Reduction and Elimination in Philosophy and the Sciences

Reduction, Sets, and Properties
Benjamin Schnieder, Berlin, Germany
1. The Traditional Debate
Some time ago, philosophers often addressed the question
about whether it is possible to reduce the category of
properties to the category of sets.
The most straightforward idea was that,
schematically speaking, we may identify the property of
being F with the set of all Fs. The straightforward problem
of this proposal was that if conflicts with the combination of
the generally accepted identity conditions for sets and
some strong intuitions on the non-identity of certain
properties: on the one hand, sets are extensional in that a
set x is identical to a set y iff x has all and only the
members that y has. On the other hand, two different
properties can contingently be possessed by the same
entities. Even if all and only those animals that naturally
have a heart naturally have a liver, the property of naturally
having a heart is not identical to the property of naturally
having a liver. And even though the property of being an
80 feet long duckbill and the property of being a 5 billion
dollar worth corkscrew are both exemplified by no entities
at all, they are not the same property.
Because of this problem, it is generally agreed that
the said straightforward identification of properties with
sets is a failure. But a very similar identification is better
off: David Lewis famously proposed to identify the property
of being F with the sets of all actual or merely possible Fs. 1
Since (presumably) no non-actual possible duckbill is a
non-actual corkscrew, properties in the second example
come out different on Lewis’s account, and since some
non-actual animals with a heart lack a liver, the properties
in the first example come out different as well.
But there are still problems with Lewis’s proposal:
first, it may seem to correspond to a too coarse-grained
individuation of properties: on his account, there are no
two properties that are necessarily possessed by the same
entities. But isn’t the property of being an equilateral
triangle different from the property of being an equiangular
triangle, despite their being necessarily co-exemplified?
However, there is a promising response available to this
third challenge: in the relevant cases, our intuitions on the
identity or non-identity of properties are not that stable,
and they are blurred by our tendency to assimilate
properties to concepts. 2 Concepts are individuated partly
via the role they play in the individuation of beliefs; for this
reason, we often distinguish between two concepts even if
they have the same extension with respect to every
possible world (someone may possess the concept of
equilateral triangles but lack the concept of equiangular
triangles because he lacks the concepts of angles). But
properties play a more worldly role such that different
properties should at least possibly correspond to
differences between things; however, in no possible world
there is any difference between equilateral and
equiangular triangles, and hence there is only one property
which is conceived of in different ways (i.e. by the
employment of different concepts).
1 See Lewis (1986: 50–69).
2 For responses along this line, see, e.g., Lewis (1986: 55), Jackson (1998:
15f., 126f.), Künne (2003: 26, et passim), and Schnieder (2004: 59–69).
But there are other problems with Lewis’s proposal.
One is that it apparently cannot provide for all the
properties that intuitively exist: 3 there is, it seems, the
property of being a set. But this property cannot be
identified with the set of all (actual and non-actual) sets,
because we know from standard set-theory that there is no
such entity. Lewis might at this place slightly modify his
proposal and hold that most properties are sets, but some
properties are proper classes rather than sets. The
property of being a set, for instance, is the proper class of
all sets. But firstly that response may seem a little ad hoc,
and secondly it only pushes the problem up to a higher
level: intuitively, there is the property of being a proper
class. But there is no proper class containing all the proper
classes. So, some properties are still missing on Lewis’s
account.
Finally, the proposal only gets off the ground if one
accepts Lewis’s form of modal realism which can provide
the required ontological resources: for the proposal to
work, there must be sets containing actual and non-actual
duckbills as members and hence there must be non-actual
duckbills (which are duckbills nonetheless). 4 Especially
due to this metaphysical burden of his approach, Lewis’s
identification of properties with sets has not a very good
reputation, to say the least.
2. A New Order: Sets as Properties
If properties cannot be reduced to sets, is there no connection
between the two kinds of entity? There is! The
main problem of the traditional debate was that it reversed
the actual order of things: instead of identifying properties
with sets, one should identify sets with properties. Sets are
properties in disguise (but not every property is a set).
The basic idea is very simple: sets are properties of
some particular sort. They are what I call identityproperties,
such as the property of being identical to Jean-
Paul Belmondo or the property of being identical to either
Belmondo or Jean Seberg.
To understand the details of the proposal, recall the
two standard ways of specifying a sets: they consist in
either (i) providing a list of the individual members of the
set—by using expressions such as ‘{Belmondo}’,
‘{Belmondo, Seberg}’, ‘{1, 2, 3, …}’—, or (ii) stating a
condition such that all and only the things satisfying the
condition are members of the set—by using expressions
such as ‘{x: x is a bear}’, or ‘the set of all duckbills’.
Focussing on these two canonical ways of specifying sets,
a recipe can be given for identifying particular sets with
particular identity-properties.
Re (i): If we specify a set by explicitly listing its
members, the set is just the property of being (identical to)
one of those entities. Thus, {Belmondo} is the property of
being (identical to) Belmondo. {Belmondo, Godard} is the
property of being (identical to) either Belmondo or Godard.
More generally, {x1, x2, …, xn} is the property of being
(identical to) either x1, or x2, …, or xn.
3 For the following point cp. Egan (2004: 49n.), and Schnieder (2004: 72f.).
4 For further criticism of Lewis’s account see Egan (2004).
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