without argumentthat everything can be classified into informativenonmodal kinds.

Consider an analogy: the attributive reading of ‘pastF’. On this reading, a past president is not somethingthat is president and once existed, but something thatonce was president. A past mountain, reduced to dustby erosion over millions of years, is no longer amountain. It counts as a past mountain when one istrying to work out how many mountains there have everbeen; it does not count as a mountain when one is tryingto work out how many mountains there are. Someonemight ask: what kind of thing is a past mountain? Toanswer by reference to traces left in the present wouldbe an error. Whether a past mountain has left anypresent traces at all is a matter of geological fact; anegative answer would make no difference to how manymountains there have ever been. Even if there are traces,they need not be the past mountain itself. The crucialproperties of a past mountain are its past properties. Toknow what it is, one must know what it was. It is some-thing that once had the properties of a mountain. If onetries to describe it without reference to the past, one mayfind little to say: it is self-identical, it lacks spatiallocation, it is not a mountain. What distinguishes a pastmountain m1 from a distinct past mountain m2? If one

is forbidden to refer to the past, perhaps one can sayonly that they are distinct. Once one is allowed to referto the past, one can say e.g. that for some place p andtime t, m1 was a mountain in p at t and m2 was not. Thatpast difference between m1 and m2 is not somehowgrounded in a qualitative difference between them speci-fiable without reference to the past. The past is notgrounded in the present.

We cannot assume that something goes for thepossible because it goes for the past; we know that theanalogy eventually breaks down. But we can use it toconstruct a hypothesis about the nature of possibleattributive

Fs. On that hypothesis, the crucial properties of a merelypossibleattributive mountain may be its modal properties.To know what it is, one must know what it could havebeen. It is something that could have had the proper-ties of a mountain. If one tries to describe it withoutreference to the merely possible, one will find little tosay: it is self-identical, it lacks spatial location, it is nota mountain. What distinguishes a merely possibleattributive

mountain m1 from a distinct merely possibleattributive

mountain m2? If one is forbidden to refer to the merelypossible (to use modal notions), perhaps one can sayonly that they are distinct. Once one is allowed to referto the merely possible, one can say e.g. that for somecontingent circumstance C, place p and time t, if C hadobtained then m1 would have been a mountain in p att and m2 would not. That counterfactual differencebetween m1 and m2 is not somehow grounded inqualitative differences between them specifiable withoutuse of modal notions. Thus in some sense the modal isnot grounded in the nonmodal, contrary to some currentpreconceptions about the relation between possibilityand actuality.

In just what sense is the envisaged defence of (BF)inconsistent with the grounding of the modal in thenonmodal? Such grounding would be a form of super-venience. Say that the modal globally supervenes on thenonmodal iff for all possible worlds w and w*, if w isexactly like w* in all nonmodal respects then w isexactly like w* in all modal respects too (every modaldifference between worlds is grounded in a nonmodaldifference). The global supervenience of the modal onthe nonmodal is consistent with the defence of (BF). Forthe latter is consistent with the assumption that allpossible worlds are exactly alike in all purely modalrespects, in which case the supervenience is trivial.

Any failure of grounding is at the level of individ-uals rather than worlds. Say that the modal locally

204 TIMOTHY WILLIAMSON

supervenes on the nonmodal iff for all possible worldsw and w* and individuals i and i*, if i in w is exactlylike i* in w* in all nonmodal respects then i in w isexactly like i* in w* in all modal respects too (everycross-world modal difference between individuals isgrounded in a nonmodal difference). Even the localsupervenience of the modal on the nonmodal isconsistent with the defence of (BF). For we can regarddifference with respect to an identity property such asbeing Napoleon as itself a failure of exact likeness in anonmodal respect, on the grounds that the property canbe expressed without use of modal operators. Thus i inw is exactly like i* in w* in all nonmodal respects onlyif they are alike with respect to the property of beingi*, and therefore only if i is i* (on the assumption thatsuch identity properties are noncontingent); then i in wis exactly like i in w* in all modal respects unless i hassome purely modal property contingently; the defenceof (BF) is consistent with the denial that individuals canhave purely modal properties contingently.

Although we can count being Napoleon as a property,we do not count it as a general property. Consequently,we do not regard a difference with respect to such aproperty as itself a failure of exact likeness in a generalnonmodal respect. The general modal locally superveneson the general nonmodal iff for all possible worlds wand w* and individuals i and i*, if i in w is exactly likei* in w* in all general nonmodal respects then i in w isexactly like i* in w* in all general modal respects too(every general cross-world modal difference betweenindividuals is grounded in a general nonmodal differ-ence). On the envisaged defence of (BF), the generalmodal does not locally supervene on the generalnonmodal.6 Presumably, for instance, Edinburgh isnecessarily a possibleattributive city and Napoleon isnecessarily not a possibleattributive city. Thus for nopossible world w is Edinburgh in w exactly likeNapoleon in w in all general modal respects. But in aworld w in which Edinburgh is a merely possibleattributive

city and Napoleon is a merely possibleattributive person,neither having any spatiotemporal location, Edinburghin w may be exactly like Napoleon in w in all generalnonmodal respects. They share all such generalnonmodal properties as being self-identical, lackingspatio-temporal location, not being a city and not beinga person; they can even share all general nonmodalintentional properties such as never being thought of.7

Some general modal differences between individualsare not grounded in general nonmodal differences,

although all such differences are grounded in general ornongeneral nonmodal differences. Once we have dis-tinguished these different senses in which the modalmay or may not be grounded in the nonmodal, wecannot dismiss the defence of (BF) by appeal to theintuition that in some sense the modal is grounded inthe nonmodal, for the defence grants several senses inwhich that is so. Only a more discriminating objectionwould damage the defence of (BF); how such anobjection would go is far from clear.

We can make these ideas precise in the framework ofa possible worlds model theory for quantified modallogic. The object-language is the result of adding thenecessity operator h to a standard first-order languagewith identity (=) but without names or functionalexpressions; ¬, & and ∀ are primitive; other connec-tives are introduced as metalinguistic abbreviations inthe usual way. In particular, e is ¬e¬. A model is atriple ⟨W, D, V⟩; intuitively, W is the set of possibleworlds, D the domain of individuals and V the functionmapping each predicate letter F to its intension VF.V interprets = normally: for all w ∈ W, V=(w) is{⟨i, i⟩: i ∈ D}. Truth at a world w relative to an assign-ment a of members of D to variables (truew, a) is definedby a standard recursion. If F is an n-place predicateletter and v1, . . . , vn are variables, Fv1 . . . vn istruew, a iff ⟨a(v1), . . . , a(vn)⟩ ∈ VF(w). The clauses for¬ and & are obvious. For any formula A, ∃vA istruew, a iff for all i ∈ D A is truew, a[v:i], where a[v:i] isthe assignment like a except that it assigns i to v. hAis truew, a iff for some w* ∈ W A is truew*, a. A is validiff given any such model ⟨W, D, V⟩, for every w ∈ Wand assignment a, A is truew, a. Since the domain ofquantification is invariant between worlds, all instancesof (BF) and its converse are valid. Since necessity isdefined as truth at all worlds, without restriction by anaccessibility relation, all theorems of the modal systemS5, in particular all formulas of the forms hA ⊃ hhAand eA ⊃ heA, are valid. The set of all valid formulasis the modal system LPC=S5 (on which Williamson1998 has more details and further references).

The formula A is nonmodal iff h does not occur inA.8 A is purely modal iff every occurrence of a predi-cate letter (including =) in A is within the scope of anoccurrence of h. That is, the class of purely modalformulas is the smallest class containing all formulas ofthe form hA and closed under ¬, & and ∃v. Allformulas are in the closure of the nonmodal and purelymodal formulas under ¬, & and ∃v. Consider any model

THE NECESSARY FRAMEWORK OF OBJECTS 205

⟨W, D, V⟩. A world w ∈ W is exactly like a worldw* ∈ W in respect of a formula A iff for every assign-ment a, A is truew, a iff A is truew*, a. If A contains freevariables, that requires w to match w* in the relevantrespect for each assignment to those variables; thiscompensates for the absence of names from thelanguage. One can easily check that any two worlds areexactly alike in respect of any formula hA, and that anytwo worlds exactly alike in respect of A and B are alsoexactly alike in respect of ¬A, A & B and ∃vA. Thusany worlds w and w* are exactly alike in respect of anypurely modal formula. Consequently, if w and w* areexactly alike in respect of every nonmodal formula, theyare exactly alike in respect of every formula. In thissense, the modal globally supervenes on the nonmodalin any given model.

Now consider local supervenience. An individuali ∈ D in w ∈ W is exactly like an individual i* ∈ D inw* ∈ W in respect of A iff for every assignment a, Ais truew, a[x:i] iff A is truew*, a[x:i*]. Roughly: we treat A asa predicate of x, and any other free variable in A as aparameter; the idea is that A is true of i in w iff it istrue of i* in w* for all such parameters. Suppose that iin w is exactly like i* in w* in respect of everynonmodal formula. Then in particular i in w is exactlylike i* in w* in respect of x = y; so where a is an assign-ment such that a(y) is i*, x = y is truew*, a[x:i*]; thusx = y is truew, a[x:i], so by the truth definition i is i*. Butthen a[x:i] is a[x:i*]; since w is exactly like w* inrespect of every purely modal formula, i in w is exactlylike i* in w* in respect of every purely modal formula.Thus if i in w is exactly like i* in w* in respect of everynonmodal formula, i in w is exactly like i* in w* inrespect of every formula. In this sense, the modal locallysupervenes on the nonmodal in any given model.

To formalize the idea of a general respect, we forbidparameters by considering formulas in which only xoccurs free. To show that the general modal need notlocally supervene on the general nonmodal, we want amodel ⟨W, D, V⟩, worlds w, w* ∈ W and individualsi, i* ∈ D such that i in w is exactly like i* in w* inrespect of every nonmodal formula in which only xoccurs free, but i in w is not exactly like i* in w* inrespect of some purely modal formula A in which onlyx occurs free. For a simple example, let the languagecontain just one predicate letter other than =, themonadic F. Let W consist of just two worlds w and w*and D of just two individuals i and i*. Let VF(w) andVF(w*) be {i} and {i} respectively. Then i in w is

exactly like i* in w in respect of every nonmodalformula in which only x occurs free, but since eFx istruew, a[x:i] yet not truew, a[x:i*], i in w is not exactly like i*in w in respect of every purely modal formula in whichonly x occurs free. In this sense, the general modal doesnot locally supervene on the general nonmodal in themodel. Of course, that supervenience does obtain insome models, for example, in the model like the onejust considered except that VF(w*) is {i, i*}. But modelsof the former kind are closer to the interpretationintended here.

Naturally, the formalizations will not achieve theirintended effect if some predicate letters are interpretedas modal or as nongeneral, or the expressive power ofthe language is too limited. But the positive super-venience results hold for all models, including those inwhich all predicate letters are interpreted as general andnonmodal, and the language is expressively adequatefor the purpose at hand. Of course, the clarity of thegeneral-nongeneral and modal-nonmodal distinctions isitself questionable. But their unclarity is really aproblem for opponents of the (BF)-based conception.For the distinctions are needed to articulate thesupervenience claim that the conception violates, notto articulate the conception itself.

In LPC=S5, objects form the necessary frameworkof the world in a precise sense. A formula is almostpurely modal iff every occurrence of any predicate letterexcluding = in it is within the scope of an occurrenceof h. Then A ⊃ hA is valid for any open or closedalmost purely modal formula A. In particular,∀xhA ⊃ h∀xhA is valid; thus the Barcan formula inthe form ∀xhA ⊃ ∀xhA (from which (BF) follows)is a consequence of such rigidity principles. The samecan be shown for its converse. For objects, there is nocontingency in whether they are, what they are identicalwith or distinct from, what properties and relations theycan or cannot have. Contingency has its place withinthat rigid framework.

As Kripke (1963) showed, one can invalidateinstances of (BF) and its converse by relativizing thedomain of quantification to the world of evaluation. Butthe resulting explanation of how (BF) could fail in theactual world w@ is not philosophically satisfying, forpart of it is that some individual in the domain of someworld is not in the domain of w@. That requires thedomain of the meta-language quantifier to containindividuals not in the domain of the object-languagequantifier, so that the latter is not being read as

206 TIMOTHY WILLIAMSON

unrestricted. But what matters is whether (BF) is validwhen the object-language quantifier is read asunrestricted. Opponents of (BF) might therefore resortto a modal meta-language for quantified modal logic (asin Peacocke, 1978) in an attempt to explain how (BF)could fail on an unrestricted reading of the quantifier.What remains, however, is less an explanation than atransfer of the assumed failure of (BF) in the meta-language to its claimed failure in the object-language.That does not make a compelling case against (BF).Although the critic can still attempt to cite intuitivecounterexamples to (BF), the conception ofpossibleattributive Fs developed above provides a system-atic response to such examples.

The critic of (BF) may point to a difference betweenthe modal and temporal cases. It is usually much harderto make singular reference to a merely possibleattributive

F than to a past F. The definite description ‘the merelypossibleattributive golden mountain’ is presumablyimproper, because there are infinitely many differentpossibleattributive golden mountains. We can single out apast mountain with spatio-temporal co-ordinates; wehave no comparable system of co-ordinates for the spaceof possibilities. The example at the beginning of themerely possibleattributive knife consisting of what isactually the handle of the knife on the right and whatis actually the blade of the knife on the left suggests thatin principle we can name at least some such objects.The critic may object that the example depends on thepossible constitution of the possibleattributive knife froman actual handle and an actual blade. Unless thecritic is attempting an implausible equation of thepossibleattributive knife with something like the set ormereological sum of the handle and the blade, theobjection seems to concede the envisaged defence of(BF) in the particular case but to question its generality.However, once it has been conceded that there are someobjects of the kind needed for the defence of (BF), itis unclear why we should not provisionally adoptLPC=S5 as the simplest systematic theory about suchobjects, and use it to answer our questions about them– as it does in favour of (BF). Perhaps the critic holdsthat every object is a potential object of singular refer-ence, and suspects that the envisaged defence of (BF)requires the postulation of objects that are not poten-tial objects of singular reference. That suspicion wouldnot be easy to verify. For even if necessarily no merelypossibleattributive F is named (de dicto), it does not followthat, for all x, if x is a merely possibleattributive F then

necessarily x is not named (de re). For x could havebeen an F; if it had been, it would not have been amerely possibleattributive F; circumstances in which x is anF may contain no obstacle to naming x. It would notbe very strange if a contingent property (such as beinga merely possibleattributive F) were incompatible with theproperty of being named; after all, the contingentproperty of being nameless is incompatible with beingnamed. In any case, the critic’s assumption that everyobject is a potential object of singular reference is notcompelling. There is no obvious incoherence in thesupposition that some object x is essentially elusive, inthe sense that it is metaphysically impossible to makesingular reference to x.9 Have we not just made singularreference to it, using the term ‘x’? No; ‘x’ was used asa bound variable, not a name (compare Berkeley on theunconceived tree). Similarly, the terms ‘m1’ and ‘m2’were used earlier as bound variables over merelypossibleattributive mountains, not as names of particularmerely possibleattributive mountains. The requirement onan object is not that it be a potential bearer of a namebut that it be an actual value of an individual variable.There is a subtler connection between the category ofobjects and the category of names: an individualvariable is a variable that takes name position.

A full defence of the modal system LPC=S5 wouldrequire consideration of many more issues than havebeen raised here (some are discussed in Williamson,1998, 1999). The elucidation of the concept of apossible F is a necessary rather than sufficient condi-tion of that defence. It allows us to understand how theBarcan formula could be correct.

Notes

* Thanks to Alexander Bird and Alberto Voltolini for helpfulcomments.1 It is not denied that something might encode the properties ofbeing round and being square in the sense of Zalta (1988).2 Even if the (obvious) point were denied, there would still be theinstance of (BF) with ‘existent mountain’ in place of ‘mountain’ toconsider.3 Perhaps the predicative reading of ‘fictional mountain’ could bedefended under the hypothesis that ‘mountain’ receives a nonstan-dard meaning in this context; that would not undermine the logicaland metaphysical claims in the text. The attributive reading ‘itemsuch that it is fictional that it is a mountain’ is not obviously correct,for there is rarely a plausible candidate to be that item.Representations are not themselves such candidates; the willingsuspension of disbelief does not seem to involve pretending of

THE NECESSARY FRAMEWORK OF OBJECTS 207

representations that they are not representations but for examplemountains. A reading of ‘fictional mountain’ by analogy with‘representation of a mountain’ (understood nonspecifically, of course,not as ‘item such that there is a mountain of which it is a represen-tation) is equivalent to neither the predicative nor the attributivereading. Incidentally, such a representational reading of ‘possiblemountain’ itself is implausible, for it does not do justice to therequirement that a possible mountain could literally have been amountain; we have only that there could have been a mountain thatit represented.4 The argument for this claim depends on the Brouwerian principlep ⊃ hep (which corresponds in possible worlds semantics to thesymmetry of the accessibility relation). The principle is valid in themodal system LPC=S5 employed later in the text.5 Linsky and Zalta (1994, 1996) give a closely related defence of(BF).6 For present purposes it does not matter whether nongeneral modaldifferences locally supervene on general nonmodal differences; ifthey do not, that may reflect a wider failure of the nongeneral tosupervene on the general rather than anything about the relationshipbetween the modal and the nonmodal.7 The example shows that the general modal does not even weaklylocally supervene on the general nonmodal, weak local superveniencebeing defined like (strong) local supervenience except that w*replaces w throughout; thus crossworld differences between individ-uals are not considered. See Kim (1984) for more discussion.8 In response to Williamson (1998), and in particular the classifi-cation of the property of being the river Inn as nonmodal, WinfriedLöffler writes:

Then the following seems conceivable: An object x may be theInn in world1 and a possible river in world2, but it does not followthat it is the possible Inn in world2. And it seems to be possiblethat x is a river in world3, but e.g., the Mississippi (1998, p. 277).

He adds that this is not excluded by the necessity of identity anddistinctness (which hold in LPC=S5). Löffler may have misunder-stood what I mean by calling a property ‘nonmodal’. Being the Innis nonmodal because it is specifiable without use of modal notionssuch as possible. Possession of a nonmodal property has modalconsequences (most simply, Fx entails eFx). Names are rigiddesignators and ‘Inn’ and ‘Mississippi’ name distinct rivers; the Innis necessarily a possible river, the Inn and not the Mississippi(∀x((x = i & x ≠ m & Rx) ⊃ h(x = i & x ≠ m & eRx)), althoughnot necessarily a river (∀x ¬hRx).9 Presumably there is not exactly one essentially elusive object,otherwise we could use the definite description ‘the essentiallyelusive object’ to fix reference to it. Thus if there is at least oneessentially elusive object, there is more than one.

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The University of EdinburghDepartment of PhilosophyEdinburgh EH8 9JXUK

208 TIMOTHY WILLIAMSON