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An argument against DavidLewis' theory of possible

worldsPeter Forrest

a b

& D.M. Armstronga b

aSchool of History, Politics and Philosophy,

Macquarie Universityb

Department of Traditional and ModernPhilosophy , University of Sydney

Published online: 28 Jul 2006.

To cite this article: Peter Forrest & D.M. Armstrong (1984) An argument against

David Lewis' theory of possible worlds, Australasian Journal of Philosophy, 62:2,

164-168

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Australasian Journal of PhilosophyVol. 62, No. 2; June 1984

D I S C U S S I O N

A N A R G U M E N T A G A I N S T D A V ID L E W I S ' T H E O R Y O F

P O S S IB L E W O R L D S

P e t e r F o r r e s t a n d D . M . A r m s t r o n g

IT h e a r g u m e n t r e s t s u p o n t w o p r e m i s s e s .

First, e v e r y p o s s ib l e w o r l d is w h o l l y d i s ti n c t f r o m e v e r y o t h e r . T h i s is

s o m e t h i n g w h i c h L e w i s in si st s u p o n . L e w i s d o e s o f f e r w h a t m a y b e t h o u g h t

o f a s a q u a l i f i c a t io n , a l t h o u g h h e h i m s e l f d o e s n o t r e g a r d i t a s s u c h :

T h e r e a r e s o m e a b s t r a c t e n ti ti e s, f o r in s t a n c e n u m b e r s o r p r o p e r t i e s , t h a t

i n h a b i t n o p a r t i c u l a r w o r l d b u t e x is t a l ik e f r o m t h e s t a n d p o i n t o f all

w o r l d s , j u s t a s t h e y h a v e n o l o c a t i o n i n t i m e a n d s p a c e b u t e x i s t a l ik e f r o m

t h e s t a n d p o i n t o f a ll t im e s a n d p l ac e s . T h i ng s t h a t d o i n h a b i t w o r l d s -p e o p l e , f la m e s , b u i l d in g s , p u d d l e s , c o n c r e t e p a r t i c u l a r s g e n e r a l l y - i n h a b i t

o n e w o r l d e a c h , n o m o r e . ( 1 9 7 3 , 1 .9 , p . 3 9 )

T h i s q u a l i f i c a t i o n , i f i t i s o n e , s e e m s n o t t o a f f e c t o u r a r g u m e n t .

S e c o n d , g i v e n a n y n u m b e r o f p o s s ib l e w o r l d s , W 1 , W 2 . . . . t h e r e ex is ts

a p o s s ib l e w o r l d , h a v i n g w h o l l y d i s t i n c t p a r t s , s u c h t h a t o n e o f th e s e p a rt s

is a n i n t e r n a l l y e x a c t ly re s e m b l i n g d u p l i c a t e o f W 1 ( h e n c e f o r w a r d ' d u p li c a te ' ),

a n o t h e r a d u p l i c a t e o f W 2 , a n d s o o n .

C o n s i d e r , f o r i n s ta n c e , t h e s e t w h o s e s o le m e m b e r s a r e W 1 a n d W 2 . T h e r e

w i l l b e a w o r l d , W B , h a v i n g w h o l l y d i s t i n c t p a r t s , P 1 a n d P 2 , s u c h t h a t P ~

internally e x a c t l y r e s e m b l e s ( is i n t e r n a l l y q u a l i t a t i v e l y i n d i s c e r n i b l e f r o m ) W 1 ,

w h i l e P 2 i n t e r n a l l y e x a c t l y r e s e m b l e s W 2 . ( T w o o b j e c t s i n t e r n a l l y e x a c tl y

r e s e m b l e e a c h o t h e r i f a n d o n l y i f t h e y e x a c t l y r e s e m b l e e a c h o t h e r i n a b s tr a c -

t i o n f r o m a n y r e l a t i o n a l p r o p e r t i e s w h i c h t h e y m a y h a v e . ) W e c a l l W B a n

' a b o v e ' w o r l d . I t i s l i k e a T h i r d M a n .

G i v e n t h e s e t w o p r e m i s s e s, w e c l a im t h a t i t f o l lo w s t h a t t h e r e c a n b e n e i t h e r

t h e a g g r e g a t e , n o r t h e s e t , o f a ll p o s s i b l e w o r l d s . W e b e g i n , i n th i s a n d t h e

n e x t p a r a g r a p h , b y m e r e l y o u tl in i n g t h e a r g u m e n t . S u p p o s e t h a t s u c h a na l l eg e d a g g r e g a t e , A , e x i s ts . C o n s i d e r t h e n a v e r y b i g w o r l d , W B , w h i c h s ta n d s

t o t h e w o r l d s w h i c h m a k e u p A i n t h e w a y a l r e a d y d e s c r i b e d . T h a t i s , f o r

e a c h w o r l d , W , w h i c h is a p a r t o f A , t h e r e w i ll e x is t a p r o p e r p a r t , P , o f

W B w h i c h i n t e r n a l l y e x a c t l y r e s e m b l e s W . F u r t h e r m o r e , e a c h P w i l l b e w h o l l y

d i s t in c t f r o m e v e r y o t h e r P i n W B , a n d e a c h P w i ll i n t e r n a l ly e x a c t l y r e se m b l e

j u s t o n e w o r l d in A . ( A s s u m i n g t h a t n o t w o w o r l d s e x a c t l y r e s e m b l e e a ch

o t h e r . I f t h is is d e n i e d , t h e a r g u m e n t m u s t b e , b u t c a n b e , r e f o r m u l a t e d . )

W B is n o t a p a r t o f A . T a k i n g ' si z e' i n it s w i d e s t s e n s e , a n y W is e x a c t l y

164



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Peter Forrest and D. M. Armstrong 165

t he s a m e s iz e a s s o m e P , a P w h i c h is a p r o p e r p a r t o f W ~ . T h e s e p r o p e r

p a r ts o f W m h o w e v e r , a r e n o t e x a c t l y t h e s a m e s iz e a s W B . F o r i n s t a n c e ,

a s w ill b e s h o w n , W B c o n t a i n s m o r e e l e c t r o n s t h a n a n y s u c h P . S o W B is

n o t a W . T h a t i s t o s a y , t h e r e is n o s u c h t h i n g a s t h e a g g r e g a t e o f a ll p o s s i b l e

w o r l d s .

I f t hi s a r g u m e n t is c o r r e c t , t h e n e q u a l l y t h e r e is n o s u c h t h i n g a s th e s e t

o f a ll p o s s i b l e w o r l d s . F o r c o n s i d e r t h i s a ll e g e d s et . U s i n g t h e s a m e m e t h o d

a s b e f o r e , w e c a n d e s c r ib e a W B w h i c h i s n o t a m e m b e r o f t h is s e t .

I n t h is a r g u m e n t , i t is c le a r t h a t t h e s e c o n d p r e m i s s is th e c o n t r o v e r s i a l

o n e. I t m a y b e t h o u g h t t h a t , g i v e n a n a g g r e g a t e o r s e t o f w o r l d s o f s u ff ic i en t ly

h ig h i nf in it e c a r d i n a l i ty , t h e r e c a n n o t b e a w o r l d ' a b o v e ' t h i s a g g r e g a t e ( s et ).

H e r e it is i m p o r t a n t t o c o n s i d e r w h a t r e l a t i o n s m u s t h o l d b e t w e e n d i f f e re n tp a rt s o f t h e s a m e p o s s ib l e w o r ld . W h a t m a k e s d i st in c t p a r t s o f a w o r l d c o -

a c tu a l? O n e o f u s ( A r m s t r o n g ) t h in k s t h a t s u c h p a r ts , o v e r a n d a b o v e a n y

internal r e l at io n s w h i ch t h e y m a y h a p p e n t o h a v e , n e e d h a v e n o r e la t io n s

b e s i d e s t h a t o f c o - a c t u a l i t y . F o r i n s t a n c e , a s s u g g e s t e d b y B r o a d ( 1 9 3 3 ,

p p . 1 7 6 - 7 ) , t w o o r m o r e w h o l l y d i s t i n c t s p a c e - t i m e s m i g h t b e c o - a c t u a l .

B o r r o w in g E d w i n H u b b l e ' s p h r a s e f o r t h e g a la x ie s , w e m i g h t r e f e r to s u ch

w h o l ly d i s j o i n t p a r t s o f a w o r l d a s ' i s la n d u n i v e r s e s ' . I f a w o r l d c o u l d c o n t a i n

a n y f in it e n u m b e r o f s u c h is l a n d u n i v e r se s , p r e s u m a b l y i t c a n c o n t a i n a n y

i n f i n i t e n u m b e r .

S u p p o s e t h is v ie w o f c o - a c t u a l i t y t o b e c o r r e c t . C o n s i d e r a g a i n , A , t h e

a lle g ed a g g r e g a t e o f p o s s i b le w o r l d s . T h e r e e x is ts a w o r l d , W m w i t h t h e f o l -

l ow i ng s tr u c t u r e . W B is m a d e u p o f i s la n d u n i v e r se s . T h e r e is a o n e - o n e

c o r r e l a t i o n b e t w e e n w o r l d s i n A a n d i s l a n d u n i v e r s e s i n W B s u c h t h a t t h e

c o r r e l a t e d o b j e c t s i n t e r n a l l y e x a c t l y r e s e m b l e e a c h o t h e r . G i v e n t h a t e v e r y

p o s si b le w o r l d i s w h o l l y d i s t i n c t f r o m e v e r y o t h e r , i t i s c l e a r t h a t s u c h a c o r -

r e l a t i o n c a n b e e f f e c t e d .

W B is n o t i d e n ti c a l w i t h a n y w o r l d w h i c h i s p a r t o f A . F o r i n s t a n c e , W Bc o n ta in s m o r e e l e c t r o n s t h a n a n y w o r l d w h i c h i s p a r t o f A , a s t h e f o ll o w i n g

a r g u m e n t s h o w s . S u p p o s e W ~ is a w o r l d w h i c h is p a r t o f A , a n d W 1 h a s

j u s t N e l e c t r o n s . T h e r e w i l l t h e n b e s o m e p r o p e r t y , F - n e s s ( i t c o u l d b e a

r e l a t i o n a l o n e ) w h i c h e a c l ~ e l e c t r o n i n W ~ m a y o r m a y n o t h a v e , a n d m a y

o r m a y n o t h a v e i n d e p e n d e n t l y o f w h e t h e r t h e o t h e r e l e c t ro n s i n W ~ h a v e

it. F o r e a c h s u b - s e t o f t h e N e l e c t r o n s i t w i ll b e p o s s i b l e t h a t p r e c i s e l y t h e

e l e ct ro n s i n t h a t s u b - s e t h a v e t h e p r o p e r t y F - n e s s . I n t h is w a y , t h e r e i s e s t a b -

lis he d t h e e x i s te n c e o f p o s s i b l e w o r l d s w h i c h a r e p a r t s o f A a n d w h i c h c o n t a i n

e le c tr o ns , w i t h a w o r l d c o r r e s p o n d i n g t o e a c h s u b - s e t o f t h e s e t o f N e l e c tr o n sin W 1 . T h e r e f o r e t h e r e a r e a t l e a s t 2 N p o s s i b le w o r l d s w h i c h a r e p a r t s o f A

c o n t a i n i n g e l e c t r o n s . T h e i r d u p l i c a t e s i n W B a r e d i s t i n c t i s l a n d u n i v e r s e s ,

t h e re f o r e W ~ m u s t c o n t a i n a t le a s t 2 N e l e c t r o n s . B u t e v e n f o r i n f in i te c a r d i n a l s

2 N is g r e a t e r t h a n N . H e n c e W 8 h a s m o r e e l e c t r o n s t h a n W ~ , s o is n o t t h e

s am e as W ~ . T h e s a m e a r g u m e n t s h o w s t h a t W B is n o t t h e s a m e a s a n y o t h e r

w o r l d w h i c h i s a p a r t o f A . S o W B is a w o r l d w h i c h i s n o t a p a r t o f A . S o

A i s n o t t h e a g g r e g a t e o f p o s s i b l e w o r l d s .

D o e s t h e s i t u a t io n a l t e r i f c e r t a i n c o n s t r a i n t s a r e p u t u p o n t h e r e l a t i o n o f



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166 David Lewis" Theory of Possible Worlds

co-actuality? One of us (Forrest) holds that for two things to be co-actual

they must be linked, directly or indirectly, by some external relation. How-

ever, we do not think that this restriction makes it any more difficult to specify

a WB which is not part of A.

Consider a demiurge who creates wholly distinct spacetime manifolds. The

manifolds would each be linked to all the others, via the demiurge, and so

would be co-actual. It seems to us, further , that this relation of creation could

hold without any further external relations holding between the demiurge

and the manifolds. In particular, the demiurge need not be spatio-temporally

related to what he creates. It seems further that the demiurge might create

an infinity of such space-time manifolds, together with objects of a non-

spatio -temporal nature , none of them externally related to each other exceptvia the demiurge. We can then remodel WB. Instead of its parts being island

universes relative to each other, let each part, otherwise isolated, be the

creation of the one demiurge. WB isistill not a world in A.

Lewis himself, however, holds that the relation of co-actuality between

contingent beings demands that they be, in some broad sense, spatio-

temporally related. He would allow disembodied minds, but they must at

least be temporally related tO other things. (Lewis: private communicat ion.

We are indebted to Lewis for commenting upon an earlier draft o f this paper.

We are also indebted to the anonymous referee for this journal.)

To meet Lewis' restriction, WB must now be a single spat io-temporal whole,

or something very like a spatio-temporal whole. The same holds for each

W which is a part of A. Lewis thinks that it is impossible that the duplicates

of the worlds which make up A could all be fitted into a single-space time.

So there cannot be a WB distinct from the worlds in A.

We think that Lewis' conception of co-actuality is unduly restrictive. Never-

theless, even granting this restriction, we still think that there is a world WB

which is not a world in A.For, given any set S of spatio-temporal worlds, there will be a, perhaps

infinite-dimensional, world WB which contains duplicates of the members

of S. Lewis objects to this that there is an upper bound to the number of

dimensions a spatio-temporal world can have (Beth omega at the greatest).

Hence, he says, there will come a stage at which it is no longer possible to

fit all the duplicates into a single world. But how can one decide such a point?

Only, it seems, by discovering whether a mathematical model can be con-

structed for the large world WB. If it can, then Lewis' restriction on the size

of possible worlds seems to be ad hoc. Now, a mathematical model can beconstructed for WB. 1 SO, even given Lewis' restrictive concept of co-actuality,

our argument stands.

In order to fit duplicates of a set S of possible worlds into a single space-time, we first requirethat for each world W in S there be a spatio-temporal possible world W* which:

(i) Contains a duplicate of W, that is, a part internally exactly resembling Wand (ii) contains some distinguished element O(W). If the Axiom of Choice is accepted, thenwe may identify W* with W itself. But even if the Axiom of Choice is rejected, we could,for instance, obtain W* by adjoining to a duplicate of W an initial event, O(W), temporally



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Peter Forrest and D. M. Armstrong 167

II

Given his premisses, then, Lewis shoul d not a dmi t either an aggregate or

a set of all possible worlds. But Lewis himse lf speaks of the 'set of wo rlds'(1973, 1.3, p. 16). Cou ld he simpl y jet ti son his belie f in a set of worlds? We

think not; he should admit a set, or at very least an aggregate of worlds.

For, i t is plausible that for an y univoca l and non- dis j unct ive predicate 'F ' ,

there is a set of Fs. Th e qualification 'uni vocal an d no n-d isju nct ive ' s required

part ly because of Russell ' s parado x. This , and other, par adoxes can be

avoided by dis t inguishing various types or orders of sets. A nd , given that

distinction, the pred icate 'is a set' is seen to be disj uncti ve, or ev en amb igu ous .

However, it seems that the only qualificat ion required is that the predicate

'is a possible world' be u nivoc al and non-d isjunc tive. Therefore, Lewis should

admit the existence of a set of all possible worlds. F or on Lewis' theo ry, there

is no hierarchy of types or orders of possible worlds. So, on his theory the

predicate ' is a possible world' shoul d be univo cal and non- disju nctiv e. Indeed

it is central to his theory that all possible worlds, includ ing the actual , are

the same sort of entity.

But perhaps the reader has some other way of ha ndlin g Russell ' s parado x

and rejects distinc tions betw een types or orders of sets. In th at case, we invite

him to check whether or not there should be a set of all possible Worlds.

And if, to our surprise, it turned out that on his theory there would be no

set of al l possible worlds, th en we rely on the int uit i on that for a ny (univocal

and non-dis ju nctiv e ) predicate 'F ' there is an aggregate of all the Fs. For

aggregates are not sets, and no p ara dox like Russell 's has ever been f ou nd

for aggregates. Con seq uen tly , even if there were n o set of possible worlds,

there should still be an aggregate of them.

We have used the 'above' world con struct ion, then, to at tack the central

part of Lewis ' theory. W e shall now supp leme nt that at tack by showing how

the 'abov e' worl d co nst ru cti on provi des a fur ther difficulty for Lewis when

prior to all events in W. (If there are events in W arbitrarily far in the past, a re-scaling oftime would provide 'room' to fit in the initial event.)

We may then construct a product space, whose structure is isomorphic to the space of allthose functions from'S to Ix: x e W and W e S], which satisfy the constraints:

(i) The function f must assign to W a member of W*and (ii) For all but a finite number of W, f must assign O(W) to W.This product space is a model for the large spatio-temporal world we are seeking. It has anatural spatio-temporal structure derived from that of the various W*. Furthermore, given

any possible world Wo, the product space contains a duplicate of Wo, namely the subspaceof all those functions which assign a member of Wo to Wo, but assign O(W) to any other W.It might be objected that, conceivably, his product space might consist of nothing but dupli-

cates of W*, and so be a collection of 'island universes' after all. The Multiplicative Axiom,which is a consequence of the Axiom of Choice, would prevent this. Alternatively, if theMultiplicativeAxiom is itself rejected, we can ensure a rich enoughspatio-temporal structurefor the product space by adjoining suitable possible worlds to the set S, before we begin theconstruction. Adjoining copies of all finite-dimensional real vector spaces would do the jobnicely.

2 This qualificationmight be required even for aggregateson the grounds that there is somethingpeculiar about a purported aggregate of quite different sorts of entity, for example a purportedaggregate of real numbers and armadillos.



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168 DavM L ew is' Theory o f Possible Worlds

h e d e v e lo p s h i s t h e o r y . T h u s L e w i s id e n t if i e s a p r o p e r t y w i t h a s e t o f p a r t i c u -

l a r s , e x i st i n g i n v a r i o u s p o s s i b l e w o r l d s . ( a h a s F - n e s s i f a n d o n l y i f a b e l o n g s

t o t h e s e t i d e n t i f i e d w i t h F - n e s s . ) B u t n o p a r t i c u l a r b e l o n g s t o t w o w o r l d s .

S o w e m a y a s s o c i a t e w i t h a n y s e t o f p a r t i c u l a r s t h e s e t o f w o r l d s t o w h i c h

t h o s e p a r t i c u l a r s b e l o n g . H e n c e , o n L e w i s ' t h e o r y , w i t h e a c h p r o p e r t y F - n e ss

t h e r e i s a s s o c i a t e d t h e s e t o f w o r l d s i n w h i c h F - n e s s is i n s t a n t i a t e d . B u t f o r

m a n y p r o p e r t i e s w e c a n s h o w t h e r e i s n o s et o f p o s s ib l e w o r l d s a t w h i c h t h a t

p r o p e r t y is i n s t a n ti a t e d . C o n s i d e r , f o r in s t a n c e , t h e p r o p e r t y being a n electron.

O u r ' a b o v e ' w o r l d c o n s t r u c t i o n s h o w s t h a t , g i v e n a p u r p o r t e d s et o f a ll w o r l ds

c o n t a i n i n g e l e c t ro n s , t h e r e i s a n ' a b o v e ' w o r l d w h i c h a l so c o n t a i n s e l ec t ro n s

b u t i s n o t a m e m b e r o f t h a t s e t.

S c h o o l o f H i s to ry , P o l i ti c s a n d P h i lo so p h y , M a c q u a r i e Un iv er sity .

D e p a r t m e n t o f T r a d i t i o n a l a n d M o d e r n P h i l o s o p h y ,

Un iv e r s i t y o f S y d n e y

R e c e i v e d A p r i l 1 9 8 3

REFERENCES

Broad, C. D . (1933), Examination o f McTaggart's Philosophy, V ol. I, C amb ridge UniversityPress.

Lewis, D avid (1973), Counterfactuals, Blackwell.



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an argument against lewis

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