Lewis on too many worlds

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Lewis on too many worlds

• Andrew Bacon,
• Oxford University

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Lewis on Worlds

• Around the 16th and 17th centuries people
  believed the earth was the centre of the
  universe.
• This picture was replaced by the idea that the
  earth was but one of many planets orbiting the
  sun.
• Later it became known that our sun was one of
  many stars in the galaxy and our galaxy one of
  many stars in the universe.

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Lewis on Worlds

• The moral there is nothing particularly
  remarkable about our world other than the fact it
  is where we happen to be living.
• In some ways the modal realism of David Lewis
  shares this quality.
• Other theories of worlds treat the actual world
  as somehow different from merely possible worlds.
  It is concrete, they are abstract.

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Some principles of Modal Realism

• The actual world is one among many concrete, yet
  spatio-temporally disconnected worlds scattered
  across the possibility space.
• Worlds that differ in content but not substance.

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Some principles of Modal Realism

• There is nothing particularly remarkable about
  the actual world, it does not provide a centre
  for the modal universe nor is it any less real or
  concrete than any other possible world.

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Some principles of Modal Realism

• The term actuality as applied to our world is
  to be treated as indexically picking out the
  world we happen to live in, and would equally
  well have picked out a different world if that
  was where we had happened to be living.

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Some principles of Modal Realism

• On ersatz theories of worlds, possible worlds are
  abstract and hence causally inert.
• Since possible worlds in Lewis theory are
  concrete there is the possibility of interworld
  causality, so we need the extra principle
• Possible worlds are causally isolated

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Ontological Economy

• Actualist theories of possible worlds are
  committed to abstract entities, states of
  affairs, propositions etc alien properties we
  would never have thought should belong to
  actuality.
• Lewis isnt committed to anything more than
  concreta.
• His theory is qualitatively, but not
  quantitatively, ontologically economic.

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A reductive account of possibility

• Unlike rival theories, Modal Realism does not
  have to take modal notions as primitive.
• For Lewis our modal concepts are reducible to the
  existence of concrete objects and their
  properties.

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Paradox in Paradise

• Although flying in the face of common sense,
  modal realism is an attractive theory and has
  resisted many of the philosophical objections
  directed at it.

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Paradox in Paradise

• I am concerned with one objection to which I
  find no satisfactory answer in Lewis writings.
• It is a semi-mathematical argument to the effect
  that the collection of worlds is too numerous.

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A detour on size

• To count the number of members a set has we need
  measure of size for sets.
• In particular, we need a measure of size which
  extends to infinite sets.

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A detour on size

• Let X and Y be two sets (sets of numbers,
  chocolate bars, whatever you like). Then
• Definition X and Y are the same size iff there
  is a one to one correlation between the members
  of X and the members of Y

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Some examples
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Examples
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Examples
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Cantors Theorem

• Theorem Let X be a set, and let P(X) be the set
  of subsets of X. Then P(X) is always bigger than
  X.
• Proof See board.

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The problem

• David Armstrong and Peter Forrest have attempted
  to show David Lewis theory of possible worlds
  inconsistent through an application of Cantors
  theorem.
• The argument relies on a principle arguably
  essential to Lewis reduction of modality. The
  Principle of Recombination.

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The principle of recombination

• To express the plenitude of possible worlds, I
  require a principle of recombination according to
  which patching together parts of different
  possible worlds yields another possible world.
  Roughly speaking, the principle is that anything
  can coexist with anything else, at least provided
  they occupy distinct spatio-temporal positions.
  Likewise, anything can fail to coexist with
  anything else. Thus if there could be a dragon,
  and there could be a unicorn, but there couldnt
  be a dragon and a unicorn side by side, that
  would be an unacceptable gap in logical space, a
  failure of plenitude. And if there could be a
  talking head contiguous to the rest of a living
  human body, but there couldnt be a talking head
  separate from the rest of a human body, that too
  would be a failure of plenitude -- Lewis, D, On
  the Plurality of Worlds, 1986, Blackwell

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The principle of recombination

• If humans and horses are possible, so are
  centaurs.
• If trout and turkeys are possible so are
  trout-turkeys.
• More generally, for any set of possibilia, there
  should be a world in which they coexist.

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Recombination and Worlds

• Worlds are possible objects
• Thus according to the principle of recombination,
  the members of any set of possible worlds are
  collectively compossible
• For every set of worlds there is a corresponding
  world containing exact duplicates of each of
  those worlds as mereological parts.

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Giganto

• In particular, there would be a world
  corresponding to the set of all worlds. A world
  which has a duplicate of every single world as a
  part. Call this world Giganto.
• Since it contains a duplicate of each world it
  must contain a duplicate of itself.

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Giganto

• Giganto contains a duplicate of itself as a
  proper part.
• This is not quite incoherent.
• Think of a magazine with a picture of a celebrity
  holding up an exact duplicate of the very
  magazine on which he or she is posing with a
  picture on the front of him or her holding the
  magazine with a picture of (and so on ad
  infinitum)

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Electrons

• Forrest and Armstrong introduce an arbitrary
  measure on the size of a world the number of
  electrons it has.
• For every subset of the set of electrons there is
  a world exactly like Giganto, except with all but
  those electrons deleted. (another appeal to
  recombination).

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Electrons

• Each of these worlds will have an exact duplicate
  as a part of Giganto.

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Electrons

• Let X be the set of electrons in Giganto. Suppose
  X has size ?.
• Let the set of subsets of X, P(X), has size ??.
  By Cantors theorem ? lt ?.
• But for every subset Y of X there is a world with
  all but the electrons in Y deleted. There are ?
  many worlds like this.
• Each of these ? many worlds contains an electron
  and is a part of Giganto. So the number of
  electrons in Giganto is at least ?. But ? lt ?. .

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Lewis Response

• Lewis restricted his Principle of recombination.
• Make the existential claim that there is some
  limit on the size of worlds, although he does not
  claim to know which.
• Armstrong and Forrests argument is simply taken
  to be a proof to the effect that some limit
  exists, rather than the inconsistency of his
  theory.

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Lewis Response

• Among the mathematical structures that might be
  offered as isomorphs of possible space-times,
  some would be admitted, and others would be
  rejected as oversized.
• This response is ad hoc.

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Classes

• Why assume that there must be a set of all
  worlds?
• There are plenty of examples of collections of
  things which are too big to form a set
• A proper class is a collection of things which
  cannot be gathered into a set.

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Some Proper Classes

• Example 1 The collection of all sets. If this
  were a set it would have to contain itself,
  something that is not permitted in well founded
  set theories.
• Example 2 The collection of sets which do not
  belong to themselves. (Russells paradox. In well
  founded set theories this is the collection of
  all sets.)
• Example 3. The set of all ordinals. If this were
  a set it would be an ordinal and would have to
  belong to itself.

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Classes

• It seems perfectly consistent to talk about such
  collections of things, even though we cant form
  a set out of them.
• Consider the following sentence involving plural
  quantification
• There are some things which are all and only the
  sets which do not belong to themselves.

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Classes

• This sentence seems consistent.
• However, if the plural quantifier is treated as
  ranging over sets (or more generally singular
  entities) it is inconsistent, since it entails
  the existence of a set of sets which do not
  belong to themselves.
• It seems natural to consider the plural
  quantifier ranging over irreducibly plural
  collections of singular objects.

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Pluralities and Classes

• Perhaps we can make sense of classes in this way
  a class is not a thing like a set is, but an
  irreducibly plural collection of things. While we
  can talk about classes in the singular,
  underneath this convenient vocabulary we are
  talking about pluralities.
• So perhaps the way to explain this problem is by
  talking about a plurality of worlds. After all,
  that is the name of the book,

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The paradox runs deeper

• Unfortunately for Lewis, once we introduce
  mereological principles, such as unrestricted
  comprehension, the paradox still lurks.
• Unrestricted comprehension, according to Lewis,
  is explicitly formulated with respect to plural
  quantification.

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Unrestricted Comprehension

• Unrestricted Comprehension Whenever there are
  some things, then there exists something which
  has all of them as parts and has no part that is
  distinct from each of them.

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The Paradox Revisited

• Let W be the class of worlds (interpreted
  plurally)
• Let ??W be the class of fusions of classes of
  worlds (guaranteed by UC).
• We define a class, f, of pairs of worlds and
  world fusions. (Each member of f is of the form
  ltx, ygt where x ? W and y ? ?W.)

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Paradox Revisited

• Intuitively f is supposed to be our one to one
  correlation between worlds and classes of worlds.
  Write f(x) y, to mean ltx, ygt ? f
• For any ltx, ygt ? f, y is given by a class of
  worlds, and we let x be the world given by that
  class by Armstrong and Forrests recombination
  method.

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Paradox Revisited

• If f really does represent a one to one
  correlation then there should be a class of
  worlds corresponding to the fusions of classes of
  worlds whose corresponding world is not part of
  that class. i.e.
• C x x ? f(x)
• Here ? is the parthood relation.

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Paradox Revisited

• Denote the fusion of a class of worlds X as ?X
• A world is a part of ?X iff it is a member of X.
• In particular consider ?C ? ?W.
• Since f is a one to one correlation there exists
  a world w ? W such that f(w) ?C.

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Paradox Revisited

• Note that w ? f(w) iff w ? ?C, since f(w) is ?C
• Also w ? ?C iff w ? C a world is a part of a
  fusion of a class of worlds iff it is a member of
  that class
• Finally w ? C iff w ? f(w). This is because
  everything in C has this property
• So w ? f(w) iff w ? f(w). A contradiction.

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What to do?

• In my opinion Lewis should restrict the
  recombination principle to sets only.
• This is a restriction, but it is not ad hoc.
• Whether or not you think there should be a cut
  off point in the world of set theoretic
  mathematical structures for candidate
  space-times, surely it is not too far fetched to
  suppose that some arbitrary collections of things
  (proper classes interpreted plurally) are too
  numerous to be gathered into one world.

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More discussion on this idea

• Hazen and Nolan discuss whether a world can
  contain proper class many objects in the context
  of hypergunk.

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• So many worlds, so much to do,
• So little done, such things to be
• -- Alfred, Lord Tennyson In Memoriam