Barwise (1989b) attempted to propose a set theory, Situated Set
Theory, not just for use in AI, but for general use. He mentioned
the problems caused by the common view of set theory with a universal
set V, but at the same time trying to treat this universe as an
extensional whole, looking from outside (which he names ``unsituated
set theory''). His proposal is a hierarchy of universes 
which allows
for a universe of a lower level to be considered as an object of a
universe of a higher level. He leaves the axioms which these
universes have to satisfy to one's conception of set, be it cumulative
or circular. There are no paradoxes in this view since there is
always a larger universe one can step back and work in. Therefore,
the notions of ``set,'' ``proper class,'' and the set-theoretic
notions ``ordinal,'' ``cardinal'' are all context sensitive, depending
on the universe one is currently working in. This proposal supports
the Reflection Principle which states that for any given
description of the sets of all sets V, there will always be a partial
universe satisfying that description.
Barwise (1989c) also studied the modeling of partial information and
again exploited Hyperset Theory for this purpose. For this purpose,
he used the objects of the universe 
of hypersets over a
set A of atoms to model non-parametric objects, i.e., objects with
complete information and the set X of indeterminates to
represent parametric objects, i.e., objects with partial
information. (The universe of hypersets on 
is denoted as
, analogous to the adjunction of indeterminates in
algebra.)
For any object 
, Barwise calls the set
where 
denotes the transitive closure of a, the
set of parameters of a. If 
, then 
since a does not have any parameters. Barwise then
defines an anchor as a function f with 
and 
which assigns sets to
indeterminates. For any 
and anchor f,
is the object obtained by replacing each indeterminate 
by the set 
in a. This is accomplished
by solving the resulting equations by the Solution Lemma.
Parametric anchors can also be defined as functions from a subset of
X into 
to assign parametric objects, not just
sets, to indeterminates. For example, if 
is a parametric
object representing partial information about some non-parametric
object 
and if one does not know the value to
which x is to be anchored, but knows that it is of the form 
(another parametric object), then anchoring x to 
results in
the object 
which does not give the ultimate object perhaps,
but is at least more informative about its structure.