Parts, Counterparts, and Modal Occurrents
Achille C. Varzi
Department of Philosophy, Columbia University, New York

(Published in Travaux de logique14 (2001), 151–171)
Abstract. The paper investigates the link between the theory of modal occurrents (where individuals are allowed to stretch across possible worlds) and Lewis’s counterpart theory (where
all individuals are world-bound but have counterparts in other worlds). First I show how tointerpret modal talk extensionally within the theory of modal occurrents. Then I show thatthe assumption that worlds be pairwise discrete is all that is needed to reconstuct the bulk of
counterpart theory (i.e., to define the basic concepts and to derive the postulates governing
those concepts) in terms of the theory of modal occurrents. Finally, I argue that this reconstruction allows us to view the indeterminacy of our modal intuitions as being part and parcel
with the indeterminacy of our criteria for individuating modal occurrents, and that this indeterminacy is naturally explained in terms of linguistic (as opposed to ontic) vagueness.

Introduction
A modal occurrent is an individual entity that stretches across possible worlds.
It is a trans-world individual. It consists of parts that are located in different
worlds just as ordinary objects consist of parts that are located at different
places, or processes and other events (so-called temporal occurrents) consist ofparts that are located at different times. If we do not think that quantification
over possible worlds is intelligible, then the notion of a modal occurrent will beunintelligible to us. (Some philosophers feel that way about the notion of a temporal occurrent.) If we think that quantification over possible worlds is intelligible but that mereological fusions should be restricted to entities existing in thesame world, then to us the notion of a modal occurrent will be intelligible but
empty. (Three-dimensionalists feel that way about temporal occurrents, at least
insofar as these are meant to include objects besides events.) However, if wethink that quantification over possible worlds is intelligible and that mereological fusions should not be so restricted, then we are well off. For then wecan make sense of modal talk even if we cannot make sense of trans-world
identity.
1The purpose of this paper is to illustrate how this can be done. First I will
lay out the basic apparatus of the theory of modal occurrents. Next I will show
how to interpret modal talk extensionally within the apparatus regardless ofwhether trans-world identity is allowed, i.e., regardless of whether distinct
worlds can overlap. Then I will show that ruling out trans-world identity by requiring possible worlds to be pairwise discrete is all that is needed to reconstruct the bulk of counterpart theory (i.e., to define the concept of a counterpart
and to derive the standard postulates governing that concept). This will throw
some new light on counterpart theory itself. Finally, I will argue that the accountallows us to see the indeterminacy of our modal intuitions as being part andparcel with the indeterminacy of our criteria for individuating modal occurrents,
and that this indeterminacy is best explained in terms of semantic (as opposedto theoretical or ontic) vagueness.
Mereological Preliminaries
I will assume a standard mereological background constructed around the binary relational predicate ‘x is part of y’, written ‘Pxy’. 1 I take this predicate to
be true when x is any sort of part of y, including an improper part, so that Pxy
will be consistent with x and y being the same. The predicate for proper parthood, along with other familiar mereological predicates, can be introduced bydefinition in the usual way:
(1) PPxy =df Pxy ∧ ¬x=yx is a proper part of y iff x is a part of y but is distinct from y.
(2) Oxy =df ∃z(Pzx ∧ Pzy)x overlaps y iff something is part of both.
(3) POxy =df Oxy ∧ ¬x=yx properly overlaps y iff x overlaps y but is distinct from y.
As axioms I will assume those of standard extensional mereology, except forthe principle of unrestricted fusion. Thus, I will assume parthood to be a reflexive, antisymmetric, and transitive relation—a partial ordering—subject to astrong supplementation principle:2

(4) ∀xPxx
Everything is part (viz., an improper part) of itself.
(5) ∀x∀y(Pxy ∧ Pyx → x=y)No two things are part of each other.
(6) ∀x∀y(Pxy ∧ Pyz → Pxz)
The parts of a thing’s parts are themselves part of that thing.
(7) ∀x∀y(∀z(Pzx → Ozy) → Pxy)If every single part of x overlaps y, then x itself is part of y.
I shall not assume the axiom of unrestricted fusion but I shall not impose any
special conditions pro or against any kind of mereological fusion, either. In thepresent context I will leave the issue open. If, given a condition φ, there is amereological fusion of all the φers, then it is unique by (7) and we can refer to it
by a definite description: 2(8) σxφ =df ιz∀y(Ozy ↔ ∃x(φ ∧ Oyx))
The fusion of all x such that φ (if it exists) is that thing which overlaps
all and only those things which overlap some x such that φ.
The notation for finitary sums is introduced accordingly:(9) x+ y =df σz(z=x ∨ z=y)
The sum of x and y is the fusion of x and y (if it exists).
Modal Occurrents
The theory of modal occurrents can be built on the basis of mereology by adding two more primitives, namely, a one-place predicate ‘Wx’ (read: ‘x is a possible world’) and an individual constant α (for ‘the actual world’). These primitives are governed by the following axioms:(10) W αα is a world—the actual world.
(11) ∀x∃y(Wy ∧ Oxy)
Everything overlaps some world.
3

(12) ∀x(Wx ∧ ∃yPPyx)
Every world includes something.
The first axiom is to fix the intended role of ‘α’ and the second axiom is to fix
the intended meaning of ‘W’ on the understanding that a world is the mereological fusion of everything in it. The third axiom is only needed to go alongwith the standard assumption that there are no “empty worlds”, an assumption
which generalizes—modally—the assumption that the domain of discourse isnon-empty. Strictly speaking this assumption is not essential and one could
allow for empty worlds just as one can allow for the empty domain; but thiswould introduce complications into the logical machinery which would only
becloud the basic picture.3 On the other hand, there is no need to require that all
worlds overlap, or that no two worlds overlap, for such requirements do notbelong to a general formal theory of worlds. For the same reason we need not
require that worlds be maximal individuals, i.e., that no world is part of another.
With this apparatus in hand, a modal occurrent is defined, quite simply, as
any object in the domain of quantification whose mereological make up does
not include any worlds:(13) Mx =df ∀y(Wy → ¬Pyx)
x is a modal occurrent iff no world is part of x.
Typically, a modal occurrent is a trans-world individual—i.e., it has fragmentsthat are too large to be mereologically included in a single world. This is how
the concept has been originally introduced in the literature (though a preferredidiom is David Lewis’s ‘modal continuant’4 ). However, our definition includes
world-bound individuals as a limit case and is therefore to be understood verybroadly. We can always introduce the limit case by definition:(14) PMx =df ∀y(Wy → ¬Pxy)x is a proper modal occurrent iff x is (a modal occurrent which is) notpart of any world y.
Given our modest assumptions concerning possible worlds, it is also possible
for a trans-world modal occurrent to be world-bound, provided there is a world
large enough to contain it all. But whether or not a modal occurrent is worldbound, it certainly has world-bound parts, the largest of which may be thought

4

of as the “actualizations” (or stages) of that occurrent in the relevant worlds.
We can make this more precise through the following auxiliary definitions:(15) WBxy =df Mx ∧Wy ∧ Pxy
x is world-bound in y iff x is a modal occurrent and y is a world
wholly containing x.
(16) WFxyz =df Pxy ∧ WBxz
x is a world fragment of y in z iff x is a part of y which is worldbound in z.
(17) WSxyz =df WFxyz ∧ ∀u(WFuyz → ¬PPxu)
x is a world stage of y in z iff x is a maximal world fragment of y in z.
Some immediate corollaries of these definitions:(18) ∀x∀y(WBxy → WSxxy)
Every world-bound occurrent is a world stage of itself (in the relevant
world).
(19) ∀x∀y∀z(WSxyz ∧ WByz → x=y)Every world-bound occurrent coincides with its own world stage (inthe relevant world).
(20) ∀x∀y∀z∀u(WSxyz ∧ WSuyz → x=u)Every occurrent has at most one world stage in any world.
Given (20), we can introduce a definite description to talk about world stages(where they exist):(21) xy =df ιzWSzxy
The unique stage (if it exists) of occurrent x in world y.
Modalities
To interpret modal discourse against the background of the formal theory ofmodal occurrents, we interpret names and predicates as usual, i.e., as objects andrelations defined on a universe of discourse. In particular, ‘ α’ will pick out anobject in the extension of ‘W’, i.e., a world. However, the intuition is that ordi5

nary names such as ‘Pavarotti’ correspond to proper modal occurrents andordinary predicates such as ‘is a singer’ correspond to sets of world stages.
(Ordinary relational predicates such as ‘is a better singer than’ will correspondto relations among world stages.) A statement such as(22) Pavarotti is a singer
will then count as true iff the actual world stage of Pavarotti is a singer. And astatement such as(23) Pavarotti might have been a ballerina
will count as true iff some world stage of Pavarotti is a ballerina.5 This intuition
is familiar from the literature on four-dimensionalism, according to which aname such as ‘Pavarotti’ denotes a temporally extended individual and a statement of the form(24) Pavarotti will be a ballerina
is true iff there is a future temporal part of Pavarotti which is a ballerina.6 In
effect, if we construed possible worlds intuitvely as temporal stages of the actual
world, then the theory of modal occurrents would yield a theory of temporaloccurrents which is compatible with much literature on the topic. (This would
be especially evident if we assumed all worlds to be pairwise discrete, fortypically a four-dimensional ontology has no room for continuants, i.e., things
wholly existing at different times.)
Keeping with this basic framework, the intuition can be generalized as follows. Let L be the given modal (first-order) language and let L* be the nonmodal language that results from L by replacing the modal operators with theprimitive vocabulary of mereology and the theory of modal occurrents. For
every formula φ of L we define a corresponding L*-formula φy ( “ φ holds aty”) by recursion:7(25) if φ = Rt1…tn , then φy = Rt1 y …tn y
if φ = ¬ψ, then φy = ¬ψyif φ = ψ • χ, then φy = ψy • χ y
if φ = ∀xψ, then φy = ∀x(E!xy → ψy )if φ = ∃xψ, then φy = ∃x(E!xy ∧ ψy )6

if φ = ■ ψ, then φy = ∀x(Wx → ψx )if φ = ◆ ψ, then φy = ∃x(Wx ∧ ψx )where ‘•’ is any binary connective and ‘E!’ is the singular existence predicate
defined by:(26) E!x = df ∃y(y = x).
Then the translation of any given L-formula φ into L* is given by the function
τ defined by setting:(27) τ(φ) = φα.
Here are some illustrative examples:(28) τ(Fb) = Fbα‘b is F’ means ‘The actual world stage of b is F’. (Thus, if b has noworld fragments in the actual world, then the statement is false.)(29) τ(∀xFx) = ∀x(E!xα → Fxα)‘Everything is F’ means ‘Every actual world stage is F’.
(30) τ(■ Fb) = ∀x(Wx → Fbx )‘Necessarily b is F’ means ‘In every world, b’s world stage is F’.
(Thus, if there is a world in which b has no world fragments, then thestatement is false.)(31) τ(◆ ∃xFx) = ∃y(Wy ∧ ∃x(E!xy ∧ Fxy ))‘Possibly, something is F’ means ‘There is a world in which something has a stage that is F’.
(32) τ( ∃x◆ Fx) = ∃x(E!xα ∧ ∃y(Wy ∧ Fxy ))‘Something is possibly F’ means ‘Something with an actual worldstage has a possible world stage that is F’.
(33) τ(∀x(Fx → ■ Gx)) = ∀x(E!xα → (Fxα → ∀z(Wz → Gxz)))‘Every F is necessarily G’ means ‘Everything that has an actual
world stage that is F has in every world a stage that is G’. (Thus, if
something x has an actual world stage that is F, then the statement is7

false if there are worlds in which x lacks a world stage altogether evenif every possible world stage of x is indeed G.)
It is worth observing that our definition of ‘modal occurrent’ (13) makes it
impossible to use this machinery to express modal discourse that involves
counterfactualizing about entire worlds. One might want to say, for example,
that this world of ours might have been better than it is, or that it might havebeen something else than a world, or that something which is not a world (e.g.,
Switzerland) might have been a world. These are not statements that can be expressed in L* unless we relax our definition so as to allow for modal occurrents
which include worlds as parts. As a matter of fact, there would be nothing
wrong with such a revision (except that ‘M’ would then have to be assumed asa primitive predicate). However, the definition given above reflects a limitationthat is common to all familiar ways of expressing the semantics of modal discourse, so I will stick to it for comparative convenience.
Counterparts
At this point it is natural to establish a link between all the things that are worldstages of the same occurrent. And the natural way of doing so is to think ofsuch things as counterparts of one another, so that our translation scheme
would effectively amount to a scheme for representing counterfactual statements
about world stages in terms of factual statements about their counterparts. For
example, on this understanding a statement such as(23) Pavarotti might have been a ballerinawould be true just in case tere is at least one counterpart of the actual worldstage of Pavarotti which is a ballerina. More generally, let us define:(34) Ax =df PPxα
x is actual iff x is part of the actual world.
(35) Ixy =df PPxy ∧ Wy
x is in y iff y is a world of which x is a proper part.
(36) Cxy =df ∃z∃u∃v(Mz ∧ WSxzu ∧ WSyzv)
x is a counterpart of y iff x and y are world stages of a common occur8

rent z. (Here the fact that we are not assuming the unrestricted fusion
axiom is relevant, for otherwise anything x in any world u would
count as a counterpart of anything y in any world v for the simple
reason that x and y would count as world stages of their sum x + y.
Without that axiom, however, we are free to think of the occurrents
that exist as being counterpart-interrelated, in fact maximally counterpart-interrelated.8 )Together with ‘W’, these three predicates form the primitive vocabulary ofLewis’s counterpart theory.9 And the appropriateness of these definitions isshown by the fact that the bulk of counterpart theory can now be derived fromour three axioms as long as we make one extra assumption—viz., that there areno trans-world identities:(37) ∀x∀y(Wx ∧ Wy → ¬POxy)No two worlds overlap.
Given this additional assumption, all the postulates of counterpart theory can beproved as theorems:(38) ∀x∀y(Ixy → Wy)
Nothing is in anything except a world. [This follows directly fromdefinition (35)](39) ∀x∀y∀z(Ixy ∧ Ixz → y=z)
Nothing is in two worlds. [From axiom (37) via definitions (1), (2),(3), and (35)](40) ∀x∀y(Cxy → ∃zIxz)
Whatever is a counterpart is in a world. [Directly from definition (36),second conjunct, via definitions (15), (16), (17), and (35)](41) ∀x∀y(Cxy → ∃zIyz)
Whatever has a counterpart is in a world. [Directly from definition(36), third conjunct, via definitions (15), (16), (17), and (35)](42) ∀x∀x∀z(Ixy ∧ Izy ∧ Cxz → x=z)
Nothing is a counterpart of anything else in its world. [From corollary (20) via definitions (17), (35), and (36)]9(43) ∀x∀y(Ixy → Cxx)
Anything in a world is a counterpart of itself. [From corollary (18) via
definitions (15), (35), and (36)](44) ∃x(Wx ∧ ∀y(Ixy ↔ Ay))
Some world contains all and only actual things. [From axiom (10) via
definitions (34) and (35)](45) ∃xAy
Something is actual. [From axioms (10) and (12) via definition (34)]There is but one noticeable difference between this reconstruction andLewis’s original formulation, namely, our axioms have the following consequence which is not provable in counterpart theory:(46) ∀x∀y(Cxy ↔ Cyx)Counterparthood is symmetric. [From definition (36)]
To rule this out we would have to introduce C as a primitive relation and replacedefinition (36) by an axiom which preserves only the implication from left toright:(47) ∀x∀y(Cxy → ∃z∃u∃v(Mz ∧ WSxzu ∧ WSyzv))
A thing and its counterparts are world stages of a common occurrent.
Then every postulate of counterpart theory would still be provable except for(43), which would have to be assumed as an axiom.
Comparisons
We have derived counterpart theory from the theory of modal occurrents, andthis may be viewed as an exercise in theory formulation. However, there is adifference between the way we would now express modal discourse in counterpart theory and the way modal discourse is expressed in Lewis’s original
version. The difference concerns the way ordinary names are interpreted, andconsequently the definition of the translation function from L to L*. For us, as
we have seen, a name such as ‘Pavarotti’ denotes a modal occurrent and a statement such as
10

(23) Pavarotti might have been a ballerina
is true iff there is some world stage of Pavarotti which is a ballerina. For Lewis‘Pavarotti’ denotes only the actual world stage of a counterpart-interrelated
modal occurrent, and (23) is true iff there is some counterpart of Pavarotti, i.e.,some world stage of that occurrent, which is a ballerina. Formally, this meansthat for Lewis the translation function τ is based on a different recursive definition of φy , namely:(25') if φ = Rt1…tn , then φy = φ
if φ = ¬ψ, then φy = ¬ψyif φ = ψ • χ, then φy = ψy • χ y
if φ = ∀xψ, then φy = ∀x(Ixy → ψy )if φ = ∃xψ, then φy = ∃x(Ixy ∧ ψy )if φ = ■ ψx1…xn , then φy = ∀z∀z1…∀zn (Wz ∧ Iz1z ∧ Cz1x1 ∧ …
∧ Izn z ∧ Czn xn → ψz z1…zn )if φ = ◆ ψx1…xn , then φy = ∃z∃z1…∃zn (Wz ∧ Iz1z ∧ Cz1x1 ∧ …
∧ Izn z ∧ Czn xn ∧ ψz z1…zn )With reference to out earlier examples, this yields the following alternativetranslations:(28') τ(Fb) = Fb
‘b is F’ means ‘b is F’.
(29') τ(∀xFx) = ∀x(Ixα → Fx)‘Everything is F’ means ‘Everything actual is F’.
(30') τ(■ Fb) = ∀x∀y(Wy ∧ Ixy ∧ Cxb → Fx)‘Necessarily b is F’ means ‘Every counterpart of b, in any world,
is F’.
(31') τ(◆ ∃xFx) = ∃y(Wy ∧ ∃x(Ixy ∧ Fx))
‘Possibly, something is F’ means ‘There is a world in which something is F’.
(32') τ( ∃x◆ Fx) = ∃x(Ixα ∧ ∃y∃z(Wz ∧ Iyz ∧ Cyx ∧ Fy))
‘Something is possibly F’ means ‘Something actial has a counterpartthat is F’.
11

(33') τ(∀x(Fx → ■ Gx)) = ∀x(Ixα → (Fx → ∀y∀z(Wz ∧ Iyz ∧ Cyx → Gy)))‘Every F is necessarily G’ means ‘If something actual is F, then its
counterparts, in any world, are G’.
There are several differences between these translations and our earlier translations in (28)–(33). How important are they?
One difference, for example, is that the truth of the translation in (30)depends on whether b has a world stage in every world, while the translation in(30') does not depend on that—that is, it does not depend on the corresponding
question of whether b has a counterpart in every world. A similar point could bemade about (33) and (33'): the former, but not the latter, depends on whether x
has a world stage (respectively: a counterpart) in every world. However theseare differences that reflect only a minor disagreement concerning the scope ofmodal statements: the translation function defined in (25) takes a modal statement to range over all possible worlds whereas Lewis’s translation function(25') takes a modal statement to range exclusively over those worlds in which
the relevant counterparts exist. This is a minor difference because it can easilybe accommodated by changing the relevant clauses in one translation functionor the other. For example, we could replace the last two clauses of (25) with thefollowing clauses and the difference would disappear:10(25") if φ = ■ ψx1…xn , then φy = ∀z(Wz ∧ E!x1z ∧ … ∧ E!xn z →(ψx1…xn )z )if φ = ◆ ψx1…xn , then φy = ∃z(Wz ∧ E!x1z ∧ … ∧ E!xn z ∧ (ψx1…xn )z )Another difference is that Lewis’s translation function (25') exploits thepossibility that an object (a world-bound individual) has more than one counterpart in some worlds, whereas our translation function (25) ignores that possibility altogether. More precisely, definition (36) allows an object to have morethan one counterpart per world, but (25) as well as its revised version (25") saythat the truth-value of a modal statement about the actual world stages of certain
occurrents x1, ..., xn depends exclusively on the properties of the possible worldstages of these occurrents—and each occurrent has at most one world stage inevery world. So, for example, according to (25) and (25") a statement such as(23) Pavarotti might have been a ballerina
is true iff Pavarotti’s world stages comprise a ballerina. These world stages are12counterparts of Pavarottiα (i.e., of that actual thing that Lewis calls ‘Pavarotti’).
They are those counterparts of Pavarottiα which fall under the same name:‘Pavarotti’. But Pavarottiα may have other counterparts besides those. He will
have as many counterparts as there are modal occurrents that share the sameactual world stage as the occurrent Pavarotti. And those are not taken into
account by our clauses for expressing modal statements in terms of modal occurrents (whereas they are taken into account by Lewis’s clauses).
If desired, however, this difference between the two translation functions
can also be eliminated. All we have to do is to emend our clauses so that (23)will be true iff at least one counterpart of Pavarottiα is a ballerina, regardless ofwhether that counterpart is a world stage of Pavarotti or of some other modal
occurrent whose actual world stage is Pavarottiα. In general, this amounts torevising (25) or (25") along the following lines:(25''') if φ = ■ ψx1…xn , then φy = ∀z∀z1…∀zn (Wz ∧ Cz1z x1α ∧ …
∧ Czn z xn α → (ψz1…zn )z )if φ = ◆ ψx1…xn , then φy = ∃z∃z1…∃zn (Wz ∧ Cz1z x1α ∧ …
∧ Czn z xn α ∧ (ψz1…zn )z )Then we would get exactly the translation function of Lewis.
So it does not look as though the choice between taking C as a primitive
relation, as in Lewis’s original formulation of counterpart theory, or defining itin terms of modal occurrents, as in our formulation, will make much difference
at all. As Lewis would say, the two theories appear to be verbal variants of one
and the same technique for expressing modal talk extensionally. New terminology is not a new theory.11
However that is not true. One important difference between the twotheories concerns the way they handle indeterminacy—the sort of indeterminacy that gets into counterfactuals about named individuals.12 And this is an
important difference insofar as this sort of indeterminacy has sometimes been
claimed to cause troubles for counterpart theory.
Suppose that we have doubts about (23). Suppose, that is, that there is indeterminacy about whether or not Pavarotti might have been a ballerina. Assuming that there is no indeterminacy as to who Pavarotti is in this world, thismeans that there is indeterminacy as to what Pavarotti’s counterparts are. Whatsort of indeterminacy is this? In the original formulation of counterpart theory it

13certainly need not be a form of ontological indeterminacy. If the counterpart
relation is understood as any old relation of comparative overall similarity, thenthe indeterminacy is merely semantic (or pragmatic): the predicate ‘C’ can suffer from the vagueness of our stipulations, and such vagueness can be dealt within semantic (or pragmatic) terms. It can be differently resolved in different
contexts. This is how Lewis himself sees the matter, and he insists thatvagueness is therefore not a problem. Still, C is the primitive relation of counterpart theory. So to say that C is vague (albeit only semantically or pragmaticallyvague) is to concede that the theory is vague—it is a theory built around a vague
primitive predicate. And to some people this is enough to say that the theory
itself—and the reductive account of modality that the theory is supposed toprovide—is seriously defective.
Consider now the theory of modal occurrents. In fact, let us distinguish
two cases depending on whether we go along with the initial formulation (25)(or its revision based on (25")) or with the later formulation based on (25'''). If
we go along with the latter, then the indeterminacy is not at all semantic (orpragmatic). The indeterminacy exhibited by a statement such as (23) is truly ontological. This is because C itself is construed as an ontological relation, in
terms of P (which is an ontological relation). If it is indetermined whether there
is a counterpart of the actual Pavarotti which is a ballerina, then it must be indetermined whether there exists a modal occurrent x such that xα = Pavarottiα andxz is a ballerina for some world z.13 So it is indetermined whether certain occurrents exist. So there is indeterminacy as to what there is. The result is thatnow counterpart theory is no longer a vague theory, but it calls for a vague
ontology.
By contrast, if we go along with the initial formulation (25) (or its revision
based on (25")), taking ordinary names to denote modal occurrents, then theindeterminacy of a statement such as (23) need not be a source of worry: it does
not make the theory vague and it does not call for a vague ontology. Simply, thestatement is prima facie indeterminate because it is indeterminate which modal
occurrent is picked out by the name ‘Pavarotti’, hence which world stages countas relevant counterparts of Pavarotti’s actual stage. And this prima facie
indeterminacy is real if some of the admissible candidates contain a world stage
that is a ballerina while other candidates do not. In fact, a supervaluational semantics suggests itself naturally here. A term is indeterminate insofar as there
appear to be many ways of assigning it a referent, all of them compatible with14

the way the term is used. Hence the truth-value of a statement containing
indeterminate terms is naturally construed as a function of its truth-values underthe various admissible ways of assigning a unique referent to those indeterminate terms. If the statement is true under all such assignments, then we may takeit to be true simpliciter; the unmade semantic stipulations do not matter because
what the statement says is true regardless (or super-true, as Kit Fine has it14 ).
This is the case of a non-modal statement such as (22), for example:(22) Pavarotti is a tenor
Likewise, if the statement comes out false under every admissible way of assigning a unique referent to its indeterminate terms, then we may regard it asfalse (or super-false) in spite of the indeterminacy. The negation of (22) would
be a case in point. It is only when the statement is true under some assignments
and false under others that there is trouble. In such cases, nothing will settle thequestion for us and the statement will fall into a truth-value gap. This seems tobe precisely the situation exhibited by a statement such as (23).
To sum up, then, we have three different accounts of the indeterminacy thatgets into counterfactuals, depending on which theory we work with. On Lewis’s
original formulation of counterpart theory, it is the theory itself that is vague.
On its closest mereological reconstruction in terms of modal occurrents, it is the
ontology that is vague. And on our initial formulation of the theory of modal
occurrents, it is just the names that we use that are vague: the indeterminacy
exhibited by a statement such as (23) is just a case of semantic vagueness on apar with many others, and can be handled like any other case.15 In this sense,‘Pavarotti’ is modally vague just as ‘Everest’ is spatially vague or ‘the industrial revolution’ is temporally (and spatially) vague. I think that these differences between the three accounts are important. And I think that they speakin favor of the theory of modal occurrents insofar as ontological vagueness andtheoretical vagueness can be objected to on independent grounds.16
A coda on existence and possibility
What are the consequences of this way of putting things for the prospects ofmodal extensionalism, i.e., the view that one can provide an extensional reductive analysis of modal discourse? Once again, one important consequence is thatby placing the vagueness of our modal discourse in the semantics of the names15we use, rather than in the underlying ontology or in the semantics of theprimitive notions around which the analysis is formulated, the account is notdirectly affected by the phenomenon of vagueness—neither more nor less thana physicalist account of the mind-body problem, for example, is affected by thevagueness of ordinary proper names. (Is the candy that Pavarotti is presently
chewing part of his body? Will it be part of Pavarotti’s body only after he has
swallowed it? Only after he has digested it? On closer look, our earlier assumption concerning the determinacy of the actual Pavarotti will have to be discharged too.) There is, however, also a seemingly negative consequence of ourway of explaining modality in terms of modal occurrents. For the appeal tomodal occurrents may be said to yield a paradox.
Briefly put, the paradox is that on the proposed reductive analysis of modality one is forced to say that all possible things exist (modal realism), but not
all existing things are possible. This follows from the fact that possibility is explained in terms of what goes on in different worlds, hence what goes on acrossworlds is not possible. The possible individuals are the world-bound stages, so
the trans-world individuals (the proper modal occurrents) are impossible. They
exist, but they cannot possibly exist. And what goes for existence goes fortruth. Let Pavarotti0 be one of the various modal occurrents that can plausibly be
associated with the vague name ‘Pavarotti’. Then the statement(48) Pavarotti0 exists
is a necessary falsehood. It is true that there is such a thing as Pavarotti0 , but it
cannot be true.
This puzzle arizes in the theory of modal occurrents but not in the original
theory of counterparts in which names designate what we have called worldstages, i.e., things that bear the relation I (in) to the things which are W (possible worlds). At least, the puzzle does not arises in the original theory as long as
we do not formulate it in mereological terms. Accordingly, we would haveanother difference between counterpart theory and the theory of modal
occurrents—and a remarkable difference to say the least.17
Now, one could fix things by changing the relevant notion of existence. To
exist in a world is not to be part of that world (i.e., to be in that world, as per
definition (36)). Rather, to exist in a world is to have parts in that world. If we
accept this, then Pavarotti0 would exist and the paradox would dissolve. Not
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only, but the difference between such terms as ‘the winged horse’ and ‘theround square’ (say) would be restored too: the former, not the latter, would denote an object that has parts in some possible worlds, so the former, not the latter, would denote (perhaps vaguely) a possibly existing thing. And what goesfor existence goes for truth. The proposition(49) the-winged-horse exists
would be possibly true, but(50) the-round-square exists
would be necessarily false.18
If this way out is accepted, then the initial paradox dissolves, and so does
the relevant difference between the theory of modal occurrents and the theory ofcounterparts. Moreover, modal realism is vindicated and modality explained
away: to be possible is just to exist in some world, i.e., to have parts that exist insome world.
On the other hand, one could object that a change in the notion of existence
is a change in the strength of the reductive analysis. For the reductive analysis
presupposes what Lewis calls the principle of Plenitude: every way that a world
could be is a way that some world is.19 Roughly speaking, this means that anything can coexist with anything, and anything can fail to co-exist with anything.
Strictly speaking, the no-overlap principle (37) prevents this from being true incounterpart theory, so what the principle really says is this: any number ofpossible individuals has duplicates which coexist in some world as well as
duplicates that do not coexist—where two things are duplicates iff they have exactly the same natural properties and their parts can be mapped onto one
another in such a way that corresponding parts have the same natural properties
too. (Never mind the details of defining “natural property”.) The principle ofPlenitude is crucially important if the account is to deliver a complete analysisof the notion of possibility in extensional terms, for it is this principle that guarantees that the analysis leaves “no gaps in logical space”. But now there istrouble. For there is no way that modal occurrents can have duplicates which
coexist. There is no way, that is, that modal occurrents can “recombine” because that would violate the no-overlap axiom. If there is a world that contains
the whole of Pavarotti0 , then that world will also include among its parts the ac17

tual world stage of Pavarotti, contrary to the no-overlap principle.20 So if Pavarotti0 is a possible thing, then it cannot recombine with other occurrents. And if
it can recombine then it is not a trans-world occurrent.
The only way out, it seems to me, is to give up the no-overlap principle itself—or else we must go back to a vague theory of counterparts.21
References
Bennett, J. (1988), Events and Their Names, Oxford: Clarendon Press.
Borghini, A. (2000), La teoria della possibilità di David K. Lewis, Tesi di Laurea,
University of Florence.
Fine, k. (1975), ‘Vagueness, Truth and Logic’, Synthese 30: 265–300.
Gallois, A. (1998), Occasions of Identity: The Metaphysics of Persistence, Change andSameness, Oxford: Clarendon Press.
Garson, J. W., (1984), ‘Quantification in Modal Logic’, in D. M. Gabbey and F. Guenthner(eds.), Handbook of Philosophical Logic, Volume II, Dordrecht: Reidel, pp. 249–307.
Heller, M. (1990), The Ontology of Physical Objects: Four-dimensional Hunks of Matter,
Cambridge: Cambridge University Press.
Hudson, H. (1999), ‘A True, Necessary Falsehood’, Australasian Journal of Philosophy 77:
89–91.
Lewis, D. K. (1968), ‘Counterpart Theory and Quantified Modal Logic’, Journal of Philosophy 65: 113–126; reprinted in Lewis (1983), 26–39.
Lewis, D. K. (1983), ‘Postscripts to “Counterpart Theory and Quantified Modal Logic”’, in
Philosophical Papers, Volume 1, New York: Oxford University Press, 1983, 39–46.
Lewis, D. K. (1986), On the Plurality of Worlds, Oxford: Blackwell.
Simons, P. (1987), Parts. A Study in Ontology, Oxford: Clarendon Press.
Van Fraassen, B. C. (1966), ‘Singular Terms, Truth-Value Gaps, and Free Logic’, Journal of
Philosophy 63: 481–495.
Varzi, A. C. (2001), ‘Vagueness, Logic, and Ontology’, The Dialogue, 1 (2001), 135–154.

Notes
1

For an overview of mereology see Simons (1987).
I will assume descriptions to be handled in standard, Russellian fashion.
3
On these matters I refer to Garson (1984).
4
The terminology is introduced in Lewis (1983: 41). I prefer my idiom because it
parallels the distinction between occurrents and continuants familiar from the literature on
temporal persistence: a temporal occurrent is an entity that has temporal parts whereas atemporal continuant is an entity which is present in its entirety at any time at which it
exists. By analogy, then, I would use ‘modal continuant’ to refer to those items which are218

present in their entirety in any worlds in which they exist—exactly the opposite of a modal
occurrent.
5This is the intuition put forward in Lewis (1983: 40–42). See also Lewis (1986:
213–217).
6
Lewis himself is a proponent of a four-dimensionalist ontology in which tensed
statements are interpreted according to this schema. See e.g. Lewis (1986: 204).
7
The first conjunct in the clause for ‘∃’ is actually redundant if descriptions are understood à la Russell. We leave it in to ensure immediate duality between ‘∃’ and ‘∀’.
8
Such occurrents correspond to what Lewis (1983: 41) calls ‘modal continuants’ andLewis (1986: 214) calls ‘*-possible individuals’.
9
See Lewis (1968).
10
The amendment of the clause for ‘∃’ is actually redundant if descriptions are understood à la Russell. Again, we include it here for the sake of duality.
11
This is claimed both in Lewis (1983) and in Lewis (1986).
12
The point is made informally in Bennett (1988: 63–64), though with reference to aslightly different way of understanding the theory of modal occurrents.
13
Let us assume that the predicate ‘is a ballerina’ involves no vagueness.
14
See Fine (1975), where a supervaluational semantics for vague predicates is developed. The notion of a supervaluation goes back to van Fraassen (1966), who used it to provide a semantics for free logic, i.e., a logic in which singular terms may lack a reference. Our
situation here is perfectly dual.
15
Heller (1990) has a similar account of the indeterminacy involved in our intuitions
about persistence and identity through time.
16
I summarize my views on ontological vagueness in Varzi (2001).
17
As a matter of fact Lewis (1983) does interpret I as P and he does accept unrestrictedcomposition, so the puzzle does affect his understanding of the theory; but it need not.
18
The point is made in Hudson (1999).
19
See Lewis (1986: 86–92).
20
This point was brought to my attention by Borghini (2000: §3.2.1).
21
Thanks to Andrea Borghini, Berit Brogaard, and Chris Partridge for helpful comments on an early draft of this paper.

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