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above).
4
But, how much ef
order to accept, along with
question in terms of a com
(·,·), the more credulity w
Grove spheres we make us
plausible manner and are m
Before introducing Defi
the field of belief revision t
for example, that 'John is at
Antarctica' given my curr
account. In this sense, we w
proposed by Grove (1988)
collection of sets of possib
Grove, 1988, pp. 158-159):
1.
S
is totally ordered by in
2. The set of possible wor
minimum of
S
. The set o
3. If X is a finite, consist
containing a model of X.
ffort, or otherwise said, how much
credulity
do we need
h
, these
simple assumptions
? Definition 2. answers t
mparison of
credulities
. The higher the value of the uti
e need. (Note that we are not defining the credulities or
se of below, but assuming that these are provided in so
making use of them.)
nition 2. we need some preliminaries. It is well known
hat some beliefs are more plausible than others. The bel
t the beach' is more plausible than the belief that 'John i
rent belief-system. We need then to take this fact i
will use in the definition a system of spheres,
S
, of the k
), that is, brought to the context of the present article
le worlds (maximal consistent subsets of
L
) such that (
d in
that
ility
the
ome
n in
lief,
is in
into
kind
e, a
(see
clusion. The elements of
S
are called spheres.
lds that contain
as a subset (models of
), |
|, is
the
of all possible worlds is the largest sphere,
M
.
tent set of literals in
L
, there is a smallest sphere i
.
n
S
Observation.
Let
S
be our
of literals in
L
. Then there
minimum e(X).
Definition 2.
Let be
,
,
potential beliefs expressed
assume an ordinal utility
(
a)
(
,
)
(X,
) (
nothing,
, to a set of be
b
) If
p p
X, then
c
) If ¬Con(
, X
), then
- adding new beliefs
credulity)
d
) If X, Y
, Con(
,
X
assumed system of Grove spheres. X a finite, consistent
is a smallest sphere in
S
containing |X|. We will call
t set
this
,⊆ Γ
not empty. Let also be X and Y any finite se
d as literals in
L
, and
S
a system of Grove spheres.
(·,·) with the following properties:
(thus
(
,
) is a lowest possible value of
- add
eliefs does not require any credulity)
(X,
) =
(
,
)
(Y,
)
(X,
) (this yields a highest possible value
inconsistent with our previous beliefs does require a gr
et of
We
ding
e of
reat
X
), Con(
', Y
) and e(|
|
|
X|) = e(|
||
Y|), then
(X,
)
(Y,
) iff #
#
,
rdinal of the set next to it, and
and
are respectively
and Y such that
p p
, and
q q
.
X
), Con(
', Y
) and e(
||
|
Y|)
e(
|
|
|X|
), then
(X,
)
(Y,
)
where # refers to the car
subset of elements of X
the
e) If X, Y
, Con(
,
4
Definition 1. owes inspiratio
version of Definition 2. Th
paper.
on to Aliseda-Llera (1997).
Rohit Parikh has a slightly diffe
he two versions will be reconciled in the journal version of
erent
f the
