Reduction and Elimination in Philosophy and the Sciences

Re (ii): If we specify a set via some condition C that
its members satisfy, the set is the property of being one of
those entities that actually satisfy condition C. Thus, {x: x is
a word on this page} is the property of being one of the
words that are actually on this page. We can specify this
set, and this property, in some alternative way by listing all
its members. Thus, {x: x is a word on this page} = {‘a’,
‘the’, ‘set’, ‘words’, ‘are’, …} = the property of being either
the word ‘a’, or the word ‘the’, or the word ‘set’, …
Similarly, {x: x is a bear}, i.e. the set of all bears, is
identified with the property of being identical to one of the
actual bears.
3. Evaluating the Proposal
To evaluate the given proposal, it should be seen whether
it can account for the generally acknowledged features of
sets. On the one hand, some central metaphysical characteristics
of sets should be accounted for:
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M.1 Sets have their members essentially and they
could not have had additional members.
M.2 Sets are extensional. For all sets x, y: x = y iff
every member of x is a member of y and vice
versa.
M.3 Sets are ontologically dependent upon their
members.
These three principles are results of the indicated identification
of sets with identity-properties (given some additional
assumptions about properties).
Re M.1: Identity-properties are necessarily
possessed by all and only those entities that possess
them. So, if a set is an identity-property, and if its members
are the things that posses it, then a set has its members
essentially and could not have had additional members.
Re M.2: Moreover, the intensional individuation of
properties (i.e. the individuation via their possible
exemplifications, as described in section 1) accounts for
the extensionality of sets: if set x and set y contain the
same members, then x and y are reduced to the same
identity-property (because x is reduced to an identityproperty
I and y to an identity-property I* such that I and I*
are necessarily possessed by the same entities—hence,
by the intensional individuation of properties, I=I*).
Re M.3: an identity property I is individuated via the
entities e, … that enter into the identities constitutive for I.
Hence, the essence or nature of e, … is prior to the
essence of I, which makes the identity property dependent
upon its constitutive entities and thereby yields the
ontological dependence of sets on their members. 5
Moreover, the semantics of designators for sets
comes out right: some of those expressions are rigid
designators (e.g. ‘{Belmondo}’) while others are flexible
(e.g. ‘{x: x is a bear}’). On the current proposal, a set
designator will be rigid if it rigidly specifies the members of
the set, because then it rigidly specifies a particular
identity-property. But if a set designator flexibly specifies
the members of the designated set, then the designator
will designate different identity-properties with respect to
different possible worlds and therefore comes out flexible
itself.
5 In this paragraph, I am mainly relying on Fine’s views on essence and
dependence; see Fine (1994 & 1995).
Reduction, Sets, and Properties — Benjamin Schnieder
A further issue to de addressed is whether the
proposal may open doors to set-theoretical paradoxes. A
threat of paradox may stem from the following observation:
M.4 Sets have a limited holding capacity, i.e., there
can be too many things of a sort be contained
in a set.
Properties, on the other hand, do not seem to have a similarly
limited holding capacity. For while we know that there
are, for instance, too many sets to form a set, there is a
property which all sets have in common, namely the property
of being a set. Analogously, there is the property of
being abstract which is possessed by all abstract objects
and therefore by all sets. But again, there cannot be the
set of all abstract objects because it would have the set of
all sets as a subset.
How exactly might the limited holding capacity of
sets pose a problem for the suggested reduction of sets to
properties? Answer: if there are identity-properties
belonging to more things than can form a set, then the
proposal should not hold that all identity-properties are
sets. But then, a criterion should be provided which
motivates why some identity-properties are sets while
others are not, and it is hard to see how such a criterion
might not be ad hoc.
Fortunately, it can be argued that there are no
identity-properties which would yield too large sets in the
first place. For which properties could that be? As we have
seen, the property of being a set cannot correspond to a
set because the latter would be too large. But that property
is not an identity-property and therefore not the kind of
property that the proposal identifies with sets. What would
be threatening is the identity-property of being identical to
one of the actual sets. But while this property may seem
innocent and thus existing at first glance, there is a reason
to deny its existence. For it was said before that an
identity-property is individuated by the entities that enter
into its constitutive identities, such that the essence of
those entities is prior to the essence of the property. Now
assume that the identity-property of being one of the actual
sets would exist. Then, on the present reductive theory, it
would be a set, and hence one of the existing sets. But its
essence should be posterior to the essence of its
constituting entities, i.e. of all actual sets. Since it would be
a set itself, its essence would have to be posterior to its
own essence—which is impossible. Therefore, the
existence of the said property can be denied. A lesson
from this consideration is that—contrary to prototypical
examples of properties—identity-properties do have a
limited holding capacity, just like sets.
The last test for the proposal will be to account for
the theorems of standard set theory. It would be desirable
if the axioms of ZF-theory could be motivated from it.
Unfortunately, the current space is too limited to go into
details here. So I must conclude with the bare promise that
the validity of ZF-axioms such as intersection can indeed
be argued for from considerations about identityproperties.
Once this is shown, the proposal can be seen
to have numerous benefits which allow the conclusion:
sets are properties.